INDIAN JOURNAL OF PHYSICS
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INDIAN JOURNAL OF PHYSICS
Voliime 72 A
Number 1
January 1998
/:a(1) I
BOARD OF EDITORS
MP
FJ^lTOR-lN-CHIEl' & HONORARY SECRETARY
S P Skn Gijma Indian A.\soaamm for The Cultivation of Science, Calcutta
CONDENSED MATTER PHYSICS
A K IUkiia Indian Association for the
Cultivation of St leiu e. CaUutta
S N liKHKRA In stituteofl'livsii .T. Lihuhimeswar
nCHAKRAVoRTY Indian Association lor the
( 'uliivationol S( letu e. Calcutta
B(;(;mosii Saha Institute oj Nucleai
rhs'sn s, ( alt ufta
S K JosHi National Physical Ixihoratory,
New Delhi
C K Majumdah S N Bose National Centre for
Basic Sciences, Calcutta
E S Rajagopai, Indian Institute of Science,
Banf>alore
NUCLEAR PHYSICS
Tata Institute of hundaniental
Researi h. Munihai
Bhahha \totni( Rescan h
Centre, Murnhai
\ SK\m\mvhih\ Department of Science
Teihnolo^v, New Delh\
PARTICLE PHYSICS
H BANhRiFi- S N Hose National ( enite lor
Basil Si uf lies, i'alcutta
I) P Roy Tata Institute of I'undaniental
Rcsearrh. Mumbai
Tahi Institute of hiindamental
Research Miimhai
RELATIVITY & COSMC^LOGY
IhnveisitY of Biiidwan.
Bin lbs an
Univcisits oj Iripura.
Af’aitala
N K Dadiiich
Inter (hiiversity Centre for
Astrononiv Astioplivsii s,
Pune
Cotton College, (iuwahati
ASTROPHYSICS, ATMOSPHERIC & SPACE PHYSICS
Tata Institute of Tundamental
Rescan h. Munihai
Indian Institute of Astrophysu
Bangalore
Indian National Si lent e
Aiademv, New Delhi
M K Das Gupta Birla Planetarium. Calc utta
K K M AHA JAN National Pfivsii al Lihoratorw
New Delhi
A K Sfn Institiiie of Radio Plis sus A
i'.lei tnmic.s, Cali utta
ATOMIC & MOLECULAR PHYSICS
S P KiiARt ( haudhars Clianin Siiii'h
ilnivetsit}’. Meerut
S C’ Mukherji'l Indian Assm union for the
( iiltivation of Si leni c, Cah ufta
OPTICS & spe:ctroscopy
G S Agahwai
Phvsit al Researi li Uihoiaiory,
Ahniedahad
Indian Assaiiation for the
Cultivation of S( lencc.Caliulki
Jadavpiir University, Calcutta
Bhahha Atomic Research
Centre, Mumbai
Indian Institute of Siience.
Bim^alote
1' N Miska
A K ScKin
PLASMA PHYSICS
K Avinash
In.sntute for Pla.sma Research,
R K Varma Physical Research Laboratorw
Gandhinaf^ar
Ahmedahad
A C Dais
Physical Research Laboratory,
Ahmedahad
STATISTICAL PHYSICS, BIOPHYSICS & COMPLEX SYSTEMS
V Balakrishnan Indtan Instituie of
Technolofiv, Chentuu
J K Biia'ITACHarjee Indian Association for the
Cultivation of Science.
Calcutta
J Das Indian Institute of Chenuval
Hifllo^y, Calcutta
Auhijit Mookrrjf.f: .S’ N Bose National Centre
for Basic S( lences, Calcutta
TP Singh All India Institute of
Medual Sciences, New
Delhi
Yashwant Singh Banaras Hindu Universiiv.
Varanasi
ASSOCUl^: EDITORS (HONORARY) ]yp
{from I ACS)
CiiriRA Bam'
D P BHA I I At'HARYYA
S ('hakravohty
Pahsaihi CiIA'I ifrjf.e
A Ghosh
K Rai D^siidah
D S Roy
Kanika Roy
S C Saha
S K Sfn
[from other Institutions)
Indkam Bosf lio\e Institute. (aU utta
Bikash Ciiakkaiiariy Saha Institute of Nudeai riiYsa s, Cah iifia
Ani i A Mein A S N Base National Centre for Hasit S( lem e\. Calcutta
STAFF EDITORS W
Dr (Mrs) K K Da riA A.wistant Sei retarv
A N Gha i ak Tei linii at ( Officer
One copy of the manuscript may be submitted through an appropriate member of the
Board of Editors. Authors may kindly see ‘Notes for Contributors’ printed at the end ot
this volume.
INDIAN JOURNAL OF PHYSICS
Volume 72, Parts A and B
AND
PROCEEDINGS OF THE
Indian Association for the Cultivation of Science, Vol. 81
Author and Subject Index
1998
(Published by the Indian Association for the Cultivation of Science
in Editorial Collaboration with the Indian Physical Society)
Indian J. Phys. 72A, 3-12 (1998)
Indian Journal of Physics
Author Index
Volume-72, Part-A (1998)
The ibllowing abbreviations are used :
(N) ‘Note’
iPl) ‘Proceedings of Condensed Matter Days— 1997, held at the Department of Physics,
Viswa Bharati, Santinikeian, India, during August 29-dJ, 1997'
(P2} ‘Proceedings of the XII DAE Symposium on High Eneigy Physics held at the
Department oj Physics, Gauhati University, Guwahati-781 014, India, during
December 26, 1996 - January I, 1997 — Part E
Author
Subject
Page
A
Abdcl-Halcs A M
See Sabry S A
141
Abdcl-Mcguid M M
See Selim Y S
155
Abraham J T
See Joseph Benny
99
Ananihanurayan B
Status of supcrsymrnelric grand
unified theories {P2)
495
B
Bakry M Y liL
See Tantawy M
73
Banerjee Sunanda
Experimental Summary-Xll DAE
HEP symposium, Guwahali, 1997 {P2)
689
Bhadra S K
Characteristics of selenium films on different
substrates under heat-treatment
201
Bhailacharyya Gautam
Precision tests of the Standard Model :
Present status (P2)
469
Bhatacharya Manabesh
See Singh W Shambhunath
133
Bhatt Kapil
See Joshi Urmi M
301
Bidadi H
See Kalafi M
43
Biswas Biswanath
See Ghosh Dipak
313
4
Author Index
Author
Subject
Page
Borgohain P
Matching of Friedman n-Lcmaitre-
Robcrlson-Walkar and Kantowski
wSachs Cosmologies
331
Bose C
Electric field induced shifts in electronic
slates in spherical quantum dots with
parabolic confinement (A^
87
Bose IneJranie
Quantum magnetism : novel materials
and phenomena iP!)
343
Byrappa K
Crystal growth, morphology and properties
of NaHMP'iO; (M = Ni, Co, Mn, Zn, >
Cd, Pb)
1
Byrappa K
Crystal growth and characterization of
(NH4)iBaCls-2H:0
259
C
Chakraborty Chailali
Influence of alloy disorder scattering on
drift velocity of hoi electrons at low
temperature under magnetic quantization
in»-Hg()sCd()2Tc(/V)
463
Chanel Navin
See Garg Ashok Kumar
189
Chatlopadhyay Biplab
An orbital anti ferromagnetic stale in the
extended Hubbard model {PI)
359
Chattopadhyay M K
See Dey P K
281
Chattopadhyay Rini
See Ght)sh Dipak
323
Choudhary M K
Sec Singh N K
241
Choudhary R B
Sec Roy P N
23
Choudhary R N P
See Singh N K
241
Choudhary D K
Foreward (P2)
i
Choudhary D K
Structure functions — selected topics (P2)
547
D
DasHL
See Phukan T
433
Das S K
Lattice relaxation in substitutional alloys
using a Green’s function {PI)
379
Das Sunil
See Ghosh Dipak
313
Datl S C
See Garg Asok Kumar
i89
Author Index
5
age
331
87
343
463
189
359
281
323
241
23
241
547
433
379
313
189
Author
Subject
Page
Deb Argha
See Ghosh Dipak
313
DcbNC
See Singh W Shambhunalh
133
DeyTK
Pulse method for measurement of thermal
conductivity of metals and alloys at
cryogenic temperatures
281
Dey Tarun K
Semiclassical theory for thermodynamics
of molecular fluids {PI)
397
Dhami A Kaur
See Dey T K
281
Dharmaprakash S M
See Suryanarayana K
307
Duttamudi G
Sticking of He** on graphite and argon
surfaces in presence of one phonon
process {PI)
455
E
HI Bakry M Y
See Tantawy M
73
HI Hagg A A
See Nouh S A
269
HI M Mashad
See Tantawy M
73
El-Ocker M M
Thermal behaviour and non-isothermal
Kinetics of Geio+,Se4oTeso-v
31
Hngineer M H
Sec Roy B
417
F
Fayek S A
See El-Ockcr M M
31
G
Ganguli S N
Results from LEP 1 {P2)
503
Garg Ashok Kumar
Thcnually stimulated dc-polari/,ation
current behaviour of poly (vinyledenc
fluoride) ; poly (methyl methacrylate)
blend system
189
Ghatage A K
Neutron diffraction study of tin-substituted
Mg-Zn ferrites
209
Ghosh Dipak
Dynamical short range pion correlation in
ultra-relativistic heavy-ion interaction
313
Ghosh Dipak
See Ghosh Dipak
313
Ghosh P K
Electron tunneling in hctcrostructures under
a transverse magnetic field {PJ)
447
6
Author Index
Author
Subject
Page
God S K
Efficiency rncasurcmcnl of a Si (l.i)
detector below 6.0 KeV using the
atomic-field brcmsstrahlung
63
Gopdiandran K Cj
Sec Joseph Benny
99
CioswaiTii K
See Bhadra S K
201
Cioswaini T 13
See Phukan T
433
Gurtii A
Physics at LEP 200 (P2)
503
H
Hale/ A F
See Selim Y S
155
Harinciranath A
Light-front QCD . present status {P2)
635
Hassaiucii A S
See El-Ockcr M M
31
Hossain 'I'
Investigation of graphitr/ing carbons
/
Indiimalhi D
from organic compounds by various
experimental techniques
225
Nuclear structure junctions [P2)
567
J
Jafry Abdul Kay uni
See Ghosh Dipak
313
Jayannavar A M
Sec Joshi Sandeep K
371
John R K
Fluctuations in high superconductors
with inequivalent conducting layers
217
Joseph Benny
Optical and structural characterisation of
ZnO films prepared by the oxidation
of Zn films
99
Joshi M J
See Raval A H
49
Joshi Sandeep K
Transport and Wigner delay time distribution
across a random active medium {PI)
371
Joshi Urmi M
Study of bismuth substitution in cobalt ferrite
301
K
Kalah M
Nonlinear light absorption in GaSe^iS^ solid
solutions under high excitation levels
43
Kalainathan S
See Rcthinam F Jesu
117
Kanjilal D
See Phukan T
433
Author Index
1
Author
Subject
Page
Kar Gupla Abhijil
See Joshi Sandeep K
371
KarT
Hardness anisotropy of L-argininc
phosphate monohydrate (LAP)
crystal (N)
83
Keller J M
See Garg Ashok Kumar
m
Khan M A
See LalHB
249
Khandhaswamy M A
Sec Byrappa K
259
Khanra Badal C
See Menon Mahesh
407
Koshy Peter
Sec Joseph Benny
99
Kouhi M
See Kalafi M
43
Krori K D
Blackholc evaporation— stress tensor
approach {P2)
621
Kumar Aksliaya
See Sharma Brajesh
107
Kundu Anjan
Quantum integrablc systems : basic
concepts and beief overview (P2)
663
Kundu K K
Sec Munshi T K
93
Kundu K K
Sec Munshi T K
391
Kuriakose V C
5£TJohnRK
217
L
Lahiri Madhuniita
See Gh()sh Dipak
313
Lai H B
On the structure and phase transition of
lanthanum titanate (iV)
249
M
Mahalanabis R K
See Munshi T K
93
Mahalanabis R K
See Munshi T K
391
Mail! A K
See Bhadra S K
201
Mallik S K
Methods of thermal field theory (P2)
641
Manoj P K
See Joseph Benny
99
Mazumdar P S
See Singh W Shambhunath
133
Mazumdar P S
See Singh W Shambhunath
233
Mcnon Mahesh
Energetics of CO-NO reactions on
Pb-Cu alloy particles (PJ)
407
Meiawe F
See El-Ockcr M M
31
8
Author Index
Author
Minkowski P
Mitra B
Mukhcrjcc C D
Munshi T K
Munshi T K
N
Nouh S A
0
Ola S B
Ota S B
Ota Smita
Ola Smita
Oza A T
P
Pandey V P
Pandey Rajiv Kumar
Pandya H N
Paranjpe S K
Paigiri Maliadcv
Palil S A
Pallabi Manjunalha
Subject
See Ananthanarayan B
See Ghosh P K
See Saha Jayashree
Disturbances in a piczo-quarlz cantilever
under electrical, mechanical and
thermal fields (N)
The problem of a composite piezoelectric
plate transducer (PI)
The effect of infrared pulsated laser on the
degree of ordering of cellulose nitrate
Inhomogcncity of vortices in 2r/ classical
XY-modcl : a microcanonical Monte
Carlo simulation study (PI)
Energy, fluctuation and the 2d classical
XV-model {PI)
See Ota S B
See Ola S B
Electrical properties of organic and
organomclallic compounds
See Rajput B S
Analysis of temperature dependence of
inlcrionic separation and bulk modulus
for alkali halides
See Joshi Urmi M
See Ghatage A K
See Borgohain P
See Ghatage A K
Stability of Ag island films deposited on
softened PVP substrates (PI)
Change in conductivity of CR-39 SSNTD
due to particle irradiation (PI)
Phukan T
Author Index
9
Author
Subject
Page
Fodder J
See Hossain T
225
Prasad TNVKV
See Rao K Sambasiva
337
Purkait Krishnadas
See Ghosh Dipak
313
R
Radwan M M
See Nouh S A
269
Rahman Md Azi/ar
See Ghosh Dipak
313
Rai S B
See Sharma Brajcsh
107
Raj B Saiijccva Ravi
See Byrappa K
1
Rajasckaran G
Perspectives in high energy physics (P2)
679
Rajput B S
Supcrsymmclrized Schrbdinger equation
for Fermion-Dyon system
161
Ram wShri
Cylindrically symmetric scalar waves in
general relativity (N)
Rao K Koteswara
See Rao K Sambasiva
337
Rao K Mohan
See Pallabi Majunalha
403
Rao K Sambasiva
Structural and dielectric studies on
lanthanum modified Ba^LiNbsOis {N)
337
Rao M Rajcswara
See Rao K Sambasiva
337
Raval A H
The role of the oxidising agent and the
complexing agent on reactivity at line
defects in antimony
49
Relhinani F Jcsu
Mechanism of gram growth in aluminium,
cadmium, lead and silicon
117
Rindani Saurabh D
New physics at c^c" colliders (P2)
533
Roy B
A new viscous fingering instability : ibe
case of forced motions perpendicular to
the horizontal interface of an immiscible
liquid pair (PI)
417
RoyPN
Tunnelling current across a double barrier
23
Roy Probir
Status of weak-scalc supersymmetry (P2)
479
Roy S K
Forward (PI)
Roy SK
Gas-surfacc scattering : A review of
quantum statistical approach (PJ)
351
RoySK
See Duttaniudi G
455
10
Author Index
Author
Subject
Page
S
Sabry S A
Equilibrium forms of two uniformly
charged drops
141
Saha Jayashree
Phase alternation in liquid crystals with
terminal phenyl ring {PI)
427
Sahay P P
Effect of interface state continuum on the
forward (I-V) characteristics of
mcial-sciniconduclor contacts with
this intcrfacial layer
57
Sahay P P
Study of forward (C-V) characteristics of
MIS Schottky diodes in presence of '
interface slates and scries resistance
287
Salrnanov V M
Sec Kalafi M
43
Sarkar C K
See Bose C
87
Sarkar C K
See Chakraboriy Chailali
463
Saiapathy M
Sec Ola Simla
421
Selim Y S
Measurements of flux and dose distributions
of neutrons in graphite matrices^ using
LR-l 15 nuclear track detector
155
Sen Gupla S P
See Kar T
83
Sen Asok K
Electronic transport in a randomly
amplifying and absorbing chain [PI)
365
Shah B S
See Raval A H
49
Shah B S
See Vaidya Nimisha
295
Shalaby S A
See Sabry S A
141
S hanker R
See GocI S K
65
Sharma Brajesh
Optical properties of Pr^'^ doped glasses
effect of host lattice
107
Shriram
Early cosmological models with variable
G and zero-rest-mass scalar fields
323
Shukla Prahodh
Mctaslability and hysteresis in random
field Ising chains 439
Singh Birendra K Scnniclassical theory for transport
properties of hard sphere fluid TP/)
385
Author Index
II
Author
Subject
Page
Singh C P
See Shriram
323
Singh C P
Quark gluon plasma — current status of
properties and signals {P2)
601
Singh M J
Sec Gocl S K
65
Singh N K
Studies of X-rays and electrical properties
of SrMo 04
241
Singh S Dorendrajit
Sec Singh W Shanibhunath
133
Singh S Dorcndrajil
Determination of the activation energy
of a iherrnoluminescence peak obeying
mixed order kinetics
233
Singh S Jnychandra
See Singh W Shambhunath
133
Singh W Shanibhunath
Determination ol the order of kinetics
and activation energy in thermo-
luminescence peak with temperature
dependent frequency factor
11
Singh W Shanibhunath
b!valuation of the trapping parameters of
TL peaks of multi activated SrS
phosphors
133
Singh W Shanibliunalh
Sec Singh S Dorendrajit
233
Sintia Siircsh S
See Singh Bicndra Singh
3K5
Sinha Surcsh S
See Dcy Tarun K
397
Srinivasan V
See Byrappa K
259
Srivaslava V P
.SVe Lai H B
249
Suryanarayana K
Defect characterization of Sr-^ doped
calcium tartrate tetrahydratc crystals
307
T
Tajalh H
Ste Kalafi M
43
Tantawy M
Multiparlicle production process in high
energy nucleus — nucleus collisions
73
Thirupathi C
See Rcthinam F Jesu
117
Tiwari S K
See Ram Shri
253
V
Vaidya Nimisna
The effect of doping on the microhardness
behaviour of anthracene
295
12
Author Index
Author
Subject
Page
VaidyanVK
Sec Joseph Benny
99
Verma R C
Heavy flavour weak decays (P2)
579
Vinod Kumar P C
See Oza A T
171
Y
Yagnik J H
See Vaidya Nimisha
295
Indian J. Phys. 72A. 13-25 (1998)
Indian Journal of Physics
Subject Index
Volunie-72, Part-A (1998)
The following abbreviations are used :
(PI) Proceedings of Condensed Matter Days-] 997, held at the Department of Physics,
Viswa Bharati, Santiniketan, India, during August 29-31, 1997
(P2) Proceedings of XII DAE Symposium on High Energy Physics held at the Department
of Physics, Gauhati University, Guwahati during December 26, 1996-January I, 1997
00. GENERAL
02. Mathematical methods in physics
02.70.-c ('omputational techniques
02.70 Lq MotUe Carlo ami Matistu al methods
InhoTTiogcnciiy of vortices in 2d classical
XK-modcl : a microcanonical Monle Carlo
smiulalion study (PI)
S B Ota and Smila Ola p 413 '
Energy, llucluation and the 2d classical
XK-inodcI (PI)
Smila Ota, S B Ota and M Satpathy
p421
03. Clasical and quantum physics;
mechanics and fields
03.20.-)>i Classical mechanics of discrete systems :
Kcncral mathematical aspects
Equilibrium forms of two uniformly
charged drops
S A Sabry, S A Shalaby and A M Abdcl-
Hafes p 141
03.65. -W Quantum mechanics
03.65. Fd Algebraic methods
Quantum integrable systems ; basic
concepts and brief overview (P2)
Anjan Kundu p 663
03 6*^ Ge Solutions of wvn'P equations . hound Males
Supersymmclri/cd Schrodingcr equation
for Fermion-Dyon system
B S Rajput and V P Pandey p 1 6 1
04. General relativity and gravitation
04.20. -ii Classical general relativity
()4 20Cv Fundamental problems and general
fotmalism
Blackholc evaporation-stress tensor
approach (P2)
KDKrori p621
04.20. Jb Exact solutitms
Cylindrically symmetric scalar waves in
general relativity (N)
Shri Ram and S K Tiwari p 253
05. Statistical physics and thermodynamics
05.20. -y Statistical mechanics
Inhomogcncily of vortices in 2d classical
XE-modcl : a microcanonical Monte Carlo
simulation study (PI )
S B Ota and Smita Ola p 4 1 3
05.40.-i-j Fluctuation phenomena, random
processes, and Brownian motion
Electronic transport in a randomly
amplifying and absorbing chain (PI)
Asok K Sen
p365
14
Suhfect Index
05.70. -a TIicrniodynamicA
0‘>7()Ce Theinunixmunu f urn I ton \ <md tquuUons
ttf xtdlf
Scmiclassical theory tor ihcmiodynamics
t)l inoJccLilar lluicls (PI)
Taruii K Dey and Suresh K Sinha
p397
05.70. -l h Phase Iransilioiis : Kincrul aspects
Inliomogeiicity of vf»rliecs in 2d classical
.Yy'-modcl : a niicrocanonical Monte Carlo
simulation study (Pf)
S B Ota and Smiia Ota p4l3
imergy, fluctuation and the 2d classical
;fY-modcl (PJ)
Srnila Ola, S 13 Ola and M Salapathy
p42l
I) THE PHYSICS OE
ELEMENTARY PARTICLES
AND FIELDS
n. General theory of fields and particles
)1.10.-z Field theory
1 1 lO.Ef Lt^mnf^uin and llaniilumiaii appnuit h
Light-front QCD : Present status {P2)
A Harindranalh p 635
II lOCih lifnatniahranon
Light-tronl QCD : Present status (P2)
A Harindranalh p 635
11.10 1 JO Notilinea} a/ nonhxul Iheones and
mode Is
Quantum integrable systems; basic
concepts and brief overview {P2)
Anjan Kundu p 663
11.10 Wx Finite Ifiupeniluie field iheoiy
Methods of thermal field theory (P2)
S Mallik p64l
ll.15.-q (fiiugc field theories
Perspectives in high energy physics (P2)
G Rajasekaran p 679
1 1 .30.-j Symmetry and conservation laws
II 'to Hv Flavor symmetries
Heavy flavor weak decays {P2)
R C Verma p 579
11 30.Pl) Snpersvmmeiiy
Supcrsymmclri/cd Schrodingcr equation
for Permion-Dyon system
B S Rajput and V P Pandey p 161
1l.55.-m \-matri.\ theory; analytic structure of
amplitudes'
1 1 5*) Hs lAUit S matin es
Quantum integrable systems; basic
concepts and brief overview
.Anjan Kundu p 663
12. Specific theories and interaction
models; particle systcmatics
12. lO.-g Unilicd field tf^eories and models
I ’ 10 Dm Unified theones and models of Kttanf’ and
elei ttoweak interai turns
Precision lesis ol the Standard Model ;
Pre.sent status {P2)
Gautam Bhallacharyya p 469
Status of supersymmetric grand unified
theories {P2)
B Ananlhanarayan and P Minkowski
p495
Experimental summary-XIl DAE HEP
symposium, Guwahali, 1997 {P2)
Sunanda Bancrjce p 689
12.38.-t Quantum Chmmodynaniics
12.38 Bx Perlmbative laknlations
Structure functions-selcctcd topics (P2)
D K Choudhiiry p 547
Light-front QCD; present status (P2)
A Harindranalh p635
Subject Index
15
|2.38.Lg Other nonperturbative ralculations
Slruclure funclions-seleclcd topics (P2)
D K Choudhury p 547
Methods of thermal field theory (P2)
S Mallik p 641
12.39.-x Phenomenological quark models
1 2 39 Fe Chiral Laaranuiuns
Quark gluon plasma-current status of
properties and signals
C P Singh p 601
Methods of thermal field theory (P2)
S Mallik p 641
1 2.60. -i Models beyond the standard model
New physics at c^c colliders (P2)
Saurabh D Rindani p 533
1 2 60 (’n Exten uonx of eUrttaweak fiaufie sermr
Prcspcctivcs in high energy physics
iP2)
Cj Rajasekaran p 679
13. Specific reactions and phenomenology
13.1 0.+q Weak and electromagnetic interactioiLs
of leptoiLS
Results from LBP 1 (P2)
S N Ganguli p 503
1 3.2S.-k Hadronic decays of mesons
Heavy flavor weak decays (P2)
R C Verma p 579
13.38. -b Decays of intermediate bosons
1 3.38. Dg Decayx of z bosons
Results from LEP I (P2)
S N Ganguli p 503
13.60. -r Photon and chargcd-leplon interactions
with hadrone
1 3.60 Hb Total and inclusive cross section
Structure functions-selectcd topics
(P2)
D K Choudhury p 547
13.60.-r Photon and charged-lepton interactions
with hadroas
13.60 Hb Total and inclusive cross sections
Nuclear structure functions (P2)
D Indumalhi p 567
13.8K.+e Polarization in interactioas and scattering
vStructurc functions-sclccted topics (P2)
D K Choudhury p 547
13.90.-i-i Other topic.s in specific reactioas and
phenomenology of elementary particles
New physics at colliders (P2)
Saurabh D Rindani p 533
14. Properties of specific particles
14.40.-ii Mesons
14 40 Cs Other mesons mth S = C - 0, mass 2 5 GeV
Heavy flavor weak decays (P2)
R C Verma p 579
i4.60.-z I.eptoas
1 4 60.pq Neutrino mass and mixinf>
Perspectives in high energy physics (P2)
G Rajasekaran p 679
14.65.-q Quarks
14 65 Ha Top quarks
New physics at cV colliders {P2)
Saurabh D Rindani p 533
Experimental summary-XII DAE HEP
symposium, Guwahati, 1997 (P2)
Sunanda Banerjee p 689
14.70.-C Gauge bosons
New physics at cV colliders (P2)
Saurabh D Rindani p 533
l4 70Fm Wbo.wns
Physics at LEP 200 (P2)
A Gurtu p 5 1 5
l4 70.Hp /.bosons
Results from LEP 1 (P2)
S N Ganguli p 503
16
Subject Index
14.80.-j other particles
14 80 Bn Standurd model-Hii^^s bosons
Kxpcnmcnial summary-XIl DAE HE
symposium, Guwahali, 1997 {P2)
Sunanda Bancrjcc P 689
14 80 Bn Sumdiud-modrl Hi^^s bosons
Physics al LEP 200 (P2)
A Gurlu p515
14 KO Cp Non standard-model ni^K\ bourns
Status of wcak-scalc supersymmetry (P2)
Probir Roy p 479
1 4 80 1 .y Supei 4 muneinc iforlners of known pat lu Ics
Status of wcak-scalc supersymmetry {P2)
Probir Roy p 479
Status of supersymmetric grand unified
theories (P2)
B Ananthanarayan and P Minkowski
p 495
Physics at LEP 200 {P2)
A Gurtu p 5 1 5
20. NUCLEAR PHYSICS
21. Nuclear structure
21 .60.-11 Nuclear-.strueture ^ 1 odcl^ and methods
Multiparticle production process in high
energy nucleus-nucleus collisions
M Tantswy, M El-Mashad and M Y El-
Bakry p 73
21.65.+r Nuclear matter
Quark gluon plasma-current status of
properties and signals (P2)
C P Singh p601
24. Nuclear reactions: general
24.85.+P Quarks, gluons, and QCI) in nuclei and
nuclear processes
Nuclear structure functions (P2) ^
D Indumathi p 567
25. Nuclear reactions : specific reactions
25.30.-C Lepton'induced reactions
25.30 Mr Muon scattering
Nuclear structure functions {P2)
D Indumathi p 567
25.70.-/ Low and intermediate energy heavy-ion
reactions
25.70 Pq Multifratiinenl emission and correlations
Dynamical short range pion correlation
in ultra-rclativistic heavy ion interaction
Dipak Ghosh, Argha Deb, Md Azi/ar
Rahman, Abdul Kayum Jafry, Rini
Chaltopadhyay, Suml Das. l^yila Ghosh,
Bi.swanath Biswas, Krishnada.'t Purkait and
Madhumila Lahiri p313
25.75.-q Relativistic heavy-iun colli.sioas
Quark gluon plasma-current status of
properties and signals (P2)
C P Singh p 601
25 75 l)w Partii le and tcsonaiu e prodm tion
»
Mulliparticle production process in high
energy nucleus-nucleus collisions
M Tantawy, M El-Mashad and M Y
Hl-Bakry p 73
28. Nuclear engineering and nuclear power
studies
28.20.-v Neutron physics
28 20 Fc Neutron absoipiion
Measurements of flux and dose
distributions of neutrons in graphite matrices
using LR-1 15 nuclear track detector
Y S Selim, A F* Hafez and M M Abdel-
Meguid p 155
29. Experimental methods and
instrumentation for elementary-particle
and nuclear physics
29.40.-n Radiation detectors
29 4().Gx Trackinff and position sensitive detector::
Subject Index
17
Change in conductivity of CR-39 SSNTD
due to particle irradiation {PI)
T Phukan» D Kanjilal, T D Goswami and
H L Das p 433
29.4().Wk Solid-state detectors
Efficiency measurement of a Si(Li)
detector below 6.0 KeV using the atomic-
field bremsstrahlung
S K Gocl, M J Singh and R Shankcr
p65
Measurements of flux and dose
distributions of neutrons in graphite matrices
using LR-1 15 nuclear track detector
Y S Selim, A F Hafez and M M Abdcl-
Meguid p 155
30. ATOMIC AND MOLECULAR
PHYSICS
31. Electronic structure of atoms,
molecules and their ions; theory
3 1 . 1 5 .-p CalculatioiLS and mathematical techniques
111 atomic and molecular physics
T 1 15 Ciy Semn iassiciil methods
Scmiclassical theory for thermodynamics
of molecular fluids {PI)
Tarun K Dcy and Suresh K Sinha
p397
33. Molecular properties and interactions
with photons
33.20. -t Molecular spectra
.33.20.Ea Infrared spectra
Vibrational spectral studies and
thermodynamic functions of 4,6-dihydroxy-
5-nitro pyrimidine
B S Yadav, Vipin Kumar, Vir Singh,
M K Yadav and Subhash Chand p 249
33.20. -t Molecular spectra
33 20.Fb Ranuui and Rayleigh spectra
Vibrational spectral studies and
thermodynamic functions of 4,6-dihydroxy-
5-nitro pyrimidine
B S Yadav, Vipin Kumar, Vir Singh,
M K Yadav and Subhash Chand p 249
34. Atomic and molecular collision
processes and interactions
34.80.-i Electron .scattering
34 80 Bin Elastii scaiiennfi of electrons hy atoms and
molec ales
Efficiency measurement of a Si(Li)
detector below 6.0 KeV using the atomic
field bremsstrahlung
S K Goel, M J Singh and R Shankcr
p 65
40. HJNDAMENTAL AREAS OF
PHENOMENOLOGY
42. Optics
42.25.-p Wave optics
42 25Ds VViiiVC propanatton, transmission and
absorption
Electronic transport in a randomly
amplifying and absorbing chain {PI)
Asok K Sen p 365
Transport and Wigner delay lime
distribution across and random active
medium {PI)
Sandeep K Joshi, Abhijit Kar Gupta and
A M Jayannavar p371
46. Classical mechanics
46.30. -i Structural mechanics of shells, plates, and
beam.s
46.30. Pa Frution, wear, adlieience, hardness,
mechanical contacts, and tribology
Hardness anisotropy of L-arginine
phosphate monohydrate (LAP) crystal (N)
T Kar and S P Sen Gupta p 83
18
Subject Index
47. Fluid dynamics
47.2().-k HydrfKlynaniic stabilily
‘17 20 Gv inMuhiliiv
A new VISCOUS fingering inslabilily : the
ease of forced motions perpendicular to the
hori/onlal interlace ol an immiscible liquid
pair (PI)
B Roy and M H Hngincer p 417
47.35.+i IlydnKtynainic waves
A new viscous fingering instability ; the
case ol forced motion^ perpendicular to the
horizonalal mlcifacc of an immiscible liquid
pan (PI)
B Roy and M H Hngincer p 417
47.55.-t N(iiihumu|{eiu'oiis flows
47 5.*' ltd Simtified
A new VISCOUS lingering instability : the
case ol forced motions perpendicular to the
hori/onlal interlace of an immiscible liquid
pair (PI)
B Roy and M H Hnginecr p 417
60 . coni)f:nsed matter :
STRUCTURE, MECHANICAL
AND THERMAL PROPERTIES
61. Structure of solids and liquids;
crystallography
6J.lfl.-i X-ray diffraction and scaticring
01 ION/ (ind [xiwdci difltcK Iff/n
Characteristics ol selenium films on
different substrates under heat-treatment
S K Bhadra, K Maiti and K Goswami
p2()l
Investigation of graphitizmg carbons
Irom organic compounds by various
experimental techniques
T Hossain and J Fodder p 225
Studied of X-rays and electrical properties
ofSrMo 04
N K Singh, M K Choudhary and R N P
Choudhary p241
Crystal growth and characterization of
(NH4),BaCls-2H20
K Byrappa, M A Khandhaswamy and
V Srinivasan p 259
The effect of infrared pulsated laser on
the degree of ordering of cellulose nitrate
S A Nouh, M M Rad wan aAd A A El
Hagg \ p 269
61.12.-q Neutron diffruciion and scaltering
61 12 Ltl Sinf’lc ( n\\i(d and powder diffi ciclu>n
Neutron diffraction study of tin-
substituted Mg-Zn fcriites
A K Ghatage, S A Patil and S K
Paranjpe p 209
61.20.-p Structure of liquids
m
vScrniclassical theory lor transport
properties of hard sphere Huid (PJ)
Bircndra K Singh and Suresh K Sinha
p 385
61.25.-r Studies of spcciHc liquid structures
61 25. Hq Muitv nudetular and polyrnei solntion\;
ptflymennelts, swelluifi
Thermally stimulated depolarization
current behaviour of poly (vinylcdcnc
lluoridc) ; poly (methyl methacrylate) blend
system
Ashok Kumar Garg, J M Keller, S C
Datt and Navm Chand p 189
61.30.-V Liquid crystals
Phase alternation in liquid crystals with
terminal phenyl ring (PI)
Jayashree Saha and C D Mukherjee
p427
Subject Index
19
61.66.-r Structure of speciflc crystalline solids
61 66.Fx Inorganic compounds
Structural and dielectric studies on
lanthanum modified Ba 2 LiNb 50 i 5 (AO
K Sambasiva Rao, K Koteswara Rao,
T N V K V Prasad and M Rajeswara Rao
p337
61.72.-y Defects and impurities in crystals;
microstnictiirc
61 72 Ft Direct observation ofdis locations and other
defei /V
The role of the oxidising agent and the
cornplcxing agent on reactivity at line defects
in antimony
A H Raval. M J Joshi and B S Shah
p49
Defect characterisation of Sr^^ doped
calcium tartrate tetrahydratc crystals
K Suryanarayana and S M
Dharmaprakash p 307
6 1 72 Ji l^oint def€( ts and deject ( lusteis
Lattice relaxation in substitutional alloys
using a Green's function (P/)
S K Das p 379
6172S!> loipuntv concentration, distribution, and
gnidienis
Lattice relaxation in substitutional alloys
using a Green's function {PI)
S K Das p 379
61.80.-x Physicfil radiation effects, radiation
chaniagc
6180F.lI y-Ray efjcLts
Change in conductivity of CR-39 SSNTD
due to particle irradiation {PI)
T Phukan, D Kanjilal, T D Goswami and
H L Das p 433
61 .82, -d Radiation effects on specific materials
61 K2.Pv Polymers, organic compounds
The effect of infrared pulsated la.scr on
the degree of ordering of cellulose nitrate
S A Nouh, M M Radwan and A A El
Hagg , p 269
62. Mechanical aod acoustical properties
of condensed matter
62.20.-X Mechanical properties of solids
62 20.Fc Deformation and plasticitv
The effect of doping on the microhardness
behaviour of anthracene
Nimisha Vaidya, J H Yagnik and S S
Shah p 295
64. Equations of state, phase equilibria,
and pha.se transitions
64.30.-ft Kquatioas of stale of specific substances
Analysis of temperature dependence of
interionic separation and bulk modulus for
alkali halides
Rajiv Kumar Pandey p 125
64.70.-p SpeciHc phase transitioas
64 70 Md Transitions in liquid ( rystals
Phase alternation in liquid crystals with
terminal phenyl ring (PJ)
Jayashree Saha and C D Mukherjee
p427
65. Thermal properties of condensed
matter
65.50.-fm Thermodynamic properties and entropy
Analysis of temperature dependence of
interionic separation and bulk modulus for
alkali halides
Rajiv Kumar Pandey p 125
Semiclassical theory for thermodynamics
of molecular fluids (PI)
Tarun K Dcy and Suresh K Sinha p 397
68. Surfaces and interfaces; thin films and
whiskers
68.10.-m Fluid surfaces and fluid-fluid interfaces
68 lOJy Kinetics
Energetics of CO-NO reactions on
Pd-Cu alloy particles {PJ)
Mahesh Mcnon and Badal C Khanra
p407
20
Subject Index
68 JJ5.-P Solid surface and solid-solid interfaces
0« Md Surfac v energy, thermodynamic properties
Gas-suHace scattering; A review of
quantum statistical approach (PI)
SKRoy p35l
Sticking of Hc^ on graphite and aigon
surfaces in presence of one phonon process
(PI)
G Dullamudi and S K Roy p 455
68.45.-v Solid-fluid intcrface.s
68 45 Da Adsorpitnn and dcsprption kinetus,
euipoKiuon and conden.uitmn
Gas-surface scalcring: A review of
quantum statistical approach {PI)
SKRoy p35l
Sticking of He"^ on graphite and argon
surfaces in pre.scncc of one phonon process
(PI)
G Dutlamudi and S K Roy p 455
68.S5.-a Thin film structure and morphology
Stability of Ag island films deposited on
softened PVP substrates (P/)
Manjunatha Patlabi and K Mohan Rao
p4()3
70. CONDENSED MATTER :
ELECTRONIC STRUCTURE,
ELECTRICAL, MAGNETIC,
AND OPTICAL PROPERTIES
71. Electronic structure
71.27, +a Strongly correlated electron .systcm.s;
heavy fermions
Quantum magnetism : novel materials
and phenomena {PI)
Indrani Bose p 343
71.28. 'fd Narrow-band systems; intermediate-
valence solids
An orbital anti ferromagnetic state in the
extended Hubbard model (PI)
Biplab Chattopadhyay p 359
71.35.-y Excitons and related phenomena
Quantum magnetism : novel materials
and phenomena (PJ)
Indrani Bose p 343
71.35 Cc Intrinsic properties of excitons, optical
absorption spectra
Nonlinear light absorption in GaSe|_^j,
solid solutions under high excitation
levels
H Tajalli, M Kalafi, H Bidadi, M Kouhi
and V M Salmanov p 43
71.55.-i Impurity and defect levels
7l5.5Jv Disordered sliuctures, atfioiphous and
fllassy solids
Electronic transport in a randomly
amplifying and absorbing chain {PI)
Asok K Sen p 365
Transport and Wigner delay time
distribution across a random active medium
(PI)
Sandeep K Joshi, Abhijil Kar Gupta and
A M Jayannavar ' p 371
72. Electronic transport in condensed
matter
72.10.-d Theory of electronic transport ; scattering
mechanisms
Influence of alloy disorder scattering on
drift velocity of hot electrons at low
temperature under magnetic quantization in
/i-HgosCdo2Tc(Py)
Chaitali Chakraborly and C K Sarkar
p 463
72.15. -v Electronic conduction in metals and
alloys
72.15. Eb Electrical and thermal conduction in
crystalline metals and alloys
Pulse method for measrement of thermal
conductivity of metals and alloys at cryogenic
temperatures
T K Dey, M K Chattopadhyay and
A Kaur Dhami p 281
Subject Index
21
72.l5.Rn Quantum localimtum
Electronic transport in a randomly
amplifying and absorbing chain (PI)
Asok K Sen p 365
72.20.--i Conductivity phenomena in semi-
conductors and insulators
72 20 Dp General theory, xcatterhif* mechanisms
Electrical properties of organic and
organometallic compounds
A T Oza and P V Vinodkumar p 1 7 1
72.80.-r Conductivity orspcciHc materials
72 KO.Jc Other crystalline inorganic semiconductors
On the structure and phase transition of
lanthanum litanatc (AO
H B Lai, V P Srivastava and M Khan
p249
72 SO.lx Polymers, oiftanic ( ompounds
Electrical properties of organic and
organometallic compounds
A T Oza and P C Vinodkumar p 1 7 1
Change in conductivity of CR-39 SSNTD
due to particle irradiation (PI)
T Phukan, D Kanjilal, T D Goswami and
H L Das p 433
72.90.-fy Other topics in electronic traasport in
condeased matter
On the structure and pha.se transition of
lanthanum titanatc (N)
H B Lai, V P Srivastava and M A Khan
p249
73. Electronic structure and electrical
properties of surface, interfaces
73.20.-r Surface and interface electron states
73.2().Dx Electron states in low-dimensional
structures
Electric field induced shifts in electronic
states in spchcrical quantum dots with
parabolic confinement (AO
C Bose and C K Sarkar p 87
Electron tunneling in heterostructurcs
under a transverse magnetic field (PI)
P K Ghosh and B Milra p 447
73.23.-b Mesoscopic systems
73 23 Ps Other electronic properties of mesoscopic
.systems
Transport and Wigner delay time
distribution across a random active medium
(PI)
Sandeep K Joshi, Abhijit Kar Gupta and
A M Jayannavar p 37 1
73.30.-fy Surface double layers, Schott ky barriers,
and work functions
Effect of interface stale continnuum on
the forward (1-V) characteristics of metal-
semiconductor contacts with thin mtcrfacial
layer
P P Sahay p 57
73.40. -c Klcctronic tran.sport in interface
structures
Effect of interface slate continuum on
the forward (1-V) characteristics of melal-
scmiconduclor contaers with thin interfacial
layer
P P Sahay p 57
Study of forward (C-V) characteristics
of MIS Scholtky diodes in presence of
interface slates and series resistance
P P Sahay p 287
73.40 Gk Tunneling
Tunnelling current across a double
barrier
P N Roy and R B Choudhary p 23
73.40. Lq Other semiconductor-to-semiconductor
conttu ls, p-n junctions and heterojunctions
Electron tunneling in hctcrostructures
under a transverse magnetic field (PI)
P K Ghosh and B Mitra p 447
22
Subject Index
73.61 .-r Electrical properlic.s of specific thin films
and layer structures
73 6 1 Al Metals and metallic alloys
Slability of Ag island films deposited on
softened PVP SUB substrates (PI)
Manjunatha Pattabi and K Mohan Rao
p403
73 61 Ca ll-Vl xcmit ondiu loi.s
Thermal behaviour and non-isolhcrmal
kinetics of Ge,o+,Sc 4 oTe 5 o_^ amorphous
system
M M El-Ockcr, S A Fayck, F Metawc
and A S Hassanien p31
74. Superconductivity
74.40.-(-k Fluctuatioas
Fluctuations in high 7, superconductors
with incquivalent conducting layers
R K John and V C Kunakosc p 217
75. Magnetic properties and materials
75.10.-b (leneral theory and models of magnetic
ordering
75 10 Ilk Classit al \pin models
Inhornogeneity of vortices in 2ci classical
XF-modcl : a microcanonical Monte Carlo
simulation study {PI)
S B Ola and Sinita Ola p 413
Energy, fluctuation and the 2d classical
XF-model {PI)
Smila Ola, S B Ola and M Satapathy
p421
75.30.-m Intrinsic propertie.s of magnetically
ordered materials
75 30Cr Saturation momenta and maf’netu
susceptibilities
Neutron diffraction study of tin-
substituted Mg-Zn ferrites
A K Ghatage, S A Palil and S K
Paranjpe p 209
75.50.-y Studies of specific magnetic materials
75 50 Gg Fennuifinelirs
Neutron diffraction study of tin-
substituted Mg-Zn ferrites
A K Ghatage, S A Patil and S K
Paranjpe p 209
Study of bismuth substitution in cobalt
ferrite
Urmi M Joshi, Kapil Bhatt and H N
Pandya p 301
75.60.-d Domain cffecis, magnetizulion curves,
and hysteresis
75 60.E) Magnetization tuives, hysteresis,
Barkhansen and related effec ts
Metastability and hysteresis in random
field Ising chains {PI)
Prabodh Shukla p 439
7S.70.-i Magnetic filins and multilayers
75 70 Kw Domain \tiu( hire
Metaslabilily and hyslcresis in random
field Ising chains {PJ )
Prabodh Shukla , p 439
77. Dielectrics, piezoelectrics, and
ferroelectrics and their properties
77.22.-d Dielectric properties of.volids and liquids
Structural and dielectric studies on
lanthanum modified Ba 2 LiNb 50|5 {N)
K Sarnhasiva Rao. K Koleswara Rao,
T N V K V Prasad and M Rajeswara Rao
p337
77.22 Gm Dielectric loss and reluKation
Studies of X-ray and electrical properties
of SrMo 04
N K Singh, M K Choudhary and R N P
Choudhary p 241
77.65, -J Piezoelectricity and electroslriction
Disturbances in a piezo-quartz cantilever
under electrical, mechanical and thermal
fields {N)
T K Munshi, K K Kundu and R K
Mahalanabis p 93
Subject Index
23
The problem of a composite piezoelectric
plate transducer (PI)
T K Munshi. K K Kundu and R K
Mahalanabis p39l
77.65. Dq Aroustoelerlnc effects and surface aenusue
ivuvf.v in piezoelectrics
The problem of a composite piezoelectric
plate transducer (PI)
T K Munshi, K K Kundu and R K
Mahalanabis p39l
77.7().+a Tyroclcctric and eleclrnculoric cffccLs
The problem of a composite piezoelectric
plate transducer (PI)
T K Munshi, K K Kundu and R K
Mahalanabis p39I
77.80.-<* Ferroflectricity and aniift'rrocleci ricity
77.80 Bh Flwse tiansitions uiul Curie point
Studies of X-rays and electrical properties
of SrM ()04
N K Singh, M K Choudhary and R N P
Clioudhary p 241
77. K4.-s Dielectric, piczoclcclric, and ferroelectric
materiaLs'
77.84 Jd Polymers: orfianK (ompounds
Thermally stimulated depolarization
current behaviour of poly (vinyledcnc
fluoride) : poly (methyl methacrylate) blend
system
Ashok Kumar Garg, J M Keller, S C
Datl and Navin Chand p 189
78. Optical properties, condensed matter
spectroscopy and other interactions of
radiation and particles with condensed
matter
78.30. -j Infrared and Raman spectra
78.30. Jw Orffunic solids, polymers
Thermally stimulated depolarization
current behaviour of poly (vinyledcne
fluoride) : poly (methyl methacrylate) blend
system
Ashok Kumar Garg, J M Keller, S C
Datt and Navin Chand p 189
The effect of inlVared pulsated Laser on
the degree of ordering of cellulose nitrate
S A Nouh, M M Radwan and A A LI
Hagg p 269
78.60.-b Other luminescence and radiative
recombination
78 80 Kn Tliermoluminesrence
Evaluation of the trapping parameters of
TL peaks of multi activated SrS phosphors
W Shambhunath Singh, S Joychandra
Singh, N C Deb, Manabesh Bhattacharya,
S Dorendrajit Singh and P S Mazumdar
p 133
Determination of the activation energy
of a thcrmolumincscence peak obeying mixed
order kinetics
S Dorendrajit Singh and W Shambhunath
Singh and P S Mazumdar p 233
78.60 Ya Other luniiticst eiice
Optical properties of Pr^"^ doped glasses,
ctfccl of host lattice
Brajesh Shanna, Akshaya Kumar and
SBRai pl07
7K.66.-W Optical properties of speciFic thin film.s,
surfaces, and low-diiiiciisiunal structures superlattices,
quantum well structures, multilayers and
microparticles
78 66Jg Amorphous semiconductor s, glasses,
nanocrystalltne rnateruils
Optical properties of Pr^^ doped glasses,
effect of host lattice
Brajesh Sharma, Akshaya Kumar and
S B Rai p 1 07
Characteristics of selenium films on
different substrates under heat- treat mem
S K Bhadra, A K Maiti and K Goswami
p20l
24
Subject Index
» CROSS-DISCIPLINARY
PHYSICS AND RELATED
AREAS OF SCIENCE AND
TECHNOLOGY
81. Materials science
8 1.05.-<t Specific materiaLs : fabrication, treatment,
testing and analysts
8 1 05 Bx Metals, semimetal.s. and alloys
Mechanism of grain growth in
aluminium, cadmium, lead and silicon
F Jesu Rcthinam. S Kalainalhan and
C Thirupathi pll7
81.10.-li Methods of crystal f;rou'th; physics of
crystal growth
8110 On (irowth from solutions
Crystal growth, morphology and
piopciiics of NaHMP 207 (M = Ni, Co, Mn,
Zn, Cd, Pb)
K Byrappa and B Sanjeeva Ravi Raj
pi
Crystal growth and charactcri/.alion of
(NH4) BaCl.v2H20
K Byrappa, M A Khandhaswamy and
V Srimva.san p 259
Bl.15.-z Methods of deposition of films and
coatings; film growth and epitaxy
81.15 Ef Vat uiim deposition
Stability of Ag island films deposited on
softened PVP substrates (PI)
Manujunatha Patlabi and K Mohan
Rao p 403
8 1 . 1 5 Gh Chemical vapor deposition
Optical and structural characterisation
of ZnO films prepared by the oxidation of
Zn films
Benny Joseph, K G Gopchandran,
P K ManoJ, J T Abraham, Peter Koshy
and V K Vaidyan p 99
8 1 . 1 5 Tv Other methods of film f>mwth and epitaxy
Mechanism of grain growth in
aluminium, cadmiumm, lead and silicon
F Jesu Rcthinam, S Kalainalhan and
C ITiirupathi pll7
81.40.-z IVeatmcnt of materials and Its effects on
microstructurc and properties
8l.40Gh Other heal and thermomechanical
treatments
Optical and structural characterisation
of ZnO films prepared by the oxidation of
Zn films
Benny Joseph. K G Gopt^handran,
P K Manoj, J T Abraham, Petiir Koshy
and V K Vaidyan p 99
Ml 40,Np Fatif’ue, conosion fatigue, emhriftlemenl,
crackinfi. fiatture and Jailine
Tlic effect of doping on the microhardness
behaviour of anthracene
Nimisha Vaidya, J H Yagnik and B vS
Shah ^ p 295
81 40 Vw pressure treatment
Electrical properties of organic and
organomclallic compounds
A T 0/a and P C Vinodkumar p 171
81.65.-b Surface treatments
81 65 Cf Surface cIeanln^, etdiin^. palhernin^
Defect characterization of Sr"^ doped
calcium tartrate tetrahydrate crystals
K Suryanarayana and S M
Dharmaprakash p 307
81.70.-q Methods of materials testing and analysis
817()Pg Thermal analysis, differential thermal
analysis (DTA), differential thenno^ravimetric analy.sis
Thermal behaviour and non-isothcrmal
kinetics of Gcio+^Sc 4 oTc 5 o,ji amorphous
system
M M El-Ockcr, S A Fayck, F Metawe
and A S Hassanien p31
Subject Index
25
Invcstigalion of graphitizing carbons
from organic compounds by various
experimental techniques
T Hossain and J Fodder p 225
Crystal growth and characterization of
(NH4)^BaCl.v2H20
K Byrappa, M A Khandhaswamy and
V Srinivasan p 259
82. Physical chemistry
82.65.-i Surface and interface chemistry
82 6S.Dp Thermodynamu s of surfaces and interfm es
Gas-surface scattering; A review of
quantum statistical approach {PI)
SKRoy p 351
Sticking of He'* on graphite and argon
surfaces in presence of one phonon process
{PD
G Duttamudi and S K Roy p 455
82 (i.S Jv Heleroficneous (alalysis at siniaces
Hnergctics of CO-NO reactions on
Pd -Cu alloy particles (PI)
Mahesh Menon and Badal C Khanra
p407
85. Electronic and magnetic devices;
microelectronics
K5.30.-Z Semiconductor devices
85 30 Kk Jum tion diode
Study of forward (C-V) characteristics
of MIS Scholtky diode in presence of
interface slates and scries resistance
P P Sahay p 287
87. Biological and medical physics
87.53.-j Ionizing>radiatlon therapy physics
87 53. Pb Proton, neutron, and heavier panicle
dosimetry, theory and alfforithms
Measurements of flux and dose
distributions of neutrons in graphite matrices
using LR-115 nuclear track detector
Y S Selim, A F Hafez and M M Abde!-
Meguid ^ p 155
90. GEOPHYSICS, ASTRONOMY
AND ASTROPHYSICS
92. Hydrospheric and atmospheric
geophysics
92.60. -e Meteorology
92 60 Jq Water in the atmosphere
Effect of rain on millimeter-wave
propagation — A Review
Rajasri Sen and M P Singh p 101
95. Fundamental astronomy and
astrophysics; instrumentation, techniques,
and astronomical observations
95 JO.-k Fundamental aspects of astrophysics
95 30.Sf Relativity and f>ravilation
Blackholc evaporation-stress tensor
approach (P2)
KDKrori p621
97. Stars
97.60. -.S Late stages of stellar evolution
97 6().Lf Blaik holes
Blackholc evaporation-stress tensor
approach (P2)
KDKrori p62I
98. Stellar .systems; interstellar medium;
galactic and extragalactic objects and
systems; the universe
98.80.-k Cosmology
98 80 Cq PuHu le-theory and field theory models of
the early Universe
Early cosmological models with variable
G and zero-rest-mass scalar fields
Shriram and C P Singh p 323
98.80.-k Cosmology
98.80 Hw Mathematical and relativi.\tic a.spects of
cosmology; quantum cosmology
Matching of Friedmann-Lemaitre-
Robcrtson-Walker and Kantowski-Sachs
cosmologies
P Borgohain and Mahadev Patgiri
p331
Indian Journal of Physics A
Vol. 72A, No. 1
January 1998
CONTENTS
Condensed Matter Physics Pages
Crystal growth, morphology and properties of NaHMP 207 (M = Ni, Co, 1 -1 0
Mn, Zn, Cd, Pb)
K Byrappa and B Sanjeuva Ravi Raj
Determination of the order of kinetics and activation energy in 11-21
thermoluminescence peaks with temperature dependent frequency
factor
W Shambhunath Singh
Tunnelling current across a double barrier 23-30
P N Roy and R B Choudhary
Thcmial behaviour and non-isolhermal kinetics of GeKUfSeaoTeso-t 31-42
amorphous system
M M liL-Oc'Ki.R, S A Fayck, F Mltawc and A S Hassanif.n
Nonlinear light absorption in GaSei_,S,^ solid solutions under high 43-48
excitation levels
M Kalafi, H Tajalli, H Bidadi, M Kouhi and V M Sai.manov
The Role of the exidising agent and the complexing agent on reactivity 49-55
al line defects in antimony
A H Raval, M J Joshi and B S Shah
Fffect of interface state continuum on the forward (I-V) characteristics 57-63
of metal-semiconductor contacts with this interfacial layer
P P Sahay
Nuclear Physics
Efficiency measurement of a Si (Li) detector below 6.0 keV using the 65-71
atomic-field bremsstrahlung
S K Gold, M J Singh and R Shankfr
Multiparticle production process in high energy nucleus-nucleus 73-82
collisions
M Tantawy, M El Mashad and M Y El Bakry
[Cont'il on wxt pa f.'t']
Notes
Hardness anisotropy of L-argininc phosphate monohydrate (LAP)
crystal
T Kar andS PSfn Gupta
Electric field induced shifts in electronic slates in spherical quantum
dots with parabolic conlinemcnt
CBusi andCKSarkar
Disturbances in a piczo-(|uart/ cantilever under electrical, mechanical
and thermal fields
T K Munsiii, K K Kundu and R K Mahalanabis
Pages
83-86
87-92
93-98
Indian J. Phvs. 72A (1), 1-10 (1998)
UP A
-- .111 inti-malioniil loiiinal
Crystal growth, morphology and properties of NaHMPjO.^ (M = Ni, Co,
Mn, Z.II, Cd, Pb)
K.llyrippa* and B.Saiijeeva Ravi Raj
Dcpailmeiil ofCieology, IJniversily uf Mysore, Manasagangolri
Mysore - 570 006, India.
Abstract Crystal growth orNaHMPp^( M-Ni, Co, Mn, Zri, Cd, Pb) has been carried out by
liYdrothcnnal technique The studies concerning the crystal growth processes and the
inoipliologY of these superiuiiic pyrupliusphates with reference to the type of cations, its ionic
radii and other theniiodynafnic charactersitics have been carried out. Similarly the impedance
spectroscopic properties of these superionic pyrophosphates have been reported
Keywords Crystal growth, morphology, hydrothermal techinque
PACS NO 81 10 Dn
1, liilitiduclioii
Phosphates foini an impuilanl group of technological materials owing to their wide range of
physical and chemical properties fhe synthesis of phosphates began in the pervious century
Supei ionic phosphates have been reported for the past 20 years or so Moreover, all the Supcrionic
phosphates reported soon aflei discovery of NASICON were all orthophosphates. Fur tlie first
lime 0111 group reported high ionic conductivity in condensed phosphates, vi^ , pyrophosphates
I \~}] Since then a lot of work is going on in this direction In the present work, the authors
have studied (he rnoiphology of these crystals in great detail, with reference to the type of
cal ion, ionic ladii and thermodynamic characteristics Ihesc studies give an insight into the
ciyslal growth of such .supcrionic compounds in general Also the impedance spectroscopic
properties of these superionic pyrophosphates are given in brief
2. Crystal (irowlh
Crystal growth CKperimenis were carried out under three different PT conditions
(i) hydrothermal crystallization under lower pressure and temperature conditions;
(ii) hydrothermal crystallization under moderate pressure and temperature conditions,
(iii) hydrothermal crystallization under higher pressure and temperature conditions
In the first case, experiments were carried out in small autoclaves(temperature
100 - 250"C, pressure 100 - 200 bars ) and smaller Morey autoclaves provided with teflon
liners The starting solutions were prepared either by dissolution of P^O, in water or by directly
taking H^PO^ followed by the addition of respective carbonates or oxides or nitrates or chlorides
of respective cations or by the introduction of respective hydroxide into Ihe HjPO^ in various
proportions (maximum filling 70%) The alkaline component of the starting materials was used
in the form of a molar solution of a a definite molarity and this solution acts as a mineralizer
Since the crystallization occurred by spontaneous nucleation, the temperature of the furnace
was slowly increased to control the rate of nucleation fhe experimental temperature range was
200-300"C and the duration was 7 to 1 0 days The experimental conditions are given in Table I
•Aiilliiir lor aMTuslinndCiicc Iv-mail IJ YRCNraKllASIKiA VSNI, Nl- 1 IN
CO 1998 I ACS
K Byrappo mJ B Swieeva Ravi Raj
T,blf I. E*pe-.men.al cond.tions for low te,nper.ture hydrothetm.l aynthe.is
Compound
Nutnent Componenls Temp
(solvent) A mineralizer ('T'C)
Pressure %fill Durabon Sis
(barn)
Colour
a) H,PO^ NaOH CdO Zr(NO,)^
(ml) ' (ml) (fim) W
V*'*'/
1)
4
S(SM)
1
05
250
S
5(5M)
1
05
250
6
5(5M)
1
0 5
250
II)
5
S(2M)
1
0 5
250
5
S(3M)
I
0 5
250
5
S{4M)
1
05
250
S
M'^M)
1
05
250
S
S(f)M)
1
0 5
250
s
M7M)
1
05
250
111)
5
^CiM)
1
0 3
250
5
S(SM)
1
04
250
5
5(fiM)
1
0 5
250
5(5M)
1
06
250
S
S(SM)
1
0 75
250
s
S(5M)
1
0 90
250
IV)
5
5(5M)
0 5
0 5
250
5
5(5M)
OTS
05
250
5
5(5M)
1 00
05
250
5
5(5M)
1 50
0 5
250
b)H,PO, NaOH Co(NO,),Zr(NO,),
(ml) (ml) (gm) (gm)
1)
5
5(5M)
1
03
250
s
5(5M)
1
04
250
5
5(5M)
1
0.5
250
5
5(5M)
1
06
250
ii)
5
5(SM)
0 5
O'?
250
5
5(5M)
0 75
0 5
250
8
2‘3
Yellowish grey
8
2-5
Yellowish grey
8
1-3
Yellowish grey
8
dissolved
8 cr>'stalline matcnal yellowish grey
8
1-3
Yellowish grey
8
1-4
Yellowisli grey
8
1-3
Yellowish grey
8
0.5
Yellowish grey
8
1-3
Yellowish grey
8
1-3
Yellowish grey
8
1-6
Yellowish grey
8
05-2
Yellowish grey
8
05-1
Yellowish grey
8
irregular
p
8
1-2
Yellowish grey
8
1-2
Yellowish grey
8
2-5
Yellowish grey
8
1-3
Yellowisli grey
8
0 5-2
pink
8
0.5-2
pink
8
1-4
pink
8
1-3
pink
8 1
pink
8 1-2
pink
80 40
70 30
80 40
85 35
85 35
85 35
85 35
85 35
85 35
85 35
85 35
QO 40
85 38
QO 40
95 45
80 32
80 32
85 35
90 40
80 32
85 35
85 35
85 35
85 35
85 35
Crystal growth, morphology and properties etc
The experiments under moderate PT conditions were carried out by using Morey autoclaves
and Tuttle autoclaves within the temperature and pressure range 150-800 bars, T“ 250-400"C
using teflon and platinum liners respectively. In some experiments the results were much superior
compared to the lower PT conditions with regard to the cFystal quality and size.
The experiments under higher PT conditions were conducted using Tuttle cold-cone
sealed autoclaves provided with platinum liners (T “700“C and P = I 5 Kbars). The results of
these experiments are different The authors obtained compounds without a proton in their
composition, like NaKeP^O, , NaCoP^O, which are isostructural to allaudite.
The crystallization processes for the formation of HNaMP^O^ (where M = Ni, Co, Mn, Pb
and Cd) crystals have been studied, based on solvent-solute interactions and the complexation
processes is described thorough the following reaction series with regard to HNaCdP^O,
3NaOH + H,PO^ -►
3C:dfNO,)^ + 2H/0, ->
Zr(NO,), + 4NaOH
2II^PO^ ->
NaNO^ + ->
Na,P0,iH,P,0, >
rd,(PO,),t ->
Cd(NO^), 4 n^p^o,
NaOII 4 H^CdPjO, >
('d(OH)2 ^ NaH3P^O, ->
Na^PO; + mp
CdjCPOJj 4 2\]p + 6NO^ T 43[()] T
4NaN03 + 2Hp4 ZrOj
NaH^Pp^^HNO,
Nall^Pp^ f Na^HPO,
fljCdPjO^ 4 2H3PO, + 2Cd(Oll)j
H,CdP,0, 4 2 HNO 3
UNaCdPjO, 4HjO
HNaCdPp,4 2II,0
3 he study of complexation process with reference to the solvent-solute interaction is
of great importance to understand the crystallization of any compound [4] including
pyrophosphates Reports of such studies are seldom found in the literature for superionic
phosphates
The crystallization process involving many chemical interactions, lead to the
rormalion ofa stable complex in the following stages (i) Acid-base interactions, (ii) Formation
of metal-aqua complexes ; (iii) Interaction between acid-base and metal-aqua complexes
A series of experiments have been carried out with several divalent and Irivalent
metals and it was found that the divalent metals enter the composition more easily than trivalent
metals However, nutrient material show that only Al” enters the composition readily forming
Na^HjAl(P^()^)^ even at lower PT conditions (P < 100 bars, T ~250”C), while others insist upon
higher temperature of synthesis |5|
3. Morphology
A number of factors such as the degree of supersaturation, type of the solvent, pH of the
mineralizer, etc aifecl the habit of crystal Habit modifications occur with significant changes
in the growth temperature and also with the presence of impurities in the growth media [6| The
pyrophosphates show a wide range of morphological variations The characteristic habits of
72A(1)-:
4 K liyriipiM and B Saiijemi Bavi Raj
some selected pyrophosphates are shown in Figures l(a-g) The habit! exhibited by these
pyrophosphates are given in Table 2 The crystal faces of most ofthe superionic pyrophosphates
ate more or less smooth and vitreous in lustre, and transparent The morphology of these
pyrophosphates varies from one another depending upon the calions present.
Table 2 Morphology of pyrophosphates |
( ompound
System
Crystal fbrm(s) 5;
Na,ll,AI(Pp,),
Monoclinic
Third order pinacoid 'j
side pinacoid
HNa('oP,0,
Triclinic
Third order pinacoid
Second order pinacoid
Fourth order pinacoid
HNaNiP,0,
Triclitiic
Basal pinacoid 1
positive and negative
third order pinacoid
HNaZnP^O,
Tricliiiic
Basal pinacoid
positive and negative
(i-HNaMnP/)^
Trichnic
Second order pinacoid
IINa('dP,0,
Tridinic
Basal pinacoid '
positive and negative
tINaPbP/),
Triclinic
Basal pinacoid
NaPePjOj
Monoclinic
Side pinacoid
positive and negative pinacoids
Na,CaMn,Pj(),
Triclinic
Third order pinacoid
Crystal growth, morphology and properties etc
5
Similarly, the morphology varies with respect to the degree of supersaluration, the
concentration of H,0, P^O, and Na^O in the system
It is interesting to observe that the morphology of superionic pyrophosphates vary with
the variation in the cation. The cations used in the present work are Al, Mn, Co, Ni, Zn, Cd and
Pb The Table 3 shows the cation properties. As evident from Table 3, the Al is the smallest ion
and it show the excellent morphology (Figure la). Crystals are well developed and highly
transparent This is followed by Mn, which gives good crystals of excellent crystal habit.
However, due to the susceptibility of Mn for the changes in the valency with sudden changes in
the experimental growth parameters, there is a tendency for the formation of polymorphic
modifications of Mn bearing superionic pyrophosphates. But both the polymorphic modifications
of Mn superionic pyrophosphates show excellent morphology with well developed habit, smooth
and vitreous surfaces (Figure lb)
Table 3. Cations In superionic pyrophosphates
FJcmcnl
Al. No.
Al. Wl.
Al. radii*
. tA) __
liniropy
at 298"K (c.u)
Al
13
65.38
182
6 769
Mn
25
54 93
1 79
7 59
Co
27
58.93
167
68
Ni
28
58 70
162
7 137
Zn
30
65.38
1 53
9 95
Cd
48
112 41
1 71
123
Pb
82
207 20
181
15 49
* ret Table of periodic properties of the elements, Sargent-Welch Scientific company
The cobalt bearing superionic pyrophosphate shows probably the best morphology (Figure
] c) The crystals are developed very well with vitreous, smooth and transparent surfaces. The
cobalt bearing superionic pyrophosphates are bigger than the other pyrophosphates It is observed
that in spite of the same crystal structure exhibited by all these pyrophosphates, they slowly lose
their morphology The crystal habit in Ni is better than in Zn, because the crystals are well
developed, but the crystals are mostly slender or rod shaped, transparent with smooth and vitreous
surfaces (Figure Id)
In case of Zn, there is a fall in the morphological development (Figure le) like crystal
habit, lustre, and transparency When we come to the Cd bearing pyrophosphate the
morphological variations are still less with the crystals loosing their size, well developed habit,
smooth surfaces and vitreous lustre and a low degree of transparency (Figure 10 When the
crystal surfaces were observed under higher magnification, they show more or less rough surfaces
with defect structures
In case of Pb bearing surpeionic pyrophosphates, it is still worse The crystals do not
have well defined crvstal habit, but instead, they look more or less rounded and clustered
NaZnHP,0
|(X300)
(b)
Figure 2. Shows the growth layers on (010) face of pyrophosphate crystals ; (a) HNaCoNiP^O?
and (b) a-HNaMnPsOy ;
K Hyrappti ‘i’>d B Sunjc'cva Kavi Raj
Plate 11(b)
(X 400)
(c)
Crystal growth, morphology and properties etc
Plate in(a)
Figure 3, Shows the macro steps in pyrophosphate crystals : (a) HNaCoP207
and (b) a-HNaMnP2C>7.
Cfystol growth, morphology ond properties etc
Plate Ill(b)
Figure 3 . Shows the macro steps in pyrophosphate crystals (c) HNaZnP2C>7 .
Crystal growth, morphology and properties etc
1
(Figure Ig). The crystals appear translucent with dull lustre and without smooth surfaces.
There is a general tendency for the crystals to become polycrystalline from Mn end to the Pb
end.
It is believed that all these superionic pyrophosphates belong to the same structure tycp,
i.c iriclinic, P, space group As the ionic size of the cations inserted into the structure increases,
there develops a general structural distortion which in turn, affects the crystal morphology. The
Table 4 shows the variation in bond lengths of these superionic pyrophosphates. As evident
from the Table 4, the difference in maximum and minimum bond lengths of Na-0 polyhedra
increases gradually towards the Zn end member. Thus towards the Pb end member, the crystals
become poorly developed The poor morphology of Cd and Pb bearing superionic pyrophosphates
is due to the changes in the bond lengths and bond angles leading to the slightly higher degree
of structural disorder as indicated by the preliminary X-ray single crystal dififraction studies.
Also it is evident from, Table 3 that the values of entropy for Zn, Cd and Pb gradually increase
and these values are quite high compared to those of Mn, Al, Co or Ni Entropy is directly
related to the structural disorder Thus, the thermodynamic properties of the cations also directly
influence on the morphology of these new superionic pyrophosphates.
The superionic pyrophosphates show very interesting surface morphology and it varies
accordingly with the growth temperature, degree of supersaturation and the cation in the nutrient
I hese variations also depend on the magnitude and anisotropy of the growth rates along different
direction
The most commonly observed surface growth features are growth layers, grwoth steps
and block structures The surface morphology of these superionic phosphates is given in Table
5 Since these crystals belong to the lower symmetry, the effect of growth temperature, degree
of siipersaturation and the impurity concentration is very well depicted in their surface
morphology
Table 5 Surface morphology of KNaCoP^O^ and HNaNiP^O^ crystals
('omposilion
Growth (crap (•€)
Common faces
Growth rate
Growth reatures
Face
HNsCoPjO,
250
(II0)(010)(I10)
V(I0I)>V(0I0)
Growth layers
block structures
(010)
IINaNiPjO,
250
(II0)(0I0)(0II)
V(II0)>V(0I0)
Macro steps
block structures
(110)
HNaZnPjO,
250
(HO) (101) (Oil)
V(II0)> V(IOI)
Macro steps
block structures
growth layers
(101)
crystal. i T ,7 1 * pyrophosphate
rZ rTf ^ ^ respresent the macro steps observed in the pyrophosphate
c^stais As the supersaturation increased, the spirals must have become rounded. The Fisure
half of the spirals/laycrs With an increase in the supersaturation and thermal
ihp G. ^ ® ''umber of macro spirals increases and a large number of thick steps appear on
ace as s own in Figure 3a The crystals obtained from experiments with surplus Na^O
g KByrappa and B San, eeva Ravi Rai
in niiitlitv mav be because of the increase in the viscoaity
and also the high solubility of pyrophosphates in H,PO,.
"" ”‘“:"K’cSS
A ’ To and Ni membOT show orderly arranged spirals and even a single Iwge spiral «faaioiially
steps and block structures Whereas the Cd and Pb end members mainly Wock “d ^
structures and highly discountinous surface structures. This is again connected with the highly
distorted polyhedra and octahedra owing to the larger ionic radii of the transitional metals and
also highly distorted polyhedra of Na-O coordination towards Cd and Pb end.
4. Chiracterization j
i
The supcrionic pyrophosphates crystals obtained were characterised using different techihques
like XKl). I'l’MA and impedance spectroscopy The single crystal X-ray data for the supei[ionic
pyrophosphates is given in lable 6 HPMA analysis is given in Table 7.
Table 6 X-ray data of superionic pyrophosphates
Compound System Space Cell parameters A
group
a
b c
p
(“)
V
A^
Z
Na/lll,(P/Vj
'friclinic
PI
8 311 (4)
7 363 (4) 4.902 (3)
81 77(2)
m 2 (4)
■
a - IINaMnl’,0,
Morioclinic
9 935 (4)
8 455(3) 13 106(4) 110 75
1029(1)
8
(1 - HNaMnI'jO,
Triclmc
PI
6657(1)
7372(1) 6517(1)
9222
1029
8
HNaCoPjO,
Triclinc
PI
6 5190(6)
6 595(1) 6 485(1)
92 07
255 97(7)
2
IINaNiP/)^
Iriclinc
PI
6 502 (3)
6418(1) 6.442(2)
91 83(1)
249 33 (7)
2
IINaZiiP^O^
Iriciinc
PI
6 509 (3)
7 250(3) 6 486(2)
92 09
260 37 (7)
2
IINaCdPp^
Triclinc
PI
6612
6 674 6 597
92 75
290 78
2
NafcP,()^
Monoclinic
II 83
12 527 6 44
114 18
870 63 (2)
-
I hc CIS rncasuremcnls were carried out using Solatroii Impedance Analyser system (Model
1 260) from I I Iz to 32 Ml Iz 'I he pellets were made by pressing the superionic polycrystalline
powder al 5 ton/cm ^ pressure The impedance data was analysed using EQUIVAl.ENT CIR-
CUl 1 ( RQUI VCRT PAS) PROGRAM [7] The complex impedance data has been analysed to
extracl the bulk resistance (Rb) and hence a c conductivity (a^
Crystal growth, morphology and properties etc
Plate IV
(b)
Figure 4. Shows etch pits and block structures in pyrophosphate crystals (a) HNaCoP'707 and
(b) HNaNiP2C>7
Crystal growth, morphology and properties etc
Tlie Figure 5 shows the Arrhenius plot ( In a^T vs 1000/T) The Arrhenius plot show single phase
with an aclivalioii energy 1 1 eV for lower temperature region up to 423 K Above 423 K - 523
K, the Arrhenius plot is not linear Non-linear Arrhenius plot may be explained In terms of interfa-
cial nicker noise due to polarization at the sample-electrode interface.
Table 7. EPMA analysis of supenonic pyrophosphates
Oxido
IINa(’ol*(),
IINaNiPn,
lINa/iiPX),
HNaMnlM),
Wi'X.
Wl%
WPI;
wia
Na/)
09.92
11.07
11.42
11.17
C^)()
30.07
(K).IO
(K).IO
(Ml.OO
/It),
(MU)O
00.00
(Ml.OO
(Ml.OO
59.4.5
58. IK
57.94
57.46
NiO
(K).04
.11 II
00.00
(Ml.OO
/nO
00.00
(M).OO
.11.63
00.00
MnO
(M).12
(M).25
00.21
12.31
Cut)
(K).04
00.00
(M) Ot)
(Ml.OO
IvO
IMM)4
00 22
00.04
(MI.04
IK).
(M).()7
00.02
00.01
00.01
foul
KM) 15
lOf 5f»
101 25
99.99
2 3
ion (k)
AckilowIcdgmeiUs ^ Aniicmus plol Tor pyrophosphates
Ihoiuilliois wish ii> thank Piol A 11 Kiilkaini, Dcpailmcnl of Applied Kleclionics, (iulbarga
linivcisilv, (hilhaiga, lot the help in impedance spcclioseopy
10
K Byrappa and B Sanjeeva Ravi Raj
Rcrercnces
1 1 1 S (jali,K Dyrappa and G S Gopalkrishna^cto C>>'.v/.,C4S 1667 (1989)
|2] K Eiyrappa^G S Gopalkrishna and S Gali^/]c//aFLj./'/iy.,63A 321 (1989)
|3] S Gall and K Byrappa^c7a CrysL, C46 0990)
[4 1 K Burger ^S()lva(^on, lomc and ( 'nmplex Formation Reaction in Non-Aqueaus Solvents
(Budapest Academial Kiaddo ) 42 (1983)
(5J K Byrappa Indian J.Phys 66A 233 (1992)
[6J R Kern in Growth of Crystals vol 8 ed. N.N. Sheftal ( New York : Consultanls Bureau
(1969)
|7| B A Boukamp Fquivalent Circuit Users Manual ( University of Twente, The Netherlands)
(May 1989)
Indian J. Phys. 72A (1), 1 1-21 (1998)
UP Ax
Lin inttrmutional journal
Determination of the order of kinetics and activation
energy in thermoluminescence peaks with
temperature dependent frequency factor
W Shambhunath wSingh
Department of Physics. Manipur College, Iniphal. Sing)ainei 79.S (AlS.
Manipur, India
Reieived 4 Apnl I4Q7, iiaepied 2^ Sppfnnhet I9<i7
Abstract : A iiicihod lo dcicrininc the order of kmeiic's and acinaiioii energy oj
t!ierniolumin«scencc peaks with icmpciatuic-dependent iieqiiency l.ieioi .s presenied I hi
niclhod uses llic peak leinpeiaiuie und/or ihe ieinperaiur'‘s eoiivspondine lo the Iao poinis ol
inlleciion of tlic peak
Kcyword.s : Thermoluiiiinesceiice. Older ol kinetics, ado ation energy
PACSNo. ; 78 60 Kn
1. Introduction
Thcrnioluminescencc (TL) glow peaks occur when the temperature ol a previously excited
crystal increases with time as a result of clcciron-hole recombination. The TL method is an
important tool for the determination of the characteristics of electron trapping states in
insulators and semiconductors. For the theoretical treatment of TL, normally it is assumed
that frequency factor which is related to the electron capture cioss section is independent of
temperature. But due to the temperature dependence of type T~*' (0 < a < 4) of electron
capture cross sections [1-^1, frequency factor will also depend on Icmpei aturc as V' (- 2 < n
^ 2) 15,61- Land [7] suggested a method of calculating trapping parameters from the
inlleciion points and maximum of TL glow curves. Garlia et cil 18 ] and Singh et al [9) pul
the method of Land [7] in a more usuablc form by suggesting a number ot expressions lor
the determination of the activation energy of TL peaks of arbitrary order of kinetics. But
unlike Land [7], they did not consider the temperature dependence of frequency factor. In
the present paper, the problem is reinvestigated by considering the temperature dependence
of frequency factor.
(n 1998 I ACS
72A(I)-3
12
W Shambhumth Singh
2. Theory
Following Fleming [6), the glow intensity of a first order and non-first order TL glow peaks
can be expressed respectively as
KT) = Cn^s
-EUkT)
exp[-£/(ir)]dr (1)
l/
and
nT) = Cn,,Vo7'"exp[-£/(*7-)][l + Uo(fc-l)//31
X Jr-' expl-£/(*r')lrfT']''''‘'""
where the symbols have their usual meanings and for ^ 1 ,
( 2 )
The temperature dependence of frequency factor is given by [5,6]
-V = sjr ^ (3)
The peak temperature of the glow peak can be obtained from the relations
£ / (kTl ) - sJl exp(-£ / (£/•„ ))lp+ a/ T,„ = 0, (£ = 1 ) (4)
anil a/7„, + EKkll,) = exp|- £/(££„,)] /^
x|l + l5„(/>-l)//3J
A r'" cxp\-E/(kT')]dT'] ^ (5)
Jr,
The integral f T'^ cxpl-£ / (kT')\dT' appearing in eqs. ( 1 ), (2) and (5) cannot be solved
analytically and therefore has been developed as
c\pl-E/(kT)]dT
cxp\-E / {kT)]dT - j\‘‘s\pl-E/(kT)]dT
= (£//:)"■*■* “du - Jm " “du
= (E/kr^^[r(-a-U2) - n-a-lu,)]
(6)
with u = Elik'F). r is the incomplete complementary Gamma function [10]. The
integral can now be evaluated numerically by using algorithm of Lentz on continued
(i action [11). This method converges rapidly and permits a very high precision.
Determination of the order of kinetics and activation energy etc
13
Eliminating Sq/P from eqs. (1) and (4), one can write [12]
= (“m
Similarly, eliminating S(fP from eqs. (2) and (5), one gets
///„ = («„/u)“exp(H„-H)[l - (8)
with F{u,uJ = (au„ + u^)u“ cxp(u„)[r(-fl-l,u„,) - Fi-a-lu)] (9)
and Im is the peak intensity.
The inflection points 7,| and 7,2 in the rising and falling sides of a TL peak satisfy
the equation.
{d^l/dp) = 0 forr = T„(/= 1.2). (10)
Eq. (10) can also be written as
lidlldu) + aid'll Idu^) = 0 (II)
with dil du = l(dFldu - a / u - \), (h=\) (12)
= -I{al u-^\)-\-(u„J uy CKp(u^-u)idG I du), {b^]) (13)
d"l/du^ = Hd^FIdu^-^alu") + (dHdu)idFldu-afu-\), {h=\) (14)
= - (dl / du){a / u-i- \) -{■ ai / + {u / u)‘^ expiu - u) (h ^ ])
) - (dG / du)(a / u + \)\,
(15)
where
G =
(16)
D = 1 - ((h-])lb)F{u.uJ.
(17)
For the case of temperature-independent frequency factor (« = 0), the eqs. (12)-(17) reduce
to corresponding expressions of Gartia et al [8].
Eq. (11) has been solved by using Newton-Raphson method [II], To a good
approximation, a plot of u,, /(«,, -«,,), u ,2 /(«„, -M 12 ) and /[w„,(m,i * w,:!] against
u,„ are found to be linear. The linear relationship is illustrated in Figure I for the pair
m,i/(m,i - M,„) and u„ for = 2 and « = 2.
Hence, one can write
+ ^1.
(18)
“m = M2/(«m-“,2) + ^2-
(19)
“m = Ml“i2 -“,2)1 +
(20)
where
u,i = £/<*T,,) and«,2 = E/ikT,^).
W Shamhhunath Smgh
Egs.()8H20)caii
aJso be written as
£■, ^ A,kTll(T„-TJ + B,kT„.
= A^kTlKT, 2-^,0 + BykT„.
( 21 )
( 22 )
(23)
The c()efficienlsy4^ and 1,2, 3) occurring in eqs. (18-23), arc calculated for different
values of a(2<a<-2) for a particular value of h. The method of linear least squares [13] has
been used to express each of the coefficients and B, as a linear function of a as
^1 = ^1/“'
= £)„, + D,ja. (25)
The coefficients Ciij,C]j ,Dq^,D^j (/ = 1, 2, 3) occurring in cqs. (24)-(25), are determined
by solving the normal eqs. [13] for the least square lines (24) and (25) and are presented in
Table I .
Table 1. Co-efficienl Cqj, C\j, and D\j occurring in equations (24) and (25)
b
J
Coj
^Oj
Ol,
1 0
1
0 9627
-0.0003
-0..5617
- 0.97.59
2
0 9626
-00003
-0 6980
- 0.9753
1 .9253
-0.0005
-0.6294
- 0.9755
1 5
1
1 5866
-0 0005
-1.1746
-0.9602
2
1 1585
-00005
- 1.1436
-0 9546
3
2.3172
-0.0010
- 1.3043
-0.9575
20
1
1.3159
-0.0007
- 1.7181
- 0.9477
2
1.3156
-0.0008
- 2.0967
-0 9398
3
2.6314
-0.0014
-1.9053
-0.9440
Determination of the order of kinetics and activation energy etc
15
3. Results and discussions
The points of inflection of numerically computed TL peaks have been evaluated by
solving eq. (1 1) with Newton-Raphson method [11], The computer code has been checked
Figure 3. Same as in Figure 2 but
for the second point of inflection
by reproducing the results of Land [7] and Gartia et al [8] (Table 2). In Figures 2 and 3,
the fractional intensities (j = 1 , 2) at the two points of inflection for first order kinetics
16
W Shambhunath Singh
Table 2. Activation energies (in cv) of numerically generated peaks reported (£,„) by Land [7]
and Gartia ei al [8] by using present set of formulas and Land [7] formula. E\, £2. £3 correspond
to the present set of formulas and E\i^ £2^ denote the activation energies calculated byusing
Land [71 formula. £(a = 0) denote the eneiigies calculated by setting a = 0
h £,n Im « Ta El E2 £3 £i(eV) fjCcV) £3(eV) £,^ £2^
(cV)
(K)
(K)
(K)
(eV)
(eV)
(cV)
fl = 0
(eV)
(cV)
1 0
0 20
97 2
0
93 4
1010
0 2000
0 2000
0 2000
0.2000
§
0
1 0
0 20
97 2
2
937
1(X)7
0 2000
0.20(X)
0.2000
0.2165
0.2165
0 2165
—
1 0
0 67
311 0
0
299.3
.322 7
0 6699
0 6700
0.6700
0.6700
0.6700
1 0
0 67
31 1 0
2
300 1
.321 8
0.6701
0 6701
0 6701
0 7229
0 7229
0 7229
—
—
1 0
1 2
.*>4^2
0
525 1
565.2
1 2000
1 2000
1 1999
1 2000
1 2000
1 0
1 2
2
2
526 5
563.8
1 2001
1 2002
1 2002
1.2927
1 2927
1.2927
—
20
0 2
%9
0
91 9
101 8
0 2001
0.2001
0.2001
0 2000
0 1998
2 0
0 2
96 9
92.3
101 5
0 2001
0 2(X)I
0 2001
0.2162
02161
0.2162
—
__
20
0,67
310 2
0
295 0
325 2
0 6702
0 6703
0.6703
0 6699
0 6709
2 0
0 67
310.2
2
296.0
324 2
0.6704
0 6704
0 6704
0 7220
07218
0.7219
_
— '
2 0
1 2
.•143.9
0
5177
.569 7
1 2(XM
1 2006
1 2005
1 2000
1.1993
20
1 2
•143 9
2
519 4
.568 0
1.2008
1.2009
1 2008
1.2913
1 2909
1 2911
—
1 0
1 0
536 5
0
513 2
559 6
1 (KXX)
1 0000
1 0000
1 (XXX)
1 0000
1 0
1 0
4K8 7
0
469 3
507 9
0 9999
1 0000
0 9999
1 0000
1,0000
1 0
1.0
448,4
0
432 1
464 7
1 0000
1 0000
0.9999
1 0000
1 00(X)
1 0
1 0
414.1
0
4(K)2
428 0
1 (KXX)
1 0000
0.9999
1 (XXX)
1 (XXX)
1 0
1 0
384 6
0
372 5
396 6
1 exxx)
0 9999
0 9999
1 .<5(xx)
1 0000
1 s
1 0
535 6
0
508 4
562 4
lOOOl
lOOOl
1 0001
-
1 S
1 0
488 0
0
465 3
510.4
1 0001
10001
10001
—
—
1 s
1 0
447 9
0
428 7
466.9
1 (XX)1
1.0002
1 0001
—
—
1
1.0
4137
0
.397 .3
430.0
1.0001
1 0001
1.0001
—
1 5
1 0
384 2
0
370 0
398 2
1 0001
1 0002
1 0002
—
—
2.0
1 0
534.7
0
504.7
.564.3
1.0002
1.0002
1 (X)02
1 0000
0.9988
2 0
1 0
487 3
0
462 2
512.1
1 0003
1 0004
1.0003
0 9988
0.9992
20
1 0
447 4
0
426.1
468.4
\.om
1 0005
1.0004
0.9999
0.9994
2 0
1 0
413 3
0
395 0
431 3
1 0004
1 (X)05
1 0005
0 9999
0 9996
2 0
1 0
383 9
0
368 1
399 5
1 0004
1 (X)06
1 0005
0 9999
0.9997
(/? = 1) corresponding to a = -2,0 and 2 are plotted against The results for other values
of a fall between the curves for a = -2 and a = 2 and cannot be distinguished in the present
scale of the figure. From Figure 2, it is seen that the fractional intensity corresponding to
the first point of inflection increases with u„ but for > 40, it does not change much. On
the other hand, the fractional intensity corresponding to the second point of inflection
decreases with increasing m^. But like the first point of inflection, this ratio is almost
insensitive to u„, and a for > 40. The same feature is observed for second order kinetics
Determination of the order of kinetics and activation energy etc
17
{h = 2) (Figures 4 and 5). In Figure 6, the variation of the fractional intensities as a function
of the order of kinetics (b) is given for = 40 and for a « ^ 2, 2. It is to be noted that the
Figure 4. Same as in Figure 2 bul fur /? = 2
inflection point in the falling side of the peak is more sensitive to the order of kinetics than
that corresponding to the rising side. Figure 6 can be used for the preliminary estimation of
the order of kinetics like the curve connecting ^ and b\5].
Now the determination of the activation energy is considered. As already noted, the
coefficients occurring in eqs. (21)-(23) for the evaluation of the activation energy depend
18
W Shambhunath Singh
on b and a. In Table 2, the activation energies of some numerically computed TL peaks
reported by Land [7] and Gartia et at (8] are presented. If the temperature dependence
of frequency factor with a = 2 is considered, there is a change in the values of 7,i and Tjj .
Figure 6. Variation of fractional intensities
at two points of mfleciion againtit order of
kinetics (h), A. B, C, D stand rt^pectivcly
for /,,//„ (« = 2), (« = -2).'fo//„ (n
= 2) and /c//„(n = -2)
II IS found lha( (here is an excellent agreement between the input values of energy (£,„) and
the energies £,( j = 1 , 2, 3) calculated by using the present set of expressions (21 )-(23)
when the actual values of a are used. But the difference in the values of T,-, and 7,^ with a
= 2, results in an error of activation energy by 1% to 8% if eqs. (23H25) are used with a =
0 rather than with a = 2. The activation energies of TL peaks (with a * 0) calculated by
setting a = 0 are denoted by £, (rj = 0). But as already noted, the inflection points are
not much sensitive to the values of a in accordance with the observations of Fleming |6|.
As a result, it is very difficult to determine a. To estimate the error for a wide range of
(10 < < l(X)) in the determination of activation energy due to the lack of information
about the value of a, the lormulae of activation energy for a = 0 are used to determine tlic
activation energies of some numerically computed TL peaks with a it 0 and in Figure 7,
thejnoportional percentile error 5 = ll£,„-£,l/£Jxl00% is plotted as a function of
- E/{kT) lor h = 2 and a = - 2, 2 (£„ is the input value of the activation energy and £, is
the calculated value of the activation energy for h = 2 by using the formula involving both
the infiection points). The values of £,„ and T„ to obtain £, are choosen such that £,„ and
u„ lies in the ranges 0.5 eV 5 £,„ S 2.0 eV and 10 S S 100. It is seen that the activation
Determination of the order of kinetics and activation energy etc
19
energy is over-estimated for the case of a = 2 and under-estimated for o = - 2. d decreases
from about 9.5% to about 3% as changes from 20 to 80. The same featuie is observed
for other values of b. So, it is observed that the temperature dependence of frequency factor
can lead to a maximum error of around 10% in the determination of activation energy by
point of inflection method.
_l
60
Figure 7. Ploi ot proportional percentile
error S against \o\ h ■- 2 A a = 2 and
B (1^-2
In Table 2, calculation of activation energies of the numerically generated TL peaks
of Land [7] and Gartia et al [81 has also been done using the formulae presented by Land
[7). Gartia et al [SJ commented that the formulae of Land do not work well. But it is
observed that in order to obtain accurate results by using Land's formula, one has to carry
on Iteration. In the present work, the iteration has been carried out using Newton-Raphson
method [11]. For h = \, rapid convergence is obtained but foi t = 2, a large number of
iterations is required. The present .set of formulae are superior to ihosc of Land in the sense
that no iteration is required and can be used for any value of b. The formulae of Land [71
are available only for = 1,2 and for a = 0.
Finally, the applicability of tfie method is considered by taking well-studied
experimental 165.5“C TL peaks of Ca-doped KCl fl4J and 320”C bluish green michroline
(K Al Si 3 QR) [ 1 5], It has been shown by Singh et al [9] that these peaks correspond to a = 0.
Using Figure 6, it is found that these peaks obey first and second-order kinetics
respectively. But since the determination of a for an experimental TL peak in accordance
with Fleming [6] is difficult, the possible shift in the determination of the activation
energies with different values of ci (- 2 < o < - 2) is presented m Table 3. The values of E,
0 = 1, 2, 3) using the present set of cxpre.ssions (2IH23) and the values of = “ 2, - 1, 0,
1, 2 but using the same observed values of 7,, and 7,2 values off? are presented in
Table 3. From Table 3, it is seen that the activation energies are over-estimated for the case
72A(1).4
20
W Shambhunath Singh
()1 a < 0 and under-estimated for^i > 0 which is opposite lo the observations in Figure 7 and
Table 2 w here there is over-estimation for a > 0 and under-estimation for a <0. This is due
10 the fact that while observing the shifts in energies in Figure 7 and Table 2, the shifted
values of the inflection points with a arc used to calculate E by setting a-0; but in Table 3,
the shifts in energies arc calculated using the same observed values of the inflection points
Tabic 3. AcMvalion energies (/ - I. 2. 3) of some expcnmenlal peaks t^),14,151 computed
with the observed values of 7',i and T ,2 and by using picseni set of expressions setting at values
ofri=-2, 1,0. 1,2
/AKs')
VK)
7-,i(K)
TaiK)
h
a
£i(eV)
£2teV)
£2(cV)
I 36
04167
438.5
426 9491
450 0051
1
-2
1 4342
I 4343
1 4342
: 36
04167
438 5
426.9491
4500051
1
-1
1.3970
1 .3971
I 3971
1 36
0 4167
438.5
426 9491
450.0051
1
0
1 3598
1 3.599
1.3599
1 36
0 4167
m 5
426 9491
450 0051
1
1
1.3225
1 32^27
1 3226
1 36
0 4167
438 5
426 9491
450 0051
I
2
1 .2852
1 28^4
1 2853
1 42
0 6667
593 0
566 562
619 0944
2
-2
1.5186
1 5182
1 507 1
1 42
0 6667
593 0
566 562
619 0944
2
-1
1
1 4695
1 4b96
1 42
0 6667
593 0
566 562
619.0944
2
0
1 4206
1 4208
1 4207
1 42
0 6667
593.0
566 562
619 0944
2
1
1 .3713
1 3717
1 3715
1 42
0 6667
593 0
566 562
619 0944
2
■>
1 3219
1 3226
1 3222
but using the expressions (2l)-(23) setting at different values of a. But in both the cases,
the niaximurn eiTor lies within 10%. The possible theoretical shifts in the values of 7’,) and
7,1 lor these experimental peaks with different values of a {- 2 <a < 2), have also been
calculated and are found to show a marginal change around 1%.
4. Conclusion
In this paper, the point of innecliori method for the determination of the activation energy
of a TL peak suggested by Land [71 and refined by Garlia et al [8] and Singh et al 19] has
been generalised for the case of lenipcralurc-dcpenucni frequency factor. The present sets
of expressions unlike those of Land [7J, do not require any iteration and can be directly
used. It is also found that in accordance with the findings of Fleming f6|, it is difficult to
distinguish TL peaks corresponding to different values of a.
Acknowledgment
Thanks arc due to Dr. P S Mazumdar and Dr. S Dorendrajit Singh for fruitful di.scussions.
Kcfcrences
1 1 1 M Ux Phys Rev. 1 19 1 502 ( 1 960)
( 2 J G Bein.'Jki Phys. Rev 1111515 (1958)
[3] P N Keating Proc Pliyx. Sor, 7K 1 408 (1961)
t4J P Kivits and H J L Hagebcuk J lAtmiu 15 I (1977)
Detemination of the order of kinetics and activation energy etc
21
[5] R Chen and Y Kirsh Analysis ofThennally Shmulaied Processes (Oxford : Pergamon) Ch 6 ( 1 98 1 )
[6] K J Fleming J. Phys D23 950 ( 1 990)
[ 7 ] PL Land J. Phys. Chem. Solid 30 1 68 1 ( 1 969)
(8| R K Gaitia. S J Singh, T S C Singh and P S Mazumdar 7. PIm. D24 1451 (1991)
[9J S D Singh, T B Singh. R K Garlia, N C Deb and P S Mazumder J Phys. D28 2536 (1995)
[101 M Abramowitz and I A Slcgun Hand Book ofMathematiial Functions (New York Dover) Ch 5 ( 1972)
[ 1 1 J W H Press, S A Teukolky, W T Vetrerling and B P Flannery Numerical Hecipes in Fortran (Cambridge
Cambridge University Press) (1994)
1 12) S D Singh, R K Gania and P S Mazunuhir Phys Star Sol. (a) 146 825 (1994)
1 1 31 E J Dudewicz and S N Mishra Modern Mathematical Sta/i.siK s (New York ' Wiley) Ch 14 (1 958)
1 14] PS Mazumdar and R K Gartia J. Phys. D21 85 1 ( 1988)
[15] A B Ahmed and R K Gania Phys. Stat Sol (a) 94 645 ( 1 985 )
Indian J. Phys. 72A (1), 23-30 (1998)
UP A
an intemaiional journal
Tunnelling oiirent across a double barrier
PNRoy
Deportment of Physics, T N B College, Bhagalpur-812 007,
Bihar, India
and
R B Choudhary
Department of Physics, S S College, Mehush (Sheilchpura)-811 102,
T M Bhagalpur University, Bhogalpur. Bihai, India
Received I August 1997. accepted 16 September 1997
Abstract : An analytical expression for tunnel current density across a double barrier has
been obtained under non resonant conditions. The denvation is based on the ideas of quantum
measurement There is a good agreement with observed results in the nature of current -voltage
and differential conductivity-voltagc characteristics.
Keywords : Tunnel current density, double barrier, quantum measurement
PACS No. : 73 40.Gk
1. Introduction
Tunnelling across a double barrier was studied experimentally by Chang ei al [\] in
which they had observed resonant tunnelling under suitable conditions. Esaki and his
coworkers [2-4] later applied the conventional theoretical models to explain their
observations in such devices which find their application in superlattices. In recent
times, Roy et al [5,6] have applied the ideas of quantum measurement model to study
this problem.
Double barrier tunnelling continues to evince interest in workers even now.
Vanhoof and his coworkers [7] have studied spatially indirect transitions due to coupling
between hole accumulation layer and a quantum well in resonant tunnelling diodes.
Kuznesov et al [8] have studied the effect of electron-electron interactions on the resonant
tunnelling spectroscopy of the localised states in a barrier. vSilvestrini et al [9] have
studied resonant macroscopic quantum tunnelling in SQUID system. Alonzo and his
© 1998 lACS
24
P N Roy and R B Choudhary
coworkers |10] have presented a tunnelling spectroscopy of resonant interband tunnelling
structures. Song [1 1] has presented a transition layer model and applied it to resonant
tunnelling in hetero-structures.
In the present study, the quantum measurement model of Roy and his
coworkers [5,6j has been used to derive an analytical expression for tunnel current
density across a double barrier. Since the earlier workers had observed negative
differential conductivity (n.d.c.) under non-re.sonant conditions in such systems, the
main purpose of this work was to find a theoretical expression which could lead to
n.d.c. effect.
The quantum measurement model differs from the conventional model
fundamentally The conventional model seems to rest upon the idea that the electron
energy must remain unaltered throughout the tunnelling process. But tunnelling of |)articlcs
IS not a continually observable process. The tunnelling particle can be reckoned only after
a definite time r measured from the instant of incidence of the particle upon the'.barrier
because of a finite time becoming necessary for potential energy estimation as required by
Heisenberg’s uncertainty relation. The electron is then able to reeover its wave or particle
shape that it once lost while making tunnelling transition. In other words, we may regard
tunnelling as a process ot quantum measurement being carried out by the barrier. Both
energy and time being conjugate variables, simultaneous and accurate estimation of ibem is
not possible because of Hci.senberg’s uncertainty relations. So, if a time r elapses in
reckoning the tunnelling process, the electron energy at the conclusion of the process must
be uncertain by fi/ r. The electron energy is expected to undergo a fluctOalion of (Vo - E)
energy around its original value E where Vq represents the height of the barrier. This idea of
quantum measurement has been successfully applied by Roy and other workers 112,13) to
different tunnel devices.
2. Tunnelling across a double barrier
The electronic wave functions in various regions of a double barrier system (Figure 1)
can be written as
V/,(x) = -oo < x<x^
(1)
[j/^ix) = JC, <X<JC2
(2)
i//-^(x) = -h JC2<JC<x-,
(3)
i//^(x) = -y x^ <x<x^
(4)
\l/^{x) = x^ <x<<«
(5)
where
(6)
and
. _ 2.-(V,-E)
>C2 44 ” ^2
(7)
Tunnelling current across a double harrier
25
Here ni* is the effective mass of the electron in the double barrier system and Vq is the
barrier height.
Figure 1. Double barrier system
Matching the wave functions and their first derivatives at different boundaries, one
finally obtains an expression for the tunnelling probability [14,15] as
l"i'' {xl+k{)(xl+kj)(xl+k^)(xl+k^,)\K,\^
whcir 1 ^1 I' “ sin^ +0^ -*,0),)
+ - <p^ ■■ k,a}^)
+ ,j„2
'/."'j) 4.0^ +A,(0,)
C 0 S 2 ^^ ,
sin( 0 , + 0 ^ -
tjWj) sm (04 -
<t>4 -
-k^W,)
cos 20 T
.Slll(0, +04 -
^,w,) sin( 0 , -
-K
+ *,W,)
- 2 cos{ 2 ( 0 ,+ 0 ,;
|}.sin( 0 , +04
- *,aj,)sin( 0 ,
+ <p>
1
- 2 cos| 2 ( 0 j~ 0 ,]
|}.sin (04 -04
-t 4 W,)sin( 0 ,
1 +*,«»,)
+ 2e~^*^‘' cos 20 ,
.sin( 0 , - 0 ^ -
-<:,(u,)sin( 0 , -
+ <:,£U,)
-le cos 20 ^
.sin( 0 , -04 +k
,W,)sin( 0,+0
4+*
,W,)], (9)
26
P N Roy and RB Choudhary
Ii IS found that a resonance is obtained i.e. Z= 1 when the following conditions are satisfied
simultaneously
(I) *1 =X2 =*? =*S'
n
which leads to 0j = ^, = P4 -
(ii) X2l^2=X4‘04
and (lii) k^O)^ = ^n + j^7C wheren = 0,l,2
3. Tunnel current density
Regions 2 and 4 of Figure I are barriers whereas region 3 is a potential well. The currejit in
this system Hows by the quantum measurement mechanism through regions 2 and 4 ani by
conventional mechanism through region 3. The principle of continuity suggests that \he
current densities through all these regions must be the same.
The one-electron tunnel current density through region 4, generated by quantum
measurement process, is given by [ 13]
sintu, r
sin w. L . .
~ (D T ^ ^ ’
m
which on lurther simplification leads to
•^01 " 2ni*x 1 ^’
T = ■ the tunnelling time.
hxl
When a group of electrons having a random phase difference amongst themselves is
incident upon the double barrier system, the diffcrentral tunnel current density is given by
112,13]
X sin cu . T . V
where p/ (£*)// (E) JE is the density of the wave group at the incident end.
The minimum phase difference at ihe transmitted end is
Tunnelling current across a double barrier
where e, is the difference in consecutive energy levels at that end. Thus, the summation of
(14) can be converted into integration as
dJ(E) = p,(E)f,iE)dE
(jlL
Ur ^
r sintu, r . ,
1 0 ). T
Ir
+■>02 jsin(<u,^T+e)j(<B,,r)
which finally leads to ( 12,131
nh
dJ{E) ^
e, r
•^oi Pi I iE)dE .
But £F, can be expressed in terms of the density of slates as
I
(Ih)
(17)
where U is Ihc volume of the electrode at Ihe transmitted end. Substituting (17) in (16),
we get
dJ(E) = ~f^(E)]pi(E)p^(E)<IE.
For absolute zero temperature, /, (£)[l - = 1 and hence
dJ(E) = IL!^j^^p^iE)p^{E)dE.
(18)
(19)
Substituting for ^qi from (13) and putting
T =
2m* _ h
, one finally obtains
dJ{E) =
\a^ p (Vq - E)^^^ dE
h^V^lK, p
After certain simplifications, it is found that
I AT, p = 4sin2(20, - tjfl), .
where X 2<02 = X, 0 }^
Thus, one gets
dJ{E) = P !!— r dE,
sin^(20j - K^(i)j^)
(20)
( 21 )
( 22 )
P N Hoy and R B Choudhary
2K
where
and
n Y
(23)
The lunncl current density can be expressed as
(/>■
J(E) = /' j
0
-Kiv..-E)'r‘
dE
sin ‘ (20^ ~ ]
(24)
where V is the applied bias. fcq. (24) has been obtained for non-resonanl conditions.
4. Results and conclusions
The integration was done numerically with the help of a computer The ciirrcnt-
voliage characteristics lor AIGaAS-GaAS-AIGaAS as double barrier .system, arc shi^wn in
FlRure 2. Currcni-vo)ia{;e thaniciensucs
Figures 2 and 3 for different ranges of biases. Negative resistance regions arc
c early seen in the.se plots. Figure 4 shows the differential conductivity-voltage
c aractenstics. One finds a good agreement so fai as the nature of these characteristics
are concerned.
j/d^ (orbit'
P N Roy and R B Choudhary
M)
Acknowledgment
The authors are thankful to Dr. D K Roy of Indian Institute of Technology, Delhi for
his valuable suggestions.
References
1 1 1 L L Chang, L Esuki and R Tsu Appl PIm Un. 24 593 (1974)
(2] L Esaki and R Tsu IliM J Res Develop 14 61 (1970)
( 3 J L L’saki and I . L ( 'hang Phys Rev Lett 33 495 ( 1 974)
|4| R Tsu ujid L Esaki Appl PIm Utt 22 562 (1973)
[5] D K Roy. N S T Sai and K N Rai Pranuina 19 231 (1982)
[6| D K Roland A Ghosh Prot IV Inf Workshop on Phys of Semuond De\ IIT Madras (India)
cds S C Jam and S Radhakrishna p 520 (1987)
f7J (’ Vanhoof. J Genoe, J C Portal and G Borghs Phys. Rev. B51 14745 (1995) j
[Hi V V Kuznesov A K Savchenko. M E Raikh and L J Glazcman Phys Rev BS4 1502 (1996) I
|9] P Silvcsirini, R Ruggiero and Y M Ovchinnikov Phy.s, Rev B54 1246 (1996) |
( I0| A ( Alonzo, D A Collins and T C Mcgill Solid State Commun. lOl(B)'' 607 ( 1 997) \
|ll| Y.SongP/ivv leit A216 183(1996)
1 1 21 I) K Kny Quanuim Met Imt, ,il liumfllmi; and m Appliralions (Philadelphia World Scientific) ( 1 986)
ID] P N Koy, P N Singh and n K Roy P/iw hil A63 81 (I 977 )
1 14| limnfllmit I’hnomena in Salids cds C Ruistcin and S Londqvist (New York . Plenum) ( l%9)
IISI HKKoyandASingh/iif ./ Mad I'lm 233039(1995)
Indian J. Phys. 72A(1). 3M2 (1998)
UP A
— an mtemational jour nal
Thermal behaviour and non-isothermal kinetics of
Geio+*Se4oTe5a-j«r amorphous system
M M El-Ocker*, S A Fayel^, F Metawe^ and A S Hassanien^
* Department of Physics. Faculty of Science, Al-Azher University,
Nasr City, Cairo
^Solid State Department, National Center for Radiation Research ahd Technology,
Nasr City. Cairo
^^Basic Science Department. Faculty of Engineering (shoubra). Benha Branch,
Zagazig University
Received 13 March 1997, accepted 19 November 1997
Abstract : Differential thermal analysis DTA at different heating rates for
Gei()+^e 4 ()Te 5 Q_( chalcogenide glasses are reported and discussed Characteristic temperatures
Tf,, 7^ and were estimated at different heating rates. Cyclic scanning technique was used to
investigate thermal induced phases during two consecutive heating-cooling cycles covering the
temperature range Tg
The effective activation energy (E), the order (n) and the rate (A) of
crys(alli 7 .ation along with the growth (m) under non-isothermal conditions are also reported
Obtained results were treated according to Johnson-Mehl-Avrami and modified Ki.ssinger
approaches.
Keywords ; Cholcogcnide glasses, thermal analysis.
P ACS Nos. : 73 61.Ga,8I 70.Pg
1. Introduction
Semiconducting chalcogenide glasses of Ge-Se-Te system have received attention because
of their important optical application in the infrared region. The glass-forming region in the
Gc-Se-Te system exists in two sections :
(i) Compositions with excess Se,
(ii) Compositions with large Ge and Te content. The GeSe 2 -Te was found to form the
boundary between these two regions. The connectedness varies from two (Se and
Te) to four (Ge) in such glass systems, where the bonding is essentially covalent.
The investigated systems were subjected to different studies [1-3].
© 1998 lACS
32
M M FJ-Ocker, S A Fayek. F Metawe and A S Hassanien
Studies of kinetics are always connected with the concept of activation energy. The
value of this energy in glasses is associated with nucleation and growth mechanisms that
dominate the devitrifications of most glassy solids. Studies of the crystallization of a glass
upon healing can be performed in several different ways, isothermal and non-isothermal.
The aim of the present work is concerned with the study of the crystallization
kinetics of Gejo+^Se^oTeso^, (x = 16.65, 13.35 and zero) system. Different methods have
been discussed to evaluate the associated activation energies applying the non-isothermal
technique Moreover, DTA data were correlated with characteristics differences between a
structurally stable materials (exhibiting switching phenomena) and a reversible materials
(exhibiting memory phenomena).
2. Experimental method
High purity (99.9997^ ) Sc, (Ic and Tc m appropriate proportions were weighted in teilica
lubes. The tubes were scaled under a vacuum of 1.3 x l()-2 Pa and heated in a furn^e at
200'-C for two hours The furnace temperature was raised up to 1()(K)''C with rate of 2Q0“C
per hour The synthesis was continued for six hours during which the molten materials Were
occasionally shaken vigorously to ensure the homogeneity of the samples. The melts were
then rapidly quenched in ice -water with the tube in a horizontal position.
The obtained quenched materials of the Ge,o+,Se 4 ()Te 5 (^, system were identified as
glas.ses due to their bright fractures except x = 0. The ingots were confirmed to be
completely amorphous or partially by X-ray diffraction and differential thermal analysis
(DTA), as shown in Figures 1 and 2.
I'iRure 1. X ray ditfraclinn patlems for (a) Bulk
rjL-|Q^^Sc4()Tcso-, with difl'crcnl composiiion.s
\ - 16 6!>. 13 3.S and ?ero
Figure 2. Differential thermal analysi.s DTA
ihermogram.s for glas.sy (jC|Q+jjSe 4 oTe 5 o^^ .sy.stem.s
(r- 16 65, 13 35 and zero) at 10®C/min
DTA thermograms were obtained using the powder of freshly quenched (as
prepared) maierial, 25 mg was put in an aluminum sample pan. The latter was immediately
introduced in its place in the DTA apparatus (Shimadzu model DT-30) and a constant
heating rate was applied. The differences AT. between the sample temperatire and that of
Thermal behaviour and non-isothermal kinetics etc
33
the reference (a-Al203) were recorded directly as a function of the furnace temperature (T)
using a double-pen recorder. Heating rates of 0 = 2, 5. 10, 20, 30 and 50°C/min were used
as shown in Figure 3.
3. Results and discussion
.1 /. Effect of composition and heating rate on thermal transition :
DTA traces at rale of 10°C/min of freshly prepared Ge, (>^^8040X650,, where x = 16.65,
13.35 and zero, are shown m Figure 2. The traces follow the known common behaviour,
where the three characteristic temperatures T,. and T„, are observed and given in Table 1 .
Table 1. Data data of the investigated glasses in Gc| 04 .,.Se 4 ()l'eso_, system, the temperatures arc
given in (°C).
r uinp
Rule
Tk
J si peak
2nd peak
T
Stan
T
max
T-c
end
T
Sian
7 ;
max
end
Tm
Stan
T'm
end
S
60
182
196
206
3,30
340
345
-
400
10
70
185
198
210
334
345
150
395
1 -- 111 6.S
20
70
190
200
212
'
350
-
-
390
10
90
190
202
220
-
365
-
-
395
SO
ISO
2IS
225
242
-
395
-
402
2
185
192
202
325
335
145
-
s
-
175
195
2(K)
330
340
345
-
405
\ - n vs
10
90
185
2CK)
210
335
335
357
380
400
20
105
2(K)
205
232
-
170
405
10
125
-
215
-
372
397
SO
ISO
215
240
275
_
390
-
-
405
Compositions of > 0 exhibit two cry. stall ization peaks and one melting peak. On the other
hand, for a = zero there is a broad crystallization peak followed by a melting peak. This is
most likely due to the partial crystallinity of the latter composition, as shown in Figure 1 .
The effect of heating rale on the characteristic temperatures was investigated at
SIX different rates for jr = 16.65 (as an example and shown in Figure 3 the data listed in
T able 1 ). The observed of a glass is increased by increasing the healing rate, inspection
v)t obtained data. For an ideal glass there, is a lower limit to this change. But for this system,
the wide range of changes in indicates that this system behaves as a normal glass.
It is worthy to mention that for jc = 16.65 and 13.35, the DTA scan at 0 = 2°C/min
shows that a .small amount of the sample material has been crystallized. The crystallization
ol amorphous material proceeds by the processes of nuclcation and growth. Moreover, the
crystallization rate is suppressed by reducing the rate of nucleation or the rate of growth.
Since growth follows nuclcation, in some cases if the nuclcation is prevented, there will be
no crystallization. However, even if nucleation occurs, the crystallization rale can still be
suppressed by reducing the rate of growth. Turnbull 14] indicates that the growth rate in
34
M M El-Ocker, S A Fayek. F Metawe and A S Hassanien
liquids with high viscosity is limited. So, perhaps at the heating-rate of 2°C/min, the liquid
will reach the maximum rate of crystallization when the viscosity is still high.
.?.2 DTA cyclic scanning :
Thermal cycling has been performed up to 450°C. For composition x = 13.35 (as an
example), the degree 450''C was chosen such that the maximum temperature is far from the
decomposition temperature
The DTA cycling has been |x;*- formed as follows :
Run (a) ; the DTA temperature was raised from room temperature up to 450°C at
heating rate 1 5“C/min.
Run (h) : lower tHe temperature down to 50“C at the natural cooling rate of the DTA
device, i.e. at an average rate of about 15‘^C/min, followed by another cycle
healing run (c) and cooling run (d). \
figure 3. DTA thermograms tor ,r = 16.65 Figure 4. DTA .scanning compositions foi x -
at ditTereni heating rates. I3.35 full and dashed lines -are the re.spectivc
heating and cooling curves at 1 5“C/ min
For composition x = 13.35 as an example, and shown in Figure 4, the value of T^,
detected on first healing run (a) forjc = 16.65 at WC and for x = 13.35 at 105X. Also,
one melting peak in the range 39()“C for x = 16.65 and 410X for jc = 13.35 and two
crystallization peaks were observed for both compositions .x = 16.65 and 13.35 in the
lange 200-360'^C. This is most likely due to segregation of two phases. In run (b), during
the first cooling, one solidification peak has been delected in the range 355-370°C, in case
ot A = 16.65 and 1 3.35. In heating run (c) (the second heating cycle), the disappeared and
a broad T, peak moved to a higher temperature for x = 13.35, 16.65 and listed in Table 2.
This is probably connected with the morphology of the sample; before run (a), the material
Thermal behaviour and non-isothermal kinetics etc
35
was in the powder form, but before run (c) it was in the solid form. I( was observed that the
growth of crystals was more readily from powdered glass than from large solid pieces. It
can be concluded that the glass forming tendency is weak for the high Te content.
Moreover, for compositions jr > 0, the phase separation appears. This is related to the Gc-To
and Ge-Se according to the chemical bonds.
Table 2. Transition temperatures during two eonsecunvc heating-cooling DTA cycles (°C) of Ihe
ternary gla.ssy fieio+xSe 4 oTe 5 o_, systems
X = 1
16 65
,35
‘ 1
T
‘ m
Run (a) 70
205
355
3t)5
105
200
360
405
Run (b)
-
no
4)5
^60
Run u )
200
400
-
240
-
4(X)
Run (d)
370
,4)5
365
The above arguments allow to conclude that the investigated compositions belong to
reversible class of materials. In other words, their structures can be changed reversibly
between two structural slates located at the border of the glass forming regions.
It IS well mentioned that chalcogcnide glas.ses exhibit many useful electrical
piopcrties including threshold and memory switching. The electrical properties arc
influenced by the structural changes associated with thermal effects and can be related to
thermally induced transitions |5|. The glasses which exhibit no exothermic peaks in the
cooling runs, display very little tendency to crystallization. This is usually belongs to
threshold switching type. But the glasses which exhibit an exothermic crystallization
peak in cooling runs, display high tendency too crystalline, and they are memory-
swiichmg type. The latter case is clearly observed for the glassy composition rich m
Te, which causes thermally induced niicrophase separation and subsequent crystallization
ofTe|6|.
x3. Crystallization kinetics :
Studies of kinetics are always related to the concept of activation energy. The values of this
activation energy in glass crystallization phenomena are associated with nucleaiion and
growth processes that dominate the devitrification of most glassy solids [6,7-91. Two basic
methods can be u.sed for knowing the crystallization of a glass upon heating, i.soihcrmal and
non-isolhcrmal. In the former, the sample is brought quickly to a temperature above the
glass transition temperature and the heal evolved during the crystallization process at a
constant temperature is recorded as a function of time t. In the non-isothermal method, the
sample is heated at a fixed rate a, and the heal evolved is recorded as a function of
temperature or time., The isothermal crystallization data is usually interpreted m terms of
the Johnson-Mehl-Avrami (JMA) transformation equation [9--12].
36
M M El-Ocker, S A Fayek, F Metawe and A S Hassanien
The evaluation of non-isolhermal activation energy for crystal growth has been
estimated by a large number of mathematical treatments based on the formal theories of
transformation kinetics. The theories differ greatly in their assumptions and in some cases,
they lead to contradictory results. Partial area analysis and peak shift analysis are the basic
method for all mathematical treatments.
(a) Peak shift analysis :
The peak shift technique is based mainly on the systematic variation in the peak
temperature of crystallization with the heating rate.
(i) The activation energy (E) for any crystallization mechanism can be calculated from
two different thermograms by the following equation [13] :
^ = I H)
p n ^
where 0 and 0 ' arc two different heating rates, and and 7 ' are the correspjjnding
temperatures of the crystallization peaks. Combinations between different healing\ rates
were made to compute the average £ values, which are listed, in Table 3 .
(ii) The Kissinger formula was used for homogenous crystallization |14] or in other
words, surface nucleation dominates and n = 1 .
In(0/r5) = -E/RT^ + constant. ( 2 )
The plots of In (0 / Tj ) w 1 / , which are shown in Figure 5, are well fitted by straight
lines From the slopes of these lines, the activation energy of crystallization E can be
estimated and arc listed in Table 3 .
Figure 5. The relation belween In ) and IO’/T^ forr= 1665 and 13.35.
(Ill) The approximation due to Mahadevan et al ( 6 J was used, where the variation in
In ( 1 / T 2 In 0 is mm-h less than that I /T,, with In 0. Therefore, eq. (2) can
be written in the form
ln0 =
Ec
RT„
+ constant.
(3)
Table 3. The thermal parameters of the ternary glassy Ge],>^rSe4oTe50-x systems for the t^%o crystallization peaks. £ values arc given
in e V / atom
Thermal behaviour ard non- isothermal kinetics etc
37
38
M M El-Ocker, 5^4 Fayek, F Metawe and AS Hassanien
A plot o( In (t> vs 1 / for Ge,o+xSe 4 f)Te,o., gives a straight line, as shown in Figure 6.
The value ol E obtained from the above method is listed in Table 3.
6-0
30
■f
I 0
00
Figure 6. The relation between Inland \()^ /T tora = lb 65 and 13 35
(iv) The modified Kissjnger-mclhod : \
In this method, the relation between the rale 0and the crysiaUizalion temperature T^,
is assumed to have the following form :
In(0"/7’j5) = + constant (4)
H the Liyslalh/aiion mechanism is precisely known and it does not change with the healing
rale, the plot of In (0" / 7^; ) v.v 1/T^, gives the value of mE. Dividing»m£ by m, the
aclivalion energy for crystal growth can be obtained as shown in Figure 7. In cq. (4), m and
n arc integers having values between one and four. When nuclei are formed on heating at
constant rale, n is equal to (w+l) 16].
Figure 7. The relaiion between
and 10 ' fT for modified Kissinger, for x = 16.^i5
and 13 35
li.0 155 170 2 0 22 241
All techniques depending on peak shift, lead to different relationships between the
temperature of maximum crystalli/.ation rate 7^. and the heating rate 0. Such methods have
the disadvantage that they neglect both the nucleation and the crystal growth rate K, the
latter being active at the involved lempierature range. The modified Kissinger method
Thermal behaviour and non-isothermal kinetics etc
39
facilitates the estimation of activation energy and rate of growth. However* the order of
crystallization is usually obtained by other technique.
3.3. (b) Partial area technique :
In this technique, the relation between sample temperature T and heating rate 0 can be
written in the form
T (l>t, (5)
where 7o is the initial temperature. The estimation of the complex activation energy of
crystallization (E) was obtained using Piloyan et a/'ji method [15] which is based on the
differential form of the model relation for (a) known as g(a) and on Borchard’s assumption
[16]. According to them the reaction rate da/dt is proportional to the temperature
deflection AT as detected by DTA. This means that the kinetic (JMA) equation [17] :
-ln(l-a) = " exp (-£■/ /?r) (6)
can be applied to non-isothermal conditions after some modifications [6,9,18-20]. To
apply JMA equation for non-isothermal technique, the region of cursory check [21]
should he considered, namely the fraction a transformed at should be= 0.60-0.63. For
Gc|o+^^Sc 4 oTc 5 (^^ glasses with > 0, values of a at satisfy this condition. Differentiation
of JMA equation yields :
da/dt = nJt,'/" exp[-£//fT]''" f(a), (7)
where £(«) = (1- a)[-£n(l-a)]"''^" (8)
Here, A'o is considered to be constant with respect to temperature. For a constant heating rate
0= dT/di, eq. (7) is separable in a and T and can therefore, be directly integrated
j(<ya/(l-a)[ln(l-a)-']"-'"'} = (9)
0 ^ 0
This yields to
[in (l-a)-'
p{x) = g{a).
(10)
where x = E/ nR7\ the behaviour of the exponential integral functions PU) for different
types of approximations has been reviewed in [22]. For a limited range, i.e. normal
temperature interval of about 100°C, P{x) = exp (jr). Therefore, eq. (9) can be written in the
logarithmic form as
where C is constant. '
40
M M El-Ocker, S A Fayek, F Metawe and A S Hassanien
Calculation of the function g(a) has been carried out by Stava and Skavara [23] for
different reaction kinetic equations. A plot of log [g(a)l against 1 /T sfiould yield a straight
line over the whole range of a(0 < a < 1) when the appropriate mathematical description of
the reaction is employed. The slope of such straight line, is used to evaluate E/ n. Typical
plots of log fg(a)| against l/T arc given in Figure 8, for the two crystallizations peaks.
According to Sharp et al I24|, it was observed that the function A 3 (a), where Hn(l-a)]'/^
= Kt, fits the obtained experimental results over a maximum range of a is almost 0.01 to
0.99 with the except of the second crystallization peak of the composition x = 13.35, where
ms in the range of 0 01 to 0.68.
According to Avrami's equation (15)
a = I -cxp(-/ft" ), (12)
which can be written in logarithmic forms as
ln[-ln(l-a)| = InA + nln/. \(13)
A plot of ln[-ln(l - Of)] against In r, as shown in Figure 9, should yield a straighf\^line
whose slope is the order of crystallization (n) and intercept on the ordinate at In k. The
value ol n, which reflects the nucleation rale and the growth morphology is correlated with
the effective activation energy (F/n), obtained from eq. (11). The values of n, k and £ are
evaluated and are given in Table 3. The plots arc linear over most of the temperature range.
FiRurc 8. Plots of log \y(a)\ irrvi/^ 10^/7' for Figure 9. The relation between In [-ln(l-a)]
the leiichon kinetics A^{a) tor v - 16 65 v.v In/ for.r = 16.65. jc = 13 35.
Al high temperature, or in regions of large crystallized fractions, a break in the linearity, or
rather a lowering of the initial slope, is seen. Breaks of similar nature have been reported
for many chalcogenidc glas.ses, a naturally occurring oxide glass and two metallic glasses
125-271. Generally, this break in slope is attributed to the saturation of nucleation sites in
the final stages of crystallization [27,28] or to the restriction of crystal growth by the small
si/x of the particles [29] In all these cases, the analysis is confined to the initial linear
region, which extends over a large range [25,26].
Thermal behaviour and non-isothermal kinetics etc
41
The partial area allows direct estimation of more parameters such as order of
crystallization and rate of activation energy by single slow scan, which realized the non-
isoihermal condition.
Conclusion
The systematic study of the thcrmoanalysis of different compositions Cieio+rSe 4 ()Te* 5 ()_,
where x = 16.65, 13.35 and zero, using DTA technique indicates that :
I . The glass transition temperature T^, rises by increasing the heating rate as a result of
increase heal capacity and the excess of the degrees of freedom that the material
posses supercold liquid stale.
2 By increasing the healing rate, the crystallization temperature increases due to the
reduction in crystal growth.
3. The two compositions jc = 16.65 and 13.35 are characterized by two crystallization
peaks revealing the formation of two phases during the heal treatment.
4. Sample of x = 0 exhibits broad crystallization peak indicating partial crystallinity.
This was confined by X-ray diffraction.
5 It IS also observed that the melting peaks arc independent of the healing rate which is
a common behaviour.
6 According to the behaviour of .v = 16.65 and 13.35 during the two cycles, we can
conclude that these compositions having the ability for memory switching.
7 Many techniques, such as single-scan and mulliscan techniques, were applied to
evaluate the activation energy (E), the order {n) of crystallization, the rate of
crystallization (k) and the order of crystal growth (m). The activation energy of
crystallization is increased by increasing the ratio of Te for the two phases.
8. The excess ol Tc increases the freedom in bonding. This is confirmed by the order of
crystallization («) which is low for = 16.65 (two-dimensional cry.stal growth) and
incrca.sed for.r = 13.35 (three-dimensional).
References
1 1 J Z U Honsova CJla.wsy Semiconduc ton (New York Plenum) Chaps 1 and .“^ ( 1 9R I )
(2) J P Dc Neufville J. Non-Cryst Solids 8-10 85 (1972)
(3] 1) J Sarrah, J P Deneufvillc and W L Haworth J Non-Cryst Solids 22 245 (1976)
|4J D Turnbull Gwi/emp Hh\s 10 47.1(1969)
15] H Friizschc and S R Ovshmsky J. Non-Crvst Solids 2 148 (1970)
16] S Mahadevan, A Giridhur and A K Singh /. Non.Cryst Solids 88 1 1 (1986)
17] H G Kif?singer Ana/. Chem. 29 1702 (1957)
(«] N Rysava. T Spasov and L Tichy J Therm -Anal 32 101 5 ( 1987)
H Yinnon and O R Uhlmann J Non-Cryst Solids 54 253 (1983)
1 10| B G Baglcy and E M Vogel J Non-cryst Solids 18 29 (I9'’'ii
42
M M El-Ocker, S A Fayek, F Metawe and A S Hassanien
fill MG Scon 7. Miller Sa. 13 29 1 ( 1 987)
1 1 21 V R V Raman and G F Fish J Appi Phys. 53 2273 ( 1 982)
f 131 mm Hafiz, A A Airier, A L All and Abotalb Mohamcd Phvs Stal Sol (■) 76 ( 1983)
114] H E Kissinger 7 Res Nat liur Stand 57 217 (I9.S6)
115] F O Piloyan, 1 0 Ryabchikov and O S Novikova Nature 212 1 226 ( 1 966)
116] H J Borchard 7 Inorfi NucL Cheni 12 252 (1966)
fl7| M Avrami 7 Chem Phvs. 7 1 103 (1941)
f 1 8] D W Henderson 7 Non-rrvst Solids 30 301 (1970)
[19] K Malusita, T Konatsu and P Yorola 7 Mater. Sci 19 291 ( 1 984)
[20] K Matusiia and S Sakka Phys Chern iilasse\ 20 81 (1979)
[21 [ M Avrami 7 Chem Ph\s 9 177 (1941)
[22] J Sestak Thernun hem Acta 3 1 50 ( 1 971 )
[231 V Slava and F Skvaru 7. Am Ceram. Sor 52 591 ( 1969)
[24] J H Sharp. G V Brindley and B N N Achar7. Am Ceram Snc 49 379 (1966)
[25] L J Shelesrak, R A Charez and J D Machenzie 7 Non-Cryst Solids 27 83 (1978)
[26] J J Burton and R P Ray 7 Non-rrs'sf Solids 6 393 (1971)
[27] P Duhaj, D Baranucok and A Ondnka 7 Non-Cryst Solids 21 41 1 (1976)
[28] J Colemcnero and J M Barandiaran 7 Non-rrvst Solids 30 263 (1978)
[29] R F Speyer and S H Risbud Phvs Chem Glasses 24 26 (1983)
IndianJ. Phys. 72A (1). 43-48 (1998)
UP A
— an intemalional journal
Nonlinear light absorption in GaSei-jcS^ solid
solutions under high excitation levels
H Tajalli, M Kalafi, H Bidadi, M Kouhi and V M Salraanov*
Center for Applied Physics Research, Tabnz University. Tabriz, Iran
Received H Januan' 1997, aicepied J9 November 1997
Abstract : Transmission, phoioluinmcscence and photoconductivity spectra of GoSc]
solid solutions have been investigated expenmentall) in the exciton resonance region at high
optical excitations The absorption edge of CjaSe|_j.S^ is caused by exciton transitions and lineai
shifts towards short wavelengths by raising the value of x in the solution The exciton peak
ilisappcars and nonlinear absorption appears in GaSe)_|S, crystals by increasing the excitation
intensity The new luminescence band appears at about 20 meV below the free exciton line at
high excitation levcU These pecularities are interpreted by means of the excilon-exciton
scattering process
Keywords : Exciton, GaScj. solid solutions, nonlinear light absorption
PACS Nos. : 7 1 3.S Cc, 7H 55.Ci
I. Introduction
Gallium Selenidc (GaSc), Gallium Sulfide (GaS) and Gallium Selenium Suliidc
(GaSci_^S,) arc III-VI semiconductors which crystalliite with a lamellar structure. The
bonding between two adjacent layers is of the Van dcr Waals type, while within the layer
Ihe bonding is predominantly covalent. Therefore, the hulk material obtains a strong-
mechanical anisotropy which allows easily to prepare thin samples using a simple peeling
procedure. The optical c-axes of crystals are orthogonal to the layers having thickne,sscs ot
--O S nm in. The exciton binding energy is equal to 20 rneV, which is close to the room
temperature thermal energy f2J. Therefore, one can observe the exciton in GaScl_^S^ at
room temperature from optical transmission experiments. Exciton absorption and
luminescence in GaSei.j^Sj, have been investigated by a number of investigators i3-7|. In
these crystals (with the exception of GaSe [8-18]), the optical absorption has not practically
been considered at high excitation levels.
'Piirmanent address : Baku Stale University, Baku-370I4S, Z Khalilov Av, 23
© 1998 I ACS
44
H Tajalli, M Kalafi, H Bidadi, M Kouhi andVM Salmanov
In the present work, the nonlinear light absorption has been investigated
experimentally in the exciton resonance region at high optical excitation levels in GaSei^j^Sjt
layered crystals.
2. Experimental method
GaSe,.j^j crystals (;r = 0, 0.05, 0.1, 0.2 and 0.25) were grown by Bridgman technique.
Thicknesses of samples were about 30-100 pm. Ohmic contacts were obtained by
deposition of high-purity indium on the surface of samples. As an excitation source, a dye
laser (PR A, LN-I07) pumped by the output of a N 2 -laser (PRA, LN-IOOO) were used. The
dye laser gave possibility of selecting different wavelengths (473-547) nm, (568-605) nm
and (594-643) nm with the resolution 0.04 nm. The pulse power was 120 kW at the
repetition frequency of JO Hz and at pulse width 1 ns. The laser light was focussed opto the
sample with the focus diameter of about 0.5 mm. Laser beam intensity was vaijied by
inserting calibrated neutral density filters. Luminescence was excited by dye laser photons
with energy more than the band gap width of the GaScj.^^ crystals = 5001 nm).
Luminescence was detected under a small angle with respect to the c-oplical axis (\t' the
crystal. Photoluminesccnce spectra were analysed by means of a diffraction grating
monochromator (JOBIN-YVON) with the reciprocal dispersion 2.4 nm/mm. The output
signal was detected by a photomultiplier and then was sent to a recorder (HP-7475A)
through a storage oscilloscope (Lc Croy 94(X)).
3. Results and discussion
Transmission spectra of GaSci.^S j crystals are shown in Figure I near the fundamental
absorption edge at low excitation intensities. These spectra show positions of the
Trcc-exciton peaks for different jc-valucs |4J. Nonlinear light absorption is observed in these
Figure 1. Transmission spectra of GaSci.^S^ (300 K) for various values of
-X 1-0,2-0 05.3-0 1,4-0.2
Nonlinear light absorption in GaScj^j^^ solid solutions etc
45
crystals at the high pumping levels. Figures 2 and 3 show transmission spectra of GaSe and
GaSeoQsSoos crystals at different excitation levels, respectively. Bleaching of the samples
takes place in the region of the exciton resonance by increasing the excitation intensity.
. /
w
605 530 H5
- X(nm)
Figure 2. Transmission spectra of
GaSc (80 K) at two pumping intensities
(inMW/cm^) 1-1 7.2-12
Figure 3. Transmission spectra of GaSeo 95 Soo 5
at different pumping intensities (in MW/cm^) I-0. 1.1,
2-2 01,3-12.
Figure 4. Exciton absorption peaks (3(X) K) versus
pumping intensities for (I) GaSe and (2)
GaSeo.gSo j.
Figure 5. Luminescence spectra (80 K) of
GaSeo 95 S 0 05 ^or various pumping intensities (in
MW/cm^) • 1-0.12, 2-1.01, 3-4.02, 4-6.03, 5-12.
Figure 4 illustrates the dependence of absorption on the excitation intensity at the
wavelength where the exciton absorption is maximum. According to Figure 4, the
absorption is constant up to the intensity Iq = (0.1-O.2) MW/cm^, and then decreases more
than three times in the region 0A5 < Io<\2 MW/ cm^.
46
H Tajalli. M Kalafi, H Bidcdi, M Kouhi and V M Salmanov
Figure 5 shows luminescence spectra of GaSeo. 95 So,o 5 various pumping levels.
These spectra include the low-energy band ( L band) besides the free-exciton peak (A= 589
nm). The cxciton peak takes place also at low excitation intensities. On the other hand, the
L band appears when Iq > 0.5 MW/ cm^ at A = 595 nm {Le. 20 meV below energy of the
free exc :ion) The peak of the L band exhibits a red shift by increasing the pumping
excitation. L emission strongly predominates at highest pumping levels. Dependence of the
L-emission on excitation levels is a square-law, while the free exciton dependency is a
linear one.
The photoconductivity spectra of GaScogSQ | at various pumping levels are shown in
Figure 6. One can see from this figure, that both exciton (A) and impurity (B)
photoconductivities are observed in the spectrum (curve 1). The exciton peak first increases
Figure 6. Pholoconduclivity spectra (300 K)
of GaSco qSq I for vanous pumping intensities
(in MW/cm^) I -1.01, 2-4 02, 3-6 0.3, 4-12.
by increasing the laser intensity (curve 2), then begins to diminish (curve 3), and almost
disappears at higher intensities (curve 4). The dependence of the exciton photoconductivity
on the pumping intensity is shown in Figure 7. It is clearly seen that the exciton
photoconductivity dcr, first increases linearly with growth of the incident intensity /q,
up to /o =1 MW/cm2, then varies according to Ao - , and at last decreases at /o >
4 MW/cm‘ (curve 1). In the case of the impurity excitation, the photoconductivity
changes Iirst linearly, and then approaches with a farther trend to the saturation
(curve 3). The concentration of impurities determined from the region of the saturation
IS equal to 1.0 x 10'“ cm ’.
It IS known that photoconductivity of nonequihbrium carriers in semiconductors is of
the form Aa ~ al^, where a is the optical absorption coefficient and lo is the excitation
intensity (19]. Dependences of a/„ on /o are shown in Figure 7 (curve 2), where values of
acotrespond to Figure 4. It is seen that, dependences of alo and Affon k are similar. Thus,
the di,sappearance of the exciton peak in the photoconductivity spectrum is caused by the
same mechanism, as in the transmission spectrum.
Nonlinear light absorption in solid solutions etc
47
Exciton-exciton collision process is one of the possible mechanisms of the nonlinear
light absorption in GaSe|_^5jf solid solutions [20,21], Disappearance of the exciton
absorption and appearance of the new band of the luminescence (at 20 me V below the free
Figure 7. Photoconduciiviiy of GaScQ qSqj
verms incident intensity . 1 , 3-exciton and
impurity photoconductivity, I-oIq.
cxciton) at high pumping levels indicate such possibility. The density of the absorbed
photons, averaged over the sample thickness, reached 3 x 10’^ cm”^ which exceeds the
oxciton density necessary for the Mott transition in GaSci.^S^ [12,22].
4. Conclusion
Transmission, luminescence and photoconductivity spectia of GaScj.^Sj, solid solution
(upto 25% S) contain only lines corresponding to free-excitons at low pumping intensities.
Thus, GaSei_jS^ present materials which are especially convenient for the study of the
interactions between excilons. Such interaction leads to disappearance of the exciton lines
and gives rise to new radiative transitions.
References
( 1 1 Z S Bazmski, C B Dove and A Moose Helv Phys Acta 34 373 ( 1 96 1 )
12 ] Landolt-Bdrnstein Numerical Data and Fundamental Relationships in Science and Technology
eds K H Hellwege and O Madelung Vol 2 p 525 (Berlin ; Springer) ( 1 983)
13] N Kuroda and Y Nishina Phys State SoL 72 8 1 ( 1 975)
[4] S G Abdullaeva. G L Belenkii, P Ch Nani, E Yu Salaev and R A Sulcimanov Sov. Phys. Semicond
9 161 (1975)
I M Schluter, J Gamas.sel, S Kohn, J P Voitchovsky. Y R Shen and M L Cohen Phys Rev. 1,3 3534 (1976)
[6] A Mercicr, E Mooscr, J R Voitchovsky and A Baldcresehi J. Lumin 12 285 (1976)
[71 EL Ivchenko. M I Karaman, D K Nelson, B S Razbirin and A N Siarukhin Phys Sol. Star 36 218 (1994)
1 81 T Ugumori, K Masuda and S Namba Phy.s. Utt A38 117(1 972)
19) T Ugumori, K Ma.<iiida and S Namba Solid State Commun . 12 389 (1973)
110] A Mercicr and J P Voitchovsky P/iy.v. Rev. Bll 2243 (1975)
48
H Tajalli, M Kalafi, H Bidadi, M Kouhi and V M Salmanov
III] I M Calalano. A Cingolani, M Ferrara and A Minafra Phys. Slat. Sol. (h) 68 34 1 (1975)
] 1 2] A Frova. Ph Schmid. A Gnsel and F Levy Solid State Commun 23 45 ( 1 977)
1 13] V Capozzi, S Caneppele, M Montagna and F Levy Phys Sfat. Sol (h) 129 247 (1985)
[14] VS Pncprovskii, V D Egorov. D S Khechinashvili and H X Nguyen Phy.s. Slat. Sol. (h) 138 K39 (1986)
[15] X Z Lu. ft Rao. B Willman, S Lee, A G Doukas and R R Aifano Phys. Rev. B36 ] 1 40 ( 1 987)
[16] V S Pncprovskii. A J Furtichev, V J Klimov. E V Nazvanova, K K Okorokov and U Y Vandi.shev
/V/v.v .Star .Sol (hi 146 341 (1988)
1 17] L Pavesi and J L Slachli PIm Rev B39 10982 ( ]989)
[18] H Bidadi. M Kalafi, H Tajalli. S Sobhanian and V Salmanov Indian J Phys. 69A 323 (1995)
[19] S M Ryvkin Photoekctm EJfects in Semiconductors (New York ; Consultants Bureau) p 3 (1964)
[20] L V Keldish hxciton in Semiconductors (Moscow ■ Nauk) p 1 8 ( 1 97 1 )
[2 1 1 C Benoit, A La Guillaume, J M Debever and F Salvan Phys Rev 177 567 ( 1 969)
(221 S S Yao and R R Aifano />Avi Rev B27 2439 ( 1983) j
Indian J.Phys.n\mA9-55 (1998)
UP A
— n n mte mational journal
The role of the oxidising agent and the complexing
agent on reactivity at line defects in antimony
A H Raval, M J Joshi and B S Shah
Solid State Physics and Materials Science Laboratories, Departincnl of Physics,
Saurashtra University, Rajkot-360 005. Gujarat. India
Received 7 November 1997, accepted 2 1 November 1997
Abstract : The (III) cleavage planes of antimony single crystals were etched in the
etchants containing malic acid, nitnc acid and distilled water. The composition of malic acid
(complexing agent) and nitric acid (oxidising agent) was varied m such a manner that the total
composition of the etchants remained the same The values of knietic parameters, such as I he
frequency factor and the activation energy, were calculated The oxidising agent and the
complexing agent mixlily the kink kinetics within the etch pits which has been discussed.
Keywords : Antimony single crystals, etchants, line defects
PACSNo. : 6172Ff
1 . Introduction
Dissolution and vaporization may be considered to a great extent as the reverse process of
crystal growth. These two processes are useful for the revealation of emergent ends of
dislocations on free surfaces. Both the processes, together in general, rely on the fact that
some extra energy is associated with dislocation lines; hence preferential attack of the
surfaces occurs. Many workers [1-8) have carried out their investigations to study the
revealation of dislocations; in order to do so, they developed new etchants and modified a
lew of them also [ 1 ,2,6-8].
In the present study, the etchants containing aqueous solutions of nitric acid and
malic acid were selected to etch (111) cleavages of antimony single crystals.
2. Experimental technique
Single crystals of antimony were grown by Chalmers method, which has been elaborately
discussed by Thaker and Shah [9]. The metal was kept in a .specially designed graphite boat.
IQQR lAPS:
50
A H Raval, M J Joshi and B S Shah
The boat is pointed at one end and flat at the other end. The tip provides the freezing of the
melt at a point and because of the constriction, very few crystals are formed. The graphite
boat was kept at the centre of a silica tube having 2-5 cm diameter and 75 cm length. The
trolley furnace, which moved along the silica tube, was prepared by the standard
techniques. Several crystals were grown under a temperature gradient of 92°C/cm and a
growth velocity of 1-5 cm/hr. The crystals were cleaved at liquid nitrogen temperature in
the conventional manner.
The etchants containing malic acid had been selected to etch the (1 1 1) cleavages of
antimony single crystals. The composition of the selected etchants are as follows :
Etchant A : 9 parts malic acid +3 parts HNO 3 + 1 part distilled water.
Etchant B 9.5 parts malic acid + 2.5 parts HNO 3 + 1 part distilled wiiter.
Etchant C 10 parts malic acid +2 parts HNO^ + I part distilled water.
The etchants were made from AR grade chemicals. For high temperature etching, the
crystals were first heated .separately and brought to the temperature of etchant before
etching. All the etchants were tested by the standard technique and found to be revealing
dislocation etch pits.
The values of activation energy and frequency factor were calculated using the
Arrhenius law :
W = ' (I)
where W is the average width of the etch pits, T is the absolute temperature, E is the
activation energy and A is the frequency factor.
Figure 1 is the photomicrograph revealing crystallographically oriented triangular
etch pits corresponding to the dislocations of the ( 1 1 1 ) [lOT) type by etching in the etchant
Table 1. The values of activation energy and frequency factor for difierenl etchants
Erchanis
Activation
energy
Frequency
factor
[AJ
9 parts malic acid
0.63 cV
5.32 X lO'^ cm/s
+
3 parts HNO^
1 part di.stilled water
IBJ
9 .5 parts malic acid
1.07 eV
1.31 X 10 ’^ cm/s
+
2 .S parts HNO 3
+
1 part di.stilled water,
IQ
to parts malic acid
0.79 eV
4.00 X lo'^cm/s
+
2 parts HNO 3
+
1 part distilled water
The role of the oxidising agent and the complexing agent etc
Plate I
Figure 1. Photomicrograph revealing the type ot etch pits produced on (III)
cleavage plane of antimony single crystals by etching in the etchant A at
for 120 seconds.
52
A H Raval, M J Joshi and B S Shah
A at 38°C for 120 seconds. Figures 2-4 are the plots of logarithm of average value of etch
pit Width W versus reciprocal of absolute temperature Tfor etchants A, B andC, respectively.
1/T V’
3 14 316 316 3 20 Xi6^k’
Figure 4. A graphical relation between the logarithm of average width of^tch
pit and the reciprocal of absolute temperature for etchant C.
The values of activation energy for the lateral motion of steps as well as the frequency
factors were found out from the plots and are tabulated in Table 1.
3. Discussions
The approach to etch pit control through the study of the kinetics of pitting must have
some relevance, since even if a thermodynamically stable shape is eventually attained, the
pits are certainly kinetic phenomena immediately following nucleation of an appropriate
hole at the defect. The two aspects of kinetics, namely, the nucleation of monomolecular
steps at the defects and the motion of steps away from the source, have been outlined by
Gilman et al [10] and Cabrera [11]. Successive nucleation of steps at the defects gives
rise (o dissolution rate at the detects. If the height of the steps is h, then Vj/h steps
arc produced in unit time. This model takes no account of the shape of the steps, and hence
that of the pit they comprise, and it is best thought of as a two-dimensional model.
Following nucleation, the steps move out by removal of atoms and can be assigned a
velocity The dependence of on the position of the steps is unknown; but as pointed
out by Cabrera [11], it is likely to be controlled by superimposed diffusion fields,
particularly close to the defect source. An analysis of moving steps in tenns of the surface
The role of the oxidising agent and the complexing agent etc
53
concentration gradient has been performed by Hirth and Pound fl2] in developing the idea
of Burton et a/ [13] for crystal evaporation. They predict that a series of steps emanating
from a source, after sufficient distance of travel, will achieve a uniform steady velocity V*
and also a uniform spacing. Such a conclusion indicates that the velocity V'*, at a sufficient
distance from the source, is independent of the rate of production of steps and hence,
is independent of Vj. Consequently, if the steady state is achieved at a small distance
comparable to the size of a pit, it will be difficult to perform an analysis, due to
ignorance of the boundary conditions. If, however, the steady state is achieved well
inside the pit, its slope and rate of widening will be independent of the type of defect.
In this connection, the results of Ives and Hirth [14] and Ives and McAusland [15] arc
quite important.
There are many kinks in the ledges and their nucleation is primarily controlled
by the effective undersaturation of the dissolving species in the solvent and the principal
effect of the etchant inhibitor is the retardation of kink motion. These two effects are in
no \\ay independent. Whereas an uninhibited solvent attacks crystal surfaces very
roughly, etchants containing inhibitor produce pits in an otherwise relatively smooth
surface. It appears, therefore, that the inhibitor produces some modification in kink
mechanism rate, in addition to its effect on kink motion [16]. Many etchants for bismuth
117,18| and antimony [6-8] have been reported which are believed to be modifying the
kink kinetics.
The present study is a part of the investigations carried out to verify the use of
different hydroxy acids as a component in the dislocation etchants. The earlier studies on
ciinc acid containing etchants [8] and dextro-tartaric acid and levo-tartaric acid containing
etchants [6] arc already reported by the present authors. The values of activation energy for
the lateral motion of ledges and frequency factors, i.e. the kinetic parameters, have been
calculated for the etchants containing malic acid for the present investigation and compiled
in Table 1. One can notice from the table that the activation energy and the frequency factor
increase initially and then decrease on increasing the content of malic acid. The activation
energy, which is defined as the difference between the mean energy of all the collisions of
the reactants and the average energy of collision in which the reaction takes place, is
thought of as a barrier to the occurrence of reactions. The greater the activation energy,
the lower the rate of reaction. This also further suggests that the reaction rale decreases first
and thereafter increases, i.e, a critical point is observed. The critical point shows a deviation
in the normal trend of reaction. This type of behaviour was reported by Shah et at in
bismuth [17]. They observed a presence of critical composition in the ethyl alcohol
containing etchants and propyl alcohol containing etchants, where the critical point
exhibited the maximum value of the activation energy in the case of the former and the
minimum value in the Case of the latter. They conjectured that at critical dilution, changes
54
A H Raval, M J Joshi and B S Shah
in the kink kinetics took place. In antimony single crystals, the citric acid containing
etchants indicated that the critical point had minimum values of the activation energy and
the frequency factor [8]. In contrast, the present study suggests that the critical point
indicates the maximum values of the activation energy and the frequency factor. Nitric acid
is serving as an oxidising agent in the composition of the etchants [19], the variation in its
composition seems to modify the kink kinetics in the ledges within the etch-pits which also
depends upon the type and composition of the complexing agent {Le. malic acid or citric
acid) so that the critical point either indicates maximum values or the minimum values of
the kinetic parameters.
4, Conclusions
(1) All the etchants exhibited dislocations of (1 1 1) [10 1] type. |
(2) Increasing the composition of malic acid and reducing the composition l^f nitric
acid in such a way that the total composition of the etchants remains ci^nstanl,
a critical point is observed where a deviation in the normal behavipur is
observed, which corresponds to the earlier results in bismuth and antimony.
However, the specific composition of oxidising agent (nitric acid) as well as the
type and composition of complexing agent (malic acid or citric acid) is responsible
for modifying the kink kinetics in the ledges in such a way, cither to indicate
the critical composition with maximum values of kinetic parameters or the
minimum values. '
Acknowledgment
The authors are thankful to Prof. R G Kulkarni for his keen interest.
References
[1] J H Wemick, J N HobMetter, L C Lovell and D Dorsi J. Appl. Phys 29 1013 (1958)
[2] V M Kosevich Phys Crystallogr. 5 715 (1%1)
13] J Shigeta and M Hiramalsu J Phys Soc Jpn. 13 1404 (1958)
[4] L C I-ovclI and J H Wemick J Appl. Phys. 30 234 (1959)
(5J V P Bhalt, A R Vyas and G R Pandya Indian J Pure Appl. Phys 12 807 (1974)
[61 A H Raval. M J Jo.shi and B S Shah Cryst. Res. Technol 30 1003 (1995)
[7] A H Raval and M J Joshi Indian J. Phys. 68A 1 13 (1994)
[81 A H Raval. M J Joshi and B S Shah Indian J. Phys. 70A 569 ( 1 996)
[yj B B Thaker and B S Shah CrysL Res. Technol. 21 189 (1986)
[ lOJ J J Gilman, W G Johnston and G W Sears J. Appl. Phys. 29 747 (1958)
[11] N Cabrera The Surface Chemistry of Metals and Semiconductors ed. H C Gatos (Wiley) plOl
(1960)
[ 1 2 1 J P Hirth and G M Pound J. Chem. Phys 26 12 16 ( 1 957)
[13] W K Burton. N Cabrera and F C Frank Phil. Trans. Roy. Soc. 243 A 299 ( 1 95 1 )
The role of the oxidising agent and the complexing agent etc
55
[ 14 ] MB Ives and J P Hirth J. Chem. Phys. 33 5 1 7 ( 1 960)
[ 15 ] MB Ives and D D McAusland Technical Report No. JI, U, S. Office of Naval Research (1968)
[16] MB Ives J. Phys. Chem. Solids 24 275 (1963)
[ 17 ] B S Shah. M J Joshi and L K Maniar Latin Am. J. Met. Mat. 7 48 (1987)
[18] B S Shah M J Joshi and L K Maniar Cryst. Latt Def Amorph, Mat. 17 417 ( 1988)
[19] J W Faust Jr. Reactivity of Solids eds. J W Mitchell et al (New York : John Wiley) p 337 ( 1 969)
Indian J. Phys. 72A (I), 57-63 (1998)
UP A
— an intemationul journal
Effect of interface state continuum on the forward
(I-V) characteristics of metal-semiconductor
contacts with thin interfacial layer
P P Sahay
Department of Physics, Regional Engineering College, Silchar-788 010.
India
Received JH March 1997, accepted 20 November 1997
Abstract : The effect of interface stale continuum on the forward current-voltage
characteristics of metal-sciniconductor contact has been examined. It is observed that with the
increase of the density of interface state continuum, the nonlinearity in the chnractenstics begins
at lower voltages where the current increases The increase of current is due to quantum
mechanical tunneling of electrons between the enhanced interface states and the metal, so as to
provide additional current paths However, at higher voltages, the current decreases with the
enhanced interface state den.sity due to the increment on the part of the applied voltage drop
across the intcrfacial layer For a particular density of interface state, the nonlineanty in the
characteri.stics begins at lower voltages if the interaction rate of the interface slates with the
ma]onty carriers is much larger than that with the minority earners.
Keywords : Metal-semiconductor contact, interface state continuum, ( I -V) charactenstics
PACSNos. ; 73 30+y,73.40.-c
1. Introduction
Although the performance and reliability of devices based on metal-semiconductor
Schottky contact depend on the energy and density of interface states, little efforts have
been done to investigate the effect of these states on Schottky (1-V) characteristics, Barret
and Vapaille [1,2] in their theoretical work of characterization of these states, considered
the interface state spectra as a set of discrete levels or narrow bands sparsely distributed
within semiconductor band gap. Later on, this concept was modified by Murel and
Deneuville [3] with the assumption that the distribution of the interface stales within the
band gap can be fitted by several localized rectangular bands. Recently, Chattopadhayay [41
look the case of discrete localized states to study their effect on the (I-V) behaviour of
metal-semiconductor contacts.
© 1998 I ACS
58
PPSahay
However, experimentally it is found that for almost all MIS tunneling devices, the
interface states are distributed in a wide continuum rather than in a few discrete levels in the
semiconductor band gap [5,6]. Therefore, the distribution of the interface states for metal-
semiconductor Schotiky contacts can still be expected to be a continuum. In this paper, the
effect of the interface state continuum on the forward (I-V) characteristics of metal-
semiconduclor contacts has been reported.
2. Theory
Figure 1 represents the energy band diagram of a forward biased metal-n type
semiconductor contact with a thin interfacial layer. Here (l)„ is the work function of the
metal; the electron affinity of the semiconductor; the semiconductor surface potential;
5, the thickness of the interfacial layer; A the voltage drop across the interfacial layer and
Figure 1. Energy band diagram of a forward biased meial-n type
semiconductor contact with a thin interfacial layer.
V„, the depth of the Fermi level below the conduction hand edge in the bulk semiconductor.
and Efp are the respective quasi-Fermi levels for electrons and holes in the
semiconductor at a forward bias voltage V applied to the junction. Applied voltage V
consists of two components : V = V, + V^, where V, is the part of the applied voltage drop
across the interfacial layer and that across the semiconductor space charge region.
Considering the energy band diagram, the voltage drop across the interfacial layer
can be written as
^ = <l>m - X - V, - V„ - V. (t)
The voltage drop across the interfacial layer can also be obtained by using charge neutrality
condition and Gauss law. Thus,
4 = ^{ftc+a + G/). (2).
where is the semiconductor space charge density; Q,,, the interface trapped charge
density and Qf, the fixed charge density in the interfacial layer.
Ejfect of interface state continuum etc
59
Taking the case of the interface state continuum throughout the band gap, the net
charge density trapped in the interface stales is given by [7]
Qjv^)=-gj [b; (£, ) + (£, )]/„ (£, , V', )dE, + g j Df, (£, )</£, . (3)
where ) and£)'/ (E^) are the densities of acceptor and donor types of interlace
states at the energy level £„ and {E„) = D"(£,) + D!/(£,). The occupation function
is assumed to be indistinguishable for both acceptor and donor types of ihc
interface states.
The occupation function of the interface slate is obtained using the Shockley-Rcad-
Hall statistics and considering the charge exchanges between metal and interface states
(8,9). Thus,
/„(£rV,)
c„[/i,+n|(£,)] + +p,(£,)] + 1 /t,’
(4)
where c„ and Cp are the average capture rates of the interface states for electrons and holes;
n, and i\ arc the quasi-thermal equilibrium densities of electrons and holes at the
semiconductor surface; rj| and pi are the densities of electrons and holes if their quasi-Fermi
levels were coincident with trap energy level Ef ; f„{Ef) is the probability of occupancy ol
slate £, by the metal electron; T, is the electron tunneling time constant between ihc
interface states and the metal conduction band.
Further,
n, = /i,exp[(£^„ - E,)/kT^,
(5a)
p, = w,exp[(£, -£^, )/*£],
(5b)
«!(£,) = n, cxp[(£, -£,)/*7'],
(5c)
Pi(£,) = n,exp[(£, -£,)/A:7'],
(5d)
where E, is the intrinsic Fermi level of the semiconductor and n, is the intrinsic carrier
concentration.
( 6 )
Since the quasi-Fermi level of minority carriers in the semiconductor is aligned with
that ol the metal at the interface, we can write
f (E)= ‘ = P,(£,)
' 1 + exp[(£, -£^,)/^r] p, + p^{E,)
With this suhslitution, eq. (4) becomes
Cn'ts + c;,p,(£,)
fAEnVj =
where
c„[/i, + fl|(£,)] + c;[p, + p, (£,))’
1
(7)
79 A< I \ fi
6 ()
P P Sahay
Taking r' / 1 .„ = a, a parameter specifying the controllability of minority carriers on the
rK-cupancy ol the interface state continuum, eq. (7) becomes
n, + apy(E,)
n, + «,(£,) + a[p, + Pi (£,)]'
( 8 )
Usually, the U ~V) characteristics of most metal -semiconductor contacts are characterized
by thermionic emission theory. Thus assuming interfacial layer-thermionic emission theory
[ I0| the iJ -VO relation for these metal-semiconductor contacts can be written as
J = /\Y“<9„cxp
for V >
3Jt7
(9)
where A* IS the effective Richardson constant; 7, the absolute temperature and is the
transmission coefficient across the inlerfacial layer. 1
The transmission coefficient may be approximately expressed as [ 1 1 1 \
( 10 )
where Xe ~ assumed to be independent of the applied voltage; m„, the
effective tunneling mass of electrons; and X( ihc effective barrier height presented by the
intcrfacial layei.
The voltage dependence of surface potential y/, can be obtained numerically from
eqs. (1-3) The values of thus calculated, can be used to obtain (J-V^ characteristics of
metal semiconductor donlact from cq. (9).
3. Discussion
The study has been carried out on any arbitrary melal/n-type Si contact where the metal has
the work function 5.0 eV, like Co. The occupation function , V^^ ) of the interface state
continuum within the band gap has been calculated with the help of eq. (8) The parameters
used here are (p„, = 5.0 eV, x = 4.05 eV, Nj = 10’^ cm"\ Nf = 5 x 10’ ' cm 1.12 eV, S
= 10 A, = 1 1 .9, e, = 3.9 and a = 0.01. The occupation function thus obtained, is used to
gel interface trapped charge density Q,, from eq. (3). Here, the interface states are assumed
to be of donor nature and uniformly distributed throughout the semiconductor band gap.
Considering the intcrfacial layer to be of oxide layer and with
and Qi - qN f , Nf being the density of fixed charges in the oxide layer, the values of y/,
have been calculated for different values of V for a given interface stale density from eqs
( 1 ) and (2) by a self-consistent iteration method. Plots of yr, vj V with interface state density
as parameter are shown in Figure 2,
As desired by Card and Rhoderick [II], the band structure of the interfacial films
formed in metal-semiconductor contacts does not resemble that of bulk Si 02 even if one
considci ihe film of thickness 26 A; .so it is unreasonable to use a value of x^ of 3.15 eV
obtained by William [12] in the case of thick film MOS devices. Card and Rhoderick
Effect of interface state continuum etc
61
calculated the effective transmission coefficients for oxide films of thickness from 8 A to
26 A. which have been used here in obtaining (J-V) characteristics of metal-semiconductor
contacts.
Figure 2. Voltage dependence of the surface potential at different values of
the density of interface stale continuum
The forward (J-V) characteristics of metal-semiconductor contact wiih interface
slate density as parameter are shown in Figure 3. The curve corresponding to D„ = 0
icprcscnts the ideal characteristic for which the logarithmic variation of current with
voltage IS linear. With the presence of the interface state continuum within the hand gap, the
Figure 3. Effect of the density of interface state continuum on the forward
(J-V) charactenstics of metal-semiconductor contact.
variation of current wkh voltage becomes nonlinear. From Figure 3, it is clear that the
nonlinearity in the characteristics begins at lower voltages as the density of interface state
62
P P Sahay
continuum increases. Ai lower voltages, the current has been found to increase with the
increase of the density of interface slate continuum. This can be understood with quantum
mechanical tunneling if electrons between the enchanced interface states and the metal,
which provides additional cunciit pdlh^. However, at higher voltages, the current decreases
with the increase of the density of interface state continuum. This is due to the increment on
the part of the applied voltage drop across the interfacial layer at higher voltages. Figure 4
Fipure 4. F-lteci of a o.i the forward (J-V) characiensiics of metal- *
sciiiiconduclor conlacl for a typical value of D,,(£p = 5 x 10*^ cm“^ eV*
shows the cficci of «(a parametei specifying the controllability of minority carriers on the
occupancy of micrfacc stale continuum) on the forward (J-V) characteristics of metal-
seniiconcluclor conlacl for a typical value of =5 x 10’^ cni"^ cV“'. It is seen that
the nonlinearity in the characteristics begins at lower voltages if the interaction rate of
the interface states with the majority carriers is much larger than thiJt with the minority
carriers.
Rcforciices
1 1 ] C’ Hiirrcl and A Vapaillc Solid-State Electron 19 73 ( 1 976)
f2) C Band ad A Vapaille J AppI Phys 50 42)7 (1979)
t M B Mulct and A Oeneuville J Appl Phys 53 6289 ( I9H2)
Ml F Chaltopadliyay Solid-Suite hlertnm 37 1759 (1994)
f S] S Kar and W E Dahlke Solid-State Electorn 15 1 2 1 ( 1 972)
'6] W E Dahlke and S Jain J Appl Phys. 59 1264 (1986)
Effect qf interface state continuum etc
63
[71 E H Nicollian and J R Brews MOS Physics and Technology (New York . Wiley- Interscience) Chap 5
p 176(1982)
[8J L B Freeman and W E Dohlke Solid-State Electron. 13 1 483 ( 1970)
[91 P P Sahay and R S Srivastava Cryst Res. Technol 25 1461 (1990)
[ 1 01 C Y Wu / Appl Phys. 51 3786 ( 1 980)
[III H C Card and E H Rhodcrick y Phys. D4 1589(1971)
[12] R William Phys Rev 140A 569 (1965)
Indian J. Phys. 72A (1), 65-71 (1998)
UP A
< an international journal
Efficiency measurement of a Si(Li) detector below
6.0 keV using the atomic-field bremsstrahlung
S K Goel, M J Singh and R Shanker
Atomic Physics Laboratory, Department of Physics, Banaras Hindu Univcrsiiy,
Vanina$i-221 005, India
Received 5 Auffust 1997. accepted 24 September 1997
Abstract : The atomic-field bremsstrahlung spectrum produced in bombardment of atoms
by an electron beam of kcV -energies has been used to determine'' the rclaiive efficiency of a
Si(Li) detector in the energy range of 2.0-'7 5 keV The relative efficiency of the detector as a
function of photon energy is obtained by normalising the observed bremsstrahlung spectrum to
the corresponding theoretical cross sections The relative efficiency is put on an absolute scale
using a calibrated radioactive source This technique is illustrated by measuring the
bremsstrahlung spectrum produced by 7.0 keV and 7 .5 keV electrons incident on (semi-thick)
targets of Ag, Au and Hf. The present method is believed to be as precise as the convent lonol
technique using calibrated radioactive sources
Keywords : Atomic-field bremsstrahlung, efficiency of a Si(Li) detector
PACS Nos. : 29 40 Wk, 34 80.Bm
1. Introduction
A precise knowledge of the efficiency of solid state detectors, for example, of a Si (Li)
detector, is important in many applications. A recent paper by Campbell and Me Ghee [1]
reviews the current state-of-the-art with regard to Si(Li) detector and provides an extensive
bibliography for the work on efficiency measurement. The approach adopted is normally
the traditional one which uses the carefully prepared, calibrated radioactive sources with
X-rays having photon energy region of interest. Efficiency calibration of Si(Li) detectors
with X-ray reference sources at energies between 1.0 keV and 5.0 keV, has been discussed
by Denecke et al [2]. This method is known to utilise the data on nuclear- and atomic-
physics processes, such as, internal conversion coefficients, fluorescence yields, and
relative X-ray intensities of the lines.
Atomic-field bremsstrahlung (AFB) is an alternative phoioi source which can be
used to replace the conventional radioactive sources. Palinkas and Sciilrfil^ [31 were the first
© 1998 lACS
66
S K Goel, M J Singh and R Shanker
to use this technique for efficiency determination of the solid-state detectors as a function of
photon energy who bombarded 10.0 keV electrons on a 10 ^igm/crn^ carbon target and
ob.served the bremsstrahlung spectrum at 105®. A few years later, Quarles and Estep [4] and
Altman et al [5] published their works in which they used the AFB technique to determine
the efficiency of a Si(Li) and a HPGe detector in the photon energy range of 2-40 keV and
15" 100 keV, respectively.
In this paper, we have demonstrated the usefulness of atomic-field bremsstrahlung as
an 'alternative' method for determining the relative efficiency of a Si(Li) detector even at
photon energies below 6.0 keV, where not many X-ray lines are available from the
conventional radioactive sources. The potential advantages of using the atomic-field
bremsstrahlung in efficiency measurement is that the theory is independent of the atomic-
and the nuclear-processes which form the theoretical basis for determining the line
intensities of X-ray fluorescence sources. In other words, it may be stated that the AFB
process can provide an 'independent' photon source with the potential of absolute
calibration to the accurate theory.
Pratt and his coworkers [6,7] have calculated the doubly differential cross sections
foi atomic-field bremsstrahlung process in a wide range of bombarding electron energy and
for all atomic numbers. When electrons with kinetic energy T bombard a target of atomic
number Z, the radiation produced may be a non-characteristic (continuum) with energy k
ranging from zero to T, the so-called kinematic 'end point' of the bremsstrahlung spectrum.
The number of photons of energy k, Ng {k) within a photon energy window Ak, detected by
a detector placed al angle with respect to the incident beam and subtending a solid angle
IS given by
Ngik) =
dkdQ
\
AkAi2£{k\
y
( 1 )
where is the incident electron beam intensity, t is the target thickness, £(/r) is the photon
energy
sections.
( d^a \
dependent efficiency of the delectcr and is the theoretical AFB
I dkdU
\{ is seen from eq. (1) that absolute efficiency measurements would require the
precise delenninalion of the target thickness and that of the which is a limiting factor in
doing so. However, the relative efficiency e(k) can be determined readily from the ratio
Ng(k)/ (d^a / dkdQ) with much more accuracy since it does not depend on t and N^,
If good relative measurements are available over a desired range of photon energies,
the relative efficiency can be placed on an absolute scale by the measurement of one line
from a calibrated radioactive source in the region of interest.
2. Experimental procedure
A collimated beam of electrons of 7.0 keV and 7.5 keV energies was obtained from our
indigenously built electron gun and was incident on semi-thick targets (150 pgm/cm^-6(X)
Efficiency measurement of a Si(Li) detector below 6.0 keV etc 67
^gm/cin^) of Ag, Au and Hf which were placed at 45° to the beam. The uncertainty in
target thickness quoted by the manufacturer is 20%. The bremsstrahlung spectrum was
observed at 90° to the incident electron beam by a Si(Li) detector (active area of 80 mm-
and thickness of 5 mm; FWHM = 250 eV at 5.9 kcV). The photons emitted from the target
reached the detector thiough a 6 p.m thick hostaphan chamber window and an air column ot
1 .6 cm. The thickness of detector's Be-window was 0.25 mil, The data was collected using
a PC-based MCA in about 2500 secs with an average beam current of about 3 nA to avoid
any pile-up events. The counting statistics on data points varied between 3-10%. A typical
bremsstrahlung energy sepcclrum produced from 7.0 keV e~-Ag collisions is shown in
Figure 1. The background photons produced from scattered electrons hitting the chamber-
wall and hostaphan window were minimised by preventing them from reaching the detector
PHOTON ENERGY (kaV)
Figure 1. Electron bremsstrahlung photon energy .spectrum for 7.0 keV
electrons incident on a semi-thick (157 pgm/cm^) silver target. Photons were
detected at 90“ to the incident electron beam direction by a Si(Li) detector
by using a suitable aperture on the Be-window. A more detailed discussions on the
experimental arrangements, data acquisition, background subtraction and analysis etc. can
be found in Refs. [8,91.
3. Determination of the efficiency of Si (Li)
Absolute efficiency of the Si(Li) detector was detennined from the data of bremsstrahlung
spectra in two ways ;
(i) by obtaining the ‘relative’ efficiency of the Si(Li) and making it on an absolute
scale,
(ii) by making use of the measured values of all parameters treated with necessary
corrections and putting them in eq. (1).
In the ‘first’ method, the number of photons, Ng^) obtained from the recorded
bremsstrahlung energy spectra with Ag, Au and Hf targets normalised to their respective
72A(4)-10
68 SK Goel M J Singh and R Shanker
theoretical bremsstrahlung doubly differential cross sections obtained from
Ref. [7] in a chosen photon energy window dk (Ak = 250 cV). The normalised data thus
obtained yield the relative efficiency of the detector as a function of photon energy. This is
so because the ratio involves only the efficiency parameter which depends on photon
energy k\ the other parameters are independent of k [see, eq. (1)]. The relative efficiency is
then put on an absolute scale in an independent way by further normalization to a standard
calibrated radio-active ^^Fe-source at a photon energy of 5.9 keV. For this, the radioactive
source is placed at the target position and the emitted Mn-K^^ line is recorded. The formula
for determining the absolute efficiency of a detector as a function of photon energy k, using
the characteristic Mn-Ka line is given by [10],
£{k) =
net area of Mn - K ^ line
liC. (t) X live time x yield x 3.7 x 10^ ’
( 2 )
where, net area is the area under the Mn-K^ line appearing at energy k, which is directly
related to the intensity of the line; is the activity of the isotope at time t in micro-
Curies, live time is actual analogue to-digital converter (ADC) non-busy time of data
collection in seconds and yield is the branching ratio fraction of the Mn-Ka line by the
source. The factor 3.7 x 10^ converts radioactive disintegrations per second to micro-
Curies, since 1 Curie = 3.7 x I0‘® disintegrations per second. The is calculated by the
following relation
/iC (0 = ;iC,(ro)exp
-0.693. decay, time
half, life
(3)
where, is the activity of the isotope at the initial time /q in microCuries.
Practically, the absolute efficiency or the total detection efficiency of the detector is broken
into two factors i,e. into the ‘geometricar efficiency [AQJAk) and the ‘intrinsic’
efficiency. The latter one depends on transmission through the detector’s Be- Window,
Au-contact and Si-dead layer. is simply the solid angle element that the front surface of
the detector subtends at the source of photon. AQj^n in the present configuration is
found to be 1 .93 x 10^ Sr. By determining the value of ^C,(f) from eq. (3) and substituting
its value into eq. (2), the E(k) is calculated. At k = 5.9 keV, the value of e(k) is found to be
1,71 X 10^.
In the ‘second’ method, the calculations for detector’s absolute efficiency E(k) are
made by using the values of measured parameters after treating tfiem wi^h prpper
corrections [see, eq. (1)] and theoretical bremsstrahlung doubly differential cross
sections [7]. Since, we have done experiments with targets which are thick enough to
arrest the impinging electrons but thin enough to transmit a substantial number of photons,
it was necessary to do corrections in the experimental data, namely, in N^k) and in N, for
the solid-stale-effects i.e. for electron energy loss, photon attenuation and electron
backscattering events. The details of various corrections are given elsewhere [11].
Efficiency measurement of a Si(Li) detector below 6.0 keV etc
69
As a result, the absolute efficiency of the Si(Li) detector at a photon energy k is obtained
using eq.(l) as
e(k) =
N'(k)
N'
' [dkda
N.AkAQ
Theorv
(4j
where, N'^ {k) = number of bremsstrahlung photons after corrections for electron energy
loss and photon attenuation in a photon energy window Ak, N ' = number of incident
electrons on the target after correction for electron backscattering events and N, = target
thickness (number of atoms/cm^).
Using eq. (4), we have determined the absolute efficiency of the Si(Li) detector in
the photon energy range of 2,0 keV to 7.5 keV from the data of 7.0 keV and 7.5 keV e~-Au,
Ag and Hf collisions. The calculated values of £(k) from this method are found to agree
Figure 2. Intrinsic-efficiency of a Si(Li) detector versus photon energy for (i)
7.5 keV electrons on Au 200 pgm/cm^ (x) and Hf . 600 pgni/cm^ (n) using
‘first’ method; (ii) 7.0 keV electrons on Ag 157 jigm/cm^ (A) and Au . 200
^igm/cm^ (O) using ‘second’ method (•) • datum corresponding to the radio-
active '^'^F^-.source Error bars on data point.s are purely statistical in nature
while the error bar on the source point corTespond.s to the uncertainty in target
thickness The solid line curve is the simple photo-absorption model for the
efficiency The drop-off at about 6 0 keV is an electron energy-loss effect in
bremsstrahlung spectra of the semi-thick targets studied in the present impact
energy range
with those obtained from the ‘first' method within the uncertainty of the target thickness of
about 20% and they are shown in Figure 2 by respective symbols for comparison.
4. Results and discussion
The absolute efficiency £(k) of a Si(Li) detector as a function of photon energy in the range
of 2.0 keV-7.5 keV is shown in Figure 2. The £(k) determined from data of each target at
arbitrarily chosen photon energy it, using the ‘first’ method (see. Section 3) is found to
agree with the normalization to the radioactive source within the uncertainty in the target
70
S K Gael, M J Singh and R Shanker
thickness which is about 20%. Furthermore, the calculated values using the ‘second’
method for detector’s absolute efficiency as a function of photon energy arc also included
m the figure for comparison. Further shown is the scaled theoretical efficiency curve based
on a simple photo absorption model using the photo-absorption cross sections of Storm and
Israel 1 12|. We have made the Chi-squares fit to the data to determine the photon energy
dependence of the photo-absorption cross section. The curve is given by,
£(k) = (1 - 0.69k'^'^^) exp [- 3.11/^^ - 1.86fg^/c'^’
- 37.96r^„/t”2], (5)
where, i„. tuf, (), and are the air gap, Be. hostaphan and gold layer thicknesses in mg/ cm^
respectively. In the above fit, the values of and r^u were taken to be 1.6 cm, 25 pm,
6 pm and 200 A. respectively. No attempt was made to fit the M-Shell edge effect in the
efficiency model The fitted efficiency curve as shown in Figure 2 by a solid line, shows a
good agreement with the experiment (X^ = 1.21 per degree of freedom). The factor
proceeding the exponential, corrects for silicon escape.
A few interesting features can be noted from the data for targets studied in the
present work. First, the data deviate from the curve from “6.0 keV upwards. This is due to
energy-loss experienced by the electron beam which suffers multiple collisions in the semi-
thick targets. This slow drop off is a characteristics of the ‘thick target’ effect. However, the
t\k) behaviour with photon energy for thin targets would show, in contrast, a sharp drop off
at the ‘end point’ or at the maximum possible photon energy. This is a characteristics of the
ihin-largel’ bremsstrahlung end points. Further, a silicon K-X-ray at about 1.7 keV may be
induced due to a monolayer of silicon on the targets due to contamination by pump dll. This
peak can be avoided by a careful attention to a good vacuum (i.e. better than 1x10“^ torr).
Second, the absolute efficiency curve calculated from the photoab.sorption data of Storm
;tiul Israel [ 121 shows a good agreement with the measured efficiency below 6.0 keV for
each target with the normalisation to the radioactive source within the uncertainty in the
target thickness of about 20%.
5. C'onclusion.s
In tills paper, wc have demonstrated the use of atomic-field bremsstrahlung for determining
the relative cfliciency of a Si(Li) detector below 6.0 keV by measuring the bremsstrahlung
photon energy spectra from Ag, Au and Hf semi-thick targets bombarded with electrons of
J 0 keV and 1.5 keV energies. The relative efficiency is placed on an^absolute scale by
normalisation to a calibrated radioactive source or to the absolute theoretical
bremsstrahlung cross sections, provided an accurate knowledge of thickness of the target
IS known. With some care, the background can be minimised. We believe this technique
to be us precise as the conventional technique using calibrated radioactive sources and to
be useful for determining e(k) for the entire efficiency curve in one run with a good
statistics.
Efficiency measurement of a Si(Li) detector below 6.0 keV etc
71
Acknowledpiient
This work was supported by the Department of Science and Technology (DST), New Delhi
under Project No. SP/S2/K-37/89. S K GocI wishes to acknowledge the financial support
from the DST.
References
[ I ] J L Campbell and P L McGhee Nud Instrum, Meth A24B 393 ( 1 986)
[2] B Denecke. G Grosse, U Watjen and W Bambyneck Nud. Instrum. Meth. in Phys Res. A286 474
(1990)
[3] J Palinkas and B Schlenk Nud. Instrum. Meth. 169 493 (1980)
[4] C Quarles and L Estep lEE Trans Nud. Sci. NS-30 1 5 1 8 ( 1 983)
[5] J C Altman, R Ambrose, C A Quarles and G L Westbrook Nud Instrum. Meth. in Phys. Res
624/25 1026(1987)
[6] H K Tseng and R H Pmtt Phys. Rev A3 100 ( 1971)
[7] L Kissel, C A Quarles and R H Pratt At Data Nud Data Tables 28 38 1 (1983)
[8] S K Goel, M J Singh and R Shanker Pramana . J. Phys. 45 291 (1995)
[9] S K Goel and R Shanker Phy.\. Rev A52 245, 3 (1995), S K Goel PhD Thesis (Banaras Hindu
University, Varanasi) (1996)
1 1 OJ Canberra Products Catalogue 7-th edn. p 34 ( 1 988)
fill S K Goel and R Shanker Phys. Rev. A54 2056 ( 1996); J. X-ray Sci. Tech. (Submitted) (1997)
f 1 2] E Storm and H Israel Nud Data Tables A7 565 (1970)
Indian J. Phys. 72A (1). 73-82 (1998)
UP A
— an ifitemationaJ joumaJ
Multipartide production process in high energy
nucleus-nucleus collisions
M Tantawy, M El-Mashad and M Y El-Bakry
Department of Physics, Faculty of Education, Ain Shams University,
Roxi, Cano, Egypt
Received 17 December 1996, accepted 26 September 1997
Abstract : We apply here an impact-parameter analysis depending on the parlon
two-Fireball model. In this model, each of the colliding hadrons is considered as a bundle of
point-like particles (paitons). Only those partons in the overlapping volume from the colliding
hadrons participate m the interaction, which are assumed to be stopped in CMS. Therefore,
two excited intermediate states (fireballs) arc produced which later on decay to produce the
observed created secondaries The parameters characterizing the muUiparticle production
process for Li^, C*^ and in nuclear emulsion have been estimated and compared with
the experimental data
Keywords : Nucleus-nucleus collisions, multiparticle production process, impact
parameter analysis
PACSNos. : 21 60.-n,2.V75.Dw
1. Introduction
li is well established that nucleons are coinposite objects consisting of a fixed number of
partons [1]. This nucleon strucluro have been used in different models [2,3] along with
other assumptions to describe hadron-hadron interactions. One of these models is the parton
two fireball model (PTFM) proposed by Hagedorn [4.5]. PTFM along with the impact
parameter analysis have been used in studying the high energy proton-proton and proton-
nucleus interactions by Tantawy [6] and El-Bakry [7]. It has also been used to study high
energy hadron-hadron and hadron-hucleus interactions by El-Mashad [8]. All these studies
show good predictions of the measured parameters. In the present work, we extend this
model to study the multiparticle production process in nucleus-nucleus high energy
interactions.
© 1998 lACS
74
M Tantawy. M El-Mashad and M Y El-Bakry
2. The model
We apply here an impacr -parameter analysis depending on the parton two fireball model to
study nucleus-nucleus interactions at high energies. The basic assumptions in this model
can be summarized as follows :
(i) The colliding hadrons are composed of a fixed number of point like particles called
parlons. These parlons can be treated as losely bound states. At high energies,
partons have negligible transverse momenta [1].
(ii) Only those partons within the overlapping volume of the two interacting hadrons,
have the probability to interact which are assumed to be stopped in the CMS.
Therefore, their CM-kinetic energy will be consumed in the excitation of the
produced two fireballs.
(iii) Each fireball will decay into a number of newly created particles (mainly poins) with
an isotropic angular distribution in its own rest frame.
It IS now clear that in this model, the mass of the produced fireballs and
consequently the number of the created particles are functions of the overlapping volume at
certain incident energy. The overlapping volume is defined by the incident impact-
parameter. Then using the above assumptions, we can investigate the multiparticle
production process in nucleus-nucleus interactions.
2. J. Impact-parameter distribution :
Let us assume that the interacting nuclei (projectile and target) at rest are spheres qI radii R\
and /?2 respectively. Then the statistical probability of impact parameter (b) within an
interval dh is given by
P(b)db = 2b<n>l{R^ +^ 2 )^
i.e. P(b)db = Ibdb j, (1)
where ro = (1.22 — > 1.5) fm and A\ and A 2 are the mass numbers of the two interacting
nuclei respectively. In terms of a dimensionless impact parameter (x) defined as X =
eq. (1) becomes
Pix)dx = Ixdx I j . (2)
If one assumes that the partons from the incident nucleus in the overlapping volume
v;ill interact with the nuclear matter of the target, then we can calculate the overlapping
volume v(j[) in the incident nucleus rest frame. Then, we can calculate the fraction of
partons participating in the interaction (z) as a function of (jr), as
Multiparticle production process in high energy etc
75
(K’-K’)'-?''!
4
where i^o is the volume of the nucleon.
From eqs. (2) and (3) we can get the z-function distribution as
Ixdz
P{z)dz =
(3)
(4)
We have calculated eq. (4) for Lf, and on nuclear emulsion.
Table 1. The values of the coefficients Q in eq. (5)
Type of
interaction
Co
C|
Cl
C3
Li^-Em
0.69
-041
0.17
- 0.03
0.0021
Li^-CNO
0.135
0.022
0035
- 0.01
0.0009
Li^-AgBr
0 143
0.039
00044
-0 0027
0 0004
Li^-C
0.112
0 071
0.0105
- 0.0057
0.00058
Li^-N
0.117
0.055
002
- 0.0081
0.0008
Li’-O
0.105
0.075
0.011
- 0.0067
0.00075
Li’-Ag
0 255
-0232
0 169
-0 039
0 0031
Li’-Br
0.309
- 0.’281
0.185
- 0.0418
0 0033
C'2.Ein
0.058
0047
00047
- 0.0016
0.00012
c'2-cno
0183
0.013
0.0068
- 0.001
0 00004
C'^-AgBr
0.063
0.048
- 0.0004
- 0.0004
-000004
c'2-c
0.068
0 105
-0019
0.0019
- 0.00007
C'2-N
0.073
0.094
- 0.014
0.0012
- 0.000037
C*2-0
0.07
009
- 0.012
0.00078
- 0.0000116
C'2-Ag
0 121
0.0023
0019
-0 0037
0 00021
C'^-Br
0.147
- 0.028
0.029
- 0.0049
0.00027
O'^-Em
0.068
005
- 0.003
- 0.000014
oooool
O'^-CNO
0.083
0.066
- 0.007
0.00044
- 0.00001
O'^-AgBr
0,24
- 0.026
0.008
- 0.0006
0.00002
u
o
0.059
0095
- 0.016
0.0014
- 0.00005
O'^-N
0.066
0.085
- 0.013
0.001
-0 00003
q
6
0.07
0.077
- 0.0099
0.0007
- 0.00002
0'®-Ag
0.104
0.017
0.0077
- 0.0013
0.00006
o'®-Br
0.125
- 0.0018
0.0123
- 0.0018
0.000076
72A(I)-I1
76
M Tantawy, M E^-Mashad and M Y El-Bakry
Since cq. (4) is noi a simple function of z, to get analytic equation for the z-function
distribution, we used the fitting procedure to the curves drawn from eq. (4) for all collisions
which yields
P(i)dz = Y^C^z^dz. (5)
A = -l
The values of he cocfncicnts Q are given in Table 1 .
2.2. Shower particle production in N-N collisions ;
After the collision takes place, the partons within the overlapping volume stop in the CMS
and their K.E changes an excitation energy to produce two intermediate slates (fireballs).
The produced fireballs will radiate the excitation energy into a number of newly
created particles which are mainly pions. We assume that each fireball will decay in its own
rest frame into a number of pions with an isotropic angular distribution plus one baryon.
The number of created pions will be defined by the fireball rest mass {Mj) and the mean
energy consumed in the creation of each pion (£).
The excitation energy from each fireball is
M f - m = Tq z(;c),
where 7J) is the kinetic energy and m is the proton mass at rest.
The number of pions from each fireball (/Iq) will be given by
T,,Z{x)
e
Z{x)Q
lE
( 6 )
(7)
where Q is the total K.E in CMS (- 2Tq), since all the experimental measurements are
concerned with the charged (shower) particles in the final state. Therefore, we have to
assume some distribution (e.g. Binomial and Poisson distribution) for the shower
particles (n^) in the final state of the interaction at any impact parameter, out of total created
particles (hq).
Accordingly, we shall investigate the probability of getting shower particles (nj
from the two fireballs as follows :
From eqs. (5) and (7) we get,
+ C_| In
”o
. ( 8 )
Let us assume different probability distributions for creation of shower charged pions from
one fireball V^« 2 ), such as, (a) binomial distribution of the form :
Multiparticle production process in hif>h energy etc
77
N\
AN-n. )
(9)
where N is ihe number of created particles from one fireball = no/2,
H 2 is the number of pairs of charged particles,
P and q are the probabilities that the pair of particles is charged or neutral,
respectively.
or (b) Poisson distribution of the form :
N
= H p"2 c-NP
Now, the number of charged particles from one fireball will be given by
n = 2/1 2 +1
Then the distribution of shower particles from one fireball will be
0(n) = ^ T(ri 2 )P(n„).
'•o
( 10 )
(in
Because of charge conservation, 0(n) at /lo = even, is equal to 0(n) at (//q + 1 ). Therefore,
we can calculate the probability of getting any number of shower particles (nj from the two
fireballs as
n j
POi^) = '^0(n)<P(n^ -H). ( 12 )
»l = l
The above equations can be used for studying all the characteristics of the shower particle
production process such as multiplicity distribution, average multiplicity, KNO-scaling as
well as the multiplicity dispersion.
2.2./. The shower particle multiplicit}' distribution :
If wc assume that the energy required for creation of one pion in the fireball rest frame (f)
increases with the multiplicity size (/?o) as
£ = uhq + b, (13)
where a and b are free parameters which can be evaluated lo give the best fitting with the
experimental data, e.g. a = 0.04 and b = 0.35 gives good filling for hadron-hadron and
hadron-nucleus interactions [8J.
Wc have calculated the shower multiplicity distribution (eq. 12) for C'^ incident on
target emulsion [{A) = 70) at P/ = 4.5 A Gev/c. The results of these calculations have
been shown in Figure 1 compared with the corresponding experimental data [9].
78
M Tantawy, M El-Mashad and MY El-Bakry
Figure I shows a qualitative agreement of predicted distributions (using binomial
disiribuiioii eq. (9) and poisson distribution cq. (10)) with the measured ones. There is some
P(n.)
deviation of the numerical values between the calculated and measured distributions.
We refer this disagreement to the unspecification of the target. Thus, we can
recalculate the shower particle multiplicity distribution for the emulsion groups CNO
and AgBr. The results of these calculations for Li^, C" and O'^ in emulsion at 4.5 A
Gev/c, arc shown in Figures 2(a-c) together with the corresponding experimental
data 19.II-I3I.
FiKure 2(b). n^-disinbuiion for Li^ with emulsion groups (CNO. AgBr) at = 4..S A Gcv/c.
For further refinement of the model predictions, we have calculated nv-disiribution
from the emulsion components percentage as follows :
79
Multiparticle production process in high energy etc
(i) For a specific projectile, the z-function distribution can be calculated for this
projectile with the components of the target emulsion separately i.e. (C-N-0-
Ag-Br).
Flfun Wb). B,-<lislribiitioii for C'^ with emulsion groups (CNO, AgBt) ul Pj, = 4.5 A Gev/c,
(ii) Using the same scheme, we can calculate the shower particle multiplicity
distribution for each projectile (Lf , C'^ 0>‘) with the emulsion components,
(lii) From the emulsion components percentage 110], we can combine these distributions
to get the final shower particle distribution for this projectile with target emulsion.
The results of these calculations for Li’, C” and O'* in emulsion at P/, - 4.5 A
Gev/c using eq. (10), are represented in Figure 3 which shows good agreement with
the corresponding cxpcrimenUil data [9,1 1-IB]-
In Figure 3, we compare our results for shower particle multiplicity distribution in
Li’-Em collisions with those obtained by the nucleon-nucleus superposition method (14). In
(his method, the multiplicity distribution is given by
( 14 )
80
M Tantawy. M El-Mashad and M Y El-Bakry
where Pp(N) is the probability for the interaction of N out of Ap projectile nucleons,
given by
Pp{N) = , (15)
and Ap, Ap are the mass number of the projectile and target respectively.
-V-
Flgure 2(c). /ij-dislnbution for 0*^ with emulsion groups (CNO, AgBr) at = 4 5 A Gev /c
2.2.2. Average shower particles multiplicity (<n^>} and multiplicity dispersion (D ) :
Using the shower particles multiplicity distribution described above with the Poisson
distribution of emission, we have calculated the average shower particles multiplicity
through relation
( 16 )
Multiparticle production process in high energy etc
81
Figure 3. n ^-distribution for Li^,
and O* ^ with emulsion
(considering emulsion components
percentage) at Pi - 4.5 A Gcv/c.
Tabic 2 shows the calculated <np» for the considered interactions together with the
corresponding measured values for comparison.
Table 2. The calculated and the n^asured values of average shower multiplicity and
dispersion parameter.
Type of
interaction
< n, > th
< > exp
^exp
Li^-CNO
4.88
2.16 ±013
2.97
Li^-AgBr
5.61
4.63 ±0.195
2.83
Li'^-Em
3.88
3.6 ±0.11
3.32
3.07 ± 0.12
C'2-CNO
6.41
5.04 ± 0.21
3.4
3.66 ± 0.15
C*^-AgBr
7.76
8.92 ± 0.25
3.5
5.17 ± 0.18
C‘2.Em
7.01
7.67 ± 0.13
6.41
7.10 ± 0.23
O'^-CNO
8.47
5.99 ± 0.41
4.28
6.16 ± 0.
O'^-AgBr
8.62
12.87 ± 0.63
4.33
10.01 ±
O'^-Em
6.7
9.6 ± 0.4
5.73
82
M Tantawy, M El-Mashad and MY EhBakry
Included in this table are also the dispersion parameters defined as
^> = |{«)^ n?)
Table 2 includes the calculated values of the dispersion D due to our predictions together
with the corresponding experimental data. From this table, we can conclude that
(i) The calculated values for <nj> and D agree with the corresponding experimental
ones only at specification of target (C-N-OAg-Br) while it is in qualitative
agreement for unspecified target.
(ii) <nj> and D increase as projectile and target mass numbers increase which reflects
that <ns> is strongly dependent on each of beam and target mass numbers.
Acknowledgment
The authors are grateful to Drs. M M Sherif, M S El-Nagdy and M N Yasin, Laboratoiy of
High Energy Physics, Physics Department, Cairo University, for providing us with the
experimental data. \
Refcrencefl
[ 1 1 R P Feynman Photon-Hadron Interactions (Reading, Massachussets : Benjamin) (1972)
[2] E Fermi Prog. Theor. Phys 5 .570 (I9.‘50)
[3] J Ranft Phys. Utt. 31B 529 (1970)
[4] R Hagedom Nuovo Cm. Suppl. 3 147 (1965)
[5] R Hagedom and J Ranft Nuovo dm. Suppl. 6 169 (1968)
[6] M Tantawy PhD Dissertation (Riyasthan University. Jaipur, India) ( 1 980) ^
[7] MY El-Rakry MSc Thesis (Ain Shams UnivcRity, Cairo, Egypt) (1987)
[8] M El'Mashad PhD Dissertation (Cairo University, Cairo, Egypt) ( 1 994)
[9] M S O-Nagdy Phys Rev. C47 346 (1993)
[10] M N Ya.<iin El-Bakry II Nuovo dm 108A 8. 929 (1995)
[11] M El-Nadi, A Abd El-Salam. M M Sherif. M N Yasin, M S El-Nagdy, M K Hegab, N Ali Moussa,
A Bakr, S El-Sharkawy, M A Jilany, A M Tawfik and A Youssef Egypt J. Phys. 24 49 (1993)
[12] MM Sherif, S Abd El-Halim, S Kamel, M N Yasin, A Hussein, E A Shaat, Z Abou-Moussa and
A A Fakeha IL Nuovo Cim. 109A 8. 1 135 (1996)
[ 1 3] Tauseef Ahmad et al Modem Phys. Utt. AS 1 103 (1993)
[14] M K Hegab et al J. Nucl. Phys. A3S4 353 (1982)
NOTE
Indian J. Phys. 72A (1), 83-86 (1998)
UP A
- an international joiiiiiul
Hardness anisotropy of L-arginine phosphate
momdiydrate (LAP) crystal
T Kar and S P Sen Gupta
Department of Materials Science. Indian Association for the
Cultivation of Science, Jadavpur. Calcuna-700 032. India
Received 23 July 1997. accepted 24 September 1997
Abstract ; Deformation characteristics of an imponant non-Iincar maicnal, L-argmine
phosphate nionohydrate (LAP) was studied by measuring (he anisotropy of Knoop
microhardness on (100) cleaved plate of LAP lor vanous loads It was found that the low load
deformation is mainly due to the slip system ( l(K)) <01 1> whereas the higher loud deformation
is dominated by twinning
Keywords : L-argininc phosphate monohydnite. Knoop microhardness
PACS No. ; 4fi.30.Pu
L-arginine phosphate nionohydrate (LAP) with the chemical formula [H 2 N]“^CNH(CH 2)3
CH|NH3rC00“H2P0^2C>, is a promising nonlinear optical (NLO) material discovered by
Chinese scieniists Xu et al fl]. The attractive features of this material arc its hi^h damage
threshold (> 15 j/cm^ at 20 ns), large nonlinearity (> 1 pm/V) and the ease with which
large crystals of high optical quality can be grown [2,3]. So, it has the potential to replace
potassium dihydrogen phosphate (KDP), the material most commonly used for frequency
conversion of infrared lasers in the harmonic frequency generation for laser fusion
experiments. As a part of our project work on this important NLO crystal, we have already
reported the growth and characterization of LAP [4], In the present note, we report (he
anisotropy of Knoop microhardness on (100) cleaved plate of LAP.
For these studies, a (100) cleaved plate of LAP grown in this laboratory, was taken
and polished with water to make the test face flat. A mhp 160 microhardness tester, fitted
with a Knoop mdentor and attached to a Carl-Zeiss (Jenaverl) incident light research
microscope, was used for the measurement of microhardness. The hardness anisotropy was
measured by applying a minimum of five indentations for each load at an interval of 15"
and over the range of 0-165". The zero degree orientation of the long Knoop indenior
© 1998 I ACS
72A(1)-12
S4
T Kar and S P Sen Gupta
diagonal is parallel lo [OlO] and was established by the macrosteps that form along the
direction [01 0] [51. The load was varied from 10 gm to 50 gm and the indentation period
was kept constani at 10 S. Owing -to the microcracks at the corners of the impression, the
maximum load applied was 50 gm. The Knoop hardness was calculated from the usual
(ormula |6j
H ^ = 0.014228 X Pjd^ gmmm“^
using the known lest load P (gm) and the measured length d (mm) of the long diagonal of
the indcnlalion
Figure 1 shows the Knoop microhardness (Hk) measured on the cleavage (100)
plane of LAP with 10 gm normal load in air at different orientations of the indentor.
Rotating the crystal from 0^ to 165'*, hardness was found lo decrease from a high of 66.40
Indentor Orientation (degrees)
Figure I. Vaiialion of Knoop iiiiciohardncKs {H^) with inilontor orienltilion
on Iht.' ( 10(1) cleavage plane ol L-aiiginine phosphate inonohydrale (I AH) at
lOgrnluail •
at 0 ’ to a low ot 31.31 at 45^^ and then there is a lurthcr incrca.se in hardness upto 150",
though no sharp maximum is found in between.
Figure 2 shows the load dependence of the Knoop microhardness for the two
indentor orientations [0I0| and [01 1 J on the (100) plane. Initially, the hardness number
(H^:) decreases with load for both indentor orientations and maintains a constant ratio upto
P = 20 gm, that is the relative microhardness apparently is independent of load in this range.
But 111 the high load region above P = 20 gm, the difference in hardness number (Hk)
between the two orientations becomes more pronounced and also the hardness number
increases for 1011] orientation upto a load P = 50 gm, but for [0I0| orientation the
variation of is slightly different from that of [01 1| orientation. The result is found
disputable with the observations made by different investigators on different crystals which
generally have shown that the microhardness cither increase or decrease with load at low
loads and subsequently attains a fixed value at higher loads. The explanation for this type of
behaviour observed here is due to the different types of slip systems that are operative with
increase in load. To gel a clear understanding of this behaviour, an additional study of
Hardness anisotropy of L-argimne phosphate etc
85
hardness anisotropy al 50 gm load was undertaken and it was observed that the variation in
hardness at this test load is somewhat different from that with load 10 gm (Figure 1 ) The
variation of Knoop microhardness (Hk) at different orientations of the indentor on cleavage
Figure 2. Vatiiilion ol Knoop microhardness (Hf;) wnh load (/^) on Ihc (100)
cleavage plane ol I. anginine phosphate monohydrule (LAP) for two dilTctcnl
orientations ol indentor OM). 135-’-^
( 100) plane of LAP for these two loads 10 gm and 50 gm is presented in Figure 5. This
shows dillcrent degrees of plastic deformation by twinning and slip al the two loads. The
Figure 3, Vanation of Knoop microhardncss {Hk) with indentor oncntation on
the (100) cleavage plane of L-anginine phosphate monohydrale (LAP) at 10
gm (O) and 50 gm (x) loads
low load deformation is mainly caused by slip where as the higher load deformation is
dominated by twinning. The low load hardness anisotropy could be explained by slip on the
( KK)) (01 1 ] system.
In conclusion, plastic deformation in organic nonlinear optical crystal LAP occurs as
a result of either slip or twinning and the relative contribution depends on the magnitude of
load.
86
T Kar arid S P Sen Gupta
References
f) Xu. M Jinang and 2 Tan Ada Chem Smica 41 570(1983)
|21 I) Eimerl, S VcKsko, L Havis, F Wang, G Loiacono and G Kennedy IEEE J Quantum Electron. QE-2S
179 (I TO)
13] .S P Vcisko and 0 Eiinerl Soi Photo-Opt Instrum En^ft H9S 152 {\%^)
|4J A Mci/uindai, T Kar and S P Sen (Jiipia Jjm J AppI PIm 34 57 17 ( 1995)
|5| G Dhanarai. I Shripathi and H 1. Hhat J. Cryst Growth 113 456 (1991)
16J H W Mf)tl Mu ro-lndentahon Hardness 7Wn/ijf (London . Butterworths) 206 (1956)
Indian J. Phys. 72A (1), 87-92 (1998)
UP A
- an international journal
Electric field induced shifts in electronic states in
spherical quantum dots with parabolic confinement
C Bose
Department of Electronics and Telecommunication Engineenng, Jadavpur University,
Calcutta-700 032. India
and
C K Sarkar
Department of Physics. B E College, Deemed University, Shibpur,
Howrah-71 1 103, India
Received JO July 1997, accepted 30 SeptembeT 1997
Abstract : An attempt is made to investigate the electric field induced shifts in electronic
states in a spherical quantum dot (QD) with an isotropic parabolic potential (PP) The
perturbation method is used to estimate the shifts of the above energy levels due to an unifomi
electric field. The energy shift of the lowest state is also worked out by the variational method,
and compared with the results obtained from the perturbation method Both the methods arc
found to yield exactly identical re.sults within the range of the applied field considered In the
case of a spherical QD with square-well potential (SWF), the ground level shift is also compared
with the above results
Keywords : Semiconductor quantum dot, electro-optic effect
PACS Nos. : 73 20.Dx, 78.20Jq
Studies of nanostructured seiniconductors exhibiting quantum confinement in all three
dimensions have been made possible by the recent progress in nanoscale lithography and
microcrystallite doping of glasses. Research on electro-optic effects in such quasi-zero-
dimensional (QOD) systems is attracting increasing attention, due to their applicability in
the field of optoelectronics [1,2]. The electric field-induced shifts in excitonic and
electronic energy levels have already been investigated for a spherical QD with SWP [3,4].
A number of both theoretical and experimental works, however, indicate that the in-plane
confinement in QDs is approximately parabolic [5,6]. These observations have stimulated
further interest in QDs with parabolic confining potential [7]. In this communication, we
shall investigate the effect of electric field on electronic states of such QDs. We shall first
derive an expression for the shifts in electronic energy levels, due to the field applied on a
© 1998 I ACS
88
C Bose and C K Sarkar
spherical parabolic QD made of a typical wide-gap semiconductor, by using the
perturbation method. The field-induced shift in the ground level of the above system will
also be derived by the variational method, and the results will be compared with that
obtained by the perturbation method. Similar results for a spherical QD with SWP will also
be compared with the above results. Since GaAs is a typical example of wide-gap
semiconductors, we will consider GaAs QDs to compute the energy level shifts.
In order to estimate the Stark shift of electronic energy levels in a spherical QD with
isotropic parabolic potential, wc assume the barrier height to be infinite for simplicity. In
absence of electric field, the wavefunction of electrons confined within a spherical QD, can
be cxpres.sed in polar coordinate as
= Rni(r)Y^jG.(l>X ( 1 )
where is the angle dependent part and Rni(r) is the radial part, which in its
normalized form can be given by 18]
RJr) =
r(n + I + 3/2)
exp(-^r2 / ).
In the above equation, j3 = m*(ol h,h = where h is Planck's constant, m* is effective
mass of electron, (0 is the parabola frequency, LJ is Laugurre Polynomial of order n
and degree ct, F is Gamma function, n (= 0, 1, 2, 3, ...) is the principal quantum
number, / (= 0, I, 2, 3, .. .) is the angular momentum quantum number, and m (= 0, ±1, ±2,
. , ±1) is the magnetic quantum number. The corresponding energy eigenvalue E„f^ is
given by
= (2n + U3/2)ho). ' (3)
Let us assume that an uniform electric field be applied in the polar direction (z>. z-
direction). For electric fields higher than 1(F V/m, the voltage drop across the dot may be
comparable to the barrier height, making the assumption of infinite barrier QD no longer
valid. We therefore, restrict the magnitude of the applied field to the value 10^ V/m in the
present analysis.
In the presence of the electric field F,., the Hamiltonian of the system takes the form
/r 1 *
H = — 7 + —ni 0)^r^ + eFr cos 0, (4)
where e is the electronic charge, p is the momentum, 9 is the polar angle_and F is the field
inside the QD. The field F is, however, related to the external field F,. by the familiar
expression
where fj and t',. are the dielectric constants of the QD and the embedding material
re.spcctivciy.
Electric field induced shifts in electronic states etc
89
The energy levels of electrons confined in the spherical QD gel shifted due lo the
applied field and such shifts are derived separately by using the perturbation and the
variational methods.
The perturbation method :
For the range of electric field considered here, the effect of the field can be treated as a
small perturbation over the original Hamiltonian. The energy levels in the presence of a
field, can be corrected to the second order in F, by applying the standard perturbation
technique. The first order correction term vanishes due to the orthogonal property of
spherical harmonics. The second order correction term, giving the shifts in the energy levels
due to the applied field, can finally be obtained as
where
with
and
and
= I
17
[ Vl ./«^2
F — F F — F
n n ^n\l-\,n
Ml
1 !w'!
/, = eF\
h = eF\
/Jr(M + / + 3/2)r(n' + / + 5/2)
-,1/2
i3r(w + / + 3/2)r(Aj' + / + l/2)
Jo n n
(/ + m + l)(/-m + 1)
(2/ + 0(2/ + 3)
1/2
(b)
(7)
( 8 )
(9)
(10)
( 11 )
The variational method :
To perform the variational calculation, the ground state trial wavefunclion for electrons in
the presence of electric field, is taken as
V/ = A^(A)exp(-^r^ / 2)cxp(--^rcos0), (12)
where A is the variational parameter and N{X) is the normalisation constant, given by
rA^^
yV(A) = (p/n)^^^exp\
The corresponding energy eigenvalue is
IhneF
(13)
£(A)
—
2m’
ph^
+ 3P
(14)
90
C Bose and C K Sarkar
where X = . With this value of A, the shift in the ground level energy (Eq) induced by
the applied field, is given as
2m*(0^
(15)
To find the effect of parabolic confinement, we compare the field induced shifts in the
ground stale energy, as calculated above, with that of an identical spherical QD having
SWP. To estimate the ground level shift in the latter case, we use the relation derived by
Nomura and Kobayashi [9], using variational technique. According to their derivation the
energy shift is given by
( 271 - - 3 )^
" I08;r^
(•FR
where (f) = ~j ^ , R being the radius of the spherical dot.
(16)
Tick! induced energy level shifts in spherical QD with parabolic confinement have'
been computed by taking material parameters for GaAs [10], The energy shifts have been
calculated lor the ground state by the perturbation as well as by the variational methods.
The computed results have been tabulated in Table 1. It can be .seen from the table that the
two methods give identical results.
Table 1. Energy shift (4£) ol the ground state electron in GaAs sphencal paiabolic QD. obtained
using the second order pcrtutbalion method and the vanational method,
Field
(V/m)
P
(nm '^)
d^penurbation
(meV)
(meV)
3 28‘)87x
-4 05876 X 10'^
-4 05877 X Kr"^
1 X lo'’
-2 53672 X UH
- 2. 5.3673 X 10*^
28987 X lo'^
-4 05876 X IQ-'’
- 4 05877 X 10“^
1 28987 X I0‘”
- 1 01469 X 10“*^
- 1 01469 X lO'-*’
1 X lO -*^
1 .11595 X 10*^
- 6.341 RI X 10“
6.34182 X lO-"'
3 28987 X It)'^
-0 101469
0 101469
1 28987 X l(V^
4 05876 X 10'*’
-4 05877 X Ur*’
X 10^’
1 .1 1.595 X 10*'^
-0 025.3672
- 0.025.3673
28987 X
- 0^*05876
- 0 405877
The shift in the ground level energy as a function of applied field is shown in Figure
I, both lor QD with SWP and QD with PP. The energy reference point in this figure is
chosen at zero field. To compare the shift in energy level in the two systems, the ground
state energies ot both the structures have been taken to be equal. Here, we note that the
energy in QD with PP is ~ hco = ^-~. The parameter jS, therefore, scales as (i.e. R ~
m R-
). The parameters in Figure 1 have, therefore, been chosen as dot radius R for QD
Electric field induced shifts in electronic states etc 9 1
with SWP and tor QD with PP. Figure 1 shows an increase in energy shift with
increase in applied field. This can be ascribed to the cnhancetl overlap between adjacent
Electric field tlO^V/m)
I'ipurc 1. A cuinpanson of the electron grouiwJ- slate eneipy shill as obuiineci
Iroiii the spherical 0I> with .SWP (solid line) and PP (biokcn line) The upper
and the lower set of curves are loi dots of radius - S nni and 10 nin
respectively The results are calculated from a variational Ircaiincnl
wavel unctions with increase in electric field It is also seen trorn the ligiire that the shifl m
ihe lowest energy level in spherical dot with SWP, calculated by the variational method, is
more than that in QD witli PP. The adjacent higher-lying levels arc closer lo the giouiuJ
level in the QD with SWP as compared to the QD with PP. This leads lo more pionounced
held induced shift in the case of SWP Energy shifts have also been compiiled lor lew
liigherdying levels ol a spherical dot with PP. The shifts, which have been fountl lo be the
same for all levels, arc due to equal mterlcvel energy .separation.
f'igurc 2 presents the variation of the ground level shift in a QD with parabolic
eonfincmcni as a function of the parameter ft r.e., effectively of the dot si/.c, (or ihree
diffcrcnt electric fields. Here, the energy reference point has been chosen at /i = 0.55
nm, which corrc.sponds to the smallest .spherical dot (with SWP) of I nm radius Fiom the
figure. It can be seen that the energy shifl increa.scs with increase in the value ot i r ,
of the physical dimension of the dot, as expected The more prominent variations arc
observed tor larger held strengths.
To sum up, the effect of the parabolic confinement is seen to reduce the held
induced shifl m the lowest energy stale of a spherical QD. as compared to a dot wilii SWP.
In addition, the parabolic confinement makes the lowering of different levels mscnsiiivc
lo level energies. 7'hc perturbation and the variational methods, employed to estimate the
ground level energy shifl, yield identical results while being com|)Liled lor a spherical QD
with PP.
72A(1) 13
92
C Bose and C K Sarkar
Figure 2. Energy shifts of ihc ground slate electron m a GaAs spherical QD
wKh parabolic confinement as a function of /r‘^2 ^hrec represcntaiivc
electric llckls ol F = 1 x 10^ V/m (solid line). *ix 10^ V/m (dashed line) and
1 ^ 10*’ V/iii (dotted line)
Acknowledgments
The work is financially supported by the University Grants Commission and the Council of
Sca-niific & Inilusirial Research, India. We arc gralcl'ul to Dr. M K Bose lor hciptiil
discussions.
Kdereiices •
111 S Schmill Kink. HAH Miller and D S C’henila F/o-.v Rev B35 8113 (1987)
\2\ DA H Miller, DSrheiiila and S.Schmilt-kink/l/Y>/ Plivs Uii 522154 (1988)
l.l| (i W Wen. J Y Lin and H X Jiang PIm Rev B52 591 1 (1995)
in C Hose huliiin J Phw 71 A (3) 293 ( 1997)
1 5 1 A Kumar. .S E Lau\ and ft Stern Pliy\ Rev B42 5 Iftt) (1990)
161 K Itruiincr. U BiKkelnuinn. C Ab.Mrcilc. M Wallhei. G Bohin. C frankle and Ci Wciinann I'lm Kn
I ell 69 3216 ( 1992 )
17| T (iaim ./ /V;v^ ( V»/u/ A/nkf’/ 8 5725 1 1996)
1 S 1 S Fhigge Prui m ut Qucmliim Met hanu s (Berlin Sprmger-Verlag) ( 1 994)
1 0 1 S Nomui a and T Kobayashi Solid State Commun 73 425 ( 1 990)
|ll)| S Adachi / AppI Phss 58K I (1985)
NOTE
Indian J. Phys. 12\ (I), 93-98 (1998)
UP A
' an inicmational |oumal
Disturbances in a piezo-quartz cantilever under
electrical, mechanical and thermal fields
1 K Munshi*
Departineni of Physics, Kharagpur College, Kharagpur- 72 1 3n.S. Midnapiii.
West Bengal. India
K K Kundu
Department of Physics, City College. Cakulla-7(K) (K)9. India
and
R K Mahalanahis
Departmenr of Malhemuiics. Judavpur University Calculla 7(K) 0.32,
India
Rei eived 6 tmcptcd 12 M ptcmhei /<>d7
Abstract ; An attempt has been made in this papci lo invcsiigaie aiialyiii ally (he
disiiirbanccs produced in a pie/, o quart/ cantilever under the mlluciuv ol Ihicc dilleicnl licMs.
17. . electrical, nicchunical and theirnal The cantilever is ennsuleu’d lo have n 'l.iiii widlh, (he
upper and lower edges of which are (ree tiom load and the shuiring loiccs having eertaiii
resuliani aie distrihuled along ,\ - 0 The expression for the elcLlnc poienlial funs lion is laken
such lhai il IS constanl on i = -ti and iKo make Ihe imeiisity - 0 along Ihc Iciigih Wc lesiini
ourselves lo the.vv plane and componciUs of elastic displacemenis along i and \ dnetiioits h,i\e
been illustrated
Keywords : Mechanical disluibances, pie/.o-elcclricily, cantilevn
PACSNo. : 77 (vS -j
The problem of investigating dislurbances in piczo-cleclric media under dilTcieni inpuis
have been studied in the liieralurc 1 1-4) from Ihc point of view of circuit Iheoiy Several
other researchers 15-*^] have extended the work to find out the disturbances in pie/o-cleclric
media. In most of the works on (he pie/,o-clccTncily, there is a (rend lo cxlcncl the elastic
problems to corresponding piczo-clcctric problems and most of Ihe workers have nii.-lc use
of classical solutions in purely elastic material. A particular area ot pie/o-cicctric problem
IS on bending of pie/.o-elcctric material and the problem ol distui bailees of pic/o-quari/
Address for correspondence ' I, Dr A L Munsbi Lane. UlliUpara, Hnoghly-712 2.‘SK.
West Bengal, India
(D 1998 I ACS
94
T K Munshi K K Kutidu and R K Mahalanabis
LUiiiilcvLM' arc of ulmosl importance to the physicists as well as to the Engineers for various
practical uses in the field of science and technology.
The present work is confined to study the displacement of a cantilever of
pie/,o-clectnc material under electrical, mechanical and thermal fields. The cantilever, in
the present investigation, is considered to have certain width, the upper and lower edges of
which arc free from load [ I Of— the shearing forces having certain resultant arc distributed
along X = t), We restrict ourselves to the .n‘ plane and finally the components ol clastic
displacements along .v and v directions have been illustrated. It is found that the
displacements arc partly linear, hyperbolic and exponential in nature.
As pic/o-clcctncity is essentially the interaction of the electric and clastic fields in a
crystal, wc must therefore, define the electrical as well as the elastic state of the crystal and
specify Its electrical state by two variables — the electric field E and the electric
displacement /) and we specily ns elastic state by two elastic variables — the stress T
and the strain S Rcfeired to a rectangular system of axes A'. K Z, the components of the
electric field and the electric displacement arc supposed to be /), (/ = 1-3). Denoting
the elastic stress and strain components by 7,, S, {i = 1-6) respectively, the stress equations
ol miction are
<n.
cVii
f)\
fh
-f
~()Z
^ dr '
iPv
-f-
+
— n
()\
' <)r '
,n\
d-\v
~ £) —
f):
^ dr
ulieie p IS the mass jici unil volume and n. v. w arc the components of elastic displacement
In free space, the cleeiric displaceiiienl D, .satisfies the Gauss's divergence equation
r)ll ()Jh do,
div /) = -T-- + — ^ — = 0 (2)
rh rJy (k
riie linear pie/o-elecli ic constitutive relations between 7'. S, E and D which describe the
mterridation among the electrical and clastic variables for pic/.o-cicctric materials are
\ ^=1.2.14.5,6. (3)
/ 1
7=1. 2, .3. (4)
I -I
where the consiani .Vj ( - ) is the elastic compliances at eonsiani electric field stiength E,
/,, - (Ij, IS the pie/o-eleclnc strain constant while ) is the dielectric permillivii) at
constant stress.
Disturbances in a piezo-quartz cantilever etc
95
Besides these, we have the relations connecting the strain components and
displacement components given by
S|
S,
Bu dv
9v du
dy dz ' ^ Sz ^
dv du
dx dy
(5)
( 6 )
Jn deriving the plane equations, we restrict ourselves to the xy plane and we represent the
stress components by
d^tj) d^0 d^ip
= 0 .
(7)
where (pis the stress function. The components of electric field are represented by
dv dv
^2 = ^3 = 0,
( 8 )
where V is the electric potential function.
For plane problems, we assume the piezoelectric relations to be
5| = + K(hT 2 + r/,,£, + p[6,
.S, = + sfj. - f pfO,
~ - 2//, I /111
- A) -f -h pfO,
--2^/,,T’6 £. + /?? fcl,
where are the ihcrmo’claslic compliances at conslani electric field strcngih /:, p'^'s
arc the thermo-pie/o-electric moduli at constant stress and <9 is the input temperature. From
CL|s. (7) -(9), we gel
d^V d^V d^^ ' d^(p d^(p '
dx^ dy^ Cl, ^ dy^dx ^
( d9 do
( 10 )
Here we choose 0 1 1 1 ), tnc temperature input as a linear function of x and y as
0 =
where a and jSare arbitrary constants. Eq. (10) becomes
96
T K MunshL K K Kundu and R K Mahalanabis
For the problem of cantilever of width 2c, the upper and lower edges of which are free from
load ?nd in which the shearing forces having a resultant P, are distributed along the jc = 0,
we assume for 0 [7,8,10) as
0 = Axy^ + Bx}\
where the constants and Pare to be determined from the mechanical boundary conditions
refciTcd to above, viz.,
+r
= 0 and -jT^dy = P.
Applying boundary conditions we get,
0 = - P I Ac^ixy^ ~?>c^xy) (12)
A I'orm of V for which the electrical boundary condition can be reasonably satisfied [8], is
given by
V" = VnyO'^-c-). (1.^)
The expression for V leads to the condition that the potential is constant on y = ± c and also
makes = 0 along the length. The constant is determined from the cqs. (11) and (1 2) as
^■0 = —(■-/’/4c'') + (14)
C], 6y^ '
The expression for V becomes
V =
— (-P/4rM + —(p2Pe~^^-^^p^ae'^’^^) v(v^-c^).
Cl, 6y'
(1-M
We get from cqs. (7) and ( 1 2)
3P
T^ = d^0ldy^ - - 7*2 0 ,
7ft = - d-0ld\dx = - — (16)
4c'
Finally, to calculate the displacement components (m,v), wc start with the eqs. (9) and (5),
namely.
clu
3P
iVv + ^e-
(17.1)
flv
3P
tnxy + +c'
(17.2)
dv
du
r 3P , ,
2d^(-p/4c^)(iy^
Tx
- 2(.v/,
- .V ) -
4(..i
-c-)/e,t
- Kpe ^Hy'^ -c^) + +Le-^)2}\
where K (= O^^Pj^p I 3) and 7 (= or / 3) are constants.
(17.3)
Disturbances in a piezo-quartz cantilever etc
Integrating eqs. (17.1) and (17.2) wc get,
3P
“ " ■ 27 **''''^' / a + e-fi'x) + f(y),
3P
- - p-W + //)) + f(x).
where /JT) and/i» are functions of jr and v respectively. Differentiating eq. (18 1) with
respect toy and eq. (18.2) with respect to. and summing and using eq, (17 3) we get
e-fiy
- c / ’; t! t.a
2c^
97
(18.1)
(18.2)
du
d/(y) 3P
P d\
vt y 2
2c^ • '2'
■ P2^\jy +
a
dfix)
clx
2(-'i1-4)p(H-.v^) - 2d2(-/>/4r’)(3y^-r3)/e,|
- Kfie-^>{y^ -c' ) + (Ke-f' y- U «' )2v,
which can be written as
dfix) 3P
4f(y) 3P p , d}, p
dy 2 c^ 2.y = 6^.
where m and ^ arc functions of jc and v respectively.
Integrating again, we gel
P
^f^dx + h.
p
2 ?
( d- ]
+ + ledy + g,
V ^11 ; .ft
Mx'n ' Tm it ^"‘1 Av) into the eqs.
. ), we gel the components of elastic displacement u and v as
u =
V =
3P
f ^ ^-ax
+ e P' x I +
2c-
^2 A
:j -'ii
,/r ^ "11
■Vll +
^11 )
3P ^ P
p''t 2 V + p'tV’ + pfOole'^y-e^y/P) + imdK + h. (20)
rcspectivll^** components of elastic displacement along . and y direction
-<fnV2l8c-^'(>-+e^)/3 + ledy + g,
(19)
98
T K Munshi, K K Kundu and R K Mahalanabis
The variation of the component of the elastic displacements along x and v directions,
have been shown in cqs. (19) and (20) respectively. As the term e is a function of y, so the
expression jedy in eq. (19) contains y-term only. We have investigated the variation of the
component of the clastic displacement u along x direction, keeping y as constant, so the
term \edy in eq. (19) will behave as a constant part. Similarly, the term ni is a function of x,
so the expression Imdx in eq. (20) contains only jc-term and it will give a constant
contribution in the variation of the component of the elastic displacement v along y
direction, keeping the value of .Ji' as constant. It is found from eqs. (19) and (20) that both
the disturbances consist ot sonic linear, hyperbolic, exponential and constant part with
diHercnl coefficients
Ktfercnci'i
|l| W t’ Mason Pip:<) pleitiu ond Thetr Applicatumx m their Ultni^onu^ (New York D. Van
Nosliand)pK4{19M))
(J.1 F I Haskins and LJ Walsh./ Annisi Stn Am 29 729(1951)
(31 M C Me guairie and W l< J Hussern J Am Ceram Sot 34 402 (1955) \
|41 M J KecIwooJ / /If .nn-f So< Am 33 527(1%!)
(51 W (j Cady Piero-clertru iiy (New York McGraw-Hill) (1959)
(hi R 1) Mindlin On I he hiualians of Motion of Pieio-ekctnt Crystals ■ Problems of Continuum Methann \
(SIAM, Philadelphia Pensylvaiiia) p 282 (1961)
1 7) ( i P.aia ./ .S( / Res 4 381(1 960)
1 8 1 I ) :< Sinha Indian ./. Theo Ph\s 9 I ( I % I )
(9) C K Hniska J Appi PIiys 72 2432 (1992)
I lOJ K R Gin Prm Nat Inst Si i 33 546 (1968)
1 1 1 1 IK Miirishi. K K Kundu and R K Mahalanabis 7 Aioust Soc Am. 96 2836 (1994)
EOStHCOMING PUBUCATIONS (B)
FEBRVABY I99S. Vol. 72. No. I
Astrophysics, Atmospheric &. Space Physics
Nighttime ionospheric electron content enhancements and
associated amplitude scintillation at Lunping
SuDHTR Jain, S D Mishra and S K Vuay
Atomic & Molecular Physics
Studies on positron-hydrogen ionization cross sections
A Bandyopadhyay, P K Dutta, K Roy, P Mandal and
NCSil
General Physics
II. Thermodynamics of Fourier-likc radiative conduction heat
currents and equilibrium temperature gradients
Christopher G Jesudason
Degeneracy of Schrddinger equation with potential l/r in
^/-dimensions
M A Jafarizadeh, SKA Seyed-Yagoobi and H Goodar/i
The hydromagnelic convective How through a vertical channel
A Marcu, M Vasiu and C Beaga
Radiation conductance and directive gain of a ferrite based
microstrip phased array antenna at X-band
Birendra Singh
Dynamical capillary instability of a compressible streaming fluid
cylinder under general varying tenuous magnetic fields
Ahmed E Radwan
Optics & Spectroscopy
Vibrational spectra of a novel selenite Cdi(HSeO02(SeO3)2
A Bindu Gopinath and S Devanarayanan
Notes
Kelvin-Helmholtz instability in the polar cusp region of the
magnetosphere
3 P Mishra and R Dwivedi
on next
72A(1)-14
The calculation of potential curve of A state of ^LiH from
experimental Data
Rehm Ai -Tuwwoi, A Bakry, M Rafi and Fayyazuddin
S-stalc energy levels and wave functions of the Hellmann potential
using self consistent zerolh order approximation
SaRMISIHA MfSRA AND BlSHNUPRfYA BhUYAN
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[5J U Fann and ARP Rao Atomic Colli.^ion.s and Spectra (New York : .Academic) Vol 1, Ch 2. Sec 4,
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17| T Atsuini, T Isihara, M Koyama and M Matsuzawa Phys Rev. A42 6391 (1990)
[111 T Le-Bnin, M Lavolle6 and P Morin X-ray and Inner Shell Proce.s.ses (AIP Conf Proc. 215)
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Indian Journal of Physics A
Vol. 72A, No. 2
March 1998
CONTENTS
Condensed Matter Physks
Opliciil and structural characterisation ofZnO films prepared by Ihe
oxidation of Zn films
Bf.nny .losRPM, K G Gopchandran, P K Manoj, J T Abraham,
Pi n,k Koshy and V K Vaidyan
Oplicdl properlies of Pr^"^ doped glasses, effect of host lattice
Brajlsh Sharma, Akshaya Kumar and S B Rai
Mechanism of grain growth in aluminium, cadmium, lead and silicon
I Jr.su Rlihinam, S Kalainathan and CThirupathi
Aiuilysis t)f temperature dependence of inlcrionic separation and
bulk modulus for alkali halides
Raiiv Kumar Pandi y
1 .valuation of the trapping parameters of XL peaks of multi activated
SrS phosphors
W SlIAMUHUNATM SiNCiH, S JoYCHANDRA SiNGH, N C Dl'.B,
MANARhsii Bhaftacharya, vS Dorlndrajit Singh and
P S Mazumdar
Nuclear Physics
l^quihhnum forms of two uniformly charged drops
S A Sarry, S a Shalary and A M ADOEL-HAn^s
Measurements of flux and dose distributions of neutrons in graphite
matnccs using LR-1 15 nuclear track detector
Y S Seum, a F Hafez and M M Audll-Meguid
Particle Physics
J^upcrsymnietrized Schrbdingcr equation for Fermion-Dyon system
B S Rajput and V P Pandey
Pages
99-105
107-116
17-124
125-131
133-139
141-153
155-160
161-169
Indian J. Phys. 72A (2), 99-105 (1998)
UP A
— an intemariona l journal
Optical and structural characterisation of ZnO films
prepared by the oxidation of Zn films
Benny Joseph, K G Gopchandran, P K Manoj, J T Abraham,
Peter Koshy* and V K Vaidyan
[3epariment of Physics, University of Kerala, Kariavaitom,
Trivandruni-695 581, India
^Regional Research laboratory, Pappanamcode, Tnvandrum-b95 019.
India
Received 4 September 1997. accepted 13 January 199ft
Alislrart : Zinc oxide films have been prepared by the post-deposition heat ireuiinent of
iinc films X-ray diffraction studies have confirmed the prefeieniial orientation of the films
along (002) plane The texture coefficient has been calculated to explain the preferential
orientation SEM studies have revealed u faceted elongated microstructure Prom the
transmission s[iectia optical bandgap has been determined as 3.3 eV
Keywords : Zinc oxide, annealing, bandgap
PACS Nos. : 8 IJ 5Gh, 8 1 40Gh
1 . Introduction
ZnO ts a mullifunciional material with a wide area of applications. ZnO films have attracted
considerable attention because they can be made to have high electrical conductivity, high
infrared reflectance and high visible transmittance. The constiluenl elements of the film are
abundantly available at low cost and are nonloxic. Aktaruzzaman etcxl 1 1) and Minami el al
(2| have reported that ZnO films are known to be more resistant to the reduction by
hydrogen containing plasma than the conventional transparent conductors tin oxide and
indium oxide. Zinc oxide is an n-iypc semiconductor with hexagonal wurizite structure and
a bandgap of 3.3 eV [3]. Pure zinc oxide is transparent in the visible region and it has low
conductivity and low infrared reflectance. The nonstoichiometric zinc oxide films are the
simplest, most economical to prepare and their electrical and optical properties are also
excellent.
© 1998 lACS
100
Benny Joseph et al
Zinc oxide films have potential applications in energy efficient windows, solar cells,
liquid crystal displays, optoelectronic devices, gas sensors, piezoelectric devices, etc. Zinc
oxide films have been deposited by different methods such as evaporation [4], spray
fiyrolysis [5], chemical vapour deposition [6], magnetron sputtering [7] and laser ablation
technique [8). Only few reports are available on the formation of zinc oxide films by
evaporation technique. In this paper, we report the deposition of zinc oxide films on glass
substrates by post-deposition heat treatment of zinc films. The structural properties of the
films and their morphological and optical properties are investigated.
2. Experimental
Thin films of zinc were prepared on glass substrates at room temperature by resistive
heating of metallic zinc under a vacuum of ~10"^ mbar. It is necessary to use a zinc source
that is almost enclosed since the emission rate has to be controlled [9]. The.se films were
subjected to post-deposition heat treatment above the melting point of zinc (693 K). As-
deposited films annealed at 723. 773, 823 and 873 K for 30 min were cooled slowly at a
rale of 9 K/min to room temperature. The X-ray diffraction studies were conducted \on a
Philips PW 1701 powder crystallography instrument using CuK„ radiation. The surlfacc
morphology of the films were evaluated by using JEOL 35C scanning electron microscope.
The optical transmission studies were performed using a Shimadzu double beam
spectrophotometer LTV 240 in the range 300-900 nm.
3. Results and discussion
The X-ray diffraction peaks of the films are readily indentiliable and their position
concide with reflections reported in the ASTM diffraction pattern file for po\*fder ZnO [10].
Figure 1 dcpicits the X-ray diffraction patterns for films annealed at various temperatures.
The presence of many peaks indicate the polycrystalline structure of the films. The
strongest diffraction peak al all annealing temperatures is along (002) cry.stal plane
with 29 = 34.4°. This pronounced peak indicate that the preferred orientation of the
microcrystals of the film is along the c-axis normal to the substrate surface. The preferred
orientation of the microcrystals was found in the case of films prepared using other
methods [1 1,12]. The other peaks observed with less intensity are (100) and (101) with
29 values about 31.7 and 36.3 degree respectively. The XRD data of the film annealed
at 723 K reveals that the films contain slight amount of metallic zinc. This indicates
that the oxidation , is incomplete al 723 K. In all the other diffractograms the phases
identified are those of zinc oxide. Each grain in a polycrystalline film normally has a
crystallographic orientation different from that of its neighbours. Considered as a whole,
the orientations of all grains may be randomly distributed in relation to some selected
frame of reference or they may tend to cluster, to a greater or lesser degree, about a
particular orientation or a few orientations. Any polycrystalline material characterised
by the above condition is said to have a preferred orientation or texture. When cold
worked material pocessed of deformation texture is crystallised by annealing, the new
grain structure usually has a preferred orientation different from that of cold wjorked
optical and structural characterisation of ZnO films etc
101
material. This is called annealing texture [13]. In the present investigation the films
exhibited a preferred orientation along (002) diffraction plane. To describe this orientation.
1
LJ
1 (d)
!
i
s
A J
V A
1
9 1
. (b) 1
|l A
(ZOO)
(101)
8
1 1 ^ L
40 36 36 34 32 30
20 (degree )
Figure 1. X ray diffraciograins of zme films
prepared at different annealing temperatures ■
(a) 723, (b) 773, (c) 823 and (d) 873 K
kMLirc cocfficicnl TC {hkt) is calculated for planes (002), (100) and (101) using the
expression f I4j
JC{hkl) =
l{hkl)ll^{hkl)
/(,(**/)
(I)
'xhcrc I is the measured intensity, Iq is the ASTM standard intensity of the corresponding
powder sample and N the reflection numbers. From the defenition, it is clear that the
deviation of the texture coefficient from unity implies the preferred orientation of the
^irowih. Figure 2 shows the variation of texture coefficient with annealing temperature for
dll fraction peaks along the planes (002). (100) and (101). The texture coefficient along the
( 002 ) crystal plane (TC (002)) increases with substrate temperature and is found to be
maximum for films annealed at 873 K. The preferred orientation along (002) is associated
with the increased number of crystallites along that plane. The values of TC (1(X)) and TC
72A(2)-2
102
Benny Joseph et al
(101 ) are gradually decreasing with increase of annealing temperature. A marked seperalion
of the high angle in the X-ray diffraction peaks of the Ka^ and KO/i line is also observed.
The crysialliie .size D along c-axis can be estimated by [ 13] :
0.9 A
Bcos9'
( 2 )
where A is the X-ray wavelength, 0, the Bragg diffraction angle and B, the full width at half
maximum (FWHM) for the film.s prepared. The lattice parameters are calculated from the
7^5 T73 823 873
TEMPERATURE (K)
Figure 2. Variation of texture coefficients PC (002). ’PC' { l(K)) and 'PC ( 101 ) ,
as a function of annealing Icinpcinture
X-ray diffractogram of the films and the values obtained arc consistent with the values
given in the ASTM data and are listed in Table 1.
Tabic 1. Microstructural parameters a.ssociated with ZnO films.
Annealing
Orain.s siz^ (nm)
Lattice puramclcis (iim)
temperature (K)
«()
'0
723
50
0 3254
0 5210
773
66
0.3253
0 5206
823
36
0 3255
- 0 5208
873
66
0 3262
05212
ASTM
0.3249
0.5205
The strongly textured thin films, presenting intense diffraction peaks with small
width at half maximum, have high resistivities [15]. Our measurements quantitatively
support this fact and the samples were of high electrical resistivity.
Figure 3. Scanning eleciran micrographs of zinc films annealed oi different tempeialures
(a) 723, (b)773, (c)823 and (d)873K.
Optical and structural characterisation ofZnO films etc
103
The scanning electron micrograph of these films were taken to evaluate their surface
morphology. Figure 3 depicits the scanning electron micrograph of the filips annealed at
723 [Figure 3(a)], 773 [Figure 3(b)l, 823 [Figure 3(c)] and 873 K [Figure 3(d)]. The
structural studies have revealed the strong intensity of the (002) peak, indicating that the
grains have c-axis perpendicular to the substrate surface. The micrographs indicate a
textured morphology with network like structures at all annealing temperatures similar to
other investigators [16,17], The surface morphology reveals a faceted elongated
microstructure. The network like structure seen on the micrograph deteriorate with increase
of annealing temperature.
The optical transmission spectra of the films were studied in the wavelength region
300-900 nm. The intrinsic absorption in a semiconductor occurs for wavelengths in the
vicinity of the energy gap. The transmission spectrum of the film of thickness -150 nm
annealed at 873 K is shown in Figure 4. The absorption coefficient a was calculated
following the Lamherst's law and was calculated as ;
a =
2.303 X A
t
(3)
where A is the optical density which was taken directly from the transmission spectrum
and f, the film thickness. Figure 5 depicits the typical variation of absorption coefficient a
with photon energy. The absorption has its minimum value at its low energy and increases
Figure 4. Transmission speclnim of zinc oxide Films prepared by the posl-
deposition annealing of zinc Films at 873 K.
with optica] energy in a similar manner to the absorption edge of the semiconductor. It
can be seen that this film shows a high absorption (a -10^ cm'*). The increase of
transmittance with wavelength in the transmission spectrum (Figure 4) may be due to the
existence of large number of levels in the forbidden gap just below the conduction or just
1 04 Benny Joseph et al
above ihe valence band. Because of the large absorption before the absorption edge,
interference fringes are absent and consequently the refractive index of the film could
not be determined.
PHOTON ENERGY (W)
Figure 5. Absorption coefficient (a) versus photon energy (hv) of
zinc oxide film.*: prepared by the post-deposition annealing of zinc
films at 873 K.
Assuming that the transition probability becomes constant near the absorption edge,
the absorption coefficient a for directly allowed transition for simple parabolic band
scheme can be described as a function of incident photon energy hv, as [ 18] ;
a cc (hv-E^yi\ (4)
where is the optical bandgap. The extrapolation of the linear portion of the graph v.v
hvio the hv axis gives the value of the band gap and is found to be 3.3 eV for the sample
annealed at 873 K in good agreement with the reported values [19,20].
4. Conclusion
Polycrystalline zinc oxide thin films are prepared by the posL-deposition annealing of
evaporated zinc films in the range 723-873 K. The grains of the films have preferred
orientation along (002) plane and the texture coefficient increases with annealing
temperature, The lattice parameters calculated are consistent with the ASTM data. Surface
morphology analysis has revealed a faceted elongated microstructure. The optical
absorption shows that the fundamental absorption starts at 3.3 eV and the transition leading
to this is a directly allowed one.
optical and structural characterisation ofZnO films etc
105
References
f 1] A F Aktaruzzaman. G L Sharma and L K Malhotra Thm Solid Films 198 67 (1991)
[2] T Minami, H Nanto and S Takata AppL Phys. Lett. 41 58 ( 1982)
[3] J Hu and R G Gordon / Elecirochem. Soc. 139 2014 (1992)
[4] H Walnabc Jpn. J. AppL Phys. 9 418 (1970)
[5] J Amovich, A Ortiz and R H Bubc J. Vac. Set. Technol. 16 994 (1979)
|6] A P Roth and D F Williams J. AppL Phys. 52 4260 ( 1981 )
[7] Y Igasaki and H Saito J. AppL Phys 69 2190 (1991)
[8] H Sankur and J T Cheung J. Vac. Set. Technol A1 1806 (1983)
[9] L Holland Vacuum Deposition of Thin Films (London ' Chapman & Hall) p 1 80 ( 1 970)
[10] Powder Diffraction File Data card no. 5-664 JCPDS (Internationl Centre for Diffraction data,
Swartmore, PA)
1 1 1 1 D Cossemenl and J M Streydio J Cryst Growth 72 57 (1985)
[12] YEUe,JBLee,YJKim.HKYang.JCParkandYJKim/ Vac.Sci Techoi AU 1943(1996)
[13] B D Cullity Elements of X-ray Diffraction (Reading, M A : Addison Wesley) p 284, 295 ( 1 956)
1 1 4] C Barret and T B Massalski Structure of Metals (Oxford : Pergamon) p 204 ( 1 980)
[ 1 5] Li-jian Meng, M Andritschky and M P Dos Santos Vaccum 44 109 (1993)
1 1 6] Yoshino, W W Weans, A Yamada, M Konagai and K Takahashi Jpn. J. Appl Phys 32 726 (1993)
[17] W W Weans, M Yoshino, K Tabuchi, A Yamada. M Konagai and K Takahashi Proceedings of 22nd
lEEEPVSC 935(1993)
[18] A Abeles Optical Properties of Solids (Amsterdam . North Holland) p 32 ( 1 992)
[19] A P Roth and D F Williams J Appl. Phys. 52 6686 (1981)
[20] Chns Ebrspacher, A L Fahrenbruch and R Bube Thin Solid Films 136 1 (1986)
Indian J. Phys. 72A(2), 107-116 (1998)
UP A
- an intemaiional joumaJ
Optical properties of Pr^*^ doped glasses, effect of
host lattice
Brajesh Sharma, Akshaya Kumar and S B Rai
Laser and Spectroscopy Laboratory. Department of Physics.
Banaras Hindu University, Varanasi-221 005, India
Received 25 November 1997. accepted 23 December 1997
Abstract : Glass base effect on optical absorption and luminescence properties of Pr^'^
doped glasses have been studied in different hosts using modified J-0 model proposed by
Kornienko et al Using modified J-0 intensity parameters ) radiative properties (A, A;' and
/J%) and life times (r,) of the emitting levels Vq. ^P] and '/)2 have been calculated and
compared for different lattices. Theoretical results for oscillator strength obtained using this
model show better agreement with experimental values.
Keywords : Pi^* doped glasses, optical properties, effect of host lattice
PACS Nos. : 78.66 Jg, 78.60 Ya
1 . Introduction
The observed electric dipole transitions of the electronic configuration 4/ ^ of
occur mainly from the excited states -^^ 0 , 1,2 ^^d ‘ 1>2 to the ground and low lying excited
states [1|. Compared to the optical 4/" transitions of other rare earths ions, the electric
dipole transitions ^Po. 1.2 — > ^^4 of exhibit short decay times of about few |j-s. In
several technical applications, .such as Scintillators, fast luminescence is required. So
Pr^"^ is best suited as a dopant in X-ray conversion detectors for modern X-ray
computed tomography [2]. Transitions from the ^Po,i ,2 levels are also used in phosphers.
The other predominant transition of Pr^ has a much larger decay time and
laser action has been observed for this transition in PrCl^ [3] and PrP 50 i 4 [4] etc. Smart
et al [5] have reported that Pr^*^ doped fluoride fibers exhibit lasing action in orange and
red regions of spectrum when pumped with a Ti-Sapphire laser at 1010 and 835 nm’s.
Upconversion has also been demonstrated for Pr^'*' ion in borate, fluoride and phosphate
glasses [6], The fluorescence quenching and decay of Pr^ ion in different glass hosts have
also been studied [7],
© 1998 lACS
108
Brajesh Sharma, Akshaya Kumar and S B Rai
The Judd-Ofelt approximation has been successfully applied to most of the doped
rare earth ions to explain their optical properties. However in the case of Pr^ it is marked
that there is a poor agreement between calculated and experimental oscillator strengths [8,9]
and in some cases negative values are also obtained for Qj parameter [10]. This is due to
reason that some of the assumptions made in this approximation, in particular that the
energy difference between excited configurations and each of the two levels involved in the
electronic transition is the same, is probably not valid in the case of Pr^*^ because 5d levels
of it are at lower energies than in the other 4/ions. To improve upon this, some changes are
needed in Judd-Ofelt formulae.
In the present work we report the absorption and fluorescence properties of Pr^'*^ in
phosphate, tellurite, oxyfluoride and Zr based heavy metal fluoride (HMF) glasses where
the results have been analysed using modified J-0 theory [11a, 11b]. A good agreement in
the calculated and observed oscillator strength have been observed. The optical parameters
for different transitions in different hosts are also compared. ’
2. Experimental
The composition of glasses are expressed by
PBK; 68P205.22Ba0.8K20.2Pr2p3
Te.NaO; 72Te0.26Na20.2Pr20,
ABCP; 33 AIF 3 . 1 1 AlPO4.30CaF2.24BaF2.2PrFi
ZrBAN; 55ZrF4. 19BaF2.5AlfS.21NaF.2PrF3
Glasses arc prepared using standard quenching technique. We weighed the glass and
compared its weight with the weight of the mixture used. A very little change in the mass
was found. This indicates that the composition of the glass is probably the same. The
experimental details related to density and refractive index measurements are given in our
earlier papers [12,13]. The stokes luminescence were obtained pumping Pr^”^ glasses with a
coherent innova 400 Ai^ ion laser is 476.5 nm). The dispersed spectra were obtained
using 0.5 m Spex monochromator. The absorption spectra were recorded using Perkin
Elmcr-551 and Carl Zeiss Spccard spectrophotometers.
3. Results and discussion
3.J. Absorption studies :
The absorption spectra of Pr^"*^ doped PBK, Te.NaO, ABCP, ZrBAN glasses at room
temperature arc shown in Figure 1 . The spectra show nine absorption bands for the three
glasses while in the case of tellurite glass only eight peaks are observed irrespective of the
fact that 13 levels arise due to 4/^ configuration of Pf**^. These bands arise due to electronic
transition from the ground state manifold to various excited levels ^Po.i. 2 » '^ 4 * ^^ 4 , 3,2
and and their positions and relative intensities changes with glass base arc also reported
by other workers ([14] and references therein). The wavelength of different absorption
Optical properties of Pr^* doped glasses, effect of host lattice 1 09
peaks in different lattices are tabulated in Table 1. The transition at 440 nm
is hypersensitive and as can be seen from Figure 1 , it has maximum Intensity in the case of
Figure 1. Absorption spectrum of doped Zr based heavy metal
fluonde (Z), tellurite (T). oxyfluonde (O), phosphate (P) glasses.
lellurite glass. The intensity of this* peak in other glasses follows the order Te.NaO > Zr
BAN > ABCP > PBK. The absorption peaks corresponding to the levels ^P|, ^Pq in the
lellurite glass have nearly same optical intensity but in other glasses do not show any
regularity. The energy of the ^Pq is nearly the same for the PBK, TeNaO and ABCP glasses
Table 1. Assignment of the peaks observed in visible, NIR absorption spectra of Pr’”*" in
difTerent glasses (wavelengths in nm)
Energy
level
Free ion
levels of
Pr3+
Pr*'*’ ion in
LaCl 3 crystal
[25J
ZrF 2 -CdF 2
glass [ 21 ]
PBK
Te.NaO
ABCP
ZrBAN
432
447
443
435
455
450
445
454
475
468
471
470
473
470
-Vo
468
488
480
485
485
485
490
'D 2
577
598
588
588
585
585
589
'C 4
1008
1020
1016
1012
1012
1011
V 4
1459
1475 •
-
1465
1460
1471
1466
V'3
1559
1.579
1570
1560
1567
1560
15-58
V 2
1930
1932
1944
1935
1932
1929
1931
^6
2401
2426
-
2488
2.500
2500
2466
hut shifts in lower energy side by almost 150 cm"* in the HMF glass due to ncphlauxetic
clfect (see ref. [15]). A comparison of the NIR spectra shows that the band involving 'C /4
level of the Pr^ appears broader in PBK glass in comparison to its width in ABCP glass but
sharper in the case of HMF glass. The line corresponding to this level is totally absent in
lellurite glass. A similar feature has been marked for and levels also. These
72 All. 2
110
Brajesh Sharma, Akshaya Kumar an4 S B Rai
observations suggest that the intensity, position and band width of the optical transitions are
host dependent. The optical density is increased when it is doped in HMF glass.
The oscillator strength if) corresponding to different transitions were obtained by
integrating the intensities of absorption bands. The absorption peaks are supposed to be
pure Gaussian shape. The oscillator sU'ength is given as [16],
= {mc^ ! ne^ N) ^ a{X)dk j (1)
which in terms of energy reduces to
/„ = 9.20x10-’£^.4v',/,, (2)
where is maximum value of molar extinction coefficient e. Av \/2 is the half width of
the line at /niax(''V 2 .
In order to get the theoretical value of oscillator strength, one calculates the Judd-
Ofelt intensity parameters. The Judd-Ofelt theory is applicable to only those cases \^^he^c
the /levels splitting are smaller compared to f-d energy gap [16]. In Pr^"^ however, \his
situation is different. As a result of this matrix elements llfy'*ll and WU^W for the and V 4
transitions lying in the NIR region are found to be quite large. This gives very large value
for 1^4 and and a small value for This predicts very large intensity for the bands near
1560 nm due to the levels. On the other hand when data for these bands are included
in the fit a negative value for 02 obtained in all the cases except the HMF glass. Carnall
et al [17,18] and Knipke [19] have also marked a negative value for Oi for in LaF 3
lattice. Kornienko et al [lla] and Goldner and Auzel [lib] proposed a modification in
Judd-Ofelt theory and called it as modified Judd-Ofell model. •
3. 2. Modified Judd-Ofelt model :
In the Judd-Ofelt theory, the electric dipole line strength (5ed) is given by
where f 22 , and are three Judd-Ofelt parameters. The values of these intensity
parameters depend on impurity ions and the host lattice. \\UH\ is called the reduced matrix
element and its value is almost insensitive to lattice environment [20|. The reduced matrix
elements calculated by Weber [16] for Pr-^"*" in LaF 3 have therefore been used to derive
the optical properties of Pr^"^ in other lattices. This assumption is however not valid in all
situations. Kornienko er u/ [lla] introduced a new formula to describe the experimental
data which takes into account the dependence of Judd-Ofelt parameters on the energy level
manifolds. The modified form of electric dipole line strength equation is
I ^2; [1 + 2a(£,,, -2£» )]
( 4 )
Optical properties ofPr^* doped glasses, effect of host lattice
111
are the modified Judd-Ofelt parameters (A = 2,4.6). a is another parameter
whose value in the case of Pr^ is [1/2 (£4/5rf-E4^1. Its value is found to be 10”^ cm“* [21].
and Eipj’ are the energies of the levels corresponding to th6 wave functions ¥0
and and is the energy of the center of gravity of configuration of Pi^ [for Pr^
this value is given to be 10002 cm’‘ by Camall etal (18)]. The Judd-Ofelt parameters 12^
thus obtained are given in Table 2. The O' values obtained here is positive in all the
three cases.
Table 2. Judd<Ofelt intensity paranieter (XJ;i x 10 ^ cm^) of in PBK,
Te.NaO, ABCP and Zr.BAN glasses.
Glasses
PBK
0.24
5.91
3.62
1.69
TeNaO
2.90
6.72
1.85
3.63
ABCP
1.15
6.36
2.82
2.25
ZrBAN
0.28
5.72
1.73
3.30
35Zn0.65Tc02
[231*
2.59
7.26
5.45
-
Li20,2B20g
[231*
0.77
3.84
3.58
-
ZrBAN
[24]*
0.84
4.79
9.13
-
ZrBA
[261*
0.06
5.05
6.92
-
Chlorophosphate
[27]*
4.38
1.86
4.15
-
*Thesc values are Q2' ^^4
Table 3. Measured (J^) and calculated %) oscillator strength (x IC^) of Pr^'*' in
phosphate, tellurite, oxyfluoride and fluoride glasses.
Transition
from
PBK
TcNaO
ABCP
Zr. BAN
fa
fb
fa
fb
fa
fb
fa
fb
500
4.88
7.21
6.22
5.12
6.99
4.21
3.30
1.58
1.29
2.10
2.85
1.92
0.58
2.00
1.15
■’^0
2.00
1.86
3.52
2.80
2.81
2.92
4.86
3.59
'D 2
2.31
1.59
1.22
0.86
2.18
1.95
2.11
1.84
'04
0.12
0.32
-
-
0.32
1.39
0.33
0.41
■V 4
1.02
1.33
1.21
2.93
1.63
2.61
2.31
2.40
5.32
5.18
6.92
6.32
4.32
5.21
3.52
3.81
^2
1.93
1.84
2.54
3.81
0.92
1.00
1.36
1.38
0.25
0.50
0.31
0.33
0.31
0.36
0.52
0.54
R.m.s. X 10®
1.06
2.12
3.12
1.52
The oscillator strengths for different absorption bands in the four glasses were
calculated using the modified value of They are compared with the experimenUl
values also in Table 3. One can see from Table 3 that oscillator strength are in general,
112
Brajesh Sharma, Akshaya Kumar and S B Rat
lower in HMF glass than the other three. This is possibly due to fact that the crystal field
affecting the Pr^ ions is smaller in fluoride glasses than in the phosphate glasses [18].
3.3. Stokes luminescence ;
As mentioned earlier fluorescence measurements have also been carried out at room
temperature using 476.5 nm radiation of Ar* laser (power 700 mW). This wavelength of
Ar+ excites the ^P\ level of Pi^ since its energy (20981 cm“’) is close to the energy of ^Pi
level (21118 cm'') [see Figure 2]. Therefore the fluorescence from to the ground state
could not be observed as this line is overlapped with the exciting line. The fluorescence
spectra of Pr^ in the four glasses are shown in Figure 3. In total nine peaks could be
observed. These lines arise due to the excited ^Pq and levels to the ground state or
low lying excited states [see Figures 2 and 3]. The wavelengths of the bands and their
assignments, relative intensities in the four glasses are listed in Table 4. The band with
maximum intensity in the emission spectrum is at 610 nm which corresponds to the
transitions 'Z >2 ^/f 4 and ^P] The other bands are found to lie at 526, 545, 642,
681, 704 and 725 nm*s. The intensities of the peaks are however found to vary from one
glass to the other so much so that in some cases some of the peaks are completely absent.
Optical properties ofPr^^ doped glasses, effect of host lattice
113
For example, ^Pq transition in the tellurite glass is completely absent but it appears
in the other glasses. Similarly in oxyfluoride glass the transition ^Pi does not appear.
Figure 3. Fluorescence spectnini of
doped Zr based heavy metal fluoride (Z).
tellurite (T), oxyfluoride (O). phosphate (P)
The bandwidths of the observed fluorescence lines also differ from glass to glass.
The transition -¥ appears sharper in heavy metal fluoride glass and its fluorescence
Table 4. Fluorescence line assignments, peak frequency (cm~'), effective bandwidths (cm~')
and relative intensities of Pr^'*’ doped in tellurite, phosphate, oxyfluoride and fluoride glasses.
Transition
PBK
TeNaO
ABCP
ZrBAN
V
Av
I
V
Av
1
V
Av
I
V
Av
1
V,
%
19047
360
36
19011
310
42
18975
330
45
18975
250
47
18858
130
17
-
-
-
18484
90
22
18484
92
19
%
17986
97
10
-
-
-
17986
90
12
17986
90
13
’Po
%
16666
350
69
16583
300
83
16722
310
80
16639
285
83
'Dj
16420
341
80
16367
290
86
16393
3.55
82
16367
270
101
%
15576
146
54
15601
140
59
15576
140
61
15576
142
70
>1
’P3
14706
178
8
14749
170
8
14684
168
13
14662
180
14
’P4
14184
98
7
14168
88
10
-
-
-
14227
100
12
’P4
13818
142
7
13778
130
9
13758
120
9
13831
105
13
yield is also found to be maximum. This is probably because the fluoride host have lower
phonon energy and hence non-radiative losses due to multiphonon relaxation giving better
fluorescence efficiency.
Emission cross section and life time :
Laser materials are generally characterised on the basis of stimulated emission cross
section, life time, branching ratio, transition probability etc., for different transitions. In
order to calculate these parameters we used the relation given by Saisudha and
Rama Krishna [IS], The radiative transition probability and branching ratio for
different transitions are given in Table 5 and stimulated emission cross section in Table 6.
114
Brajesh Sharma. Akshaya Kumar and S B Rai
Table 5. Radiative- transition probabilities and branching ratio of the ^P\ and ^^2 excited
states of Pr^ in phosphate, tellurite, oxyfluoride and Zr based fluoride glasses.
Transition
energy
PBK
Te.NaO
ABCP
Zr. BAN
A
p
A
A
p
A
p
^ D 2
16420
1212
0.307
1011
0.266
1320
0.386
1518
0.460
14378
40.0
0.010
32.0
0.008
30.0
0.008
10.0
0.003
12282
750
0.190
850
0.224
750
0.219
690
0.209
^F2 16672
920
0 233
890
0.234
630
0.184
530
0.160
103(X)
130
0.032
230
0.06
115
0.033
95.0
0.028
V 4 9825
770
0.195
6.50
0.171
420
0.122
320
0 097
'r ;4 7001
120
0 03
130
0.034
150
0 043
130
0.039
>0
X 16666
3042
0.156
3146
0 1.54
2930
0 159
3295
0.166
V 2 15454
16022
0.826
17021
0.833
15048
0.821
16022
! 0.809
V 4 13583
220
0.001
120
0006
240
.0.013
330
y ).016
‘G 4 10773
65.0
0.003
95 0
0.005
67.0
0.003
102
9005
'D 2 3768
32.0
0.02
46.0
0.002
40.0
0.002
50.0
6.003
>1 -»
'’W 4 21118
120
0.004
132
0005
125
0.004
136
0 005
18690
10452
0 3.59
98.56
0 359
10280
0.37
11350
0 404
'X 16593
80.0
0.002
102
orxM
100
0003
95.0
0 003
^2 15991
5830
0,20
5680
0 207
5490
0 197
5620
0.200
Vi 14616
12361
0.425
11362
0414
11463
0 413
10560
0 376
V 4 14125
70,0
0,003
80.0
0002
82.0
0 002
80 0
0 003
'C 4 11318
20 0
0.006
20.0
0.007
27.0
0.(K)9
• 20.0
0007
'O 2 4312
150
0.005
153
0006
162
0 006
156
0.005
Table 6 . Stimulated emission cross section of three intense bands and total
transition probabilities of final state ’D 2 , different glasses
Glasses
Transition
Cross section
CTx lO'^^cni^
Total transition
probabilities {lA)
Phosphate
V2
42 3
3942
^Po
-> ^F2
22.1
19381
V 5
31.0
29083
Tellurite
V 2
V 4
43 15
3792
■Vo
^ V2
20.16
20428
-» ’W 5
32.10
“27385
Oxyduoride
‘02
-* ’H 4
40.36
3415
26 02
18325
28.20
27729
Fluoride
'02
^ ’ 1/4
50.36
3293
%
24.19
19799
^Pi
-» ’H,
36.5
28026
Optical properties of doped glasses, effect of host lattice
115
The cross section values for -4 'F 2 (^^1 ^^5 (525 nm) and
(609 nm) transitions in the HMF glass are 24 x 36.5 x l(^2^ qnd 50.36 x 10~2^ cirr
respectively. The large cross section Indicates that Pr doped HMF glass looks more
promising for laser transitions. The radiative life time of the excited states and 'Z >2
were also calculated and compared with the results given by Lakshman and Suresh Kumar
[14] and Bunuel et al [22] in Table 7. These values are in reasonable agreement with the
experimentally estimated values.
Table 7. Radiative life time (r,) of tluoFeceni levels *D 2 . and Vi in different hosts
Inn
Fluore.sccnt
level
PBK
tf (US)
Te NaO
r,(us)
ABCP
r,(lis)
Zr.BAN
rr(Ps)
ZnNa.P
Trdis)
(141
Zn.F2 CaF2
T,(ps)
[22 1
PrU
'«2
253
263
292
303
369
^exp = 245
51
48
54
50
47
Teal -27
34,
36
36
35
47
4. Conclusion
Using modified J-0 theory by Kornienko et al [11a], the intensity parameters, oscillator
strength, transition probabilities and life time of the different levels of ion doped in
PBK, Tc-NaO, ABCP and HMF glasses have been calculated. These calculations show that
among the four glasses selected, Zr based HMF glass is superior laser material.
Acknowledgment
Authors are grateful to the Council of Scientific and Industrial Re.search, Govt, of India for
financial assistance.
References
in G C Aumullar, W Kosiler, B C Grambaierand R Frey J. Phys Chem Solids SS 767 (1994)
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|3j L Pauling and M D Sliappell Z. Knst. 75 128 (1930)
(4] H Forest and G J flan J. Electrochem. Soc. 116 474 (1969)
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(1991)
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[71 L Weterkamp, G F West and H Tobben J Non-Crysl Solids 140 35 (1992)
[8J B R Judd Phy. Rev 127 750 (1962)
[9] G S Ofelt ./. Chem. Phys 37 51 1 (1962)
116
Brajesh Sharma, Akshaya Kumar and S B Rai .
(10] M J Weber, T E Varifms and B H Matsingcr Phy. Rev. M 47 ( 1 923)
[11a] A A Kornienko, A A Kaminoki and E B Dunina Phys. Stat. Sol. 157 267 (1990)
(Mb] P Goldner and F Auzel J. Appl. Phys. 79 7972 (1995)
{ 1 2] Brajesh Sharma, J Vipin Prasad. S B Rai and D K Rai Solid State Commun. 93 623 (1995)
[ 1 3] Brajesh Sharma, S B Rai, D K Rai and S Buddhudu Ind. J. Engg. Mater. Set . 2 297 { 1 995)
114] S V J Lakshman and A Suresh Kumar J. Phys. Chem. Glasses 29 1 46 ( 1 988)
[15] MB Saisudha and J Ramakrishna Phys. Rev B53 6186 (1996)
1 161 M J Weber J. Chem. Phys 48 4774 (1968)
1 17] W T Camall, H Crosswhite and H N Crosswhite Energy Level Strt4Cture and Transition Probabilities of
the Tnvalent Unlhanides rn LaFj (Argonne National Laboratory, Illinois) (1978)
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[20] C K Jorjensen Adsorption Spectra and Chemical Bonding in Complexes (New York ■ Pergamon) (1962)
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[24] J Sanz, R Cases and R Alcala J Nonlinear Cryst. Sol. 93 377 (1987)
[25] A Singh and P Nath Central Glass Ceram Bull 30 6 (1983)
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Indian J. 'Phys. 72A(2), 117-124 (199g)
UP A
— an international journal
Mechanism of grain growth in aluminium, cadmium,
lead and silicon
F Jesu Rethinam*, S Kalainathan** and C Thinipathi*
* Department of Physics, Sacred Heart College. Tinipattur-635 601,
Tamil Nadu, India
Department of Physics, Vellore Engineering College, Vcllore-632 014
Tamil Nadu, India
Received 7 November 1997. accepted 27 November 1997
Abstract : A simple model has been developed to evaluate the mechanism of grain
growth in AI, Cd, Pb and Si for various annealing times and annealing temperatures based on the
diffusion of atoms from one grain to another. Our numerical values of grain size have been
compared with the available experimental results. There exists a fairly satisfactory agreement
between them The above model has been extended to calculate the grain size distribution as a
function of annealing time and annealing temperature. The results are discussed in detail.
Keywords : Groin growth, grain size, annealing temperature
PACSNos. ; 81.05.Bx. 81.15.Tv
1. Introduction
Polycrystalline metals have been used in all engineering application. Recently the
study of mechanical properties has moved away from the processes which occur inside
the individual grain to those which are governed by the boundaries between the grains.
Most of the properties such as high temperature creep, superplasticity, recrystallisation,
yielding and embattlement all depend strongly on effects at grain boundaries. There
ard numerous variables which can affect the structure and properties of a grain boundary.
These include the crystallographic parameters which describe the orientation of the
adjoining crystals and the interface between them. The grain boundary is an internal
.surface, in a single phase material, across which there is a discontinuity of crystal
orientation. It shares in common with interphase surfaces, the property of a free energy
per unit area [1]. At elevated temperatures, it is able to move under driving forces
by invading one grain and enlarging the other in its wake. Recently, powerful theoretical
© 1998 lACS
72A(2)-4
118 F Jesu Rethinam, S Kalainathan and C Thirupathi
and experimental techniques have been developed to investigate the grain boundary
migration.
The phenomenon of grain boundary migration occurs when a grain boundary is
subjected to a driving force sufficient to cause motion of the boundary. The resultant
boundary migration will be such as to cause a reduction in the free energy of the
system. Boundary migration is a thermally activated process and hence the rate at
which a boundary migrates under the influence of a given driving force is strongly
temperature dependent [2-5]. In this paper, a kinetic model was proposed to understand
the grain growth mechanism in Al, Cd, Pb and Si for various annealing time and
temperature.
2. Theory
Our kinetic model is based on the diffusion of atoms from one grain to another. Using
this, we have determined the grain size distribution in Al, Cd, Pb and Si. The driving
force for the movement of atoms from one grain to another is related to the j^rain
boundary energy and radius of the grain. The difference of chemical potential beti^een
the two adjustant grains, which is the driving force for the grain growth is given by [6]
D =
Fa^E
r
( 1 )
where F is a constant related to the geometric shape of the grains (for spherical grains
F = 2), E IS the grain boundary energy, a is the lattice constant and r is the grain size.
From basic rate theory, the net rate of atomic transfer across the boundary (from
lattice sites of one grain to those of a neighbour) is given by [7]
where is the self diffusion constant of the atoms across the grain boundary, k is the
Boltzmann's constant, T is the annealing temperature and W is the thickness of the
grain boundary. The rate of boundary motion is given by the product of the net rate of
atomic transfer across the boundary and the thickness. Therefore, the grain growth
rate becomes
f (i?)} ®
Exjtanding the exponential, neglecting the cube and higher powers of D/kT and using
eq(l)
Fa^E
where
2kT ’
Mechanism of grain growth in aiuminiunt, cadmium, lead and silicon 1 19
Integrating eq. (4)
r
2D^ U/W = ((r-£.)2 /2 + 2^r-L) + log(r- fc)]. (5)
^0
where t is the time of annealing and ro is the initial grain size. Expanding the right hand side
of eq. (5), applying the limits and neglecting the cube and higher power of L, since L is very
small compared to the value of Tq, we get
r =[1"^ + AD^UjW +rl+lLr^)''^ -L. ( 6 )
It is noted that the grain size is directly proportional to the square root of annealing
time as reported earlier [6-12]. This model has been extended to evaluate the grain size
distribution for different annealing time and annealing temperatures. By introducing a
dimensionless variable known as relative grain size, V = r/r„ in eq. (4), and simplifying
where
where
dr
(4D^ L/W)
(7)
1/ / dr^^ , r^r is the average grain size and dr represents the time.
G = i4D^LIW)dtldrl and H = L/r^^.
The eq. (9) has been solved to determine the steady state grain size distribution
during normal grain growth mechanism, using the Hillert's [9] approach. The whole
distribution of the individual grain size during the steady state is given by
P(V) = jexpl-P\if/2)/[dV/dT]. (10)
where p is a constant (P= 2 for two dimensional system, /J = 3 for three dimensional
system) and
V
yr = jdV/(-dV/dT). (II)
0
From eqs. (9) and (11)
-GV + G/f
)]
V
0
+ r.
( 12 )
Y =
V
2G f dV
3 J (V^^CV + GH)
0
where
120
F Jesu Rethinam, S Kalainathan and C Thirupathi
Numerical integration method has been followed to determine the value of Y. Substituting
the values of dVIdx and v^from eqs. (9) and (12) respectively in eq. (10), we get
PiV) =
(V’ -GV+Gtf)[(V’ -GV + GH)/GH]^'3 exp(/3y/2)'
(13)
Eq. (1 3) gives the grain size distribution.
3. Results and discussion
Eqs. (6) and (9) have been used to evaluate the numerical values of grain size distribution
for different time of annealing and annealing temperature.
\
Figure 1. Variation of grain si]^ vs
annealing time in AI for different
annealing temperatures.
90 60 90 120 190 IBO
Tim* (min)
Figure 2. Variation of grain size vs
annealing time in Cd for different
annealing temperatures.
Mechanism of grain growth in aluminium, cadmium, lead and silicon 1 2 1
Figure 1 shows grain size V5 annealing time in A1 for different annealing
temperatures. Experimentally reported results [13] are also plotted in the Figure. Figure 2 is
Figure 3. Variation of grain size annealing time in Pb for different
annealing temperatures.
Figure 4. Variation of grain size vs annealing time in Si for different annealing
temperatures.
plotted between grain size and annealing time in Cd for different annealing temperature
with the experimental values [14]. Figure 3 is drawn for grain size and annealing time in Pb
for different annealing temperatures. Experimental points [15] are also shown in the Figure.
Figure 4 is plotted between grain size and annealing .lime in Si for various annealing
temperatures and experimental results [16]. From the numerical results it can be concluded
122
F Jesu Rethinanu S Kahinathan and C Th^mpathi
that grain size increases with annealing time and annealing temperature and there is a fairly
satisfactory agreement between the experimental and our theoretical results.
Figure 5. Variation of groin size distribution vs relative grain size in A1 for
different annealing temperatures for three hours of annealing time.
Figure 6. Variation of grain size distribution vs relative grain size in Cd for
different annealing times for the annealing temperature of 389.5 K.
Figure 5 shows grain size distribution vs relative grain size in A1 for different
annealing temperatures and for three hours of annealing. Figure 6 is drawn grain size
distribution vs relative grain size in Cd for different annealing times and for the annealing
temperature of 389.5 K. The dotted line shows the Hillert's result. Figure 7 is drawn for
grain size distribution and relative grain size in Pb for five different annealing temperatures
and three minutes of annealing time. Figure 8 is plotted between grain size distribution and
relative grain size in Si for different annealing times and annealing temperature of 1548 K.
From the results, we observe that the grain size distribution gradually increases with
increases of relative grain size, attains a maximum and with further increase of relative
Mechanism of grain growth in aluminium, cadmium, lead and silicon 123
grain size, the grain size distribution decreases. The maximum point also increases with
annealing time and annealing temperature. There are no experimental reports in the
literature for the grain size distribution.
Figure 7. Variation of grain size distribution vs relative gram size in Pb for
different annealing temperatures and three minutes of annealing time.
Figure 8. Variation of grain size
distribution v.t relative grain size in Si
for different annealing limes and
annealing temperature of 1 548 K.
4. Conclusion
From the numerical results, we conclude that the grain size and their distributions in
materials like Al, Cd, Pb and Si, increases with the increase of annealing times and
annealing temperatures. It is due to the large number of atoms segregated at the grain
boundaries.
Acknowledgment
The authors thank the University Grants Commission, New Delhi and Rev. Fr. P
Soundararaju, S.D.B., Principal, Sacred Heart College, Tirupattur for the financial support.
One of the authors (S Kalainathan) is very grateful to The Chairman, The Vice Chairman
and The Principal of Vellore Engineering College for their constant encouragement to carry
out this research work.
124
F Jm Retfmm, S Kalauu^m and C ThinpaM
ReTeitnca
[1] W W Mullins Am Mm/. 4421 (1956)
[2] C J Simpson, W C Winegasrd and K T Aust Grain Boundary Structure and Properties eds G A
Chadwick and D A Smith (London ; Academic) p 201 (1976)
[3] D J Jensen Am. Metallurg. Mater, 43 41 17 (1995)
|4] D C Vanaken, P E Krajewski, G M Vyletel, J E Allison and J W Jones Metallurg. Mater. Trans.
26A 1395(1995)
[5] V Y Gortsman and Q R Bininger Scr. Metallurg. Mater. 30 577 (1994)
[6] S Kalainathan, R Dhanasekaran and P Ramasamy Thin Solid Films 163 383 (1988)
[7] C V Thomson J. Appl. Phys . 58 763 ( 1985)
[8] L Mei, R River, Y Kwart and R W Dutton J Electrochem. Soc. 129 1791 (1982)
[9] MHillen Am. Mm/. 13 763 (1965)
[10] S Kalainathan, R Dhanasekaran and P Ramasamy / Crystal Growth 104 250 (1990)
[11] S Kalainathan, R Dhanasekaran and PRamasamy J. Electron. Mater. 19 1135 (1990)
[12] S Ka]ainathan, R Dhanasekaran and P Ramasamy J. Mater. Sci. : Materials in Electron. 2 98 (1991)^
[13] S S Iyer and C Y Wong / Appl. Phys. 15 4594 (1985)
[14] E A Grey and G T Higgins Scr. Metal. 6 253 (1972)
[15] CFBoIlingand WC Winegard Am. Mem/. 6 283 (1958)
[16] G C Jam, B K Das and S P Bhattacheijee Appl. Phys. Lett. 33 445 (1978)
Indian J. Phys. 7ZA (2). 125-131 (1998)
UP A
—’Han imemational jo urnal
Analysis of temperature dependence of interionic
separation and bulk modulus for alkali halides
Rajiv Kumar Pandey
Department of Physics, G. B. Pant University of Agriculture & Technology,
Pantnagar-‘263 145, Uttar Pradesh, India
Received 29 January 1997, accepted 16 December 1997
Abstract : A thermodynamic analysis of Anderson-Griineisen parameter is found to
yield useful relations for estimating the temperature dependence of intenonic separation riT)
and bulk modulus These relations can be used to predict liT) and BjiT) upto melting
temperature of eight alkali hahde solids. The results are compared with the available
expenmental data and are discussed in the view of recent research in the field of high
temperature physics
Keywords : Thermodynamic properties, equations-of-staie, inorganic compounds.
PACS Nos. : 64.30 +t, 65.50 +m
1. Introduction
Various efforts have been made to understand the thermodynamic properties of solids
or materials under the effect of high temperature by many workers [l-lOj. In previous
studies, the temperature dependence of the thermodynamic properties, viz., temperature
dependence of interionic separations, bulk modulus and cubical thermal expansion of solids
from static lattice to the melting temperature have been studied and discussed by various
expressions developed on the thermodynamic approximations and best fit relations [5,9,10].
Kwbn et al [11] have investigated the thermal properties of KCl by using modified
Einstein model. Such study required appropriate form of potential energy and huge
cuinputation. The adequate knowledge of temperature dependence of bulk modulus is very
necessary for understanding the thermoelastic and anharmonic properties of solid. The
expressions for temperature dependence of interionic separation and bulk modulus have
been developed with the assumption that the thermal expansion coefficient depends linearly
© 1998 lACS
7M(2)-5
126
Rajiv Kumar Pandey
on temperature [9,16]. This can be justified from the work of Spetzler et al [12] and other
workers [13,14]. The cubical thermal expansion coefficient (oO is related to the density of
solids assuming that Anderson'Griineisen parameter ^is independent of temperature above
Debye temperature 0^ [15,16]. The validity of this assumption has been discussed in
Anderson et al [15-20]. This assumption is widely used for predicting interionic separation
of alkali halides from static lattice to melting temperature because many of them have
Debye temperature near to room temperature.
The aim of present paper is to develope relations for temperature dependence of
interionic separation r(T) and bulk modulus B-jiT) by using thermodynamic relations and
under following approximations :
(a ) Anderson-Gruneisen parameter &p remains independent of volume [ 1 7-2 1 ] .
(b) Anderson-Gruneisen parameter 5f is volume dependent [20-22] .
The present paper is an effort in a such direction. The method of analyses is
described in Section 2. The calculated values are compared with each other '^and
experimental values. The results and discussions are given in Section 3.
2. Method of analysis
Anderson-Gruneisen parameter is very important and useful quantity for developing an
understanding of anharmonic properties of ionic solids. The Sp is defined as [23,24]
5
T “
(1)
where a and Bj are cubical thermal expansion coefficient and Bulk modulus, respectively.
These are defined as
The Maxwell thermodynamic relation is given as
r dal 1
&
1
[dPjr ■ fir
[dTj
using eqs. (1), (2) and (4), we get following relations :
da ^ dV
V
= adT.
( 2 )
( 3 )
( 4 )
(5a)
and
(5b)
Analysis of temperature dependence of interionic separation etc
127
2a, Expression for temperature dependent interionic separation [r(T )] :
Integrating eq. (5a) under approximation that Sr is unchanged with change of volume
[17-19], we get well known Anderson relation [9] as
a
(6a)
where Oq is the value of a at V= Vq. The eq. (6a) is strictly based on the assumption that Sr
IS independent of volume. However, in view of recent studies [20], Sr has been found to
decrease with n = (V/Vq) according to the following relation [20-22]
5^ +1 = An,
(6b)
where A is constant for a given crystal. A is determined from the initial condition, viz., at
V = Vq,A = Sr 1. This relationship has been widely used. Putting eq. (6a) in eq. (5a) and
integrating, we get the following relation [25,26]
^ = -^exp[A(V/V„-l)]. (6c)
Now putting the value of a from eq. (6a) arid (6c) in eq. (5b) and integrating, we gel
following relations
■rr = [' - (7a)
0
^ = [i -/i-'{in(i-/)a„(r-r„))}]. (7b)
In eqs. (7a) and (7b), we put /^q)[^ 1'E]/ r^l and we get the expression for r{T)
under approximations :
i_
r(T) = r,[l - 5,.a(,(7 -r„)]'3«, (7c)
and KT-) = ro[l - .4-'{ln(l-y4ao(7-7o)}]. (8)
2b Expression for temperature dependent bulk modulus BjiT ) :
Starting from the assumption that Sp is independent of temperature (T) above Debye
temperature Qp, [20] and putting the value of a from eqs. (6b) and (6c) in eq. (1), we get
following forms :
dBj SjOL^dT
X l-5^cro(7'“7'o)
dB^ -Sja^dT
Bj ~ [l-A->ln(l-/loOfp(r-To)l][l-Aao(r-ro)]‘
and
(9b)
12B
Rajiv Kumar Pandey
Integrating eqs. (9a) and (9b), we get following expressions forS 7 <T) as :
B^(T) = B„[l-ao5^(r-ro)].
(10)
Bt(T) lnn-/lOo(T-r„)l]-'^ .
(11)
The expressions from the eqs. (7c), (8), (10) and (11) are used to compute intcrionic
separation r(T) and bulk modulus Bt(T) at different temperatures Le. from room
temperature to melting temperature.
3. Application, results and discussion
In order to demonstrate the applicability of these expressions i.e. eqs. (7), (8), (10) and (1 1)
reported here, we calculate the interionic separation r{T) and bulk modulus Bj(T) as u
function of temperature (i.e., from room temperature to melting temperature). The Debye
temperature Qq of solids considered in the present study, have values near to room
temperature except for LiF. The eqs. (8) and (10) work well above Debye temperature
From application point of view, we have studied only eight alkali halides with ISaCI
structure in present study. The values of dj, A and kq at room temperature are used as
input parameters which arc given in Table 1 [8,25]. We use eqs. (7) and (8) to compute
inlerionic separation at different temperatures of these alkali halides. A comparison of the
Table 1. Values of parameters at room temperature [8,24].
Crystal
'■o(A)
ao(l(r*K-')
Sj
A
7’„(K)
LiF
2.013
0.999
6.15
7 15
6^)5
114.3
NaCl
2 820
1.190
5 95
6.95
240
1050
KCI
3.146
1.110
6.29
7 29
175
1043
KBr
3 289
1.160
5.88
6 88
148
1006
Kl
3.525
1230
5 83
6 83
117
957
RbCi
3.291
1.030
6.73
7 73
156
990
RbBr
3 445
1.080
6.64
7.74
1.32
950
Rbl
3.668
1.230
6.53
7 53
105
913
calculated values and the exprimental data [5,26] of r(T) at different temperatures (upto
melting temperatures) is given in Table 2(a-d) along with experimental data [5,25] for the
sake of comparison. It is clear from the calculated values of r(T) that the calculated values
of r(T) of all the eight alkali halides are in good agreement with the experimental values [5].
The value of r(7) for every alkali halides computed by expression [eq. (7)] is slightly lower
than computed values by eq. (8) for the entire range of temperatures because of the
assumptions (a and b) which are used to develop eqs. (7) and (8). At melting temperature
Tn of some of the ionic solids, the calculated value of r(T„) by using eqs. (7) and (8), show
significant difference with the experimental data. The values of bulk modulus Bt{T) are
computed by using eqs. (10) and (1 1) for the entire range of temperatures. A comparison
Analysis of temperature dependence ofinterionic separation etc
129
is presented in Table 2(a-d) for the case of NaG. The experimental data for NaCl [26] are
available. The calculated and experimental values are found to be in good agreement with
each other. In case of LiF also, we have good agreements with experimental and calculated
values above and below Debye temperature.
Table 2<a). Calculated value of interatomic separation KT) ink and bulk modulus ByiT) in unit
of (X l(r^) GPa at different temperature. The experimental data are taken from [5,25].
LiF NaCl
KT)
BjiD
K77
Bt<T)
Eq.7
Eq.8
Exp.
Eq.lO
Eq.ll
Exp.
Eq.7
Eq.8
Exp.
Eq.lO
Eq.ll
Exp.
300
2.013
2.013
2.012
665.0
665.0
665.0
2.820
2.820
2.820
240.0
240.0
240.0
400
2.020
2.020
2.019
624.1
624.1
2.832
2.832
2.831
223.0
223.0
224.1
500
2.027
2.027
2.028
583.3
585.2
2.844
2.844
2.845
206.0
205.9
205.0
600
2 035
2.035
2.037
542.4
542 3
2.858
2.858
2.860
189.0
188.9
188.0
700
2.044
2 044
2.047
501.6
501.2
2.873
2.873
2.877
172.0
171 8
174.0
800
2.053
2.054
2 058
460 7
459.9
2.890
2.890
2.894
155.0
154.6
156.0
2.094
2.096
2.101
320.6
315.5
2.942
2.945
2.945
112.6
110.5
119.0
Table 2(b).
KCl
KBr
KT)
BjiT)
r\T)
B-iiT)
Eq7
Eq.8
Exp.
Eq.lO
Eq.ll
Exp
Eq.7
Eq.8
Exp.
Eq.lO
Eq 11
Exp.
300
3 146
3 146
3 146
175 0
175 0
175 0
3.289
3 289
3 289
148.0
148.0
148 0
400
3 158
3.158
3.158
162.8
162 8
3.302
3 302
3 302
137.9
137.9
•500
3 171
3.171
3.170
150 6
150.6
3.317
3.317
3.316
127.8
127.8
600
3.184
3 186
3.185
138.4
138.3
3 332
3 332
3.331
117.7
117.7
700
3 201
3.203
3.200
126.1
126 0
3.349
3 349
3.346
107 6
107 3
800
3218
3.219
3 220
1139
113.6
3.368
3.368
3.364
97 5
97 3
Tn.
3.270
3.273
3.259
84.2
82.9
3.413
3.416
3.401
77 0
76.1
Table 2(c).
K1
RbCl
KT)
BiiT)
KD
B-jiT)
Eq.7
Eq.B
Exp.
Eq.l0
Eqll
Exp
Eq.7
Eq.8
Exp.
Eq 10
Eq.ll
Exp.
300
3.525
3.525
3.525
117.0
117.0
117.0
3.291
3.291
3.291
156.0
156.0
156.0
400
3.540
3.540
3.540
108.6
108.6
3.303
3.303
3.302
145.2
145.2
500
3^56
3.556
3.556
100.2
100.2
3.315
3 315
3.316
134.4
134.4
600
3.574
3.574
3.578
91.8
91.8
3.329
3.329
3.331
123.6
123 5
700
3.594
3.594
3.592
83.4
83.3
3.344
3.346
3.346
112.8
112.6
800
3.616
3.616
3.612
75.1
74.8
3.361
3,362
3.364
101.9
101.7
7'm
3.656
3.658
3.646
61.9
61.2
3.399
3.400
3.399
81.4
80.6
130
Rajiv Kumar Pandey
Tabk2(d).
RbBr Rbl
KD
Bt<7)
KD
BjiT)
Eq.7
Eq.8
Exp
Eq.lO
Eq.ll
Exp.
Eq.7
Eq.8
Exp.
Eq.lO
Eq.ll
Exp.
300
3.445
3.445
3.442
132.0
132.0
132.0
3.668
3.668
3.668
105.0
105.0
105.0
400
3.458
3.458
3.457
122.5
122.5
3.684
3.684
3.683
96.7
96.7
.500
3.472
3.472
3.472
113.1
113.1
3.701
3.701
3.699
88.1
88.1
600
3.487
3.487
3.487
103.6
103.6
3.720
3.720
3.716
79.7
79.7
700
3.504
3.504
3.502
94.1
94.0
3.741
3.742
3.734
71.3
71.1
ROO
3 523
3.523
3.518
84.7
84.4
3.765
3.766
3 753
62.8
62,5
3.555
3.5.57
3.544
70.5
69.9
3.797
3.799
3 776
53.3
52.7
To summarise, the assumption that the Anderson -Gnineisen parameter 5t remains
unchanged with temperature (7) above the Debye temperature (Qd), leads to simple
relations for finding the temperature dependent interionic separation and bulk modulu^ for
alkali halides. For best explanation of interionic separation at melting temperature i.e., r(7„,)
for ionic solids the Debye-Lindemann criterion of melting calculation may be useful and
meaningful over the methods based on approximations (a and b). The good agreement
between theory and experiment obtained in the present work confirms the internal
consistency of thermodynamic relations from which most useful relations of temperature
dependence of interionic separation and bulk modulus have been derived.
Acknowledgments
The author would like to thank Dr. B R K Gupta for encouraging to look into this problem.
The author is grateful to the referee for his comments which have been found useful for
revision of this manuscript. He is also thankful to Mr. Madan Singh and Miss. Vandana Pal
for discussions and help.
References
[11 A M Sherry and M Kumar Indian J. Pure. AppL Phys. 29 612 (1991)
[2] M P Verma and B Dayal Phys. Stat. Sol, 3 901 (1963)
[3] M P Verma and B Dayal Phys. Siat. Sol. 6 6545 (1964)
[4] J L Tallon J. Phys. Chem. Solids 41 837 (1980)
[5] K K Srivastava and H D Merchant J. Phys. Chem. Solids 34 2069 ( 1 973)
[6] LLBoycrP/iyi. Rev. B23 3673 (1981)
[7] S C Kim and T H Kwon / Phys. Chem. Solids 52 1 145 (1991)
[8] A M Sherry and M Kumar J. Phys. Stat. Sol. 52 1 145 (1991 )
[9] M Kumar and S P Upadhyay Phys. Stat. Sol. (b) 181 55 (1994)
[10] M Kumar Physica B205 175 (1995)
[11] T H Kwon, S D Kwon, Z H Yoon. Y K Sohn and S C Kim Physica B183 75 (1993)
[12] H Spetzler. C G Samis and R J Connell J. Phys. Chem. Solids 33 1727 (1972)
Analysis of temperature dependence of interionic separation etc
131
[13] R Boehler and G C Kennedy i Phys. Chem. Solids 41 1019 (1980)
[14] F D Enk and J G Dommel J, Appi Pkys. 36 839 ( 1 965)
[15] 0 L Anderson Phys. Earth Planet. Inter. 45 307 (1987)
[16] 0 L Andenon. A Chopelas and R Boehler Geophys. Res. Lett. 17 685 (1990)
[17] J Shanker and M Kumar Phys. Stat. Sol. (b) 179 351 (1982)
[18] M T Yin and M LCohen Phys. Rev. B26 5668 (1982)
[19] N Dass and M Kumari Phys. Stat. Sol. (b) 127 103 (1985)
[20] 0 L Anderson, D Isaak and H Oda Rev. Geophys, 30 57 (1992)
[21 ] M Kumar Solid Stat. Common. 92 463 ( 1 994)
[22] M Kumar Physica B212 391 (1995); 205 175 (1995)
[23] 0 L Anderson J. Geophys. Res. 72 366 1 (1967)
[24] M P Madan J. Appl. Phys. 42 3888 (1971)
[25] A Dhoblc and M P Verma Phys. Stat. Sol. (b) 136 497 ( 1 086)
[26] S Yamamoto, I Ohno and 0 L Anderson J. Phys Chem. Solids 48 143 ( 1987)
Indian J. Phys.llK (2), 133-139 (1998)
UP A
^ an interoational journal
Evaluation of the trapping parameters of XL peaks of
multi activated SrS phosphors
W Shambhunath Singh“, S Joychandra Singh**, N C DebS
Manabesh Bhattacharya^, S Dorendrajit Singhs
and P S Mazumdai®* ^
“Deparimenl of Physics. Manipur College. Imphal, Singjamei-795 008,
Manipur. India
*’BaUistics Division. Manipur State Police Forensic Laboratory,
Pangei Yang(Jang-79S 1 14, Manipur, India
‘^Center for Theoretical Studies of Physical Sciences. Clark Atlanta University.
Atlanta, Georgia 30314, USA
^Department of Theoretical Physics. Indian Association for the Cultivation of Science,
Jadavpur. Calcutta-700 032, India
^Department of Physics, School of Science, Manipur University, Chanchipur,
lmphal-795 003, Manipur, India
^Present Address •
r^epartment of Physics. Acharya Prafulla Chandra College, Sajirhal,
New Barrackpoit:-743 276, North 24 Parganas, West Bengal, India
Received 2 May 1997. accepted 4 February 1998
Abstract : In the present paper we suggest a relation connecting the symmetry factor (Pg)
and order of kinetics (h) so that one can directly determine b once Pg is known. We also extend
the work of Gartia et al for a wider range of b values (0.4 ^ ^ ^ 4). As rough check of the
theoretical methods we determine the trapping parameters namely activation energy, order of
kinetics and frequency factor of thermoluminescence (TL) peaks of multi activated SrS
phosphors reponed by Kao et al. The present values of activation energy differ appreciably from
those of Rao et al obtained by using sonK: crude methods
Keywords : Thermoluminescence, order of kinetics, activation energy
PACS No. : 78.60.Kn
1- Introduction
Thermoluminescence is the light emitted from an insulator or a semiconductor as a result of
heating after exposing it to some ionising radiation. TL is an indispensable tool for
exhaustive study of the nature of thermal stability and concentrations of both electron and
hole trapping centres in luminescent materials. It has found important applications in dating
(g) 1998 lACS
72A(2)-6
134
W Shambhunath Singh el al
and dosimetry fll. The shape, position and intensity of a TL peak are related to various
trapping parameters namely order of kinetics (b), frequency factor (j) and activation energy
(E). The various methods for determination bf trapping parameters have been reviewed by
Chen and Kirsh f 1 ], McKeever [2], Kirsh [3] and more recently by Gartia eta/ [4].
In a recent work Rao et al [5] have presented TL data of multi activated SrS
phosphors. They evaluated activation energy of TL peaks recorded by them. But in spite of
recent developments in the analysis of trapping parameters of TL peaks they use outdated
and crude methods of Urbach [6] and Randall and Wilkins [7]. For example, it has been
shown by Chrislt)doulides [8] that Urbach method was meant as a very rough guide and as
such it is of limited accuracy. It gives an activation energy value which may be wrong by
Lipio a factor of two either way. They [5] also assume mono molecular kinetics (b = 1) and a
frequency factors = 10^ seer* without any justification. It is well known [1,2] that a TL
peak can be characterised by a parameter called the symmetry factor defined by
where is the peak temperature, T|, Ti arc the temperatures at which the intensity is ecjual
to half of the maximum intensity on either side of T„, (J 2 > T,). It is well known [1-4] that
-0.42 lor b = I and -0.52 (orb- 2. In Table 1 , we calculate ^ for TL peaks recorded
Tabic 1. Trapping parameters £, s and b of TL peaks of niulli activated SrS phosphors reported by
Rao et al [5] by using the present method Samples of series A contain Gd in fixed amount and
concentrations of Cu and Mn are varying Similarly in senes B, concentration of Mn is fixed while those
of Gd and Cu are varying. In series C, Cu concentration is fixed and those of Mn and Gd are vaned
No
Sample
name
Tm
("K)
h
(cV)
^£5
(eV)
(eV)
s
(s"')
K)
SrS : Mn
.367
0 467
1 48
0500
0 548
0 524
1 22x 10^
22
SrS Cu
359
0.400
0 95
0.375
0 426
0 396
2 61 X itr*
24
.SrS Cu.Mn
343
0 552
2.42
0.626
0.624
0 625
1 53 X 10*
25
SrS .Gd •
345
0 333
0.51
0 410
0.411
0.411
7 64x 10^
Al
SrS : Gd, (Cu, Mn)
363
0 487
1 67
0.386
0 432
0409
2.87 X 10"^
A2
do
343
0.585
3 05
0 804
0.777
0.788
4.94 X 10‘^
A3
do
350
0 500
1.79
0.579
0 607
0,.594
3 29 X 10^
A4
do
3.57
0.510
1.90
0.701
0718
0710
1 15x 10*^
A5
do
358
0.491
1 70
0.574
0 609
0 592
1.95 X 10^
A
do
360
0 480
1.60
0 622
0.660
0.641
9 23 X 10^
A6
do
356
0.421
1 11
0421
0 474_
0.445
1.54 X 10-^
A7
do
375
0 433
1.20
0464
0 520
0 490
2.86 X 10^
B2
SrS.:Mn,(Cu,Gd)
344
0.477
1.57
0.646
0 681
0 663
5.84 X 10*
B5
do
360
0.420
1 10
0.500
0.552
0-523
1.88 X 10^
C'
SrS : Cu, (Mn, Gd)
360
0.480
1.60
0.622
0.660
0.641
9.23 X 10^
C.3
do
355
0.426
1 14
0531
0.582
0.554
7.03 X 10^
C5
do.
362
0.378
0.80
0.491
0.530
0.507
1.03 X 10^
C7
do
373
0.435
1.22
0.445
0.501
0 471
1.66x10^
Evaluation of the trapping parameters ofTL peaks etc
135
by Rao et al [5]. It is seen from Table 1 that values range from 0.333 to 0.585. So it is
not appropriate to make the mono molecular assumption as above by Rao et al [5]. In the
present paper, we analyse the TL peaks recorded by Rao et al [5] by using the peak shape
method of Gartia et al [9].
We also suggest a method for the estimation of the order of kinetics (b) from
which is more objective than the conventional method of Chen [10] and extend the work of
Gartia et al [9] for a wider range of b values namely (0.4 < ^ < 4),
2. Theory
The equation for the first order = 1) and general order {b* I ) TL peaks can be written as
18,9,11,12]
'/L = e*P(“», - “ + (b = I) (2)
and ///„= exp(u„ - [(A-l)/fc]F(u, (fr ^ 1) (3)
with /■’(«,«„) = exp («„){£j(«„)/u„, - £j(u)/u}, (4)
where / and are respectively the TL intensities at any temperature T and peak
icmpcrature 7),,. £ 2 ( 14 ) is the second exponential integral [13|. The variable u is defined as
u = E/kT, similarly = EfkT„.
The eqs. (2) and (3) can be solved by an iterative technique [8,1 1,12] to determine
the half intensity temperatures T\ and T 2 for which l/l^ = l/2. Knowing half intensity
temperatures Ti and T 2 , symmetry factor can be calculated from cq. (1). It has been
observed by Gartia et al [9,11] that Pf. is a function of u„ and b but the dependence of p^ on
h IS much more stronger than that on u„. This point has also been observed by Chen [ 10]
who instead of considering the dependence of p^, on considered its dependence on the
activation energy E and frequency factor s separately. Chen [10] has also presented a
graphical method for the determination of the order of kinetics (b) from symmetry factor p^,.
Moreover, both Gartia et al [I \] and Chen [10] considered values of b between 0.7 and 2.5.
But Kirsh [3] considered an order of kinetics range from 0.5 to 3.0. In the present paper, we
consider the range of b values between 0.4 and 4.0. For most of the observed TL peaks u„
ranges from 20 to 40. As a result, we have calculated the average p^^ for 20 40 for
each value of b between 0.4 and 4.0. Finally, the average Pg has been expressed as a
quadratic function of b (0.4 ^b^ 4.0) using the technique of non linear regression 1 14] as
Pg = 0.2453420 + 0.1858256 b - 0.02441 83 (5)
In Figure I, we exhibit the quadratic plot of Pg against b. Knowing Pg, b can be determined
from eq. (5). For a particular Pg, eq. (5) being quadratic in b will give two values of b. One
has to take the value of b which lies between 0.4 and 4.0. We feel that the present n)elhod
136
W Shambhunath Singh et al
for the determination of the. order of kinetics h is more objective than the graphical method
of Chen [lOj. We also see that for /? = 1 , -0.42 and for ^ = 2, -0.52 as expected.
04 13 22 31 40
Figure 1. A plot of average as a function of b illustrating their quadratic
relationship for 0.4 $ h < 4.0
Again following Gartia et al [9], it has been found that a good linear correlation
exists between the following pairs of variables :
/(«l -“»,))
and /[«„,(«: -« 2 )])
with w, = EjkT^ andwj = EjkT^ so that one can write
“« = /("l -“«) +^T- (6)
“« = Cg«2/(M„ -Hj) +0^. (7)
/«„(«, -Mj) +£>„, (8)
where the coefficients Cj and D, {j = T, ft (o) occurring in eqs. (6^8) are dependent on the
order of kinetics {h). The eqs. (6-8) can be recast in the following forms
Er = C,kTllr^D^kT„, (9)
Es = CgkTllS+D,kT^. (10)
E. = CJTllm^D^kT„, ( 11 )
where r=T„-T,, 5 = T, - r„ and o) = T, - f, .
Evaluation of the trapping parameters ofTL peaks etc
137
By using the method of least square regression [14] each of the coefficients C, and
D, can be expressed as a quadratic function of b
- ^0./ + + C2yh2; (12)
Dj = D^l+D^jb + D^.b^. (13)
The coefficients Q; and Djty (k = 0-2) are presented in Table 2.
Table 2 . Coefficients Cf^j and Dhj ik = 0 - 2 J = x, 5 , to) occurring in eqs ( 12 - 13 )
J
Coy
C|,
^2j
^0/
Oi/
^2/
T
0 6967
0 3869
-0.0390
-0 7419
-1 1511
0.0758
6
0.1593
0 6454
-0.0336
0 2080
-0 4920
-0.0707
0)
0.8561
1 0312
-0.0725
-0.5221
-0.6187
-0.0250
The present coefficients are somewhat different from those of Gartia et al [9]
because in the present work, a wider range of b values have been used. Now knowing h for
a particular from cq. (5), one can evaluate Er, and E<y from eqs. (9-11). Once the
activation energy and the order of kinetics are known, one can determine the frequency
factor .V from the relations (1-4).
.9 = /3[£/(*r„',)]exp(«„). (*=1) (14)
^ = Pl[h{kTi /£)exp(-«„, )-((>- 1) iexp|-£/«:r)]</7-), {b*\) (15)
T,<
where /I is the heating rate, Tq is the initial temperature. The integral J c\pl- E/(kT)]clT
appearing in eq. (15) cannot be solved analytically and therefore has been developed as
L, T,
j exp[-£/(*r)]dr= J exp[-£/(Jt7)lrfT- j txp [-EI{kT)\dT,
To 0 0
= (E/k)
J[exp (-«)/u^]</u - J[exp (-«)/u^ ]rfu
“O
= (£/*) {£3 («„,)-£j(«o)},
(16)
where u„, = E/kT„ and mq = ^ 2 («) have been calculated by following the technique
outlined by Mazumdar et al [12] and Gartia €tal[\\].
3. Results and dlsciission
The suitability of the present peak shape method has been judged by applying it to a
number numerically computed TL peaks. It is seen from Table 3 that there is a good
138
W Shambhunath Singh et al
agreement between the input and the computed values of trapping parameters. In Table 1 ,
the trapping parameters of TL peaks of multi activated phosphors recorded by Rao et al [5]
have been evaluated. As expected the values of activation energies are very much different
from those obtained by Rao et al [5]. The order of kinetics is not one as assumed by them
and the frequency factor s calculated using the average of ’ Es and Eqj ’ is also widely
different from the value 10^ assumed by them.
Table 3. Trapping parameters of some computer generated TL peaks by using the present
method.
Input values of
Calculated values of
E
(eV)
s
(r')
h
T
(“K)
b
(cV)
^8
(eV)
^ti>
(eV)
s
(s-')
1.6
lO'^
07
555.6
0.364
0.70
1 600
1.600
1.6(X)
1 04 X 10*"'
1.6
1.0
555 3
0417
1 07
1 610
1 676
1.639
243 xl|0’^
1 6
lo'-^
1.5
554.9
0 476
1 56
1 603
1 635
1 619
1 50x\o*“'
1 6
lo'^
20
5,54.5
0.517
1.97
1 ,593
I .578
1..586
7.35 X 1^’-
4. Conclusion
In the present paper, we extend the peak shape method of Gartia et al 19) and suggest a
formula for the direct computation of the order of kinetics from the observed value of the
symmetry factor. By applying the present method to TL peaks of multi activated SrS
phosphors reported by Rao et al (5J, it is found that the trapping paramctcrs^compuied by
them using some crude methods are widely different from that calculated by using the
present method.
Acknowledgments
The authors thank Professor R K Gartia for fruitful discussions. One of the authors (MB)
would like to thank the Council of Scientific and Industrial Research, India for providing
partial financial support. The authors are grateful to the referee for his critical comments on
the paper.
References
[ 1 1 R Chen and Y Kirsh Analysis of Thermally Stimulated Processes (Oxford Tergamon) Chap 6 ( 1 98 1 )
[2] McKeever SWS Thermolununescence of Solids (Cambridge : Cambridge University Press) ( 1 985)
[3] Y Kirsh Phys Stat Sol (a) 129 15 (1992)
[4J R K Gartia. S D Singh. P S Mazumdar and N C Deb Indian J. Phys. 71 A 95 (1997)
[5] A P Rao, V G Machwe and A S Mehta Indian J. Pure Appl. Phys. 34 937 ( 1 996)
[6] F Urbach Weiner Ber. lla 139 363 (1930)
[7] J T Randall and M H F Wilkins Proc. Roy. Soc. A184 364 ( 1 945)
Evaluation of the trapping parameters ofTL peaks etc
139
[8] C Christodoulides J. Phys. D18 ISOl (1985)
[9] R K Gaitia, S J Singh and P S Mazumdar Phys Stat. Sol. 114 407 (1989)
[ 10 ] R Chen J. Electrochem. Soc. 116 1254 (1969)
[ ] I ] R K Gaitia, S J Singh and P S Mazumdar Phys. Stat. Sol. {a) 106 291 ( 1 98B)
[12] PS Mazumdar, S J Singh and R K Gartia J Phys. D21 815 (1988)
1 13] M Abramowitz and 1 A Stefan Hand Book of Mathematical Functions (Dover : New York) Chap 5
(1965)
[14] E J Dudewicz and S N Mishra Modem Mathematical Statistics (New Yoric Wiley) Chap 14 (1988)
Indian J. Phys. 72A(2), 141-153 (1998)
UP A
— an international journal
Equilibrium forms of two uniformly charged drops
S A Sabry', S A Shalaby* and A M Abdel-Hafes**
‘Faculty of Wamcn, Ain Shams University, Department of Mathematics.
Cairo, Egypt
'''Faculty of Education, Ain Shams University, Deportment of Physics,
Cairo, Egypt
“Faculty of Engineering, Ain Shams University, Depanment of Physics
and Mathematics Engineering, Cairo, Egypt
Received 26 September 1996, accepted 7 July 1997
Abstract : The equilibrium form of two separate drops, assuming their forms to be
deformed spheroids, is considered. The saddle point shapes of a single drop, assuming it to be a
deformed form of two touching equal spheroids, are obtained. Numencal computations to get the
equilibrium form are carried out as illustrative example.
Keywords : Detonned ellipsoids, saddle point shapes, equilibrium forms
PACS No. : 03.26 +i
1. Introduction
In the context of fission and fusion of different nuclei, the problem of finding the
equilibrium forms of charged drops and description of the saddle point shapes have been
the subject of many authors [1-15]. In the field of heavy ion physics, this work based on
macroscopic models, (such as the liquid drop model), makes it possible to determine the
energy needed to overcome the interaction barrier between nuclei.
The description of saddle point shapes of a uniformly charged drop or rotating by a
dcforpied ellipsoid of revolution has been considered in previous works [1-3] by using a
number of deformation parameters about an ellipsoid of revolution. For small values of the
deformation parameter, when the neck thickness of the saddle point shape is small, the
description of the drop by one deformed ellipsoid fails and even the Swiatecki results [4]
are doubtful.
7 ^ 7
© 1998 lACS
142
S A Sabry, S A Shalaby and A M Abdel-Hafes
As an alternative one should consider two touching deformed ellipsoids to describe
the saddle point shape.
In this paper we shall first consider the equilibrium form of two separate drops,
assuming their forms to be deformed spheroids. Next we follow a similar method to find the
saddle point shape of a single drop by taking it to be a deformed form of two touching
spheroids. The trial is made taking into consideration all the possible deformation
parameters expressed through the two parameters Ob and a\ defined in the text. Moreover,
the mutual interaction between the distortions of the two neighbouring nuclei, and that
between the distortion of nuclei and the original ellipsoidc representing the other nuclei are
considered. This in turn, is expected to give better results in determining the equilibrium
form for the considered system.
2. Description of the method
It is required to find the equilibrium forms of two separated uniformly charged drops of the
same charge density p, and distance /i between their mass centers (Figure 1). Wc consider
for simplicity the forms to be axially symmetric about the line joining their mass centers.'
Figure 1. Diagram for the two unifonniy
charged drops in the form of spheroids
having a common symmetry axis.
Since for separated drops, they approximately lake the forms of oblate spheroids,
when they are far enough, we shall consider the forms to be slightly deformed ellipsoids of
revolution. This is also owing to the fact that this approximation worked well in finding the
saddle point shapes of a single drop [1] and for a rotating drop [2].
Using elliptic coordinates (m, v) to express the position of any point with respect to
either ellipsoids, the deformation of the surfaces is expressed by the following relations ;
r 9 ^
Ml = M.
(“o
(“o »
= «„(!+ 4(v)),
= «'(!+ 4'(v)).
( 1 )
The parameters in eq. (1) are considered small.
We shall express all energies in terms of the surface energy of a sphere having the
same volume as the sum of volumes of the two drops. Also we express the dimensions of
Equilibrium forms of two uniformly charged drops
143
the length in terms of the radius R of such a sphere. If a, b\ a', 6 'are axes of the original
ellipsoids (a, a' along the symmetry axis), then we have :
ah^ = V,
a'fc'2 = V' = l-V,
(Note that 1 /Wq = ]-h^la^ = \~V/a^\ ^ ^ - \-V' / a'^ ).
From the constancy of volume, and the position of the center of mass of each drop,
the deformation parameters ttp , a, (or ) can be expressed in terms of O 2 , ...
(or as follows :
= -T y ^ ,,
u ^ n.n n n
n,n'=2
3
«i = -4“o
n,n'=2
where
= 2(— — — dv
= •){'«. " '.2m)(«o Q^m («0 ) “ “0 22m *“o >) '
m=0
_ I v(3m5 -v2)
2 _ w 2^2 n" ' n
PJv)P 4v)dv
J (Uo^-v2)2
m=:0
Here the bracket <n, m, 1> stands for the integral
+1
(n,m,l) = Jf„(v)P„(v)/>,(v)rfv. (4)
-1
^nd , Q ' are the Legendre function of the second kind and their first derivative.
3. The total surface energy of the drops
The total energy ^ of the two deformed spheroids can be expressed as the sum of several
contributions (i-iii).
/ The total surface energy ^ ^ ' of the two drops :
can be expressed up to the second power in ^v) as :
+ Xcs(r.)a„ + i5;DS(n.n')a,a„.,
(5)
144
S A Sabry, S A Shalaby and AM Abdel-Hafes
where
j ^V(“o +«^'o2o('o))- (6)
^ _, V^“o ^
= fT^(")('oe,('o)-'oC:('o)).
1 ^ f
DS{n,n') = J
r 1
v“o 'o
5/n(>-'n) , , ,
P„ (v)/>„. (V) Ml - ^ ^ )/•:(<- (V)
and
= \-ul
a. The self coulomb energy of the two spheroids ' ) :
(Jj can be expressed up to second power in a„ as :
= C ■*• Icc(«)a„ + i^CC(«.«')a„a„.
where
:(0) _ (ab^)^
=
“oGoK)'
CC(n) - — ^2 mqQq(Uq Q “o22(“o^5^n.i j’
CC(n) = 0 for « = 1 , n > 2,
DC(n,n') = |a’fc^[u(,Qo(Mo) - UoS 2 (Up)Pj(Uo)]£^^,
+ ^“’k^(“o)e-K) + ^^«oe2(«o)
(7)
( 8 )
\(9)
( 10 )
( 11 )
( 12 )
Hi. The mutual potential energy between the two drops :
This is the sum of three contributions (a+fc-K:) :
(fl) The mutual potential energy between the original spheroids.
This is already given as :
ihzz'
4z'(5+z'-5z)
1 + z-z
z+z’+l
4z(5 + z-5z') ,
l + z'-z
w'Qq{w')
(14)
EquilWriumform of two uniformty charged drops
145
where
and
y'2 1
Z' =*'^ = -H-
y, = ilpl. ^
2* 2x' ^ 2xx'
-fr2, y'2 = a'2 -b'2.
( 15 )
All the obtained expressions are functions of «g and where = 0 ^ is
always positive.
{b) The mutual potential energy ^ ^ ^ ^ between an original ellipsoide and the
distortion of the other ;
For example, between the left ellipsoide (a\ b\ b') and the distortion of the right
ellipsoide we have :
+1
^md =2try’ -v^)(0(m',v'))„,„^ Mpd + {2«o(0(M',v'))n
+ (“o
^ v')"'
, du ^
\dv,
( 16 )
where ^ {u', v') is the potential of the left spheroid at any point (u\ v') outside, and is
expressed as :
5 aX^
y'
(^0^“'^ - G2(«')P2(v'))
( 17 )
The/two, v) is the expression after transforming w', v' to a = uq, v, the surface of the right
spheroid. Similarly,
du
An
^(Gi(«')-G2(«')^(v'))^
Sv' 5a'b'^ , ,
(i»)
In order to compute the integrals in eq. (16), we first use the relations between u', v'with
1 aspect to the left ellipsoid and the coordinates u=uo,v with respect to the right spheroid ;
v' = — r7(A + av),
y u
y'^u'^ = |('•+^),
r = (A + flv)2 + i^(l-v2) + y'2, (19)
and ^2 3 ^2 _ 4 y'^(/i + av)^.
146
S A Sabry, S A Shalaby and A M AbdeUHafes
y u
g(Uo,v) = — r-7[(Go(H') - “ Sw'Cj («'))(*»' + a)
y u
+ 7 (Gi(«') - G 2 («')/’ 2 (v') + 3u'Q^(u')ma + hv)
- 2y'%(/i + flv))|.
Now expressing ^ ^ expansion in , we obtain
( 20 )
( 21 )
Cm(n) = J/(ttj,v)PJv)rfv.
-1
Dm(n.n') =
( 22 )
On the other hand, the mutual potential energy between the right spheroid (a, b, b) and the
distortion of the left spheroid is
(23)
where given by the same expressions (22), on replacing Mq."* b, h
by UQ,a\b\-h.
(c ) The third contribution will be the mutual potential energy between the two
distortions.
This is obtained in the form :
+1
-1 «
1
n,n'
where 0 s(m', v') is the potential of a surface deformation defined by eq. (1) of the left
spheroid at any point outside and is given as
'2
4;r
( 25 )
Equilibrium forms of two uniformly charged drops
147
Expressing (26)
+1
wefind (27)
Thus for even n\
D^(n,n’) = \5a^a'^u;P^.{Uo)j^u'Q^.{u')P^,{v')P„{v)dv
and for odd n',
+1
= l5a’a'^u'J“^,(«')jj2^.(«')P^,(v')P^(v)di'. (28)
-I
Thus, the total energy of the considered system of two deformed spheroids expressed in
lemis ora 2 ^ can be written as :
^ + ^C(n)a„ + ^C'(/i)a; + ^ ^D(n.rt')a„a^,
n=2 n-2 n,n'^2
+ 1 ^D'(n.n»;;. + 1 ^D"(n,n')a„a;., (29)
/i.n'=2 n,n'-2
where 5'"’ = (30)
A' being the fissionality parameter, defined as half the ratio between the coulomb energy to
rhe surface energy of a sphere of volume equal the sum of volumes of the two drops and of
ihe same charge density.
Also C(ri) = CS(n) + 2X(CC(n) + Cm(n)),
C'(n) = CS'(n) -f 2X(CC'(/i) + Cm'fn)), (31)
D(w, n') = DSM{n, n') + 2X(DCM(n, n') + Dmm(n, n')),
D\n,n') = n') + 2X(DCM'(/2, n') + D'mm(/T, ')), (32)
and D"(n,n') = 2 XD„^,(n.n'). ^33)
The equilibrium form of the drop is obtained by minimizing the total energy ^ as
given by eq. (29) with respect to all its parameters. First we minimize with respect to the
5>niall deformation parameters •••.
For this we write eq. (29) including 'a domain comprising of all a, , a[ which we
(in short) express as . Eq. (36) is then equivalent to
( . (<« + xc'(.)«j *
(34)
148
S A Sabry, S A Shalaby and A M Abdel-Hafes
The values of a J corresponding to the minimum are obtained from the relations
or aj = -'^{DT(i,j)y'cJU), (35)
J
where ' is the reciprocal of the total matrix :
where the blocks D,D\D" are matrices whose elements are given in eqs. (31)-(33).
Substituting for these values of aT in eq. (34) we obtain the values of § corresponding to
the equilibrium values of aj as
'Seq = C^in)C^(n'). \(37)
n.n'
^eq thus a function of only two parameters a, a' and thus the equilibrium (or
saddle point shapes) corresponding to given values of a, a' are obtained by finding two
values of a, a' which make (a, a') minimum or maximum.
4. Computations for mirror symmetrical drops
In this paper we carry numerical computations to find the equilibrium shape of two separate
equal drops having mirror symmetry with respect to a plane perpendicular to the common
symmetry axis and bisecting the distance h between their main centers. In this special case
we set :
a = a', b = b\ = (-)" a„.
(38)
We thus have
^ = 4«» + ]£c(n)o„ + |
n=2 n.#i'=2
(39)
where ^ <«) = + 2x(2e> + C).
(40)
C(n) = 2CS{n) + 2X(2CC(n) + 2Cm(n)).
D(n, n) = 2DSM{n^n) + 2X{2DCM{n, «') + 2Dmm(n,n'))
The obtained results from which the equilibrium form can be drawn, taking into
consideration that the two drops are mirror symmetrical drops — are given in Table 1. The
two coordinates indicated in the table are x and y where x is the distance measured from the
Equilibrium forms of two uniformly charged drops 149
center of mass of one of the two equal mirror symmetrical drops along the symmetry axis
joining their centers and y is the corresponding distance measurec^, in a perpendicular
direction to the symmetry axis. The obtained equilibrium form is as shown in Figure 2.
Table 1. Computations for mirror symmetrical drops.
y\
-0.5621084 D
+
00
0.0000000 D
00
^).5337352D
+
00
0.4007342 D
■¥
00
-0.4887200 D
+
00
0.5585813 D
00
-0.4327050 D
+
00
0.6682536 D
+
00
-0.3702614 D
+
00
0.7480404 D
+
00
-0.3049863 D
00
0.8057159 D
+
00
-0.2395696 D
+
00
0.8460850 D
+
00
-0.1758252 D
+
00
0.8727363 D
00
-0.1 146992 D
+
00
0.8885413 D
+
00
-0.5629797 D
-
01
0.8957376 D
+
00
0.0000000 D
+
00
0.B958S06 D
+
00
0.5530602 D
-
01
0.8895987 D
+
00
0.1 108934 D
+
00
0.8768694 D
+
00
0.1678863 D
+
00
0.8567840 D
+
00
0.2270633 D
00
0.8277811 D
+
00
0.2887884 D
+
00
0,7876066 D
+
00
0.3530164 D
+
00
0.7330691 D
+
00
0.4193035 D
+
00
0.6593016 D
+
00
0.4867865 D
+
00
0.5576308 D
+
00
0.5541256 D
+
00
0.4071745 D
+
00
0.6194200 D
+
00
0.0000000 D
00
Figure X The fonn of separate drops at a distance A = I ISS between the
centers of mass.
72A(2)-8
150
S A Sabry, S A Shalahy and A M Abdel-Hafes
S. Two touching drops
In order to express the saddle point shape for a single drop when fissionality parameter X is
small, we consider the single drop as a deformation of two touching equal ellipsoids of
revolution and having mirror symmetry (for X = 0, the saddle point shape is two touching
equal spheres).
In this case, instead of applying the condition of constancy of the position of the
center of mass we apply the condition
^(-1) = 0 for the right ellipsoid, 1
zl(+l) = 0 for the left ellipsoid. J
This can be achieved on using the expansion
(43)
0 fi=l
where S„(»') = t(/’,(v) + /’„.,(v)) for the right spheroid, \
1
= 2 (“^n ^n-\ spheroid.
The condition for the invariance of volume then becomes
P\ = -T X
em(n, n') = 4 j
75„(v)5^,(v).
-iK-'')
Evaluating as before, the surface energy, the coulomb energy, and the mutual potential
energy between the two deformed touching spheroid drops one can get finally these
quantities expressed in terms of as :
where DSm.j) = DS(i.j) - ^C^(i)em(i,j). (47)
In this case (for each spheroid)
CS(n) = j
.(2«„^-1-v2)5„(v),
DS(n, n') = j ^ j (v)S,, (
-I “o >' K-v^) J
+ (l-v^)r(v)S'.(v)-
(49)
EquiUbriumform <^two untformly charged drops 151
And em (i,j) is as given by cq. (45).
Also we have. ^,DCM(n, «')/»„ P,., (50)
where DCM(n,n') = DC(n,n')-^CC(l)em(n,n'). (51)
The coefficients of expansions are in this case :
+1
CC(n)= (52)
-I
DC(n,n’) = ja^b^[uQQf,(uQ) - UgP^(u^)Q^(u^)\em(n,n')
+1
+ ■J“oC2 (“o J 5, (*’)S„.(v)dv
+ T“’ "o (“o )G, (“O ) J^n (V)rfv
+l
+ “o^,-|(“o)Cn-|(“o) j^»-|('')5„'('')dv . (53)
-1
6. The mutual potential energy between an original spheroid (the left one) and the
distortion of the right touching spheroid
Following eq. (21), we obtain in this case,
r r
^nui =2!ta^j‘lv X^"'^»(‘')(^(“'’’''>)o +|20(«',v')o
(«o-''')f^0(«'.v')'l 1 1 ul V o o . , .o , .
= ^Cm(n)^, + iY^Dm(n,n')P^P^.,
where Cm(n) = 2na^ j (0(M^ v'))j^5^(v)dv,
-1
Dm(n. n') = 2za^ j 1 20(«'. ^
■SAv)S.{v)dv.
X
(55)
152
S A Sabry, S A Shataby and A M AbdeUBitfes
3,0(u \ V
As v') and ^ given by cqs. (17), (18) we find in this case :
Cm(A) = |/(uQ,v)5Jv)rfv,
-I
+1
Dm(n,n') = ^a^a’b'^ J
(“o
- »') + 2ul y'-^
Finally, the mutual potential energy between the two distributions in this case is
(56)
-1 - \
where 0^ (u \ v') (for left spheroid) in the considered case of two touching spheroids is
Itpfesented as :
0,(“'.‘")= (58)
' \ 1
Thus.
(59)
Where DMD(n,n) - ^ j5Jv)rfv[-M' )G„,( m')P^,(v')
n.n' _i
+“i^„'.,(“i)e„'-,(“J)p-.,(v')]- ( 60 )
Similarly, on substituting for (or P[) in terms of ^2 * ^3 * " ' P '2 * ‘
[eq. (54)], the following expression in terms of , ^ 3 , ■ - :
n5=2
where now DMM(n,n') = DM(n, n') - —Cm(l)em(n,n').
Adding the expressions for and one gets the total energy of the system.
The case of mirror reflection is obtained by setting as before ;
( 61 )
(62)
a^a\b^b\ P: =(-)"i9„.
In order to find in this case the saddle point shape, the same procedure as described in the
case of two separate drops will be followed.
Eguilibrium forms of two un^mly charged drops
153
Rcfcrencfs
( I ] A Sabry Fhysica 101 A 223 ( 1 980)
[2] A Sabry Workshop II, U.IA , Antwerp 251 (1980)
[3] F Abu- Alia PhD Thesis (Ain Shams University, Egypt) ( 1 989)
[ 4 ] S Cohen and W J Swiatecki Ann. Phys. (N.Y.) 72 406 (1963)
[5] LonlReleighF/u/.^nj?.2816t(1914)
[6] S Chandrasekhar Proc Roc. Soc. (London) A286 I (1965)
[7] G Lcandcr Nurl Phys. A219 245 ( 1974)
[8] W J Swiatecki Phy.s Rev. 101 651 (1956)
[9] N Caiman and M Kaplan Phys. Rev. C45 2185 (1992)
[10] N Carjan and J M Alexander Phys. Rev. C38 1692 (1988)
[III A J Sierk Phys Rev. C33 2039 (1986)
1 12] K T R Davies and A J Sierk Phys. Rev. C31 915 (1985)
[13] N Caijnn. A J Sierk and J R Nix Nucl. Phys. A452 381 (1986)
[14] J P Lcstonc Phys. Rev Lett 67 1078 (1991 )
[15] T Wada, Y Abe and N Caijan Phys. Rev. Lett. 70 3538 (1993)
Indian J. Phys. 72A(2), 155-160 (1998)
UP A
m imeiMlioMl journal
Measurements of flux and dose distributions of
neutrons in graphite matrices using LR-115
nuclear track detector
Y S Selim, A F Hafez and M M Abdel-Meguid
Department of Physics. Faculty of Science, Alexandria University,
21511 Alexandria, Egypt
Received 5 August 1 997, accepted 2 January 1998
Abstract : Attenuation of fission neutrons has been studied in graphite blocks of
different dimensions. Fast, epithermal and thermal neutron fluxes were measured at different
depths inside graphite blocks. The fast neutron group was measured using bare LR-llS
cellulose nitrate nuclear track detector, while fast, epithermal and thermal neutron groups
were obtained using LR>1IS type B detector. But, the fast and epithermal neutron groups
were detected using LR-115 type B detector which was shielded against thermal neutron flux
by two 1 mm thick Cd-foils. Moreover, the build up factors of the three neutron groups and
the fast neutron absorbed dose rate distribution inside the graphite medium were calculated
os well.
Keywords : LR-115 solid state nuclear track detectors, graphite moderator, ^^^Cf neutron
source
PACS Nos. : 28.20.Fc, 29.40. Wk. 87.53.Pb
1. Introduction
As is well known, the measurement of the neutron flux distribution in some media is very
important in many fields such as radiation protection dosimetry, neutron therapy, shielding
siudies and nuclear reactors construction.
The study of flux and absorbed dose distributions of neutrons in graphite is of great
importance, since graphite is often used as a reflector, moderator, or shield component of
the fast neutrons in fusion and fission reactors [1,2].
The application of solid state nuclear track detectors (SSNTDs) to neutron dosimetry
is increasing mainly due to their high degree of reproducibility, long-term stability,
insensitivity to beta and gamma rays, low cost and the ease to handle [3].
61998IACS
156
y S SelinU, A F Hafez arid M M AbdeUMeguid
One of the most appropriate SSNTDs for dosimetric applications is cellulose nitrate
LR-1 15 type n (Kodak-Path6, France) nuclear track detector.
This detector can be used in two modes, either as a bare detector, or with an
external converter. In the case of a bare detector exposed to fast neutrons, the neutrons
can undergo elastic collision and (n, oO reactions with C, N and O nuclei constituting
the detector material. In the second mode, the detector is covered with (n, converter
for the detection of thermal and epithermal neutrons, the most suitable (n, oO convener
is the lithium tetraborate (Li 2 B 407 ), for which the (n, a) cross section is very large (3840
bams for and 950 bams for ^Li) [4]. The alpha particles and the recoil nuclei
resulting from these reactions leave etchable tracks in LR-1 15 detector. If the tracks are
completely etched through the sensitive cellulose nitrate layer, the tracks appear as
bright holes [5-7].
Some Measurements on flux and depth dose distributions of neutrons inside water
phantom using SSNTDs have been done. For example, using CR-39 polymeric t^uclear
track detector and D-T neutron source [8]. Sayed and Adnan [9] also used LR-1 15 ^pe 11
cellulose nitrate nuclear track detector to measure fast neutron depth-dose distriliiution
inside water phantom from Am-Be neutron source.
The aim of the present work is to measure the flux distribution of fast, epithermal
and thermal neutrons in blocks of graphite of different dimensions. The detector used in this
investigation was LR-1 15 nuclear track detector.
2. Experimental methods
The investigations were carried out using ^^^Cf neutron source of average energy
2.16 MeV. The detector used was cellulose nitrate sheet, LR-l 15 type II (C 6 HBCyM 2 )
density of 1.52 g cm"^ and 12-13 pm thick on 100 pm polyester base. The detectors used
were obtained by cutting the cellulose nitrate sheets into pieces of size 1 .5 x 1 .5 cm. These
detectors were classified into three groups. The first, was the bare group detectors which
were used to detect only the fast neutrons. In this case the track holes of a-particles that are
produced through the (n, a) ‘*B reaction whose (2- value is -0.16 MeV only can be
formed and observed in the LR-1 15 detector because the threshold neutron energies for
(n, a) ^Be and (n, a) are greater than the. energy of the ^^^f neutron source.
The second group of detectors was coated with Li 2 B 407 as (n, a) converter (LR-1 15
type B) to detect the fluxes of the fast, epithermal and thermal neutrons, while the third
group was LR-1 15 type B detectors too, but was shielded for thermal neutron flux by two
I mm thick Cd-foils. By this way, all neutrons below a cut-off energy, Eej, of about 0.5 eV
are absorbed while all neutrons above this energy pass the cadmium foil without
appreciable capture.
The medium under investigation was consisting of 100-200 graphite blocks, each
block of dimensions 5x5x1 cm. This medium was considered to be composed mainly of
carbon of density 1.6 g cm'^. The graphite blocks were arranged in different parallel
rectangular shapes of dimensions A x B x C cm.
157
Measurements of flux and dose distributions of neutrons etc
The assemblies of the three detector groups were placed axially within the graphite
blocks. The firat assembly of detectors wm put in front of the neutrdn source at a distance
of about 0.5 cm. l)uc to the low neutrons yield (^lO* n.ser*) from the closed source
the experimental arrangement was left for different exposure times lasted from 10 to 20
days, this experirnental arrangement was located 1 m above the floor in the center of an
irradiation roorh of dimensions 8mx5mx4min order to avoid the contribution of
neutrons scattered from the floor and surroundings.
After irradiation, the detectors were removed and cleaned with running water and
then dried. The LR-1 15 detectors were etched chemically in 2.5 M NaOH solution at 60®C
for two hours to give an average residual thickness of cellulose nitrate red layer of about
5-6 um. After etching, the detectors were washed in distilled water and treated with B
solution (50 cm^ distilled water + 50 cm^ ethyl alcohol) then washed by distilled water
again and dried, the (detectors were scanned and the track density was evaluated using an
optical microscope with magnification 500x.
The neutron flux 0 was calculated through the relation :
where p is the net track density (track-holes/cm ^), / is the exposure time in seconds and S is
the neutron sensitivity (tracks/neutron), this sensitivity was found to depend on the neutron
energy and the residual thickness of LR-l 15 detector. For ^^^Cf neutron source, the value of
.S was calculated by us and found to be (2.2 ± 0.7) x 10“^ iracks/neutron according to the
equation given by Medvecky [10]. this value of S was found in good agreement with the
experimental results obtained by Sawamura and Yamazaki [11]. Whereas for thermal and
epithermal neutrons the values of the neutron sensitivity S were (4.5 ± 1) x IQ"^ and (1 ±
0.3) X 10^^ tracks/neutron, respectively as reported by P^lfalvi [12].
It was observed that, within a bad geometry medium, the flux of neutrons of a
certain energy E and at a certain depth r is somewhat higher than that expected from the
exponential absorption law, i.e. ah excess of neutrons is built up at this depth. Thus, the
■ M ' ' ' , ' ' I i ■ ''U I ' . ' '
expected neutron flux at a depth r inside the medium can be given by Selim et o/ [ 13] :
0(£,/-) = B(E.r). 0o(£)exp(-r/A). (2)
where ^(K) is the initial flux of neutrons at energy E, X is the relaxation length and E(E, r)
'S the build up factor.
The fast neutron absorbed dose rate (Gys~') was calculated using the formula [14] :
D = 1.6xlp->3 0feA/cSr, (3)
where 0 is the iheasiired neutron flux, E = 2.16 MeV is the average neutron energy of
neutron source (15]^ /V = 5 x 10^^ nuclei/kg for '^c, C7= 1.7 barn is the total cross
section of g^d /= 0.142 (/*= 2A/(A + 1)^. where A is the atomic mass of the nucleus
that received the Cinergy transferred) is the fraction average energy transfer to scattered
nuclei [14]^
158
r S Si?Um, A F Hafez am! M M Abdel- Meguid
3. Remits
Figure I shows the variation of the measured fast natron flux with depth in rectangular
graphite blocks of different dimensions.
Depth r (cm)
Depth r (cm)
Figure 1. The variation of fast neutron flux
with the depth in graphite blocks of different
dimensions.
Figure 2. The variation of epithermal
neutron flux with the depth in graphite
blocks of different dimensions.
The thermal and epithermal neutron flux distributions determined by the LR-115
detectors at various depths in the considered medium are represented in Figures 2 and 3.
Depth r (cm)
Figure 3. The variation of thermal neutron
flux with the depth inside graphite blocks of
different diinen.sions.
Depth r (cm)
Figure 4. Measured build-up factors of fast
neutron in different orran^ments of graphite
blocks.
Measurements afflux and dose distributions of neutrons etc
159
Based on eq. (1) the build up factors for fast, epithermal and 'thermal neutrons were
detcnnine(} from the ratio of the measured neutron flux to the theoretical neutron flux ie. in
the absence of the build up factor. In this work, we calculated the values of the relaxation
lengths and were found to be 7.76, 2.76 and 2.94 cm for fast, thermal and* epithermal
neutrons, respectively [161.
Figure 4, shows the variation of the build up factor as a function of the depth inside
the graphite blocks for fast neutron.
<a) (b)
Figures 5(a, b). The vanation of fast neutron dose rate as a function of the
depth inside the graphite blocks.
Figures 5(a,b) represent the variation of fast neutron dose rate, as calculated from
eq. (3) with depth in the medium under study.
4. Discussion
LR-115 cellulose nitrate nuclear track detector was used to measure fast, epithermal
and thermal neutron flux distributions from ^^^Cf source in graphite blocks. The influence
of the medium size on the distribution of the three neutron energy groups were carried
out as well.
It is clear from Figure 1 that the fast neutron flux in the different arrangements of the
graphite blocks decreases with increasing the depth inside the attenuation medium. On the
other hand, the flux distribution of thennal and epithermal neutrons in the graphite matrices
show maximum values around 3 cm inside the graphite blocks as shown in Figures 2 and 3.
This means that the neutrons are accumulated in this domain. So the measurements enabled
us to study an important factor known as the build up factor. In Figure 4 it was noticed that
the build up factor increases with depths in the medium under study, then it may exhibit a
iiaturation that depends on the dimensions of the medium, its constituents, the leakage of the
neutrons and their energy.
The results obtained in this article also enabled us to calculate the distribution of the
fast neutron dose rate in different blocks of graphite as represented in Figures 5(a,b). These
results are in good agreement with previous published data [12,17-19].
I uv
I j ^ciirii, n r fiu/c«. imu iri in nvuv^t'/ric^Miu
References
[ 1 ) J R L Lamarsh Introduction to Nuclear Engineering (N^ Yoik ; Addisoo-Wesley f^biishui Company)
(1975)
[2] A E Proflo Radiation Shielding and Dosimetry (New Yoi^c ; John Wiley & Inc.) (1979)
[3] S A Durrani and R K Bull Solid State Nuclear Track Detection, Principles, Methods and Application
(Oxford ; Pergamon Press) (1987)
[4] G Dajko and G Somogyi Techniques of Radiation Dosimetry Solid State Nuclear Track Dosimetry
Chapter 11 (New Delhi : Wiley Eastern Limited) (1985)
[5] Y Cheng, J Lin, B Zhang, L Zheng and R Lu Nucl Instrum. Meth. B52 68 (1990)
[6] A Dragu Nucl. Tracks Radial Meas. 19 461 (1990)
[7] A F Hafez and G 1 Khalil Nucl. Instrum. Meth. BM 107 (1994)
[8] S E El-Chennawi PhD Thesis (Faculty of Science, Alexandria University. Egypt) (1989)
[9] A M Sayed and N R Adnan 2nd National Conference on Pure and Applied Biophysics Sciences and 1st
Egyptian British Radiation Protection Symposium. (Cairo, Egypt ) (1985)
[10] L Medveezky Acta Physica Academiae ScienSarum Hungaricae 52 357 ( 1 982)
[.II] T Sawamura and H Yamozaki Nucl. Tracks. 5 271 (1981)
[12] JPklfalviA/uc/. 7'racLt9 47(19B4) \
[13] Y S Selim, A M El-Khatib, W M Abou-Taleb and M A Fawzy The Arabian J. Set Engg. 8 377 (1983)
[14] H Cember Introduction to Health Physics (Oxford, U.K. : Pergamon) (1985)
[15] FHFr6hner/V«c/.Sd.£/igg. 106 345(1990)
[16] D Ecullen and P K Mclaughlin International Atomic Energy Agency (IAEA) NDs Vienna ( 1 985)
[17] R A Rashed and M A Ibrahim Egypi J. Phys. 12 135 (1981)
[18] AM El-Khatib, A Abdel-Naby, A F Hafez and M A Kotb J. Medical Res. Insl Egypt 13 133 (1992)
[19] W A Kansouh PhD Thesis (Faculty of Science, Ain Shams University, Egypt) (1996)
Indian J‘ rnys. lA^ Klh lOl-lW (1998)
UP A
— an igterartifflial jounwd
Supf^iPUi^eti^uE^SchK ei|iiatioii ^
Fennipiir-DyQn system
B S Rajput and V P Pwdey
Department of Physics, Kumaun University,
Nainital-263 001, India
Received 24 September 1997, accepted 25 Navendter 1997
Abstract : Supersymmetrized SchrOdinger equation for Fpnnion-Dyon system hps been
obtained by dimensional reduction of supersymmetrized four-dimensional harmonic oscillator
and it has been inteipreted as an ensemble of two Schrddtn£cr And one Pauli’s equation each
describing the motion of an electrically charged particle in the field of a Dyon with different
magnetic charges.
Keywords : Schrfidingcr equation, supersymmetry, Fermion-Dyon system
PACS Nos. : 1 1 .30.Pb. 03.65.C3c
1. Introduction
Monopoles and dyons became the intrinsic parts of all current grand unified theories [1]
with the enormous potential importance in connection with their role in catalyzing proton
dacay [2,3], the quark confinement problem of QCD [4,5] and RCD [6,9] and CP- violation
in terms of non-zero vacuum angle [10]. The dyon-fermion dynamics has been worked out
by various authors [2, 1 1-13] and it has been shown that the ncuure of dyons is strongly
perturbed by fermionic sector which couples with them. In our recent paper [13], wc have
undertaken the study of dyon-fermion bound states and showed that in dyon-fermion
system the fermion moves on a cone with its apex at the dyon and axis along its angular
momentum. It has also been shown that the exact solution of Dirac equation for such a
system is not possible due to the presence of terms vanishing more rapidly than r* in the
potential of the system.
Keeping in view the symmetry of Schrddinger equation for a fermion in the field of
monopole [14,15] and the difficulties faced [16] in supersymmetrizing the Pauli’s equation
© 1998 lACS
162
B S Rajput and V P Pandey
of a fermion in the field of dyon, in the present paper, we try to obtain the
supersymmetrized solution of Schrddinger equation of fermion in dyonic field by the
dimensional reduction of supersymmetrized generalized four-dimensional Harmonic
oscillator that we can derive the modifications in our earlier results of eigen values and
eigen functions of fermion-dyon system as the result of supersymmetrization. The
supersymmetrized Schrodinger equation for this system has been interpreted as describing
the quantum dynamics of a supermultplet of two spin-0 and one spin-l/2 particles in a
Coulomb field. It has also been interpreted as an ensemble of two Schrodinger and one
Pauli’s equation each describing the motion of an electrically charged particle in the field of
a dyon with different magnetic charges.
2. Dimensional reduction of four-dimensional harmonic oscillator to fermion-
dyon system
Let us consider the motion of a fermion of mass (m = l/2) and charge in the field of a
dyon carrying generalized charge
in terms of electric and magnetic charges €2 andg 2 respectively. Its Schrodinger equation
may be written as [13]
(2.1)
where
H -n2
" W w’
(2.2)
with
II
1
K = e^e,,
(2.3)
P = e^g^.
In eq. (2.3), V is the potential of the field of dyon. Rescaling by a(r =
be written as follows :
ca).eq. (2.1) may
(2.4)
with
=(rv^ -r + PVr),
for a =
and
= (rv7 +r +
for a = ^JE ,
(2.5)
where
<
II
1
(2.6)
Equation (2.4) may be written in the following specific form :
wi,’'V=[?(pt + -^ + r]v^= f fp-
(2.7)
Supersymmetrized Schrddinger equation etc
163
The corresponding angular momentum operator may be written as follows :
+ (2.8)
with = 1(1 + /) + =/'(/' + !). (2.9)
Solution of eq. (2.7) has been obtained in the following form in our earlier paper 1 1 3] :
(2.>o)
where ^ 0) are the spherical harmonics for a fermion-dyon system and u{r) is
given as follows in terms of confluent hyper-geometric functions :
with
u(r) = (ar)" exp {-ar)F^a - 2a, 2ar
a = -j + [(/'+ 1/2)^ - (<’|S2)^]'^^.
( 2 . 11 )
(2.11a)
The corresponding bound stale energy of fermion-dyon system has been obtained in the
tollowing form [13] :
(e,Cj)2
ll'(l' + l)-efg^ + 1+ 7 + "
( 2 . 12 )
where n is an integer and we have chosen fermionic mass m = 1 /2, Equation (2.4) is
equivalent to the Schrddinger equation for the four-dimensional oscillator [141
HV=^¥. (2.13)
wilh the constraint
X\f/ = -i-^y/ = Pij/ (2.14)
a(o
where
H= -r
’ 1
1
r2
[dr dr\
sine de[ 96) sin^ 0dip\
+
^2
+ 2cos0
do)^
- I
(2.15)
Solution of eq. (2.14) may be written as
^ = V'(r, e, 0. (0) = (7, 6, 0), (2.16)
^here o) is the angular velocity in the domain (0, 4;^. The condition for single- valuedness
ol 0 requires that P should be quantized in half integral units :
p 1
^ = «|S2 =
t64
B S Rajput and V P Paridey
The system described by eqs. (2.13) and (2!l4) possesses S<)(4, 2) ^pectram
symmetry. The angular momentum operator corresponding to Hamiltonian of eq. (2.1-5)
operates through the following relation
£>= exp[/PtB]t,(}r)v^. (2.17)
where
= e _ P.
y. - S.
rsin
in^d ’
(2.18)
showing that the projected SU(2) generators have a P-dependent contribution.
Choosing
’*'l)
or in spherical coordinates
V, = 0, V, =0, = -g^cosB. \
as the potential for a monopole of strength gi, this of eq. (2.18) may rcaiily be
recasted in the form given by eq. (2.8) for fermion-dyon system. Then Hamiltonian of
eq. (2.13) for four-dimensional harmonic oscillator reduces to that of eq. (2.7) for the
motion of a fermion through the field of a dyon. This reduction takes place under the
projection
k:R^ xS^ (2.19)
imposing the mapping
such that ~ \^\ 1^ I ^2 1^ = r> 0,
( 2 . 20 )
( 2 . 21 )
( 2 . 22 )
U I
Taking a 2-sphere of radius r and projecting the point r— of C -> 5^ through
north pole, we get
Zj ■ r-rj
or ^ = »,(Zo) =Z.<Ti*Zj,
which is Kustaanheiitio-Stiefel (K~S) trarisfoitnatlbn with o; as standard Pauli matrices.
Hiks projection defines i principal fibre bundle
M-' R^ xS^
with 1/(1) as the structural group.
Supersymmetrized Schrddinger equation etc
165
3. Supersymnietrized Fermion-Dyon system
In order to supersymmetrizc the system described by eq. (2.7), let lis start with the
supersymmetrization of generalized four-dimensional harmonic oscillator described by
cq. (2.13) and then perform the dimensional reduction through eqs. (2.19) and (2.20). To
meet this end, let us choose the supercharges
e =
(3.1)
(3.2)
where IV is a real function of Zan Z and r\ satisfies the following Clifford algebra ;
(3.3)
{t]^ , = 2Sab fora, 6=1,2.
Choosing W(Z, Z) = A In (Z^^ Z^ ),
where X e R, the domain of Clifford algebra of eq. (3.3), we get
= ~{Q.Q*}
where
+ ■=A—(.X-C)-2X-
Z..a'Z,S.
c = z.
7 If ^-1
^7 ^ dZ 4 J
and
(3.4)
(3.5)
(3.6)
(3.7)
This // has a n = 2 conformal supersymmetry. In addition to the operators Q, and H, let
us also construct the following operators :
and
D=^.
z -i-+z -L
+ 2
S=‘Z„T]^,
(3.8)
72A(2).|o
166
B S Rajput and V P Pandey
These generators satisfy the Osp (2.1). structure relation and remain invariant under the
SU(2) action generated by the operators
Operator C, given by eq. (3.6), commutes with all the Osp (2,1) generators and also with J.
As such, the full invariant algebra of this problem is Osp (2,1) © SU(2) © U(l). Within this
algebra, the Hamiltonian of eq. (3.5) for supersymmetried four-dimensional harmonic
oscillator may be generalized in the form of the following operator
R = = H + (3.10)
The constraint (2. 14) may then be written in the following form
= I (3.11)
showing that every component of ylr transforms according to the same U(l) represeniation
But the generators (?, (t, S and 5^ do not commute with x hence the supersymmetry
of R will be lost under the projection involving this constraint. Thus, we modify this
constraint to the following equivalent condition
eijf = IPy/, (3.12)
where the operator C, given by eq. (3.6) commutes with every element of Osp (2,1)
© SU(2) algebra and it will affect the projection without breaking any symmetry of the
four-dimensional system. Condition (3.12) may be understood as the supersymmetrized
version of the constraint (2.14). Then eq. (2.16) may be obtained in the following
supersymmetrized form
(^- (3 >3)
where X = ?
Then eq. (2.17) is supersymmetrized into the following form
I ^(r, «.♦,«) = «p ' [p - V^r.B.0y <3.15)
where J. is SU(2) generator given by eq. (3.9) and is given by the following
supersymmetrized form of eq. (2.8) :
(3.16)
with
(3.17)
and as the potential of a Dirac monopole of unit strength. In eq, (3.16) 5. is the spin
matrix given by eq. (3.7) with
'll =
*12 = ^‘[y' +'yM' (3>8)
Choosing chiral basis
(3.19)
which is fully reducible form showing that the system under consideration comprises two
spin-0 and one spin- 1/2 particles. Then equation
reduces to the following supersymmelrized version of eq. (2.7)
(3.20)
where = r[/7, - (P-\/2Z)V°f + r
(X-P)^ -PL+ \/AZ^ 2Xr‘S‘
^ r " r2 ■ ■
Rescaling by a, eq. (3.20) may be written into the form given by eq. (2.1) where
«D = [P, -K/\x\
[(A-?)2 -PX+ 1/4X2]
(3.22)
The spectrum of this hamiltonian may be obtained in the form of following supersymmetric
generalization of eq. (2.12)
£ =
e}e^
4[yjjU + l)-efgl + l/4(^-A)2 - l/m-x)a+ 1/2 +«]]
= E
n.J,a,x'
(3.23)
where x stands for chirality, j{j + 1) gives the eigen values of the operator with J given
by eq. (3.16) and a denotes the eigen values of the operator A given by
A = i[Q.S^]^i[Q\S]. (3.24)
The eigen functions corresponding to these energy eigen values (3.23) satisfy the following
equations :
J'^\i,m,a,X,n) = i(]*\)\j,m,a,x,n),
ij|;, m, a,;f. n) = n^j,m,a,x>f^.
^\j,m,a,x.n) = a\j,m,a,x,n).
y ’ m, a, X, n) = x\j> X’ ")■ '3-25)
The eigen slates [j, m, a, x * «) obviously belong to a representation of Osp (2,1) 0 SU(2)
Setting A= P, the two lower components of eq. (3.22) become
H, =(p-e,V)2 -
1*1
i*r
(3.26)
which is the Hamiltomian of Pauli’s equation for the spin -.1/2 particle of charge e\ (and
mass = 1/2) in the field of dyon carrying electric and magnetic charges £*2 and g 2
respectively. For this case, eq. (3.23) reduces to
^2
n,j,a
4[V/(; + l)-e?«| + 1/4+ l/2(l-a) + «]]^
(3.27)
which is Pauli’s generalization of eq. (2.12).
For Hjy given by eq. (3.22), the supersymmetric equation for the fermion moving in
the field of a dyon may be written as follows :
(3.28)
where eigen values E are given by eq. (3.23) and the corresponding eigen functions satisfy
eq. (3.25).
uttyc/j/mmctrtica ocnroouiger equotton etc
169
4. Conclusion
The eq. (3.28) can be interpreted as descrilnn^ the quantum dynamics of a supermultiplet of
two spin-0 and one spin- 1/2 particles in a Coloumb field. It can also be looked at as an
ensemble of two Schrddinger and one Pauli equation each describing the motion of a
particle with electric charge-e 2 in the field of dyon with electric charge and with magnetic
charges respectively equal to - \l2)le^ . {e^g^ +l/2)/e, andgj- Taking as
the electric charge of the point particle sitting at the origin, we can fix the electric charge of
the supermultiplet to be ^ 2 . From the coupling to the potential VP , we see that the spin-0
particles must be assigned magnetic charges equal to g 2 ± l/2ei while the spin- 1/2 particle
will have its magnetic charge equal to g 2 -
References
[1] CP Dokos and T N Tomaras Phys. Rev. D21 2940 (1980)
[2] C G Callan (Jr) Phys. Rev. D26 2058 (1982); D252141 (1982)
1 3 1 V A Rubakov JETP. Lett. 33 645 ( 1 98 1 ); Nucl. Phys. B203 3 1 1 ( 1 982)
[41 S Mandelstam Phys. Rep. C23 245 ( 1 976); Phys. Rev. D19 249 (1979)
1 51 G t Hooft Nucl. Phys. B138 I (1978)
|6J B S Rajput, IMS Rana and H C Chandola Prog Theor. Phys. 82 153 (1989); Can. J. Phys. 69 1441
(1991)
|7| B S Rajput, H C Chandola and S Sah It. Nuovo Cm. 106A 509 (1993)
I8| B S Rajput, H C Chandola and J M S Rana Phys. Rev. D43 3550 (1991)
|9| B .S Rajput and J M S Rana Int J. Theor. Phys. 32 357 (1993)
[101 E Witten Phys. Utt. B86 283 (1979)
[11] B S Rajput, H C Chandola and R Bist ll. Nuovo. Cim. 104A 697 (1991)
1 12] B S Rajput. V P Pandey and H C Chandola ll Nuovo Cim. 102A 1507 (1990), Can J. Phys. 67 1002
(1989)
1 13] B S Rajput and V P Pandey ll. Nuovo Cim. 11 IB 275 (1996)
1 14] E D'Hoker and L Vinet Nucl. Phys B260 79 (1985)
[ 1 5] S Durand, J M Lina and L Vinet Utt. Math. Phys. 17 289 ( 1 989)
[16] E D'Hoker and L Vinet Comm. Math. Phys. 97 341 (1985); Phys. Rev. Utt 55 1043 (1985)
APRIL 1998, Vol 72, No. 2
Review
Effect of rain on millimeter-wave propagation — a review
Rajasri Sen and M P Singh
Astrophysics, Atmospheric & Space Physics
A comparative study of thermodynamic nature of the atmosphere at
Dum Dum, Calcutta (22.38’’ N. 88.28° E) on thunderstorm and
favourable fair-weather days
Sarbari Ghosh, Anandamoy Manna and Utpal Kumar De
An analysis of lightning channels, charge structure and associated
atmospheric radio noise
ABBHATTACHARYYA,MKCHATrERJEE AND R BHATTACHARYA
General Physics
Dipole moment and molecular polarization of some alcohols in
carbon-tetrachloride solutions
S L Abd-El Messieh, A L G Saad and K N Abd-El-Nour
Optics <£ Spectroscopy
The infrared and Raman investigation of fumaronitrile and
maleonitrile
J Marshell
Statistical Physics, Biophysics di Complex Systems
Enzyme crystal structure in an organic solvent
SujataSharma and Tei Singh
Notes
Association of large geomagnetic storms with solar flares during
solar cycle 22
SCDubey and APMishra
Study of intermolecular interactions in binary liquid mixture by
ultrasonic velocity measurements
PLRMPalaniappan, APichaimuthu and ANKannappan
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f5] U Fano and A R P Rao Atomic Collmons and Spectra (New York Academic) Vol 1, Ch 2, Sec 4.
p 25 (1986)
[7] T Atsumi, T Isihora, M Koyama and M Matsuzawa Phys. Rev. A42 6391 (1990)
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eds T A Canson. M O Krause and S Manson (New York : AlP) p 846 (1990)
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international school on powder distraction
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Diffraction’ :
• Stale-of-art of Powder Diffraction by X-ray, Neutron and Synchrotron radiations
0 Instrumentation & Data acquisition
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0 Structure Solution from Powders
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0 Phase Identifications & Quantitative Studies
0 Powder Data Base of ICDD (USA)
' 0 Applications in Materials Science-Epitaxial Films, Multilayers & Surface Structures from
Glancing Angle Measurements
Speakers
Speakers of this School will be drawn from various countries including India. A tentative
list is as follows :
R A Young (USA), R J Cemik (UK), L B McCusker (Switzerland) I G R Tellgren (Sweden),
Ron Jenkins (USA), F Izumi (Japan), H Toraya (Japan), R J Hill (Australia), E J Mittemeijer
(Germany).
K Lai (NPL, India), S Lelc and 0 N Srivastava (BHU, India), T N Guru Row (IISC,
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(SINP, India), S P Sen Gupta (lACS, India).
Technical Lectures from Philips (Holland), Enraf-Nonius (Holland), Rigaku (Japan), MRC
(LSA), Huber (Germany) are being arranged.
Professor EN Baker, President. lUCr (New Zealand) and Dr R Chidambaram, Vice-President,
lUCr (India) arc also expected tQ attend the inaugural session. ,
RegS$tnUiaa:
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Correspondence
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List of International Union of Pure and Applied Physics
(lUPAP)
SPONSORED CONFERENCES 1998
At the meeting of the Executive Council of lUPAP held in Paris,
France September 26-27, 1997, sponsorship by the Union was extended
Ic) international conferences to be held in 1998 as listed in this News
Bulletin.
The upper limit of the registration fee for lUPAP sponsored
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that the following standard declaration should be published by
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is available through the Secretariat or on the lUPAP Web site at
http: / / www.physics.umanitoba.ca/IUPAP
m
lUPAP Conferences Approved for 1998
CxkA.
No.
Name
Acronym
Place of
Conf.
Date of
Conf.
Organizer
Fa3i,etc.
C2.I
2“* Int. Conf.
on Exotic
Nuclei and
Atomic
Masses
ENAM98
Belaire MI,
USA
98/06/23-27
B.M.SheiriU.CN.Davids
National Superconducting
Cyclotron Lab,
Michigan State University
East Lansing, Ml 48824,
USA
Phone : 1-517-333-6322^
Fax 1-517-353-5967
E-mail :
http://www.nscl . msu.edu/
conferences/E NAM98
C2.2
1998 Conf
on Precision
Electro mag‘
netic Mea-
surements
CPEM98
Washington
DC. USA
98/07/06-10
N. B. Belicki, K. H. Magruder
NIST. Bldg 220,
Room B 162, Gaithersburg,
MD 20899-0001. USA
Phone : 1-301-975-4223/
975-2402
Fax ; 1-301-926-3972
E-mail : norman.belick 1
nist.gov katherine.
magruder@nist.gov
http'./Zwww.eeel. nist.gov/
cpem98
C3.I
20"’lnr.Conf
on Statistical
Physics
STATTWS
20
Pans,
France
98/07/20-25
A. Gcrvois,
Service de Physique
Theorique, CEA/Saclay
F9 11 91 GIF-sur- YVETTE
Cedex, France
Phone +33(0)1690881 14
Fax. +33 (0)j 69 08 8120
E-mail . staiiphys@spht
saclay.cea.fr \
http://www. IpitTjussieu fr/
statphys 1
C4 1
18“’ Int Conf.
on Neutrino
Physics and
Astmphysics
Neutrino98
Tokayama,
Japan
98/06/4-9
Y. Suzuki
Kamioka Observatory,
ICRR, University of
Tokyo, Higashi-mozumi,
Kamioka, Gifu 506-12,
Japan
Phone +8I-57$-.5-96()l
Fax;+81-578-5r2l21
E-Mail : suzuki@sukai07
ierr u-tokyo. ac.jp Nu98@
suketlo icrr.u-tokyu acjp
hltp://www-sk,iciT.
u-tokyo. ac.jp
C6 1
3"' Int
Symp. on
Biological
Physics
Santa Fe,
NM, USA
98/09/20-24
H Frauenfelder,
Director for Non-linear
Studies. Los Alamos
National Lab. Los Alamos.
NM 8754.S, USA
Phone l-.505-665-2.S47
Fax : 1-505-665-2659
E-mail frauenleldcrijf'
lani gov
http://\N'WW isbp.lanl gov
C7 1
Joint meeting
of 16'“ Int Conf.
on Acoustics
and Acousti-
cal Society
of Amenca
Seattle. W A,
USA
98/06/20-28
L. A. Crum,
Applied Physics Laboratory
Univ. of Washington.
1013 NE 40'“ St
Seattle WA 98105, USA
Phone +1-206-.S43-I30()
Fax ; +I-206-543-678.S
E-mail . Ias@apl
Washington edu
http;//
C8 1
24“' Int Conf.
on Physics
of Semicon-
ductors
ICPS 24
Jerusalem,
Israel
98/08/2-7
M. Heilblum
Condensed Matter Physics
Dept.,
Weizmann Institute of
Science,
Rehovol, 76100, Israel
Phone . +972-8-934 38%
Fax +972-8-9.34 4128
E-mail Heilblum@wis
weizmann ac.il
hiip7/htip;//physics
technion.ac.il/-icps 24/
C8.2
2“'' Workshop
on
Opcoetecironic
Materials
and their
Applications
Havana,
Cuba
98/1 1/2-6
M. S. Colina
Physics Faculty,
University of Flavana,
Cuidad de La Hubana,
Cuba
Phone : +537-704270
Fax +537-333758
E-mail : opto@rmq.uh
edu.cu
http://
ciai
3*^ Int. Conf.
on Excitonic
Processes in
Condensed
Maner
EXCON98
Boston MA,
USA
98/II/2-5
W, M. Yen.
Dept, of Physics and
Astronomy,
Univ Of Georgia,
Athens, GA, USA
Phone : +1-706-542-2491
Fax : +1-706-542-2492
E-mail . wyen@hal.
physast uga.edu
http://www.physasl.uga
(Hi)
CODf-l
No.
Name
AcroDym
Pticcof
Conf.
Date of
Conf.
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ckk3
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on phonon
Scattenng in
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Matter
PHONONS
'98
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EOrrOR-IN^CHlEF & HONORARY SECRETARY
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CONDENSED MATTER PHYSICS
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Indian Journal of Physics A
Vol. 72A, No. 3
May 1998
CONTENTS
Condensed Matter Physics
Electrical properties of organic and organometallic compounds
ATOza and PCVinodkumar
Thermally stimulated depolarization current behaviour of poly
(vinyledene fluoride) ; poly (methyl methacrylate) blend system
Ashok Kumar Garg, J M Keller, S C Datt
ANTD NaVINChAND
Characteristics of selenium films on different substrates under
heai-ireaiment
S K Bhadra, a K Maiti and K Goswami
Neutron diffraction study of tin-substituted Mg-Zn ferrites
A K Ghat AGE, SAP atil and S K Paranjpe
Fluctuations in high Tc superconductors with inequivalent
conducting layers
R K John and V C Kuriakose
Investigation of graphitizing carbons from organic compounds
hy various experimental techniques
T Hossain and J Podder
Determination of the activation energy of a thermoluminescence
peak obeying mixed order kinetics
S Dorendrajit Singh, W Shambhunath Singh
AND P S Mazumdar
iiiudies of X-rays and electrical properties of SrMoOa
N K Singh, M K Choudhary and R N P Choudhary
l^otes
On the structure and phase transition of lanthanum titanate
H B Lal, V P Srivastava and M A Khan
y ndrically symmetric scalar waves in general relativity
ShriRam and SKTiwari
Pages
171-187
189-199
201-207
209-216
2F7-224
225-232
233-240
241-247
249-252
253-258
172
A T Ozja and P C Vinodkumar
electrical resistivities are supposed to vary because of variations in the band gap and
therefore, high pressure studies on these complexes are important to carry out.
Organic charge transfer complexes and inclusion compounds of iodine were
prepared using standard methods of mixed solutions [6-10].
Low temperature and high pressure measurement on resistivity were carried out
using continuous flow cryostat and Bridgemann anvil apparatus [1 1-1 Sj.
Results of DC/AC resistivity measurements on compounds like a-cyclodextrine-
KI-I 2 . 4 H 2 O, amylose-iodine and similar other compounds like DTN-I 2 * anthracene-TNB
(TNB-trinitrobenzene), (coumarin) 4 -KI-^, pyrene-2l2 ^tcare plotted in Figures (1-6).
Figure 1. Pressure dependence of resistivity of amylose-iodine.
This paper is devoted to detailed analysis and plausible interpretations based on
theoretical models related to resistivity of similar compounds under different physical
conditions like pressure and temperature. Section 2 describes plausible theoretical models
that explain the experimental results. Section 3 reviews the other theoretical models. The
temperature characteristics of the resistivity of various compositions of benzidine-iodine
complexes are discussed in Section 4. Conclusions are drawn by highlighting the main
results in Section 5.
An increase followed by the flattening of the resistivity peaks are observed
in a-cyclodcxtrine-KI-l2.4H20, amylose-iodine and (coumarin) 4 -KI-^ under high
Electrical properties of organic and organometailic compounds
173
pressure [16]. Fiat peaks observed in DTN and DTN-I 2 (see Figure 2) are similar to
those in CdS, cerium and graphite [16]. Symmetric peaks observed in anlhracene-TNB
Figure 2. Expenmenlal (E) and theoretically ( 7 ) fitted plots according 10
eq. (19) for the two inclusion compounds :
i;i) (x-cylcodextnric-KM 2 4t^O . A = 24 49516, (b) Amylose-iodine A = 1 76575, h = -0 00060
h ■- -0 00059 (k bar)"', c = 0 01122 (k bar)"', (k bar)"', r = 0 02719 (k bar)''. K - -0 882.19
K=- 0 60515 (k bar)"' (kbar)"'
Figure 3. Pressure dependence of resistivity of a pellet of an iodine complex.
pyrene-2l2 (see Figure 4) are also similar to those in metals like Ca, Sr, Neodynium,
Zn and Cd [16]. Two peaks as observed here in a-cylcodextrine-KI-l2-4H20 and amylose-
174
A T Oza and P C Vinodkumar
Figure 4. Pressure dependence of resistivity of DTN and DTN-I 2 (DTN = dithionfiphlhalene)
Figure 5. Pressure dependence of resistivity of anthracene'TNB and pyiene'2l2>
Electrical properties of organic and organometallic compounds
175
iodine at 1000 Hz are also similar to the resistivities of cesium and europium [16].
Amylose-iodine prepared by different methods are designated as types I to IV and differ in
dimensionality.
PRESSURE (Kilo bars)
Figure 6. Pressure dependence of resistivity of (counuirin)4-KI'i2-
2. Analysis based on theoretical models
An increase in electrical resistivity of organic polyiodide chain complexes at high
pressures was qualitatively discussed earlier [15]. However, the observation of decreasing
Mil and in some cases, the second increase at high pressure were never discussed before.
The exponent conductivity as (T - (P - PcY failed to show percolation mechanism of any
dimensionality in the high pressure range. There is a possibility of transformation of a
direct band gap to indirect band gap at high pressures. In this case, crossing of valleys is a
plausible mechanism for the interpretation of high pressure results as the case for
elemental and compound semiconductors (Ge, Si, GaAs, SiC etc) [17,18]. The effective
mass of charge carriers changes when the crossing of valleys occur. It is a dimensional
cross-over and dimensionality increases. This is because charge carriers along different
crystallographic directions are mixed up [19]. Flat peak observed in DTN and DTN-I 2 at
''cro frequency and in a-cyclodextrin-KI-l 2 and amylose-iodine at 1000 Hz (Figure 2)
reveal crystallographic effect because such crystallographic effect is observed in the
generalization of Blackmann approximation supporting crossing of valleys as a mechanism
of crystal structure effects [20,21],
176
A T Oza and P C Vinodkumar
Two or three dimensional phonons are involved at high pressure and electron-
phonon interaction leads to a change in dimensionality of conduction. Direct band gap
semiconductors become an indirect band gap semiconductors involving phonons at high
pressures.
The symmetric peaks observed in the dc resistivity versus pressure for anthracene
TNB and pyrene-2l2 and the ac resistivities at 1000 Hz for a-cycIodextrin-KI-l2.4H20
and amylosc-iodine under relatively low pressure range seemed to obey the expression
given be
=A + BP*^{\-P*)\ (1)
Po
where P* is the pressure range from where the symmetric peak is prominent, while ui
relatively higher pressure range, the flat peaks observed in the ac resistivities at lOOfJ H/ ol
the a-cyclodextrin-KI-l 2 and amylose-iodine and the dc resistivities observed ;in DTN and
DTN-I 2 seem to obey
=A + BP*'l^(l-P*yi^. ( 2 |
Po
It would be interesting to see the physical basis for the typical pressure dependence
on resistivity of these organic compounds. For ionic compounds like pyrene-2l2 and
anthracnc-TNB the conduction current is due to real part of the electric polarizability
given by
Rcal[J] = /V[(ua,"]£o » (3)
a
cop I m*^
(4)
where p is the damping parameter proportional to efi and m* is the effective mass [20 1
From the basic relation that J - oE, the resistivity can be expressed as
[(a)2 -0)2)^ +{o)l}l(n*y
Here, (u’s are the phonon frequencies generated under the external pressure and it is
proportional to The effective mass of the charge carrier also varies with
external pressure. According to the non-degenerate perturbation theory for coupled
bands for complex systems, the effective mass has shown to vary as [21] l/w*-
{2fi^ I )l AE, where a corresponds to the variation in the average lattice spacing
under external pressure. We assume here that the change in lattice spacing at moderately
Electrical properties of organic and organometallic compounds
177
low pressure range may be proportional to p. Thus, the change in m* under a pressure
difference/? is quadratic. Putting it back in the expression given by eq. (4) leads to
p _ +c^p^
Po Ae^p^
II
Ae^
p
P^(Po-P)^+^.
(7)
The leading term in eq. (7) is similar to eq. (1), providing a theoretical basis for symmetric
peak observed in these compounds.
The expressions given in eq. (2) corresponding to flat peak can also be found to arise
from the basic definition for resistivity,
where n is the density of charge carrier and p is the mobility. The mobility p is^ given
hy 122]
where A is the mean free path, vis the collision frequency, v is the average velocity related
10 the group velocity for the acoustic phonons because under relatively high pressure,
strong electron-phonon coupling form a condensate and travel with the same velocity.
Using the dispersion relation for acoustic phonons we get
ne^ A V 2
coi -
( 10 )
where a is the change in the mean lattice spacing under pressure. The carrier concentration
n IS given by the Boltzmann distribution
n = nQ exp{-e^ /kT), (11)
here is the activation energy of the compound. We may show a logarithmic variation in
f,; with pressure (at a later stage) providing a direct variation of n with pressure. Thus,
change in mean lattice spacing assumed to go linearly with pressure and the effective mass,
V and 0) to vary with pressure in the same way discussed before, leads to the form
- p'^^(Po
Po
d'' expected from eq. (2).
For detail analysis of conductivity variations by pressure, we consider the most
general expression for conductivity as given by eq. (8) as
(T = nep .
(13)
178
A T Oza and P C Vinodkumar
Under the external pressure, the effective change in the conductivity due to the respective
variations in n and ^is given by
da
dp
+
(14)
The variation in n under pressure is through the variation in the activation energy
through eq. (11). Similarly, the variations in mobility is through the parameters in the
eq. (9) defining the mobility.
Using eq. (1 1), eq. (14) can be written as
da
dp [ ^P ^P
(15)
The variation of resistivity with pressure is given by
dp _ 1 da
dp (j^ dp
= po exp
/Jq
pJcT dp
(16)
\
1
(I7j
Here Hq is available electrons in the valance band which does not change with pressure;
rtQ changes in such manner that resistivity decreases unless pressure leads to a back
transition from conduction band to valance band when pressure increases. Mobility is
related to mean free path and collision frequency which can change with pressure [22],
Even diffusion coefficient (D) can change with pressure. Mobility depends on pressure
through the collision time as well as m* given by eq. (9).
The dependence of pressure will also attribute upon whether electron gas is
degenerate or non -degenerate and is very complicated. Due to the lack of trust-worthy
theoretical description of electrical properties of complex organic and organometallic
compounds under pressure, we fit the experimental data for crystals of a-cyclodextrin-Kl-
I 2 . 4 H 2 O, amylose iodine and (coumarin) 4 -KI-^ by an analytical expression :
= A -I- \ - a1 cxp(-cp). (18)
Po + J
The constants A, By b and c are fitted parameters. This expression can also be
written as
P~Po
Po
= A +
Kjp-AIK)
{\^bp)
exp
-c{p-AIK)e-^l^y
(19)
where K-B-Ab (Figure 2).
For pellet, A = 6 = 0 and we get (Figure 4)
P“Po
Po
= Kp exp(-cp).
( 20 )
Electrical properties of organic and organometallic compounds
179
which can be thought to arise out of Maxwellian because p = (\/3)mni^. This dependence
of pressure on the resistivity has been found experimentally (see Figure 7). In the case of
single crystals, it is a shifted Maxwellian weighted by (1 + hp) term in the denominator.
This denominator term can be understood using a charge carrier residing on a harmonic
oscillator of lattice vibration.
Figure 7 , Conductivity v.v pressure for(eoumann)4-KI-l2
We consider that the mean free path is limited to the wavelength of charge density
wave generated by the strong electron-phonon coupling and collision frequency is
independent of pressure as it can be replaced by natural frequency of charge density wave.
In this situation, mv = hk and X= \/k gives
P =
hk^
ne^ V
2m*(£-V)
1
hne^v
(21)
Now, the momentum is lost to the screened Coloumb potential developed in a distorted
lattice at high pressure because of strain. Then
■ 2m; £
2m:
exp - 2,{r„ +ro)'|
nQC^vh
hnQC^v 1
J_
( 22 )
where r„ ^na\ a being lattice constant, A is Thomas-Fermi screening constant and Tq is a
constant. Within tight binding method, we find that m* is also independent of pressure in
180
A T Om and P C Vinodkumar
ihe required range. Substituting a = “*■ ^1 P "Pol expanding the exponential m
~ - Pip - get exactly the required form of shifted Maxwellian. Here, the
first term is independent when the total energy is constant by conservation law, then
P-
2m* OC g-Unao^*^h)g-kfuiQa{|^-p^^)
— r^p-Po)} — : ; — !•
”0 1'" [noo {• + “(p - Po )} + *’]
(23)
the change in the kinetic energy of charge carriers is related with Peierls transition at high
pressure. The activation energy changes by 0.07 eV at high pressure which gives the order
of Peierls gap found in one-dimensional conductors.
Now, consider a non-equilibrium process at high pressure in which both density of
slate and also the Fermi level change under pressure [15].
£0 + —An = £® +
“ dN “
An
c(E-fjy
(24)
where An is change in number of states, C is a constant and /i is the Fermi level using
dnjdE = D{E) - C|£-/i|" for n = I for a one-dimensional system. Now to find the total
change, we integrate over p which gives
E. + ^[ln|£| - ln|E-£o |]
= + ^[\nPp- \np(p-po)].
(25)
where P 'lS a constant relating pressure and energy. This is obtained from supply of elastic
energy as a work done by pressure. When this energy goes over exponential gives
KP
p = Po exp(£„ IkT) = exp(-cp).
where
A: = Poexp(£«/ir)-^^
and
c = Of£, /kT
and
b = —■
Po
(26)
This is the required form.
Thus there are two effects in single crystals : (1 ) due to a shift in kinetic energy one
gets shifted in Maxwellian velocity, (2) the (I + bp) term in the denominator arises out of
electronic or ionic polarizability. From eq. (19), we obtain
Electrical properties of organic and organometalUc compounds
181
Now comparing eq. (27) with eq. (17), we identify
=cp-^ ln(l + 6p).
providing the required logarithmic pressure dependence. From eq. (28), we further gel
-r— = «/ C -f
dp L
•bp
We identify from eqs. (17) and (27)
^=Kp-A
and
_ K
-^2 dp
(28)
(29)
(30)
(31)
Thus, a consistency between the eq. (17) and fitted analytical expression has been obtained.
Thus for these compounds the mobility goes as \liKp-A). The values of
{clE^ ldp)j are calculated with Eg = lE^ for a-cyclodextrin-Kl-l2-4H20, aniylose-iodine
and (coumarin) 4 -KI-l 2 , keeping other parameters unchanged. The values forT, X and L are
tabulated (see Table 1) which should be compared with the similar values for elemental and
Table 1. Pressure coefficients of energy gap from pre.ssure dependence of
reshstivity for polyiodide chain complexes
Complex
(3E,/Sp)t
X lO^’eV
-cm ’^ /kg
r
X
L
ut-Cyclodcxlriii-Kl-l2.4H20
0.432
-0.036
-
Amylose-iodiiie (Type 1)
0672
-0.060
-
Amylose-iodine (Type M)
0.321
-0.207
0 104
(Coumarin)4-KM2
0.80
-0 042
-
compound semiconductors [17]. These values are one or two orders of magnitude less than
those for elemental and compound semiconductors. However T, X and L remain conduction
band minima related with K = 000, 100 and 111 valleys. Within the localisation model the
activation energy is given by
1
p{e,)r'
(32)
'^here p(£^) is the average density of slates at Fermi level and R is the hopping distance.
It may be realized that the linear plus logarithmic dependence and activation energy
(cq 28) on pressure arise through the distortion in the density of stales in the energy bands
^^1 ihe sample compound. For example, the cosine potential leads to logarithmic fluctuations
iiciivation energy through density of states [23].
182
A T Oza und P C Vinodkumar
3. Alternative physical mechanisms
In this section, we would like to see the possible physical mechanisms supporting the
plausible physical processes described in Section 2. For weakly disordered one-dimensional
alloy, density of state for one type site is given by
1
^[4,^
(33)
for £ lies between £a-'^ and + 2/. Here, is the energy of a site and t is the hopping
matrix element for site-diagonal disorder [24]. Conductivity should be proportional to
the density of slates and pressure changes elastic energy in a linear fashion as work is
done. Also resistivity will be proportional to mean localization length. Localization with
energy [24] and resistivity with pressure change alike. This shows that pressure leads to a
change in elastic energy which tracks a band gap.
On the other hand, Interpretation of contraction for Slater orbitals\[251 is also
applicable because the pressure dependence fits functional form of Slater orbitals which m
described by
R{r) (34)
/i ' and Oo are constants.
The resistivity is related with charge density e\R\^ as follows
1 1
( 35 )
Using the linear coefficient of compression in as r= rQ[l + (X(p-Pq)] where a is
coefficient of compression and for 5p (7 hybridization along iodine chains, n-5 gives
P= exp(2/iV/ao)
= + “(P-Po)]’" exp
2^ Vo
(1 + a(p-po))
(36)
Here we may consider Bohr radius oq as independent of pressure
[l Of(p - Pq )]'*' = [l - 8a(p - Po )]• Then the resistivity can he written as
p - {A + Bp)e~^P ,
(37)
where
l + 8a r2/i'ro
— rnr e^P
B =
epN
-8a
epN'^rl
(l-otpo)
exp
apo)
S^VoC
and
(38)
Electrical properties of organic and organometallic compounds
183
The constants A and B contain mobility fd which is in general, pressure-dependent, but we
neglect this dependence here. Then, we get the resistivity expression almost similar to the
turm we obtained from experimental fit.
Alternative interpretation comes from Gunn effect in semiconductors which has been
observed in a large number of compound semiconductors. Under hydrostatic pressure, the
intcr-valley energy seperalion can be made less than the energy band gap and the Gunn
cited dominates at all pressures. Here, pressure changes wave vector k and E versus k is
tracked because this dependence is similar to resistivity pressure in crystals.
We also prove here that lever rule applicable to (coumarin) 4 -Kl-l 2 can be derived out
ol a dielectric constant formula in solid state physics, i.e.
£{(0) =
(
(ol -0)^^
(0^ -0)^ ^
(39)
because a? - p. Now, the band gap is related with dielectric model within Penn’s
model [27 1 as
£-1
4me^
' ]
2 ■
m* £2
4Ef sj
(40)
Tonsidering the band gap to be much larger than h(Op where tw,, = 4;me^ /m* is the
plasma Irequcncy, the first term can be neglected. Similarly, /4E/, is also a small
quantity for £y » Then the third term can be neglected. The second term gives
The dielectric constants of the materials were measured to be 10-20. Neglecting unity,
£l = ^ g ( Pl-p \ (42)
£2 ln((T, /(To) '‘[pt-PJ
which proves lever rule. For R<Rd and for one-dimensional system
£ = i _ ^ . (43)
^ Rr,N(E) ’ N{E)
'''here A/j is concentration. This shows that is proportional to
-pIPt -p).
Now in the case of benzidine-TCNQ, pressure dependence of resistivity is analyzed
give the following relation
Inp ?=
(44)
184
A T Oza and P C Vinodkumar
Therelorc, In p v.s ln( p - ) is found to be a straight line (Figure 8a). For semidhnductor
is very small so In p is plotted.
(45)
0-1
Cb)
Ji
z
X
o
iooo/t CK"')
0 1 ^
ZZO 240 260 280
T CK)
Figure 8. Pressure (a) and tempcranire (6 & c) dependence of resistivity of benzidinc-TCNQ
(in Figure b curve I for /? = 65 k bars and curve II for p = 70k bars).
If there is a fluctuation in activation energy arising from density of states, then
+
dE.
-An = £0 +
-An.
dn ' ■ N(E)
N(( 0 ) - = |/V(to)d£B- (o)-ao)^''5and for phonons.
(45)
4. Temperature dependence of resistivity
The study of temperature dependence of resistivity of various compositions of benzidine-
iodine complex led to determination of activation energies of all compositions [28].
Electrical properties of organic and organometallic compounds
185
Temperature dependence of conductivity of benzidine-TCNQ above 60 kilobar
shows a conductivity peak (Figures 8b and c) obeying
(47)
T" = T/Tq and A is a constant. Actually, it is a beta distribution related with Bernoulli
trails for hopping of charge carriers. This relation can be derived from solid state physics
as follows.
From the dispersion relation for acoustic phonons like
Q)= 0)^s\n^ka,
(48)
it IS easy to see that [30]
^ 2/2
js-jK-”’)
(49)
.ndihai D(0)) . —{tol -
(50)
The conductivity is proportional to Z)(tU),
a = B— efiicol
(51)
where fj is mobility and is a constant. Now m(0^ A j ^hcre A is the amplitude of
vibration. Thus, is proportional to temperature T. The mobility p = ekvj m* v. X is
mean tree path which is independent of temperature in kinetic theory or is limited by
wavelength of charge density wave. Only collision frequency is w - using kinetic
theory. This leads to
cr=4r*‘/2
where 7*= T/Tq .Tq and (p„ are related directly.
Alternative proof '.
Above beta distribution in temperature can also be derived from absorption coefficient. For
disordered material, it can be given by [29]
a= 0.115
{hv-E,)
1/2
(52)
For a crystal
a = AE,{hv-E,)''^. (53)
Now a= ootjc/An, B] = e = I + I and = f>7tnoe^ I^f ■ A is a
screening constant [29]. Bp is proportional to temperature T and therefore, n | = BlypT .
186
A T Oza and P C Vinodkumar
Then a = .4(£® -t,r)(*BT-*flro)''^
replacing hv by k^T and by kgTfj and = £® - kgT.wt get
BA(E^^ -kgT)(kgT-kgToy/^ ^
a= —
Vf
where T* = TJTq .Tq and are related. This is the required beta density.
5. Conclnsion
We have studied in detail, the high pressure behaviour of electrical resistivity of organic
and organometallic conductors. The beta density behaviour of resistivity pressure of
these compounds are physically interpreted using some of the basic theoretic^ models and
a general expression for the pressure dependence on resistivity of such compounds are
obtained. Other existing alternative models are briefly reviewed and have shown that under
some assumptions, most of these models can be deduced to the general theoretical
framework discussed in Section 2. The temperature dependence of resistivity of these
compounds are also found to have similar beta density behaviour.
References
11] J S Miller and A J Ep.stein in Progress in Inorganic Chemistry Vol 20 cd. S J Lippard (New York
John Wiely)pl (1977) •
[2] J J Andre, A Bieber and F Gautier Ann. De Phys 1 145 ( 1 976)
[3] P A Lee, T M Rice and P W Anderson Solid State Commun. 19 703 (1974)
f4] J Bardeen Solid State Commun. 13 357 (1973)
[5] D Allendcr, J M Bray and J Banlecn Phys. Rev. B9 1 19 (1974)
[6] D S Acker and W R Hertler J. Amer. Chem. Soc 84 3370 (1962)
[7] C Hsu PhD Thesis Temple University, Philadelphia, Penisilvania, USA (1975)
[81 B P Baspalov and V V Titov Ress. Chem. Rev. 44 1091 (1975)
[9] A A Berlin. L I Borsalavskii, R Kh Burschtein, N G Mateva and A 1 Schurmovakaya Dokl. Nauk USSR
136 1127(1961)
[10] S Doi, T Inabe and Y Matsunaga Bull. Chem. Soc. Jpn. SO 837 (1977)
[11] AT Oza Indian J. Cryogenics 10 62 (1985)
[12] A T Oza Phys. Stat. Solidi (b) 114 K171 (1982)
[13] A T Oza Czech. J. Phys. 33 1 148 (1983)
[14] A T Oza Bull. Mater. Sci. (India) 7 401 (1985)
[15] A T Oza Phys. Stat. Solidi (a) 80 573 (1983)
[16] H G Drickamer Solid State Physics Mol 20 eds. F Seitz and D Turnbull (New York ; Academic)
pi (1865)
[17] J J Pankove Optical Processes and Semiconductors (Engelwood Cliffs. New Jersey : Prentice-Hall)
p 22 (1971)
( 54 )
(55)
Electrical properties of organic and organometallic compounds
187
1 18 ] T E Slylchousc and H G Drickamcr J. Phys. Chem. Solids 7 210 ( 1958)
[ 19 ] A L Edwards and H G Drickamcr Phys. Rev. 134A 1628 ( 1 964)
[20j S Wang Solid State Electronics (New York ; McGraw Hill) p 278, 668, 67 1( 1960)
f 2 i| A 0 E Animalu Intermediate Quantum Theory of Crystalline Solids (Engelwood Cliffs, New Jersey •
Prentice-Hall) p 197, 340 (1977)
1 22] G Yepifonov Physical Principles of Microelectronics (Moscow . Mir Publisher) p 1 80 ( 1 974)
[23] S Das Sharma, He Song and X C Xic Phys. Rev. B41 5544 (1990)
[24] Micheal Plischke and Birger Bergrsen Equilibrium Statistical Physics (Engelwood Cliffs. New Jersey
Prenticc-Hall) p 303, 305, 42 ( 1 989)
[25] F A Cotton and G Wilkinson Advanced Inorganic Chemistry 3rd edn (New Delhi : Wiely Eastern)
p 96 (1972)
1 26] H Frohilch Proc. /?ov. Soc A223 269 ( 1 954)
1 27] D R Penn Phys Rev. 128 2093 (1952)
[28] A V Nalini, A T Oza, Anilkumar and E S Raj Gopal Proc. Nucl Phys Solid State Phy.^ic.s Symposium
(Ahmedabad. India) December 27-31 19C p4l (1976)
[29] C Kiticl Introduction to Solid State Physics 4th edn. (New York John Wiley) p 175, 208, 710 (1974)
Indian J. Phys. 72A(3). 189-199 (1998)
UP A
an intemationo] journal
Thermally stimulated depolarization current
behaviour of poly (vinyledene fluoride) :
poly (methyl methacrylate) blend system
Ashok Kumar Garg'*, J M Keller"* , S C Datt" and Navin Chand*
"Department of Postgraduate Studies and Research in Physics, Ram Durgavati
Vishwavidyalaya, Jabalpur-482 001, Madhya Pradesh, India
^Regional Research Laboratory (CSIR), Bhopal-462 026, Madhya Pradesh. India
Received 5 September 1997, accepted 20 February J99H
Abstract i The blend system of semicrystalline poly (vinyledene fluoride) (PVDF) with
amorphous poly (methyl methacrylate) (PMMA) has been investigated in detail, for thermally
stimulated depolarization current behaviour Bilaterally, aluminized solvent cast blend samples
of various wt% composition, PVDF • PMMA 100 00, 90 : 10, 80 • 20 and 70 30 were
thermally charged with field ranging from 50 to 125 kV/ cm at temperatures from 50 to 1 10°C.
The poling field, temperature and composition dependence of the short circuit thermally
stimulated depolanzation current (SC-TSDC) thermograms of such samples (electrets) suggest
that the relaxation originates from the orientation of dipoles and the motion of charge corners in
the blend system. The results also show that the electrets of such blends, however, in comparison
to the two component homopolymers store more charge.
Keywords : Short-circuit thermally stimulated depolarization (SC-TSDC), electrets,
antiplasticization, anomalous current.
PACS Nos. : 61 .25.Hq, 77 84.Jd, 78.30Jw
1- Introduction
In recent years, considerable interest has been shown to the study of polymer blends.
Many techniques including thermal analysis and scattering methods have been applied to
look at microscopic and macroscopic phenomena with regard to crystallization, morphology
^nid interfacial properties [1-5]. TSDC is a powerful technique with sensitivity comparable
dynamic mechanical and dielectric measurement. For the standard TSDC experiment,
^hich is comparable to dielectric loss measurement, the low equivalent frequency
( ('rre.sponding author : Dr.~J M Keller. 948/ 1 , Near Sonia Appartment, South Civil Line,
Jabalpur-482 001 , Madhya Pradesh. India
190
A.siwk Kumar Garf>, J M Keller. S C Dart and Navin Chand
(slO"^ Hz) [6] and high sensitivity makes TSDC quite useful for the study of amorphous
relaxation in crystallizable blends. Further, for semicrystalline materials, the low equivalent
frequency offers one additional advantage; the glass transition temperature Tg, is shifted to
low temperature and the glass transition of the purely amorphous phase can be studied
without inducing crystallization.
PVDF and PMMA is one of the few known miscible polymer blend. Several
TSDC and dielectric studies of PVDF : PMMA blends have been reported [1 -5,7,8];
however, there are certain discrepancies between the results reported [9,10]. The
interpretation of results in case of blend is extremely difficult due to complexity of
polymer relaxations which is further magnified in blends by the complex morphology.
Furthermore, blends obtained by different methods, i.e. melt mixed, solvent cast, etc ,
differ considerably in morphology [10]. Most of the experimental studies have been
concerned with the melt mixed blends while only a few studies have been undertaken
on solvent cast samples. \
In this paper, we report results of short circuited TSDC measurements on solvent
cast PVDF : PMMA blends (upto PVDF content of 70 wt%). The result.^ have been
discussed with respect to the correlation between structural as well as dynamical properties
and electrical properties of heterogeneous polymer.
2. Experimental
PVDF material (product of Aldrich Chemical Company, Inc. ly, 270-D tiD^® 1.4200.
d 1.740) and PMMA (low molecular weight) (obtained fiom Wilson Laboratories,
Bombay) were used in the present investigation. Polyblcnds were solvent cast on plane
glass plates kept in an air oven by dissolving the two polymers in required wt% in then
common solvent dimethyl formamide (DMF) (LR Grade) at 60°C. Films of various wt'yi
composition. PVDF : PMMA : : 100 : 00, 90 : 10, 80 : 20 and 70 : 30 so obtained and dried
at 6()°C for one week, were subjected to room temperature outgassing for 24 hours at
pressure of 10'^ torr. Aluminium electrodes were deposited on both sides ()f these
samples over a central circular area of 3.6 cm diameter. For SC-TSDC measurements, the
samples were thermally polarized at temperatures 50, 70, 90 and 1 10°C with fields oi
50, 75, 100 and 125 kv/cm using a stabilized DC power supply (model HV 4800 B
from Electronic Corporation of India Ltd., Hyderabad). The required voltage was applied
for 45 min at the desired temperature and than the sample was cooled to room temperature
in 45 min with the field still on. The total time of polarization was thus adjusted to 1.30 hr
in each case. The TSDC run was performed by reheating the polarized sample (eleclrct)
at a linear rate of 3^ min"' and the discharge current was recorded by a Kcithicy
6I()°C electrometer.
IR absorption spectra of the samples were also recorded using a Perkin-
Elmer spectrophotometer (model 1720 X) to yield information about morphology of
the samples.
Thermally stimulated depolarization current behaviour etc
191
3. Results
(/ ) Effect of poling temperature ;
Figure I shows thermograms for pure PVDF samples polarized with a field of 50 kV/cm
1,1 Liiffcrent temperatures, 50, 70, 90 and IIOX. It is clear ihal the thermograms
arc characterized with two peaks at 70 and 122°C and a third peak is observed ai IbO'^C.
0-5 oh
Figure 1. TSDC therinogiums obtained for pure PVDF polarized with a field of
.SO kV/cm al different temperatures . — o — 50, — •--- 70, —□ — 90 and —A™ I I0"C.
Hie thermograms for 70®C show only a hump at 78^C and a sharp peak at I20X; in
case of the sample polarized at 90°C, two very well-developed peaks are observed al
1 10 and 140°C; and the samples polarized at 1 i0°C exhibit a broad hump around 1 10 and
peak at 140°C. The thermograms for eleclrcts prepared with the higher fields (not
shown), do not show any peaks except for the sample polarized with the field of 75 kV/cm
'^hich exhibits peaks at 95 and 140°C for polarizing temperature of 70°C. It is also
e\idciu from ihc figure that the initial value of the current is high showing existence oi
a peak below 30T.
Figure 2 shows TSDC’s for 90 : 10 blend samples polarized with a field of
50 kV/cm. The thermograms exhibit probably two overlapping peaks in the temperature
intervals 85-98, 78-102 and 122-140^^0 in the case of 50. 70 and 1 lOX polarizing
temperatures. In the case of 90' C, the two peaks are well separated. The profile of th.
192
Ashok Kumar Garg, J M Keiler, S C Dost and Navin Chand
peaks increases in magnitude and shifts towards higher temperatures with increase in the
polarizing temperature. The thermograms for the samples polarized with the higher fields
(not shown) are characterized with an initial hump followed by a peak, probably two
overlapping peaks, in the temperature interval 95-140°C that show increase in magnitude
Figure 2. TSDC ihermograms obtained for 90 . 10 blend with the field of 50 kV/cin
at different temperatures : — o — 50, — • — 70, — a — 90 and — i!V— I lO^C
Inset shows dependence of peak temperatures (fmaxl composition of blends.
and shift towards higher temperature with increase in the polarizing field. However, for the
electrets obtained with the fields 1(X) and 125 kV/cm the peak current magnitude is
reduced for the higher polarizing temperatures.
TSDC’s for the 80 : 20 blend samples polarized with the field of 50 kV/cm at
different temperatures are shown in Figure 3. In this caseTalso, the thermogi ns exhibit a
peak in the temperature range 95-140®C which shows a shift towards higher temperatures
with increase in the forming temperature. The peak magnitude decreases with increase in
temperature for low fields; however for the higher forming fields, the peak magnitude
increases again with increase in the forming temperature. Again, for the electrets obtained
at polarizing temperature of 90°C, two separate peaks are seen. Surprisingly, in the case of
the highest forming temperature 1 10°C, the current is found to exhibit anomalous behaviour
and flows in the direction which is same as the charging current.
Thermally stimulated depolarization current behaviour etc
193
TSDC’s for 70 : 30 blend samples polarized with a field of 50 kV/cm at different
lemperatures are shown in Figure 4. All the thermograms arc characterized with a single
Figure 3. TSDC thermograms obtained for 80 . 20 blend with the Held of 50 kV/cm
ai different temperatures : — li — 50. — • — 70, —90 and —A— 1 1 0°C
peak However, the magnitude of the peak current is reduced with increase in the forming
U'mpcraturc. Again, in some cases at the low forming fields, the thermograms show
inomalous behaviour. A clear anomalous peak at 1 10°C is observed in case of the electrets
obtained with a field of 50 kV/ cm at the temperature of 70°C.
(n) Effect of poling field \
Typical field dependence of the TSDC thermograms of various blends is exhibited in
figure 5. It is evident that the magnitude of the peak current increases for moderate fields;
however, it decreases for higher fields.
('ll) Effect of wt% composition of the blend :
fhe dependence of TSDC thermograms on the wt% composition of the various blends
polarized with the forming field of 50 kV/cm at the forming temperature of 50°C is shown
J'' inset in Figure 5. It is evident that the magnitude of peak current l,„ is minimum for
pure PVDF samples at all the fields. As the wi% of PMMA is increased in the blend, upio
wt7, , ihg current is increased. However, for maximum wt% of PMMA {i.e. 30 wl%
ihc present investigation), the peak current magnitude is decreased for all the field
194
Ashok Kumar Carg, J M Keller, S C Datr and Navin Chand
values. Similar results are observed for the blends polarized at temperatures of 70, 9o
andllO°C.
I
Figure 4 . TSDC tlierinograins obtained lor 70 • blend with the field of 50 kV/cni
at diffcient letnperatures : — i.h- 50, — • — 70, — a — 90 and —A - 1 lO’C
4. Discussion
Persistent polarization in thermally charged specimens may arise due to variou!)
mechanisms, the important among which are dipolar polarization, space charge polarization
or truiislation and trapping of charge carriers at microscopic distances or accumulation near
the electrodes and inlerfacial or Max well- Wagner effects, i.e. the trapping of charge carriers
at phase boundaries. The charge originated in TSDC due to dipole orientation or trapping ol
charges in defect or dislocation sites is known to give rise to a uniform polarization which
IS heterocharge. On the other hand, space charge built-up by migration of ions over
microscopic distances gives a non-uniform heterocharge, whereas trapped injected space
charge results in a non-unifomi homo- or hetero-charge, depending upon the work function
ol the metal electrodes.
In the present investigation, the thermally stimulated discharge current, in general, is
(ouihI to flow in the normal direction, i.e, opposite to the charging current; however in some
cascv, for certain part of the discharge cycle, it exhibits an anomalous behaviour and flows
in ihe same direction as the charging current. Thus, pruces.ses involving heterocharge
lormaiion are mainly responsible for polarization in the blends.
Thermally stimulated depolarization current behaviour etc
195
Relaxation processes in the crystalline polymers are related to molecular motions of
amorphous and/or crystalline chains. Three distinct relaxations have been observed in
PVDF : (i) oCc : Crystal relaxation from 20 to 160°C at I to 10^ Hz, (ii) : Amorphous
iclaxation from -66 to 0°C at 10 to 10^ Hz which is due to micro-brownian motion of the
mam chain in the amorphous phase and (iii) j3 : Amorphous relaxation from -66 to -47'^C at
ID (0 10^ Hz. The TSDC cycle in the present investigation has been carried out at
temperatures above the room temperature, Le. 30°C which is much above the temperature
nf amorphous relaxations. Nevertheless, the high value of initial current observed in all the
thermograms does point towards the existence of at least one relaxation peak at some
temperature below 30°C.
Figure 5. TStXT ihermograms obtained tor SO : 20 blend polarized at temperature 90*’C
with different ricld.s . —a— 50, — 75, —a — 100 and —A— 125 kV/cm.
Inset shows the dependence of peak current (/|nox^ composition of the
various blend.s ■ — o — 70 . 30, — • — 80 : 20, — □ — 90 10 and -^A— 100 : 00.
The thermograms for the pure PVDF electrets obtained with the low polarizing fields
ol 50 and 75 kV, are characterized with peaks at 95-1 12 and 140-I60°C for the moderate
P^^lanzing temperature of 70°C while the thermograms for higher fields of 100 and 125 kV
tio not show any peak.
The low temperature peak observed in the present investigation may be the a^-dipole
relaxation peak occurring in the crystalline phase of PVDF. This relaxation has been
observed by several workers for phase II (or non-polar a-phase) PVDF [1 1,121. From the
196
Ashok Kumar Garg, J M Keller, S C Datt and Navin Chand
IR absorption spectra shown in Figure 6, we can infer that the PVDF crystals in our solvent
cast samples are also mainly of phase II. The absorption band at 488, 532, 616, 766, 796,
856 and 976 cm'' arc the characteristic bands associated with phase II structure.
Figure 6. Infrared absorption spectra
for pure PVDF and 70 .30 blend.
The high temperature peak observed in the present investigation, is probably an
additional relaxation process. PVDF is a scmicrystallinc polymer consisting of lamellae
crystals and amorphous regions. The amorphous regions reside mostly between the
crystalline lamellae. Sasabe et al [13] have reported the possibility of such an additional
peak in (he frequency interval below the a^. relaxation and at high temperature. They have
assigned it to an interfacial polarization at crystalline-amorphous boundaries or to the
rubbery How of the polymer chains. Similar relaxations at low frequencies and high
temperatures have also been observed in other polymers and biological material, viz.
PET [14], PMMA [15], Nylonc [16], all of which have been attributed to a charge, build-up
at the interfaces in the bulk or close to the electrode-dielectric interface.
Since PVDF is a semicrystalline polymer, the charge storage and transport in it is
expected to be dominated by various localized levels in the amorphous regions and also at
the crystalline amorphous boundaries. Further, since it is a polar polymer, the probability of
the presence of intrinsic charge carriers in it is also sufficiently high, particularly at high
temperatures. Incidentally, in heterogeneous heieroelectrets of PVDF, these charges will
mainly pile-up at the phase boundaries. They are supplied there by unequal ohmic
conduction currents within the two phases (Maxwell-Wagner charging). These carriers^re
also likely to be trapped in different trapping sites leading to space charge effects which
fundamentally influences all the charging and transport processes. Their high concentration,
often enables them to contribute discemly to the SC-TSDC.
The magnitude of the peak current is found to decrease with the polarizing
temperature except in the case of the electret obtained with the highest polarizing field of
125 kV/cm at the highest temperature 1 10®C of the present investigation. This shows that
in addition to dipolar orientation, space charge and trapping effects are also operative in the
present case. Yano et al [17] also have attributed the observed high value of static
permittivity eg, in PVDF to ionic space charges.
Thermally stimulated depolarization current behaviour etc
197
In general, the decay of space charges in heterogeneous system is ascribed to ohmic
dissipation alone; any motion of charges is neglected. They are considered to be neutralized
by opposed carriers replenished at the phase boundaries by the unequal ohmic conduction
current (M-W discharging). The occurrence of interfacial space charge polarization,
requires that there be enough carriers of a sufficiently high mobility which is expected near
Tg when ohmic conduction is sufficiently high.
The thermograms of the blends are expected to reflect the electrical properties of the
PVDF crystals as well as the crystalline-amorphous boundaries in addition to those of the
amorphous regions. The thermograms for the blends show probably two overlapping peaks
in the temperature range 88-140®C which increase in magnitude and .shift towards higher
temperatures with increase in temperature. The increase in magnitude and shift towards
higher temperature of the peaks is due to an increase in the total polarization at higher
temperatures. Astonishingly, for 100 : 00, 90 : 10 and 80 : 20 eleclrets, the two peaks are
well-separated for the electrets obtained with the polarizing field of 50 kV/cm at the
polarizing temperature of 90°C. PMMA exhibits and of-relaxation at the glass transition
temperature 95-105°C. It is difficult to distinguish the contribution of the PVDF molecular
orientations, since both PVDF and PMMA exhibit maxima in the same temperature
range. However, it can be concluded that the lower temperature part/peak is as.socialed
with the o^-relaxation of PVDF and PVDF/PMMA molecular motions at their common
glass transition temperature. The polarizing temperature of 90®C being close to the Tg of
PVDF : PMMA blend, the molecular motions as.socialed with it manifest their contribution
clearly as a separate peak. Further, the polarizing field being sufficiently low, this peak
IS not masked remarkably by any other relaxation proce.ss occurring in the blends and hence
is clearly seen. Increased random molecular motions in case of 70 : 30 blend probably
overcome the dipolar orientation processes resulting in appearance of a single peak in
this case.
We have plotted the T^ax of the two TSDC maxima against the blend composition
as shown in inset of Figure 2. It is evident that the T^ax of the two peak is shifted
linearly towards the higher temperature with tjie increase in PMMA content upto
30 wi% studied in the present case. This indicates that upto 30 wt% of PMMA, the two
homopolymers PVDF and PMMA form compatible blend. Even if some phase separation
lakes place as reported by others, the system is atleasi not multiphase. The onset of
mobility of the dipoles both of PVDF and PMMA at their glass transition temperature
corresponding to a linear shift of the relaxation peaks towards the high temperature,
implies that the addition of PMMA produces hardening in PVDF, raises its Tg and thus
acts as an antiplasticizer for PVDF [18].
The magnitude of the current is minimum for pure PVDF and as the wt% of PMMA
is increased, the magnitude of current increases. Thus, the variation of structure and the
poling condition influences the magnitude of the TSDC current aS expected for the
appearance of interface charges. In addition to the formation of interface charges, owing to
the different conductivities, the polarity of the crystalline regions may also cause trapping
198
Ashok Kunuir Garg, J M Keller, S C Datt and Navin Chand
of charge carriers at the interface. The strong dipole polarization of the crystalline regions
enhances the carrier trapping at the crystalline-amorphous interfaces as suggested by a
number of authors [19,20].
Since no electrode dependence has been reported in PVDF [21] and because the
space charge effects have been observed [22], then the high temperature relaxation-process
may be due to motion of space charges trapped at the crystalline-amorphous boundaries. A
change in the relative contribution of the dipolar and interfacial processes with the PMMA
wt% concentration and shift of Tg towards higher temperature may be responsible for the
observed shift of the peak to higher temperature side.
The anomalous TSDC’s flowing in the same direction as the charging current
observed under certain charging field and temperature conditions for the blend samples
can be understood to be due to space charge formation and partial blocking of the metal -
polymer contact as suggested in literature [6].
Considering one type of carriers, electrons for example, we may have a distribution
of charges just after charging. The concentration of trapped charge carriers is highest near
the charging electrode and decreases with distance towards the other electrode. Usually,
the carriers move towards the nearest charging electrode (outflow) and recombine with
their image charges on the electrode. This results in the observed normal TSDC in various
cases. However, if the charge carriers return rate towards the nearest electrode exceeds the
charge carrier exchange rate at the electrode, the carrier will diffuse and inflow towards
the farther electrode. Such suppression is considered to be due to partial blocking of the
other polymer-metal contact. Under such conditions, the observed current becomes
anomalous. Obviously, diffusion only becomes significant for large gradients which may
be found particularly in hetero-electrcts of amorphous and semicrystalline blends of
PMMA : PVDF.
In the pre.sent ca.se, the number of shallow traps with small detrapping lime is
considered to increase with increasing wt% of PMMA; hence the total amount of space
charge will be relatively greater at low temperature charged samples with higher wt% of
PMMA, than the samples with small wt% of PMMA charged at the same temperature.
The easier release of carriers from such traps is supposed to cause a high return rate of
carriers from such traps to the nearest probing electrode resulting into the partial blocking
of the electrode which leads to anomalous current for such samples. The transfer of
charges from shallow to deeper traps for high temperature charged samples results in
reduced return rate of the carriers released from the shallow traps causing the current
to remain normal as opposed to the anomalous peak observed for the low temperature
charged samples.
Acknowledgment
The authors are thankful to Prof. S K Nema, Director, Macro-molecular Research Centre,
Rani Durgavati Vishwavidyalaya, Jabalpur, for providing the IR spectra facilities.
Thermally stimulated depolarization current behaviour etc
199
References
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[6] J Van Turnhout Polymer J 2 173 (197l).T/irrm«//v Stimulated Dtxcluirye of Polymvt Eltitreh
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Indian J. Phys. 72A (3), 201-207 (1998)
UP A
— an international journal
Characteristics of selenium films on different
substrates under heat-treatment
S K Bhadra*, A K Maiti*^ and K Goswami+
^Central Glass & Ceramic Research Institute. Jadavpur,
Calcutta-7m) 0:i2. India
■^Department of Physics. Jadavpur University, Jadavpur,
Calcutta-700 032. India
Received 13 January I99H. accppled 10 February I99fi
Abstract : The structural changes on successive heat-treatment of vapour grown
amorphous selenium films on different substrates, have been studied by observing scanning
electron micrographs. X-ray diffractograms and Raman spectra Crystallization rate of a-Se films
on aluminium substrate is found to be fa.ster than glass or quartz and the films on quartz, heat
treated at .‘>0°C for 40 minutes, show characienstic mctastablc structure like dendritic growth
with partial molecular hexagonal phases At this stage, the Raman active bands are not observed
*and on further high temperature annealing, the Raman spectrum shows more symmetric Se-
rnoleculai siructuie with disappearance of the dendritic features
Keywords ; Selenium film, substrates, structure
PACSNos. : 6l.l0.Nz,78.66.Jg
1. Introduction
Chalcogenides have wide range of applications in material research. Several structural or
photo-chemic'al changes have been observed in amorphous chalcogenide elements and
compounds, viz, through ageing, photo-crystallizations etc [1,2]. Selenium is one such
chalcogen which undergoes unstable morphological transitions from amorphous to
crystalline phases. Generally, it is known that vapour-grown Se thin films crystallize to
spherulitic patterns in hexagonal form composed of helical chains [3]. During transitions,
the different phases of the a-Se films reveal that the structure, growth processes and
other properties depend on the method of preparation, deposition conditions (vapour
temperature, rate and angle of deposition), physical condition of the substrates and
vacuum environment [4,5].
© 1998 TAGS
202 S K Bhadra, A K Maiti and K Goswami
In early works [6-8], it has been observed that the deposition on a particular
substrate is by no means absolute, rather the vapour conditions and the temperature of the
substrates appear to be important. On critical assessment both for normal and heat-treated
a-Sc films, it has been ascertained that the substrate-film interface plays a dominant role
during the transition phenomena [7]. In the present experiment, the treatment of a-Se
specimens of similar film dimension on quartz and other substrates at lower thermal energy
reveals a characteristic dendritic growth only on quartz substrate. The microstructures were
analysed through Scanning electron microscopy (SEM), Raman spectroscopy and X-ray
diffraction (XRD).
2. Experimental procedure
Pure Se pellets (99.999%) obtained from Johnson Mallhey & Co. are used for making thin
films. Substrates of soda glass slides, Al foils and z-cul quartz crystals (obtained from a
single block of synthetic quartz crystal-mirror polished) are used for depositing Se films
and were cleaned using trichloroethylene, acetone and ethyl alcohol. All the substrates
prepared, had been exposed to atmosphere before Se-coaling were being taken place.
Se pellets were melted slowly in a filament wound molybdenum boat under a vacuum of
2..S X I()-^ Pa, when the substrate is placed 5 cm above the heating source (Polaron P-150
Vacuum Coating Unit). The temperature of the mass was gradually increased to 350X and
allowed to deposit with a rale of 5 nm/scc~’ for a desired time onto the substrates.
Several .specimens of Se films grouped as G-I, A-I, Q-I for thickne.ss 1 10 nm; G-II,
A“ Il, Q-II, for thickness 220 nm and G-IIl, A-lIl, Q-IIl, for thickness 330 nm were
prcparcil on gljjss, Al and quartz respectively, under identical environment of vacuum and
vapour temperature, while the substrates were kept in room temperature. The deposition
rale and thickness were recorded by Edward FTM5 thickness monitoring unit housed in the
vacuum chamber, the thicknesses of the films were further verified by ellipsometric
technique with a variation of ± 3 nm. All the films were heat-treated batch wise between
.SO'^C and 105°C for 30 to 40 mins, and they were returned to room temperature by
decreasing the temperature in steps of about l°C min“’. The SEM is performed (10-12 KV)
using Cambridge IS1-60A instrument and XRD of the heat-treated Se film is obtained at
room temperature using Philips diffractometer, model PW-1050 at CuK„ radiation
(wavelength = 1 .514 A). The Raman spectra arc recorded at room temperature using spectra
physics spectrometer (Stabilitc-2017) equipped with water cooled detector and 5145 A
radiation from an argon ion laser with power -l(X) mW.
3. Results
The scanning electron micrographs (SEMs) which were taken in quick succession are
shown in Figures l(a-0. Most characteristic features for amorphous to crystalline transition
of the films by isolhennal annealing were observed for the thickness 1 10 nm.
Ckaractefistics of selenium films on different substrates etc
Plate I
Figure 1. SEM .showing : (a) G^I Se-filnis (110 nm) deposited on
glass substrate after heat treatment for 30 minutes at 75°C. (b) after
heat treatment of G-I for 30 mins at 90®C, (c) A-1 films (110 nm)
deposited on aluminium (Al) substrate after heat treatment for 30 mins
at 75°C. (d) Q-I films (110 nm) deposited on Z-cut quartz substrate
after heat treatment for 40 mins at sd’C, (e>) magnified structute of (d)
and (f) after heat treatment of (^I for 40 mins at 90"C.
Characteristics of selenium films on different substrates etc
203
The specimen G-I when heal treated below 65°C, remains amorphous in nature;
but at 75°C, the micro structures show presence of segregated worm loopings (Figure la)
with few spherulites of circular front. The same specimen when heat treated above
90°C, shows sufficient distinct spherulites in the transparent zone, indicating completion
of crystalline phase (Figure lb). The XRD (Figure 2) shows that the structure is
hexagonal one (100). Similar phenomena are delayed in case of specimens G-II
and G-IIl.
Figure 2. XRD showing G-I films after heat treatment tor .10 mins al 90“C
Onset of crystallization has been observed in A-I specimens when heal treated al
65'^C for 30 mins; al 75°C. decorated spherulites with few disjoined chains arc observed
(Figure Ic). When a virgin sample of the same lot is directly treated al 75°C, the
microstructure is identical to that of stepwise heat treatment up to 75X. For higher
thicknesses (A-11 and A-III specimens), the crystallization process starts al higher
temperatures but the transition is very fast.
The Q-I films on quartz al the ambient have been found to be amorphous as in the
case of glass and Al substrates when examined through XRDs (figures not shown). When
the (J-I specimen was heal treated at .50°C for 40 mins, some distinct snowflake of ice like
micro-structures are observed (Figure Id). A magnified SEM of one such zone (Figure le)
is identified clearly as dendritic in appearance. This pattern is characteristic and is absent
when tried with other specimens including Q-II and Q-III samples. At 75‘^C, numerous
liny crystallites together with some smaller dendritic growths have been observed. On
further heal treatment, the equilibrium condition for stable phase appears similar to that of
other substrate films of same thickness but the sizes of crystallite are found to be smaller
(Figure 10-
The Bragg reflection peaks (Figure 3a) of Q-I heat treated at 50°C show high
intense peak al £/ = 3.68 A which indicates the monoclinic phase of Se with larger molecular
separation. During stepwise heat treatment (15°C), the intensity of 3.68 A peak starts
decreasing and finally at 90®C, the Bragg peaks (Figure 3b) show phase transition towards
stable state of Se which are identified as (100) and (lOl) crystal planes. Figure 3(c) shows
(100) and (200) faces of the single crystal quartz substrate. The Raman spectra of Q-I films
204
S K Bhadra, A K Maiti and K Goswami
at various stages were recorded but only the active bands are observed when the specimen
was annealed at 90‘'C (Figure 4).
Figure J. XRD showing (a) Q-l films after heat treatment for 40 mins
at 50T. (b) Q-I films after stepwise heat treatment to 90°C for 40 mins and
(c) blank Z-cut quartz substrate
Figure 4. Ramon vspectra of Q-l films . (a) after heat treatment for 40 mins
at 50®C and (b) after stepwise heal treatment to 90®C.
The dendritic feature did not occur in Q-II and Q-III specimens. A Q-III film on
heat treatment at 65*^0, shows swollen surfaces with clusters of indistinct pattern in SEM
micrographs. Above 75°C, the microstructure of the same specimen (Figure 5) starts
flattening with increase in sizes. Above 105®C, mixture of spherulites and swollen surfaces
have been observed (figure not shown).
4. Discussion
Before the work of Audiere et at [9,10], it was believed that the amorphous Seg has
a mixture of Se^ < g and whose proportion is the controlling factor in determining the
Characteristics of selenium films on different substrates etc
Plate II
Figure 5. SEM showing Q-lll film (330 nm) after stepwise heat tieatment
10 75T for 30 mins.
Characteristics of selenium films on different substrates etc 205
properties and they showed that Se vapour at the source temperature of 385°C quenches
into long and short species in a manner to produce a mixed deposit of two amorphous
materials. During annealing at a temperature greater than the glass transition temperature,
nuclealion commences only at the long species sites and crystallinity gradually develops by
using up the relatively mobile short species. In the present case, an adequate mixture of
long and short species are observed when the annealing is performed on G-1 (Figure la)
where the source Se vapour temperature was maintained at 350°C, the results corroborate
with Audiere et al [10]. It appears from Figure Ic that even under lower degree of heat
treatment (65°C), the process of crystallization is accelerated by changing the sub.sirate
horn glass to Al. Heavens and Griffiths [11] suggested that the substrate has a
profound effect on crystal growth kinetics and morphology. According to Griffiths
and Fitton [12], the spherulitic crystals of Se in contact with soda glass, grow fa.ster than
in the bulk with different microstructures. While studying the microstructurcs by
considering the effect of recrystallization and grain growth under high electron beam
irradiation, the present a-Se films deposited on various substrates show different
morphological transitions.
The dendritic feature oKserved in Figures 1(d) and 1(e) arises from the differences in
ihc .solute content at a faster rate than the movement of the solidification isotherms [12J.
The How initially dispersed in many paths since the flowing warm material starts to
dissolve the already solidified materials. The face growth rate lags and the corner growth
increases to iTtaintain the growth rate more or less steady. In order to attain the forced
growth rate, the corner penetrates further while the faces grow slowly in depleted part of
the material in the film [13]. When Se film heat-treated at 5()°C and returned to room
temperature, the warm Se cools on a highly polished substrate, the stem of the dendrite
and the branches grow from its sides which »arc directed along the fast growing
crystallographic axes. The symmetry of each flake may be due to haxagonal symmetry of
Se and the substrate topology. Morphological instability occurs when the film subsequently
heat treated at 75°C and further heat treatment coalesces the dendritic feature and the
formation of the crystallites are similar in appearance to that of films on other substrates
(Figure If).
The Raman shift at 237 cm"* shown in Figure 4 corresponds to trigonal state of
Se [14], other bands are characteristics of quartz crystal. But no active Raman bands have
been found during the melastable transitions, it may be due to the presence of large
asymmetry in the molecular structure. Since trigonal Se has highly active Raman bands
along the c-axis and the incident laser polarised parallel to the plane of scattering, the Se-Se
stretching vibrational modes at 250 cm”* arc observed to be absent. Dendritic nature of
Se-film and bulk has been reported by sevCTal groups [15,16] and they explained the growth
characteristics as metaStable slate of Se. While the typical nature of their growth pattern
72A(3)-6
206
S K Bhadra, A K Maiti and K Goswami
observed on a quartz substrate is quite uncharacteristic, Griffiths et a/ [17] observed that Se
film grown on (100) cleavage face of MgO has the structures which remain completely
amorphous even exercising large temperature of heat treatment; the same is not true
when the film is grown on different crystal planes of single crystal of KCl and KBr.
They inferred that there is marked contrast with the structure of the film on different
substrates and specially MgO has shown inhibitory effect on Se-crystallization. In a
previous communication, Bhadra et al [7] emphasized that the sticking of Se-atoms on
quartz is very high compared with glassy substrate, and the phenomena may attribute to
peculiar orientation of the lamellae in the film-substrate interface as happened in the
present case.
According to Kotkata [18], the crystallization time is shorter in case of thin films
than those of bulk Se. It is evident from the results of Figures l(a-f) that for isothermal
annealing of a-Se films upto a particular thickness, quasi-homogeneous growth persists. As
the thickness increases (Figure 5), the crystallization process becomes delayed and the
mixed slate prevails until a high activation energy be applied. Non-occurrence of dendritic
growth for higher thicknesses on quartz is not clear; perhaps for a critical thickness of the
a-Sc film, the intcrfacial growth becomes relatively slower and the behaviour of amorphous
to crystalline transition in this state is substrate dominant.
Acknowledgment
Authors wish to thank Dr. A K Chakraborly, Central Glass and Ceramic Research
Insliluie and Dr. S K Nandi, Jadavpur University for their help in undertaking XRD and
Raman spectra.
References
[1] N F Mott and E A Davis Electronic Processes in Non-crystalline Materials (Oxford ■ Clarendon)
p. ^18 (1979)
[2] P Andonov, J. Non-cryst. Solids 47 297 ( 1982)
1.1] M Kowarada and Y Nishina Jpn. J. Appl. Phys. 14 1519 (1975)
14] K S Kim and D Turnbull Appl. Phys. 44 5237 (1973)
[5] G Gross. R B Stephens and D Turnbull J. Appl Phys. 48 1 1.39 (1977)
[6] S Choudhuri, S K Biswas, A Chowdhury and K Goswami J. Non Crysi. Solids 46 17] (1982)
[7] S K Bhadra. A K Maiti, R Bhor, D Talapatra and K Goswami J. Mater. Sci. Lett. 13 525 (1994)
18] D C Campbell in The Use of Thin Films in Physical Inve.aif>ations ed. J C Anderson (London :
Academic) p 36 (1966)
[9] J P Audiere. Ch Mazieres and J C Corballes J. Non-Cryst Solids 34 37 (1979)
[10] J P Audiere. Ch Mazieres and J C Carballes J. Non-Cryst. Solids 27 41 1 (1978)
[11] OS Heavens and C H Grifriths Acta. Crysta. 18 532 (1965)
[12] C H Griffiths and B Fitton in Physics of Se and Te ed. W C Cooper (Oxford : Pergamon) p 163
(l%9)
Characteristics of selenium films on different substrates etc
207
[ 13 ] B R Pamplin Crystal Growth (2nd edn.) (Oxford ; Pergamon) p 485 ( 1 980)
[14] ' G Lucovsky, A Mooradian, W Taylor, G B Wright and R C Keezeer Solid Slate Comniun. 5 113
(1967)
[15] M Ozenbas and H Kalebozen J Crysr Growth 78 523 ( 1 986)
[16] J C Bnce Crystal Growth Process (London : Blackie) p 1 39 ( 1 986)
[17] C H Griffiths and H Sang m Physics of Se and Te td W C Cooper (Oxford . Pergamon) p 135
(1969)
[18] M F Kotkaia J. Mater Sci. 27 p 4847. 4858 ( 1 992)
Indian J. Phys. 72A(3). 209-216 (1998)
UP A
— an intenia tional joum al
Neutron diffraction study of tin-substituted Mg-Zn
ferrites
A K Ghatage, S A Patil and S K Paranjpe*
Department of Physics. Shivaji University, Kolhapur-416 004.
Maharashtra, India
Solid Slate Physics Division, Bhabhu Atomic Research Centre.
Trombay, Munibai-40() 085, India
Received 5 September 1997. accepted 17 February 1998
Abstract : The magnetic behbviour ol tin-subslituted Mg-Zn ferrites has been studied by
powder neutron diffraction and magnetic measurements The net magnetic moment (n/}) and
Curie temperature decreases with increase in tin content The experimental values of np
obtained from neutron diffraction and magnetic measurements are found to be less than those
calculated with Neel's model GilJeo's model has also been used to calculate n/j and T, values
These values ore higher than the experimental ones. It is concluded that canted spin airangeinent
is favoured in these ferrites.
Keywords : Ferntes, magnetic structure, neutron diffraction
PACSNos. : 61 l2.Ld,75 50.Gg.75.30.Cr
1. Introduction
A wide variety of ferrite materials has been developed for application in electronic and
microwave industries. The cubic spinel structure permits to substitute cations selectively
on the octahedral and/or tetrahedral sites. This helps in modifying electrical and magnetic
ordering in these systems. Extensive work has been done by various workers to upgrade
the properties of ferrites by substituting different types and amounts of impurities. It has
been reported that by addition of small amount of tetravalenl ions like titanium or tin,
electrical and magnetic properties of basic Ni-Zn and Mg-Zn ferrites are significantly
influenced [1-3]. An anomaly in magnetisation was observed in tin substituted Mg ferrite
[4], It was also stated that for lower contents of tin, it occupies only B-site and as tin-
content increases, it occupies both A and B sites [3]. In order to understand the cation
distribution and its effect on the magnetisation behaviour, we have carried out neutmn
©19981 ACS
210
A K Ghaiage, S A Patil and S K Paranjpe
diffraclian, low field ac susceptibility and magnetisation measurements on tin substituted
Mg-Zn ferrites.
2. Experimental
The .samples ol the senes Mgo 7 +^no 3 SnjrFe 2 _ 2 x 04 (with x = 0.0, 0 . 1 , 0.3) were prepared by
ceramic method. AR grade MgO, ZnO, SnC) 2 , Fe 20 i were weighed in required proportions
and mixed thoroughly. These powders were pre-sinlered in air at 800®C for 10 hours,
then milled and pellets were made and finally sintered at llOO'^C for 24 hours in air and
(hen slowly cooled. The formation of single phase was confirmed by X-ray diffraction
using CuKji radiation. Magnetisation measurements were carried out on high field
loop tracer. AC-susceptibility (Xac) was measured using low field ac-susceplibilily
technique, and Curie temperature was obtained from the normali.sed susceptibility versus
temperature plots.
Neutron diffraction measurements were carried out on polycrystalline samples using
the position sensitive detector based powder diffractometer at Dhruva reactor at BARC.
Monochromatic neutrons having wavelength of 1.094 A were used for the experiment. The
samples were packed in cylindrical vanadium containers. Diffraction profiles were recorded
at 300 K in the angular range (29) 10° to 100°. The patterns showed single phase
compounds except for a small unidentified impurity phase for x = 0.3 sample. The X-ray
patterns, however, did not give any indication of such an impurity pha.se. The data were
analysed using Rielveld profile refinement technique for both chemical and magnetic
structures [5,6|.
3. Results and discussion
As the compounds are magnetically ordered at 300 K, only higher angle data, where the
magnetic contribution to the Bragg peaks is negligible, were first refined to get the chemical
structure [7]. For the system under study, the occupancies of some of cations were fixed by
considering their site preference. It is well known that Zn has strong preference for A-sitc.
In the refinement, therefore, all the Zn ions were put on the A-site. The oxygen position
parameter (m), the isotropic temperature factor (B) and lattice constant (a) were varied in
addition to the profile half width parameters. The site occupancies of Fc, Sn and Mg were
varied independently within the constraint of satisfying the stoichiometry of the system.
The fitted and observed profiles for the composition x = 0.0 are shown in Figure 1 , along
with the difference plot.
The results of the refinement are summarised in Table 1 . It is observed that the
lattice parameter a increases with increase in Mg and Sn content, which may be a
direct consequence of the larger ionic radii of the Mg and Sn (i.e. 0.78 A and 0.69 A,
respectively) as compared to Fe (0.67 A). The oxygen position parameter was found to
be almost constant (u s 0.258) for all the samples. From u-parameter it is concluded that the
Neutron diffraction study of tin-suhstituted Mg-Zn ferrites
211
Tabic 1 . I’aranielcrs obtained from Rieiveld profile refinement, for Mgo 74.^n() 3Sn^’c2_2r04
system
A
Haiumeler
0
01
0 3
Lattice
8.385 (8)
8 423 (8)
8 460 (6)
constant a (A)
Oxygen
0.2579
0.258
0.2582
parameter u (A)
Temperature
0.50
0.43
0 33
factor B (cm^)
Cation
(ZnojMgo 056^^0.644)
(Zno3Mgo i38Feo562)
(Zno_3Mgo266Fco,424)
distribution
fMg0 644P‘=1.356l
[Mgo.662Sno.lFci.238l
[Mgo 734 Sno 3Feo 966)
hA
08148
0 7807
07178
0.8198
0.7995
07515
Rp
3 75
3 85
4 76
^wp
A.l
4.84
604
^exp
inA
3.6
281
Rb
3.42 4.62
From magnetic structure analysis
5.93
2.1(1)
1.85 (6)
1 22 (4)
Free ion
2.35
1.99
1.44
-
1.99(6)
1.65(6)
0.83 (6)
Free ion
2.49
2.19
L65
212 A K Chatage. S A Patii and S K Paranjpe
coordination of Fe ion is very little affected by changes in the compositional parameter.
From the cation distribution it is observed that Sn occupies B-site. Mg ion, however,
occupies both the sites.
The A-site magnetic moment was deduced from the intensities of (220) and (422)
leflections. The structure factors of both these reflections have contribution from A-siie
magnetic moment only. The B-site moment was obtained using (222) reflection, the
intensity of which depends only on B-site magnetic moment. These values were used as
initial parameters for magnetic profile refinement. The values obtained after refinement are
almost same as those derived from the intensities. The fitted profile with magnetic
icllections for one of the compositions x = 0.0 is shown in Figure 2. The values of the
magnetic moments obtained are given in Table 2. The magnetic moments determined from
hysteresis and from neutron study for different values of x are also given in Tabic 2. These
values are in good agreement with each other and decrease with increase in jr.
The magnetisation behaviour in ferrites is explained on the basis of Neel's molecular
field model. Considering the magnetic ions on both tetrahedral (A) and octahedral (B) sites,
the possible exchange intefactions are the two intrasite (AA and BB) and one intersiCe (AB).
Amongst the three magnetic interactions, the intersite AB interaction is the strongest. In this
Neutron diffraction study ^tin-substituted Mg-Zn ferrites 2 1 3
model collinear arrangement of magnetic moments of individual site is presumed i.e. the
magnetic ions on each sublattice are ferromagnetically aligned with an opposite alignment
of intcrsite moments. The net magnetisation is the vector sum of octahedral (B) and
tetrahedral (A) site magnetisations. The ng values are calculated using Neel's two sublattice
model with suitable correction for the Brillouin function and using the cation distribution
Table 2. Magnetic moment and Curie temperature from various methods for Mg(j 7 ^fZn()
SnjFc 2 _ 2 j 04 system.
content
X
Magnetic moment (ng)
Curie temperature (K)
From
neutron
From
magnetic
measurement
From
Neel’.s
model
From
Gilleo's
model
From
susceptibility
From
Gilleo's
model
0
1.88
1.71
3.58
34
590
600
0 1
1.45
1.54
3.38
3
520
577
0.3
0 45
0 54
266
1 78
450
522
obtained from neutron data. These values are given in Table 2. The ng values show a
decreasing trend with x. The experimental ng values arc lower than the theoretical ones
indicating a deviation from the collinear arrangement of moments.
4. Gilleo's model
The magnetic moment and Curie temperature were calculated using the Gilleo's model
[8,9]. With the assumption that magnetic moment actively participates in ferrimagnetism
only when it interacts with two or more magnetic ions in different coordination. Gilleo has
proposed a statistical model which neglects intrasublattice interaction and thus no canted
spin is considered.
If K is the fraction of ions which are replaced by nonmagnetic ions at one site;
for each ( l-^), the probability of one ion being linked with other m ions is
The probability E that an ion is linked with no or almost one magnetic ion is
E = P^(m) = n/f"-'
In the present system
W„Fe 3 .„ 04 ,
where m is the amount of the nonmagnetic cation M. The cation distribution is given as
The magnetic moment is given by
72A(3)-7
214 A K Ghatage, S A Patil and S K Paranjpe
where Af«» = 2 x 3(l-Aro)[l -£o(^/)]
and = 1 X 5(l-*r,)[l -£,(^ro)].
Here, K, * xm (at A site) and £o s (I -jc) m/2 (at B site),
n^g s 12 and ngg =6,
and EQ{K,)=^6K-5K.E,(KQ) = nK^'
ThuSa calculated magnetic moment values are given in Table 2 and are higher than
the observed values.
The theoretical calculations to estimate tuc Curie temperature for spinel ferrites are
given by Gilleo [8], According to the model the Curie temperature is proportional to the
number of active linkages per magnetic ion per formula unit and is cast in the form :
T, = 3x(l-jc)[l -£o (£,)][! -£,(£o)] (3- m)ro/2[;c(l -£„(£,)
+ ( 1 - j :)(1 -£,(£ 0 )].
where Tq is a constant. Milligan et al [10] have evaluated Tq for MgFe204 with cation
distribution Mgo.iFeo.9[Mgo9Fei.i], and Curie temperature 440°C. The Tq was calculated to
be 961 X. The calculated Curie temperatures (r^.) are given in Table 2. The theoretically
calculated values are higher than those obtained from experiment, the difference being
large for ferrites with higher Sn-concentration.
5. Non-collinear structure
From Table" 1 , it is observed that the A-site moment is close to the free ion value. The B-sile
moments on the other hand, are smaller than their estimated free ion values, suggesting that
the B-site moments are noncollinear. The occurrence of localised canting has been reported
in tetravalent and zinc mixed ferrites [1 1]. A canting of Yaffet-Kittel [12] type on the B-site
and spatial ordering of the transverse spin components of the magnetic moment should give
rise to the (200) reflection [6], which is purely magnetic in nature. None of the systems
studied here show this reflection. However, the absence of (200) does not rule out the
possibility of a canted structure as has been shown in many cubic spinels like^nCr2jr
Ga2_2x04 [13] and Zn^CO|_jrFeCr04 [13]. Such a behaviour can be explained on the basis of
a long range ferrimagnetic ordering of the longitudinal component with the A-site moments,
and a disordered normal component. The system under study could have similar behaviour.
6. ac-susceptibUity
The temperature dependence of the normalised ac susceptibility for the series of samples
is shown in Figure 3. From the nature of these plots, it is observed that Xk remains
almost constant in all the samples until the temperature reaches nearer to Curie temperature.
Neutron diffraction study of tin-substituted Mg-Zn ferrites
215
The normalised ac susceptibility Xoc drops rapidly to zero at Curie temperature. However,
tailing effect is observed for higher content of Sn.
Figure 3. Vanation of suKcepribility with temperature for Mgo 7 +rZno 3 Sn^e 2 _ 2 i 04 .system.
A polycrystallinc magnetic material consists of three types of domain slates, viz.
multidomain (MD), singledomain (SD) and supcrparamagnet (SP). It has been observed
that for MD gamples Xac does not change appreciably with temperature and drops off
.sharply at Curie temperature. For SP samples Xac decreases with temperature and become
zero at Curie temperature. For SD samples Xac increases and shows a maxima at blocking
temperature T/, and then decreases to zero at Curie temperature. Murthy and Nandikar [ 1 3]
have explained the magnetic behaviour of ferrites on the basis of shapes of Xac curves. From
these observations and based on the concept given above it can be concluded that the
samples under study contain MD stales. It is also noted that tailing in Xac curve near Curie
temperature is due to canting in ferrites. Our samples show tailing effect for higher content
of Sn. This observation supports our conclusion of the presence of a canted magnetic
ordering in these materials.
7. Conclusion
The neutron diffraction and magnetic measurements on the series of Sn substituted Mg>Zn
ferrites show that the magnetic moment on the octahedral site is reduced and can be
explained on the basis of a canted spin arrangement.
Acknowledgment
Authors (AKG, SAP) are thankful to lUC-DAE Facilities (Indore) for providing
imancial assistance to carry out this work. We are thankful to Dr. K R Rao, Ex Head, Solid
216 A K Gliata^iie, S A Hatil and S K Paranjpe
Stale Physics Division, Bhabha Atomic Research Centre, Mumbai, for the encouragement
and Dr. V Ganeshan (lUC-DAEF, Indore) for X-ray diffraction work.
Kcrcrences
f I ] Usha Varashney and R K Puri IEEE. Trans Maj^n, 25 3109 (1989)
(21 A R Das, V S Ananthan and D C Khan J Appl. Phys. 57 4189 (1985)
[.^1 S S Suryavanshi, S R Savant and S A Paul Indian J'Pure Appl Phys. 31 500 (1993)
(4) DR Sagar, Prakash Chandra. S N Chattarjee and P Kishan Pror ICF-5 India, Aihmces m Fernuw
cd C M Snvasiava and M J Patni (New Delhi , IBH and Oxford) (1989)
fS| MM Rielveld Am Cryvi 22 151 ( 1967)
16] HM Rielveld y Appl Crysi 2 65(1969)
17] R A Young and D B Wiles J. Appl Cryst. 10 262 (1982)
|H] M A Gilleoy Phys Chem. Sr;/iy,T 13 33 (I960)
|9| Chen Yang and HcRui-Yuny Mag. Materials 116231 (1992)
1 lOJ W Milligan. Y Tumai and J T Richardson J Appl Phys. 34 2093 (1963)
fill K C Snvastava, D C Khan and A R Das Phys. Rev B4l 1 25 1 4 ( 1 990)
1 12| NS Saiya Murlhy, M G Naicra, S I Yous.se(f, R J Begum and C M Snvasiava Phys Rn 181 969 (1969)
I n I C R K Mufthy and N G Nandikar Pramana 13 473 (1979)
Indian J. Phys. 72A (3), 217-224 (1998)
tJP A
— an imi^otional journal
Fluctuations in high Tc superconductors ivith
inequivalent conducting layers
R K John and V C Kuriakose
Department of Physics. Cochin Universilv of Science and Technology,
Kochi-682 022, India
Received 5 September 1997, auepted 4 March 1998
Abstract ; The fluctuation contribution to the London penetration depth A,
pnraconductivity parallel to the nb-plane and to the c-axis (cr' ) and the fluctuation
specific heat (Cp) ol layered high-'/'^ superconductors with inequivalent conducting layers
are calculated using a Lawrencc-Doniach (LD) free energy functional proposed by
Buhicv.skii and Vagner |)] Dimensional cross over (DCR) occurs near The specific
temperature dependence of (7^ differs qualitatively from that of ag/j . The fluctuation
contribution below T^. to the London penetration depth is anisotropic in the rib-planc for
YBaCuO compounds
Keywords High temperature superconductors, fluctuations, Lawrence-Doniach
model
PACS No. : 74.40.+k
1. Introduction
Several experiments point to the importance of fluctuations in the thermodynamics of
high temperature superconductors (HTSC). The effect of fluctuations in HTSC’s has
been observed in magnetization, conductivity, current-voltage and specific heat
measurements [2-5] and is quite pronounced owing to the small coherence length ^ -^lO A,
high transition temperature -100 K and layered structure. Since fluctuation effects are
more pronounced in lower dimensions it is possible to explore the dimensionality of the
fluctuations in the layered superconductors. Paraconductivity data from single crystals of
YBaCuO [2,3] exhibit dimensional cross over from 2D to 3D near T^. Baraduc and
Buzdin [6] extended the LD model to the YBaCuO system by introducing two different
coupling constants among the Cu02 layers and has predicted DCR above T^. in
paraconductivity measurements. Theodorakis and Tesanovic [7] attributed the positive
© 19981ACS
218
R K John and V C Kuriakose
curvature of the upper critical field H^ 2 (T) of HTSC’s near to the DCR. These authors
considered the fact that most of the layered superconductors contain not only
superconducting (SC) layers but also non-superconducting (NSC) layers. The
Josephson coupling between neighbouring SC and NSC layers makes the order parameter
non-/.ero on the NSC layers as well, through a proximity effect as observed by Briceno
and Zcill (8| in Bi 2 : 2 : 1 : 2. Consequently they proposed different order parameters
for the mequivalent layers and have shown that the spatial variation of the order
parameter from layer to layer in materials whose NSC layers are in proximity of SC
layers gives rise to the positive curvature of Ht. 2 . Bulaevskii and Vagner [1] also
employed a similar model to study the magnetic critical fields and anisotropy of vortex
siruclurc in HTSC. In the case of YBa 2 Cu 307 crystals an elementary cell consists of two
types of conducting layers : two i.sotropic Cu02 planes (SC) and one layer with CuO
chain (NSC). If the coupling of the inequivalent layers is strong enough, effective
averaging of the superconducting characteristics of the layers takes place and wc obtain
the standard model. If on the other hand, the coupling between identical planes is
stronger than that between the inequivalent planes we have a model with two weakly
coupled order parameters yr, and (see Figure I). \(f\ and y^ describe the multiple Cu02
Figure 1. Superconducting and non-isupercon-
ducting plane.s in YBuCuO Shaded area repre.sent.s
Cu02 double layers and dotted lines the CuO
chain layers The relationship between the order
parameter on the SC layer and that on the NSC
layer induced by proximity effect is schematically
indicated
Y2,n
layers and metallic layers respectively. The scenario is the same in bismuth and
thallium based superconductors also as they contain multiple Cu 02 layers separated
by metallic layers (BiO and TIO layers respectively). Like the CuO chain layers in
the Yttrium compounds, the BiO and TIO layers in these compounds act as
charge reservoirs, dope charges into the Cu 02 layers and enhance the interlayer
coupling.
In the present paper, we calculate the fluctuation contribution to the London
penetration depth, parallel and perpendicular paraconduclivity and fluctuation specific heat
based on the LD free energy functional proposed by Bulaevskii and Vagner and study their
specific temperature dependence.
Fluctuations in high superconductors etc
219
2. Fluctiiatkm contribution to the London penetration depth
The free energy expression considered in ref. [1] is
F.
.AL
2mu
+ 02
2
+ t
\dp.
( 1 )
are the order parameters for layers i = 1, 2 in the unit cells numbered by the index n.
Subscript 1 refers to the multiple Cu02 layers and 2 to the metallic layers, p = (x, y) and z is
the axis perpendicular to the layers. Xn = ^ ^ characteristic
distance between the layers.
Let us write
a, = aiCT-T,) = a,TT,
and 02 = a 2 (T-Tc) = 02
(T-T )
where T = j, -- . For simplicity we assume the same bare critical temperatures for both
‘ c
the inequivalent layers. Vn is the gradient parallel to the layers. Ah „ and A^,^ are respectively
the components of the vector potential parallel and perpendicular to the n-th layer, t is the
coupling coefficient between the neighbouring inequivalent layers, mu is the effective mass
of the Cooper pairs in the isotropic CUO 2 planes. The anisotropy of the effective mass due
to the chain structure is taken into account in the CuO planes.
We can calculate the fluctuation contribution to the London penetration depth below
Tc by writing
V'l.n
'- + 0
I, /I
hi
represents the equilibrium value of the order parameter at r< and ^ represents the
fluctuation contribution. In the calculation of the fluctuations in high-Tc superconductors for
which {(T) « X{T), we may treat A as constant. This is because the characteristic length
scale for changes in A is of the order of ^(7) whereas the same for ip fluctuations i$ of the
220
R K John and V C Kuriakost
order of ^(T). After performing Fourier transformation, the fluctuation contribution to the
free energy can be written as
+ ■^(^2,* 02, -It +02.K02.-«) + ^+|02.rf ]. (2)
whc C* = ., + + 2r{l-rcos[z. ±f ]}.
and R = K{q, k). q is the inplane wave vector and k is the c-axis wave vector. 0 \s the angle
which the inplane wave vector makes with the x-axis.
I A |2
We have set
I I
. This introduces an additional phase term which does
not affect the derivation of the final result. The general expression for the fluctuation
contribution to the free energy is
= -rinjexp[-(5//.„(0.A)/r]Df (3)
Taking ^F^as an effective hamiltonian,
Ffl = -rinjexp[(-5F,X0i.*.02.«.A)/7']D0,*D^2,jf (4)
The additional superconducting current due to fluctuations is
(5)
Performing the functional integration in (4) over the real and imaginary parts of
and 02 If,
= -^Shnff^r2(C*C. -a})+ lnK^T^(D^D. -a|)]. (6)
^ If
The London penetration depth is given by the expression
Fluctuations in high Tc superconductors etc
221
Since we are interested in finding the linear response, only the vector potential A is
considered to be small. Neglecting terms in second and higher powers of A as well as t,
changing summation over if in (6) into integration and using eqs. (5) and (7), the fluctuation
contribution to the London penetration depth can be calculated as
' ■ J^U/J Ksltlj ^ [4 - 2 |T|)
,(N + ^) 2(r,+|T|)
+ 2 — j-j — In —
W 3'-j
l = x.y, M=-rMgde, M;' = £2*^
WJO « »
i = (-!)>-■ and r, =
ite^dT,
^ f 2 |t| +
1,7‘mj M + r^ In——
;=IJ L ^
At large rvalues (r » r).
^'T,. 8|t| ^ y2.
n.)
At large r, specific temperature dependence of SXj^ is different from that of SX~^ because
of the presence of an additional term linear in It I in eq. (11).
3. Fluctuation specific heat
If we consider Gaussian fluctuations above the quartic terms in (1) can be neglected.
Setting A = 0, the Fourier transform of eq. (1) is performed. The order parameter v ^2 of the
I V'lA 1^
NSC layers arises through a proximity effect. Let us therefore put where
V'l.^ and ypiji are the Fourier transformed quantities of v'l „ and V'z./i respectively.
+ 2/(1 -5) - 2t6 cos kd
+ 2/.
* L '
where
£,
= a. + ' ■ ■
2m , 1
and
= 2A#.
72A(3)-8
222
R K John and V C Kuriakose
Change in thermodynamic potential
O-Qq = -7lnjexp|
Fluctuation specific heat
+e:|r2,Kr)
<iV\.KdV2.K- (13)
(14)
Changing summation into integration and performing the integration overX,
T,
d
a, mu
+4r|r(l-5) T+2r2
(15)
where r, = —V and r, =
I a^T^ i
The cross over between 3D and 2D regimes is characterised by the parameters r and
5. For r(l - 5) » Tthe specific temperature dependence of becomes 3D.
4. Parallel and perpendicular paraconductivity
The calculation of paraconductivity in this model is straight forward and is done using the
lime-dependent Gifizburg-Landau (TDGL) equation. Following [6] and [9] parallel
fluctuation current can be obtained as
^ ~ 4 Id
^ K
g| q(q E) ^ «2 (liq-E)
Ml
(16)
Performing integration over K, we obtain the fluctuation contribution to the conductivity
parallel to the layers
nhd
I
+4r,T(l-5) (^ + 2r2)
(17)
The perpendicular conductivity is calculated using the approach of ref. [10]. Terms in
higher powers of t are neglected.
=
32^3
m,|
(T^ +4r|T(l-5) (T+2r2)2
( 18 )
The specific temperature dependence of CT^ is thus different from that of a' .
Fluctuations in high 7^ superconductors etc
223
5. Discussion
In YBa 2 Cu 307 , the situation of two strongly coupled superconducting Cu 02 layers weakly
coupled to the non-superconducting CuO chain layers is realized. The calculations based on
the free energy functional (1) describing this situation explain the observation of
dimensional cross over in paraconductivity and fluctuation specific heat measurements.
There is a clear difference between the temperature dependences of a and a' . For T » r,
(he leading term in a';, has a T‘* dependence where as cr' has a dependence. The
(ernperature dependence of cr' for t » r is appropriate for a OD fluctuation regime. The
dimensional cross over in the fluctuation regime of Cfl and d'/, lakes place exactly at the
same temperature as that for a'. However, similar results were also obtained by Baraduc
and Buzdin [6] by considering strong coupling between the two Cu02 planes in the same
elementary cell and weak coupling between cells. They have ignored the influence of the
chain layers where as the inclusion of the NSC layers is crucial to the calculations in this
paper. Qualitatively, both models give the same temperature dependence for fluctuations.
However, the magnitude of the cross over temperature and the fluctuation contribution are
delermined by the inequivalency of the layers combined with weak interlayer coupling. The
two models diverge in the determination of the fluctuation contribution below to the
London penetration depth and the positive curvature of the upper critical field H ^.2 [12). The
presence of CuO chains in YBaCuO compounds is responsible for the anisotropy of the
Huctuations in the a^-plane. In ref. [6] the effect of the chain layers is ignored and as a
result the authors obtain isotropic fluctuations in the ab-phne. Therefore, the measurement
of fluctuation c'ontribution to the London penetration depth in YBaCuO single crystals will
lest the validity of the free energy functional (1). The model could be extended to thallium
and bismuth based compounds also. Like the CuO chain layers in YBaCuO compounds the
dniiblc BiO and TIO layers in bismuth and thallium superconductors respectively act as
charge reservoirs, dope charge carriers into the Cu 02 layers and enhance the interlayer
coupling. However due to the isotropic nature of the BiO and TIO layers, the fluctuations
will be isotropic in the aib-plane.
References
ft) L N Bulacvitkii and I D Vagner Phys. Rev. B43 8694 ( 1991 )
[2] B Oh, K Char, A D Kent, M Naito, M R Beasley, T H Geballe, R H Hammond and A Kapitulnik
Phys Rev. B37 7861 (1988)
[3] T A Friedmann, J P Rice, John Ciapintzaki.s and D M Ginsberg Phy.s. Rev. B39 4258 (1989)
14] S E Inderhees, M B Salamon, Nigel Goldenfeld, J P Rice, B G Pazol and D M Gin.sberg Phys Rev. Lett.
60 1178(1988)
(3] W C Lee, R A Klcmm and D C Johnston Phys. Rev. Lett. 63 1012 (1989)
[6] C Baraduc and Buzdin Phys. Lett. A171 408 (1992)
[7] Stavros Theodorokis and Zlatko Tesanovic Phys. Rev. B40 6659 (1989)
[81 G Briceno and Z Zcttl Solid State Commun. 70 1055 (1989)
224
RKJohn andVCKuriakose
[9] A A Abrikosov Fundamentals of the theory cf metals (Amsterdam ■ North-Holland) ( 1 988)
I lOJ C Baraduc, V Pagnon, A Buzdin, J Y Henry and C Ayachc Phys. Lett. A166 267 (1992)
[11] D £ Farrell, J P Rice, D M Ginsberg and J-U Liu Phys, Rev. Lett. 64 1573 (1990); L Matsubara,
H Tanigowa, T Ogura, H Yamashita and M Xinoshita Phys Rev. B4S 7414 (1992)
1 1 2] R K John and V C Kuriakose (submitted)
[I3| R A Klemm Phys. Rev. B41 2073 (1990)
1 14) B 1 Ivlev and N B Kopnin Phys. Rev. B42 10052 (1990)
Indian J. Phys. 72A (3), 225-232 (1998)
UP A
— an international jour nal
Investigation of graphitizing carbons from organic
compounds by various experimental techniques
T Hossain and J Fodder
Department of Physics, BUET, Dhaka. Bangladesh
Received J5 January 1998. accepted 7 February 1998
Abstract : Graphite making organic compounds such as polynuclear aromatics, high
rank coal always pass through a liquid or plastic state-structural transition of optical anisotropy,
called carbonaceous mesophose. the life time of which is limited by its hardening to a semi-
coke, X-ray analysis show,s that the intcr-laycr spacing of graphitic carbons decreases with
increasing temperature and becomes 3.354 A or nearly so in the graphitization temperature
range 2.50()‘’C to 3000°C Sensitive tint technique of polanzed-light microscopy has been found
most suitable to study the initial formation of spherules, their coalescence and the growth of
mosaic texture dunng the mesophasc period Differential thermal analysis (DTA) trace having
an initial large endotherm with activation energy of the order of 60 K col/mole or above, has
been proved to be an another effective tool for detecting graphitizable organic materials and
in determining the mesophasc intervals A sharp fall Id resistivity with temperature is found to
be an another indicator for the graphitizable organic materials exhibiting semi-conducting
behaviour.
Keywords : Carbon graphitization, XRD, DTA
PACSNos. : 61 10. Nz, 81 70. Pg
1. Introduction
During the heat-treatment of carbon containing materials to high temperatures, the removal
of non-carbdn atoms, usually oxygen, hydrogen, nitrogen or sulphur, as well as some
carbon constitutes the process what is known as 'Carbonization’. This process follows
a rearrangement of order within the remaining carbon atoms which may ultimately
develop a three-dimensional order very close to the well-defined structure of pure graphite
is termed ‘Graphitization’. In fact graphitization occurs in the temperature range 25(X)°C
to 3000°C.
X-ray analysis [1] shows that the carbon-atoms in graphite are arranged in layers.
Bach layer is a continuous net-work of planar, hexagonal rings; the carbon atoms within a
© 1998 lACS
226
T Hossain and J Fodder
layer are held by strong covalent bonds 1.415 A long. The different layers, 3.354 A apart,
are held to each other by weak forces of Van der Waals’ type. In the graphitic carbons, the
apparent inter-layer spacing decreases with increasing temperature [2].
Organic solid materials ultimately producing synthetic graphite usually pass through
a fusion stage during carbonization. This is one but not the only condition for the
graphitizability of organic solid compounds. Many workers [3-7] have demonstrated the
formation of carbonaceous mesophase in the temperature range 35(>-600°C as precursor to
graphilizalion. This mesophase is a liquid or plastic-state structural transition in which the
large lamellar molecules formed by thermal cracking and aromatic polymerization become
aligned in a parallel array to form an optically anisotropic liquid crystal, the life time of
which is limited by its hardening to a semi-coke.
In the initial stages of nuclcaiion, the carbonaceous mesophase appears as small
spherules suspended in the optically isotropic matrix and as carbonization progresses with
increasing temperature and time, the growing mesophase spherules, being denser than the
isotropic parent phase, sink to the bottom of the container. While sinking two or more
spherules coalesce to produce larger droplets, eventually leading to a bulk mesophase as
shown in Figure 1. When viewed microscopically with cross polarizers, the bulk mesophase
usually displays a complex ensemble of extinction contours. The polari zed-light extinction
•EfORC CONTACT
JUST AFTER CONTACT
Figure 1. Rearrangement which appear to
SHORT TME AFTER CONTACT occur when two spheres coalesce.
TYPE OF COMPlfX INTERNAL
STRUCTURE FORMED WHEN
COMPOSITE OFTWO OR MORE
SPHERES CONTRACTS TO ONE
large spheres
contours display nodes and the characteristic Maltese Cross patterns. Using sensitive tint
technique of polarized-light, changes in pleochroism for coalesced and for deformed
mesophase forming mosaic texture are observed.
Differential Thermal Analysis (DTA) as a technique to identify organic compounds
producing synthetic graphite was adopted by some workers [8,9]. For graphitizable organic
materials, endothermal processes of transformation with effective activation energy over
60 K cal/mole are generally seen to occur in the initial stage of the DTA trace (Figure 2).
In the Case of an organic compound under heat-treatment, two competing reactions are
often found to occur; Cross-linking producing an exothermic reaction and chain stripping
and associated reactions, which produce endothermic peaks. The second type often allows
ENDO AT — EXO
Investigation of graphitizing carbons from organic compounds etc
227
the formation of oriented aromatic rings giving rise to graphitizing carbons. Again, the
appearance of an exothermic reaction having activation energy as low as 20 K cal/mole,
somewhere in the initial polymer decomposition reaction, ensures that the resulting carbon
has non-graphitizing properties (Figure 3).
0 too 300 m 700'' C 0 200 400 600 600*
Kinure 2. Thennograins of some graphitizable organic Figure 3. Thermograms of some non-gruphitizable
inalcnals organic materials
The temperature interval of the carbonaceous mesophase may be a few degrees or it
may be tens of degrees and so very difficult to locate. A combination ot differential thermal
analysis and polarizcd-light micrography [10,1 1] has proved a valuable approach to identify
graphitic carbons and for the determination of mesophase interval.
A sharp fall in resistivity [12] is found to be observed indicating semi-conducting
nature in the case of organic compounds ultimately producing artificial graphite during
carbonaceous mesophase transition, This is usually preceded by random resistivity change
due to the emission of various entrapped hydrocarbon gases formed by thermal cracking
and aromatic polymerization.
2. Experimental
2. 1. X-ray analysis :
The experimenlal details of X-ray analysis have been described elsewhere [2]. X-ray
diffractogram of sy'ntheUc graphite derived from North-Western Bangladeshi coals in the
228
T H os sain and J Fodder
temperature 2700X has been shown in Fi-gure 6. This diffractogram resembles that of pure
graphite indicating that the north-western Bangladeshi coals give rise to synthetic graphite.
A recent study of the fneasurement of inter-layer spacing with increasing temperature
undertaken by the autht)rs in the case of pyrene has shown that the inter-layer spacing
decreases with increasing temperature indicating that it is graphitic in nature. The
dilTractograms obtained for pyrene arc depicted in Figure 4.
FiKure 4. X-ray diffraciogranis of Pyrene heat-treated at different temperature ■
(a) raw sample, (b) at 4 1 0°C/ 6 hrs, (c) at 440®C/6 hrs, (d) al 470°C/6 hrs.
2.2. Differential thermal analysis (DTA ) :
Details of the DTA technique have been described elsewhere [10]. Selected aromatic
samples, which have not yet passed through the carbonaceous mesophase due to prolonged
heating al a certain lempcralure, are subjected to heat-treatment in the Stanton
Differential Thermal Analyser. The DTA traces having large endotherms at the
beginning (Figure 5) indicate that they are all graphitizable in practice. The DTA traces
of the partially carbonized samples are also useful for the determination of mesophase
interval.
Investigation of graphitizing carbons from organic compounds etc
229
2J. Polarized- light micrography :
The technique for micrographic preparation of samples has been described elsewhere [11].
0 0 ^ 1-6 2*4 3 2 4-0 4 ^ 6-6 6-4 ( mV )
Figure 5. DTA (races of partially carbonised aromatic organic compounds.
Samples so prepared are observed and photographed with a Reichert polarizing
microscope using reflected polarized- light. Colour photographs of the mesophase spheres
and of subsequent heat-treated samples are usually obtained by High Speed Ektachrome
35 mm reversible film. The coloured mesophase spheres having characteristic Maltese
crosses identifying graphitic carbons are produced by the insertion of a gypsum plate
inclined to the Analyzer at an angle of 45° and placed between the analyzer itself and the
sample under observation. The analyzer and polarizer remain cross with respect to each
other. This is the so-called Sensitive Tint Technique.
2.4. D-C conductivity measurement :
The details of the technique of sample preparation for resistivity measurement have been
described elsewhere [12].
The resistance is measured by standard dc bridge reading to the nearest microvolt at
a heating rate of 2-3°C min"' in the temperature range 105-700°C. The temperature is
measured by a calibrated iron-constantan thermocouple.
3. Results and discussions
A comparism of the X-ray diffractogram of synthetic graphite with that of pure graphite
(Figure 6) will always ensure identification of organic compounds ultimately producing
synthetic graphite. The inter-layer spacing calculated for synthetic graphite obtained from
north-western Bangladeshi coals resembles that of pure graphite. Again the inter-layer
spacings recently calculated from the diffractograms (Figure 4) at different heat- treatment
230 T Hossain and J Fodder
fcmpcrijfurcs of pyrene in (he mesophase region were found decreasing indicating the
criiena of an organic compound producing synthetic graphite.
Charucicnslic ot the DTA traces obtained for the different aromatics (Figure 5) is
ihc presence oC an iniiial large endotherm due lo meliing which is then followed by small
lluctuaiions before a smooth trace, DTA traces trf naphthalene, anthracene and chrysene
almost show the same trend of behaviour. The temperature at which all the fluctuations
terminate, is nothing but the temperature of complete coalescence during mesophase
transition of a particular sample. This can be verified by polarizcd-light microscopy, by
viewing through it a sample heat-treated to similar temperature. For example, the polarized-
light photomicrograph obtained for chrysene (Plate 2) agree quite well with its respective
DTA trace. The temperature at which the mesophase spherules start to devetop in the
sample has not been ascertained by DTA. The polarized-light photomicrograph (Plate 1)
obtained for chrysene shows the temperature at which the mesophase spherules start to
develop.
A sharp fall in resistivity with temperature (Figure 7) is found to be observed
Indicating semi-conducting nature by Bangladeshi coal-peats during carbonaceous
mesophase transition. Irregular variations in resistivity usually occur due to the evolution
of various entrapped hydrocarbon gases during heat-treatment and due to the rearrangement
ol the atoms in the molecules ot the sample. Because of the rearrangement of the atoms, the
energy gap increases and the balance electrons need more energy to jump from balance
Investigation of graphitizing carbons from organic compounds etc
Platt 2. Mosaic formation in chrysene at 530*0 for 5 hrs.
Investigation of graphitiiing carbons from organic compounds etc 23 1
bands to unfilled conduction bands. As a result the resistivity increases with temperature at
the initial stage. Above this stage, the sample starts decomposing and ordering of the
niolecules in the parent material begins causing a gradual decrease in the energy gap which
Figure 7. Resistivity vaiiiUion with
Icmpeiature
gives rise to intrinsic conduction in (he sample. The more the molecules are ordered
struciurally, the more the conduction becomes significant and hence, the resistivity
decreases continuously with increasing temperature indicating the semiconducting nature of
the sample during graphiiization.
4. Conclusion
The criteria of organic compounds producing, artificial graphite may be summari/,cd as
follows :
(a) The inter-layer spacing of synthetic graphite obtained from organic materials heat-,
treated in the graphitization temperature range 2500'’C-3(X)()"C will be 3.354 A or
nearly so. In the graphitic carbons, the apparent inter-layer spacing decreases with
increasing temperature.
(b) Organic materials ultimately producing synthetic graphite, always pass through a
carbonaceous mesophase foimation accompanied by temporary liquefaction or
plasticizing of the materials in the temperature range 350-600X. In this liquid-state
structural transition, large planer molecules become aligned in a parallel array to
form an optically anisotropic liquid crystal. The growing mesophase spherules, the
bulk mesophase and also the plastic flow patterns generally show characteristic
Maltese Crossps and nodes wheif viewed under sensitive tint technique of polarized-
light microscopy.
232
T Hossain and J Fodder
(c) For graphitizable organic materials endothermal processes of decomposition with
effective activation energy of over 60 K cal/mole are generally seen to occur in the
initial stage of the DTA trace.
(d) An organic compound, displaying a sharp fall In resistivity with increasing
temperature in the mesophase region, ultimately produce carbons semiconducting as
well as graphitic in character.
Kefercnccs
Ml G E Bacon Acta Cryst 3137(1 950)
(2) C R Kinney Proc. 2nd Carbon Conf, (Buffalo) (1955)
[3] J D Brooks and G H Taylor Carbon 3 185 (1965)
|41 J Dubois. C Agace and J L White J Metallography 3 337 ( 1 970)
[5] H Honda, H Kimura and Y Sanada Carbon 9 695 ( 1 97 1 )
[6] H Marsh f«p/ 52 205 (1 973)
[7] C A kovac and I C Lewis Carbon 16 433 (1978)
[8] D Dollimoro and G R Heal Carbon 5 65 (1967)
[91 N A Lapina and V S Ostrovskii Thermal Analysis 2, Proc Fourth ICTA (Budapest) 407 (1974)
1 1 0] T Hossain J Bangladesh Acad, Sci 7 57 ( 1 983)
[11] T Hossain and J Dollimore J. Thermochim, Acta 108 2 1 1 ( 1 986)
[12] T Hossain, N Zaman, ABM Shohjalal, A Hossain, T Hossain and N Z Ara Ahmed Thermochim Acta
189 235(1991)
Indian J. Phys. 72A (3), 233-240 (1998)
UP A
- an mternaiional journal
Determination of the activation energy of a
thermoluminescence peak obeying mixed order
kinetics
S Dorendrajit Singh and W Shambhunath Singh*
Department of Physics, Manipur University. Canchipur,
lmphal-79.S 003. Manipur, India
* Department ol Physics, Manipur College, Singjainei.
lmphal-795 tK)8, Manipui, India
and
P S Mazumdar
Acharya Prafulla Chandra College, New Barrackpur 743 276,
West Bengal, India
Received 9 September 1997. luvepted 10 March 199H
Abstract ; A method (oi the determination of the activation energy E of a
thermolumincscencc (TL) peak obeying mixed order kinetics by using a set of expressions, is
presented The method has been applied to the numerically generated mixed order and general
Older TL peaks and expeii mental TL peak of BcO
Keywords : Thcrnioluininescence, mixed order kinetics, activation eneigy
PACS No. : 78 60.Kn
1. Introduction
Thermoluminescence (TL) is often used for the spectroscopic studies of trap levels, in
pariicular, for the evaluation of the trap depth (or activation energy) [1]. The study remains
aciive because of its application in dating and dosimetry [2,31. A large number of TL peaks
can be explained in terms of the three parameters (activation energy £, frequency factor s
and the order of kinetics h) formalism. In order to study TL peaks obeying general order
tGO) kinetics whose shape factor lies between 0.42 and 0.52, Chen [4] used the
i^mpirical relation given by May and Partridge [5]
/(f) = -jinidt = s'n^ exp (-E/ikT)), (1)
© 1998 1 ACS
234
S Dorendrajit Singh, W Shambhunath Singh and P S Majumdar
where t (r) is the intensity of emission at lime t, E (cV) is the activation energy, .v' the
pre-cxponenlial factor having a dimension of sec ', k (eV the Boltzmann
constant, n (cm’^) the concentration of trapped electrons at time t and T the absolute
lempcraturc.
In spite of the extensive use and application of GO kinetics by a number of research
workers it, however, lacks the physical basis and a more physical mixed order (MO)
kinetics has been developed by Chen et al [6] from the set of three differential
equations [7]. The first order equation [8] and the second order equation (9) can be derived
from these differential equations. Yossian and Horowitz (lOJ have successfully applied
MO kinetics both to the synthetic TL glow peaks and to isolated peak 5 in Lif- : Mg, Ti
(TLD-100) following post irradiation annealing at 165°C and commented that MO kinetics
is a viable alternative to GO kinetics in the intermediate range (I <b <2). Chen et al |6]
have also presented a graphical picture of variation of with a{a = /JoA^'o + c), where
fiQ IS the initial concentration of trapped electrons and c is the concentration of trapped
electrons or holes not taking part in TL process in the temperature range being
considered) by choosing a certain value of u,„ (a„, = E/{kT„^), where 7^,, is the temperature
at peak intensity /,„) and reported that is relatively a strong function of or (0 < a < 1 )
and a very weak function of They have also applied the half intensity peak shape
formula [4] for determination of activation energy E to synthetic glow peaks generated
using MO kinetics equation and shown that the values of E are within 3% of the
given values.
In this paper, the dependence of on is taken into account in obtaining (he value
of a from vi’ a curve by taking average of//^, for different values ofu,„ (20 < u,„ < 40)
We aJsfT present a .set of peak shape formulae involving the parameter a for the
determination of activation energy E for TL peaks obeying MO kinetics unlike those of
Chen [4] used by [6] which do not require the information of a The set of formulae has
been applied to numerically generated MO TL peaks to obtain E and found to be in good
agreement with the input energy. Recently, Sakurai and Gartia [11] used the exact solutions
of the basic differential equations involving thermally disconnected traps to fit the
experimental peak of BeO to determine the five important intrinsic trapping parameters,
namely, activation energy, frequency factor, retrapping probability, recombination
probability and concentration of the disconnected traps. In the present_paper the
applicability of MO kinetics is discussed as an alternative model to analyse the
experimental TL peak of BeO irradiated with X-ray (5 minutes).
2. Theory
Following Chen er al [6] the intensity of a TL peak obeying MO kinetics can be
written as
( 2 )
with
l{t) = -dnjdt = .v'Vi (/7 + c)exp (-£/(kr))
s" = sA„INA„ ,
(3)
Determination of the activation energy etc
235
where .y is the frequency factor fr'). N the total concentration of traps, and A„ arc
icspcctivciy the probabilities (r') of recombination and rctrapping. The solution of eq. (2)
lor a linear healing rate /9is given by
.v'V^aexp
(ci"/)3)J^ exp(-£/(*r'))d7''
exp(-£ /(*£))
)
|cxp
1 exp(-£/tO-'))dr'
2
uheic /[) is the initial temperature when/i = n^.
The condition for maximum intensity is given by
~ + (( v"//3)cxp((-fc /(A'r,„ ))j [^exp|(cj"/^)j^^"exp(-£7(*7-')yr'|-a
= 2{cs" / P)c\p(-E/(kT„ ))cxp|(f.v"/j3)j^"' cxp(-t I (kT'))dT'^. (5)
Delining dimensionless quantities u = E/kl\ = E/KTq, end u„, = ElkT„^ and expressing
ilie lem[)eraturc integral in terms ol second exponential integral ( 12], one can write eqs. (4)
and (5) as
aexp[((\v"£/ (pk)\[Ei(u)/ u - E 2 (mq )/ jlexpC -m)
/ = — (6)
|cxp|(a "E / iftk )) (£‘2 («) / M - £2 ) / «o )] - «}
and
2 "" ,
exp
(■i "£ '1 f ^2 (m„, ) E-i(uo)
= 2-^exp(-i(,„ )exp
pIc ) V "n
cs"E]( E2iu„,) E2 {uq)
pk
- a
(7)
Now, ihc I'raciional intensity x(x = ///„,) can be expressed as
exp[(«''£ / (pk)) (£2 (u)/u- El (Mq )/ Mq )]
X = B exp(M„, -«)-
|exp[(« "£ / (^O) ( £2 («) / « - £2 (“0 ) / «o )] - «) ^
with B = 4(c.i" ! P)^ a\(kl E)'^Ulcxp(2u„)-(cs" ! p)'^\'\
( 8 )
(9)
Using Newton-Raphson method [13], we can calculate from eq. (7) and
temperatures Tj on the rising side (T'J < ) and on the falling side (7^ > 7'^ ) of
TL peaks at fractional intensity x from eqs. (8-9). A plot of the pairs of variables
/(m; - w^)], [u^,w; /( m„, -m;)] and lu„,u;u^ / u„{u; -u^)i where
= £ / {kT’ ) and u* = El (kT^ ) for values of u„ in the range 10 < $ 40, are
•ound to be linear so that we can write
u
m
= C| («;/(«; -«„)) + £»,,
(10)
236
5 Dorendrajit Singh, W Shambhunath Singh and P S Majumdar
( 11 )
( 12 )
and u„ = C^(u;u* /u„(tt; -u*)) + Dj.
A linear plot of as a function of u~ / (u~ ) for a= 0.5 and x = 0.5 are shown in
Figure I . Similar results have been obtained for other pairs, namely lu„,u^ / (u^ - )]
and lu„,u^u^ /u„(uj )] and different values of a. The linear plot has been
obtained for x = 0.2 and x ~ 0.8 also.
Figure 1. Variation of
against u„ for a = 0.5.
Eqs. (10-12) can be recast in terms of activation energy and temperatures as
£, = C,l:r^/(r„-r;)+D,A:T„. ( 13 )
£2 = Cj*r2/(r;-r„)+Djtr„. (14)
and £, = C,tr2 /(t; - 7 ;) +D,*r„. (15)
m
The coefficients Cj and Dj {j = 1-3) for a particular value of x occurring in eqs. (10-15)
depend on a By using the method of non-linear least square regression [14], each of the
cueffieienis Cy and Dj can be expressed as a quadratic function of a(0 < a£; 1) as
Cj = Cjo +Cjia+Cj2a^, (16)
and Dj = Djq + Djfa+ Dj^a^. (17)
The coefficients Cy* and Dy* (/ = 1-3, ^ = 0-2) occurring in eqs. (16-17) for jc = 0.2, 0.5, 0.8,
are presented in Table 1.
Table 1. Coefficients Cy* and Dy* (j = 1-3, k = 0-2) occuring in eqs. (16-17) forjc = 0.2, 0,5 and
0.8 respectively.
X
J
<^j0
Cj\
Cj2
DjO
%
0.2
I
2.5063
0.6373
-0.2518
-3.0022
-1.1847
0.3214
2
1.4640
-0.1729
1.6549
-0.0261
2.0608
- 1.8059
3
3.9569
0.4113
1.4255
- 1.7553
0.5179
-0.1439
0.5
1
1.4411
0,5256
-0.1947
-2.0163
1.1869
0.2053
2
0.9861
0.3234
0.5607
-0.3134
1.7857
-2J585
3
2.4246
0.8274
0.3824
-1.3170
0.4891
-0.9844
0.8
1
0.7368
0.3503
-0.1139
- 1.4845
- 1.0331
-0.0315
2
0.5858
0.3376
0.0868
-0.3091
0.7503
-1.7033
3
1.3222
0.6839
0.0237
- 1.0390
-0.0522
-0.8698
Eqs. (13-15) can be used to determine the activation energy of a TL glow curve.
Determination of the activation energy etc
237
3. Results and discussion
The activation energy of a TL peak can be calculated using eqs. (13-15) not only at a point
where x = 0.5 but at any other suitable points where jc = 0.2 and 0.8 also, using the values of
C,x and (/ = 1-3, k = 0-2) from Table 1 and eqs. (16-17) to obtain C^and Dj {j = 1-3).
Mixed order TL peaks have been generated numerically with £ = 1.0 eV. j = 1 0*® i*' ,
/V = no = 10'® cm"^, lO-*^ s~^ and /? = 1.0 ®C j"' and different values of a. Using
eqs. (7-9), we have calculated and T + 3 . The values of <5 = and
tu = Tq 5 ” 7’q 5 are presented in Table 2. The values of £ 1 , £2 and £3 obtained by using
the present set of expressions (13-15) for = 0.5, have been applied to the generated peaks
and are presented in Table 2 along with the values of £ and £ir calculated using Chen’s
peak shape formulae [4] and initial rise method [9]. From Table 2, it is seen that the values
Table 2. Activation energies E\ (cV), £2 (cV) and £3 (eV) of numerically generated MO TL
pcaks(£in= 1.0eV.r= lO'V. = KtV. N = no= l0'®cm‘^ jS = I 0°C .r' and
different values of a, using the present set of expressions (13-15) and Chen's formula [4J. tm is
the activation energies obtained by u.smg initial nse method.
£,n a T^CC) SCO (OCO £,(eV) £2(cV)£3(eV)£|(eV)£2(cV) £3(eV)£,R(eV)
(eV) Present Chen
I.O
0.1
141.4272
15.1250
35.6503
0.4243
0.9973
0.9964
0.9968
0 9839
0.9854
0 9928
0.9890
1.0
02
151.4806
16.6453
38.6441
0 4307
lOOlS
0 9992
1.0008
0.9696
0.9835
0.9833
0 9877
1 0
03
157.6403
18.0741
41.2074
0 4386
10027
1 0026
10032
0 9586
0.3831
0 9767
1.0000
1.0
0.4
162 1039
19.4828
43.6937
0.4483
1.0020
1.0046
1 0037
0 9519
0 9848
0.9736
0.98.56
10
0.5
165.5575
21 2601
46.2173
0 4600
1.0002
1 0037
1.0023
0 9.503
0 9885
0 9744
0 9860
1 0
0.6
168.2938
23.1268
48.8105
0.4738
0.9981
0.9934
0 9951
0.9547
0 9932
0.9790
0 9860
1.0
07
170.4603
25.1581
51.4298
0.4892
0.9981
0.9934
0 9951
0.9649
0.9975
0 9790
0 9861
1.0
0.8
172.1469
27.1922
53 9005
0.5045
0 9981
0.9981
0 9951
0 9790
1.0002
0.9946
0.9858
10
0.9
173.4261
28.8256
55.8208
0.5164
1.0028
1.0077
1.0059
0 9919
1.0007
1 0008
09857
of £ obtained by using the present set of expressions arc more accurate than those
obtained by using Chen’s formula and initial rise method. But the present expressions
require the prior knowledge of a To find a for an experimental peak, one has to calculate
the shape factor and derive a from the vj a curve [ 6 ]. Chen et al [ 6 ] have presented
Idg vs a curve and observed that the curve is modified slightly with the change in £ and s"
ie. with since can be found for a particular £ if s" is known. In Figure 2, we have
presented the variation of li^versus a for u„ = 20 and = 40. The value of decreases
slightly by around 3% when changes from 20 to 40 in agreement with the observation of
Chen et al [ 6 ]. But it is observed that a increases appreciably by around O.l to 0.2 when
changes from 20 to 40 for a paiticular value of ^g. For a TL peak since we do not have prior
knowledge of u„ to find the accurate value of a from iig vs a curve, the average values of
fig for u„ B 20 and 40 are obtained for different values of a and are plotted as a function of
238
S Dorendrajit Singh, W Shambhunath Singh and P S Majumdar
a (solid line in Figure 2), The average vs a curve can now be used as a preliminary
estimation of a to find E. Talcing into account of the error in the estimation of a by this
method, we have found that the possible error in the evaluation of the activation energy E
docs not exceed 3%. It is to be noted that for many experimental TL peaks, u„ lies between
20 to 40 except for a limited number of peaks.
Figure 2. Variation of (atx = 0 5) as a
function of a. (a) - • - • - for - 20, (b)
for u„ = 40, (c) correspond to average
of the values of fig at u„ = 20 and u„ = 40.
0.5 a
To show that MO kinetics model is a viable alternative to the GO kinetics model,
we have generated numerically GO TL peaks with £ = 1.0 eV, s = 10'^ s‘', 1.0 °C r'
and different values of b (I < ^ < 2) and computed the values of and fig (at
X = 0.5) using the expressions of Gartia et al [15] and also used the present set of
expressions to obtain £i, £2 and £3 (Table 3). The values of 5 (5 = Tq 5 ~ ) and
£U (<u = Tq 5 - 7q , ) of the numerically generated GO TL peaks are given in Table 3. In
computing the values of £, we have used Figure 2 (solid line) to obtain the values of a from ^
Tabic 3, Activation energies £1 (eV), Ej (eV) and £3 (eV) of numencally generated GO TL
peaks (£=10 eV, s = 10*^5''* and different values of b, calculated using present set of
expressions (13-1.^)
£
(eV)
5
(r>)
b
6
CC)
(0
(•C)
(eV)
£2
(eV)
£3
(cV)
1.0
10*3
1.1
11.8
27.3
0433
1.0398
1.0230
1.0323
1.0
10'3
1.2
12.7
28.5
0.446
1.0530
1.0345
1.0445
1.0
io‘^
1.3
135
29.6
0.457
1.0559
1.0403
1.0485
1.0
10'3
1.4
14.4
30.7
0.468
1.0523
1.0374
1.0450
1.0
io‘’
1.5
15.2
31.8
0.478
1.0457
1.0296
1.0376
1.0
lo'^
1.6
16.0
32.9
0.487
1.0370
1.0157
1.0261
1.0
10'’
1.7
16.8
33.9
0.495
1.0285
1.0055
1.0165
1.0
I0l3
1.8
17.6
34.9
0.503
1.0202
0.9985
1.0088
I.O
10 '^
1.9
18.3
35.9
0.511
1.0120
0.9940
1.0025
the values of (x = 0.5). The values of £ 1 , £2 and £3 agrees well with the input value £.
Hence our present set of expressions can be used as an alternative method for finding the
value of activation energy.
Finally, the applicability of MO kinetics model and the present set of expressions for
determining the activation energy, is discussed by taking the well-studied experimental TL
peak of BcO (T„ 3 160. TC) [1 1] irradiated with X-ray (5 minutes). Sakurai and Gartia [1 1]
Determination of the activation energy etc
239
fitted the peak with their numerically generated peak and obtained the activation energy
as 1 .09 eV (Table 4), Now the present method for the determination of the activation energy
Table 4. The values of activation energies iij and £3 (eV) of the experimental TL peak of
BcO, £nnocf ^ respectively the activation energies of the peak obtained by curve lltting
with MO kinetics and numencal method [11].
Tm
^0.5
n.5
£2
£3
^moef
m
("C)
(°C)
(jr*0.5)
(eV)
(eV)
(eV)
(eV)
(cV)
160.1
135.6
18S.6
0.510
1.0418
1.0374
1 0397
I 0410
109
using the expressions [13-15] and half intensity points Tj 5 , , has been applied to this
peak. The value of a of this peak used in the computation of the activation energies is
observed from the Figure 2 (solid line) using the value of (at x = 0.5). The values of
Fq , , Tq 15 , (at .jc = 0.5), £|, £2 and £3 are given in Table 4. The experimental peak of
BeO can be fitted with a MO kinetics peak using the values of £^ 0^1 = 1041 eV, a= 0.85,
5 = 6.67 X s~\ A„ = An = s~\ N - fiQ = 10'® cm“^ (Figure 3). The values of
£i, £2 and £3 are in good agreement with the value of £mocf and lies between the value of E„
TCC)
Figure 3. Curve fitting of experimental TL
peak (full circle.**) of BeO {T„j = 160 TC) with
MO kinetics (continuous line) (£ = 1.041 eV,
s = 6.67 X 10'®r'. /!;„ = = KTV, ^ = hq =
10'® cm"'^ and ot = 0.85),
obtained by Sakurai and Gartia and 0.98 eV obtained for the same peak by curve fitting
with GO model.
4. Conclusion
In the present paper, we have derived a Set of expressions for the determination of
activation energy of a TL peak obeying mixed order (MO) kinetics involving the important
MO parameter a which can be determined from the value of shape factor at half
intensity points. We have also applied the method to the experimental TL peak of BeO and
obtained the activation energy. A comparison of the value of activation energy thus
obtained, is made with the value obtained by using curve fitting technique with mixed order
kinetics as well as general order kinetics model. It is observed that the values obtained in
the present paper is comparable with the value obtained by Sakurai and Gartia [ 11 ].
Acknowledgment
The authors are thankful to Prof. R K Gartia for fruitful discussions. One of us
(W Shambhunath Singh) likes to acknowledge the financial support from the University
Grants Commission. India.
240
S Dorcndrajit Singh, W Shambhunath Singh and P S Majumdar
References
f 1 1 R Chen and Y Kirsh Analysis of Thermally Stimulated Process (Oxford : Pergamon) (1981)
[21 M J Ailken Thermoluminescence Dating (New York * Academic) p 204 (1985)
[31 K Mahesh and I) R Vij Techniques of Radiation Dosimetry (Delhi ■ Wiley Eastern) (1985)
14] R Chen Electrochem. Soc. 116 1254 (1969)
15] C E May and J A Partridge / Chem. Phys. 40 1401 (1964)
[6] R Chen, N Kristianpoller, Z Davidson and R Visocckas J. Lumin. 23 293 ( 1 98 1 )
[7] A Halperin and A A Braner Phys. Rev, 117 408 (1960)
1 H] J T Randall and M H F Wilkins Proc. Roy. Soc. 184 366 ( 1945)
[9] G F J Garlic and A F Gibson Proc. Phys. Soc. 60 574 (1948)
[10] D Yossian and Y S Horowitz Radiation Measurement 27 465 ( 1 997)
[11] T Sakurai and R K Gania J. Phys D29 2714 ( |996)
[121 R K Gartia, S D Singh and P S Mazumdar Phys. Stat Sol («) 138 3 1 9 ( 1 993)
[13] W H Press, S A Tcukolsky, W T Vctterling and B P Flannery Numerical Recipes in Fortran (Cambridge
Cambridge University Press)
[14] E J Dudcwic? and S N Mtsra Modern Mathematical Statistics (New York : Wiley) ( 1 988)
1 15] R K Gartia, S J Singh and P S Mazumdar Phy.s. Stat Sol (a) 106 291 (1988)
Indian J. Pkys. 72A (3). 241-247 (1998)
UP A
— an intemaiional journal
Studies of X-rays and electrical properties of SrMo 04
N K Singh, M K Choudhary and R N P Choudhary*
l>5partmcnt of Physics, H D Jain College, Ara-802 301. India
Department of Physics, Indian Institute of Technology,
Kharagpur-721 302, India
Received 7 November 1997, accepted 20 March 1998
Abstract : A polycrystalline sample of SrMo04 was synthesized by high-temperature
solid-state rcaciioii technique Preliminary X-ray stuoy has been earned out to check the
formation of compound and to determine its preliminary crystal data. The variation of ac
conductivity and dc resistivity with temperature has also been studied. Measurements of
dielectric constant (€) and loss tangent (tan ^ as a function of frequency (4(X) Hz- 10 kHz) and
temperature (-180® to 3(X)°C) show that the compound is a linear dielcctncs
Keywords : Solid state reaction. X-ray diffrailion, dielectric constant and dc resistivity
PACSNos. ; 77.22Gm.61.10.Nz,77 80.Bh
1. Introduction
Since the discovery of ferroelectricity in BaTiO*, in 1945 [1 ), a large number of oxides of
different structural families have been examined [2,3] in search of new materials for
device applications. It has been found that each member of an oxide family has some
interesting structural and physical properties in spite of some similarities in their chemical
formula or coitipositions [4-7]. Among all the oxides studied so far, some molybdates
and tungstates, such as Gd2(Mo04i [8], PbMo04 [9], PbWQj [10] etc., have interesting
ferroelectric and related properties. Some molybdates and tungstates of the general
formula ABO4 (A = alkali ions; B = W, Mo) have very unusual successive phase (Le.
commensurate^incommensurate) transitions [11] in wide temperature range, with high
electrical conductivity and dielectric loss [12] and low dielectric constant [13]. This has
attracted us to synthesize and study structural, electrical and spectroscopic properties of
different structural families of tungstates/ molybdates, such as pervoskite, tungstan bronze
(TB) [14], spinel [15} and scheelite [I6]. SrMo04, suitable for laser applications [17],
© 1998 1 ACS
242
N K Singh, M K Choudhary and RNP Choudhary
belongs to the scheelite structural family with the space group Uja [16]. Detailed
literature survey on this compound suggests that except a few studies [18-20], not
much work have been done on it. Therefore, we have carried out systematic studies
on structural and electrical properties of the compound for the better understanding of
its structural and physical properties and to check the existence of ferroelectric properties
in it.
2. Experimental
The SrMo04 sample was prepared from strontium carbonate SrCOj (99% pure,
M/s. Bugoyne Ltd.) and molybdenum oxide M0O3 (999.5% pure, M/s. BDH Ltd.) in
desired stoichiometry by solid state reaction technique. These component compounds
were mixed in a agate-mortar for 2 h and calcined at 725°C in a platinum crucible for 20 h.
The calcined powder was ground and recalcined at 800“C for 18 h. The calcined
powder was gfound again to make fine and homogeneous powder which was uniaxially
cold pressed into pellets (diameter = 10 mm and thickness = 1-2 mm) at a pressure of
4.5 X 10^ kg/m^ using a hydraulic press. The pellets were then sintered at 825°C for 14 h.
The quality and the formation of the compound were checked with an X-ray diffraction
(XRD) technique.
For preliminary structural studies, an X-ray diffraclogram was recorded at room
temperature by a Rigaku X-ray powder diffractometer (Miniflex, Japan) with Cul^„
radiation (A = 0.15418 nm) for a wide range of Bragg angle 29, (15° ^29^ 90°) at the
scanning rate of 2°/min. To measure the dielectric constant, the flat surfaces of the
pellet ‘Sample were electroded with high purity and ultrafine silver particle paste.
Measurements of dielectric constant (e) and loss tangent (tan 5) of the sample were carried
out both as a function of frequency (400 H/. to 10 kHz) and temperature (-1 80°C to 300°C)
by GR 1620 AP capacitance measuring assembly in small temperature interval (-8°C).
Measurement of dc resistivity was done both as a function of temperature (room
temperature to 325°C) and biasing electric field (1.5-8 kV/m) by Keithley 617
programmable electrometer. Existence of spontaneous polarisation in the compound was
checked using laboratory made Sowyer-Tower circuit.
3. Results and discussion
The sharp and single diffraction peaks (Figure 1) in the X-ray spectra (XRD) suggest
that the compound was formed in a single phase. Lattice parameters and d- values of the
compound were calculated for different crystal systems and unit cell configurations
with observed <i-values of strong, medium and low-intensity reflections using a
standard computer program "powdin”. Finally, lattice parameters and crystal system
were selected on the basis of minimum EM (= d^ - d„i) which was consistent and
very much comparable with those of the calculated values obtained here and
Studies of X-rays and electrical properties ofSrMo04
243
vjIucs reported in JCPDS [21]. The refined lattice parameters are : a = 1 1.4377 (10) A
and c = 12.0316 (10) A (estimated error in the parenthesis). A very good agreement
Ix'ivvcen observed and calculated ^-values (Table 1) suggests the correctness of the selected
c ell and structure.
Tabic 1 . Compari.son of some observed and calculated d-valucs (in A) of some
reflect lon.s of SrMo04 at room temperature
kill
^oba (^)
rfcal(A)
i/h
.102
3 2203
3 2203
100
004
3.0079
3 0071
67
41 1
2.7023
2 7031
28
2 4616
2 4601
6
304
2.3636
2 3615
9
334
2.0081
2 0076
34
600
1 9013
1 9063
15
306
1 7712
1.7748
29
640
1.5883
1.5861
20
730
1.5011
1.5018
6
660
1.3474
1 3479
3
823
1 3107
1.3108
10
428
1.2960
1.2964
13
90%
1.2424
1.2434
4
664
1.2296
1 2301
21
419
1.2043
1.2043
15
717
1 1780
1 1779
7
1000
1.14.38
1.1438
8
1012
1.1179
1.1183
4
though it is not possible to determine the space group from the limited powder data, the
reported space group I4i/a has been confirmed with some systematically absent reflections
and physical properties. The particle size of the compound calculated using Scherrer's
^nation [22], was found to be 363 A, which was consistent with those observed from the
particle size analyser.
244
N K Singh, M K Choudhary and RN P Choudhary
The dielectric constant (e) and loss (tan S) decrease with increase in frequency at
room temperature (Figure 2). At low frequencies, all the polarizations exist but with
increase in frequency some of the polarizations vanish. Therefore, the dielectric constant
and loss decrease with increase in frequency. This suggests the normal behaviour of a
dielectric. Variation of e of the compound with temperature (-180°C to 3(X)°C) at 10 kHz
shows the linear dielectric .behaviour of the compound. Below room temperature
(upto -180°C), the values of e and tan 5 were found to be about 10 and 0.005 respectively,
Figure 3. Variation of dielectric constant (e) and lo.^s (tan ^ of SrMo 04 with
temperature at 10 kHz.
(therefore, not shown in Figure 3) which are almost constant and linear. Variation o
dielectric loss with temperature at a frequency 10 kHz shows a almost constant valu<
Studies of X-rays and electrical properties ofSrMo 04
245
(in tan 6 ) with a small anomaly at 230®C. The slow increase at low temperature is due
to lattice ionic polarizability and the faster increase in high temperature region is due to
space charge polarization. Similar behaviour in € and tan 8 has been observed in many
molybdates and tungstates studied recently by us [23,24]. The electrical conductivity a of
the sample was calculated from the dielectric data using formula CT = tu € ge tan 8, where
6o = dielectric constant in vacuum and (o ® angular frequency. An activation energy
calculated from the formula O’ = a© exp (-EjK^T) (Kg = Boltzmann constant) was
found to be 0.03 eV (Figure 4).
Figure 4. Variation of ac conductivity (In a)
of SrMo 04 with inverse of temperature (10^/7)
at 10 kHz
Study of field dependence of dc resistivity (Figure 5) shows that the resistivity
decreases with increase of electric Held. This may be due to ionisation of gases and
Figure S. Variation of dc resistivity of SrMo04 with Figure 6 . Variation of dc resistivity (In a) of
<^pplied electric field at nxnn temperature. SrMo04 with inverse of absolute temperature ( I / 7 ).
246
N K Singh, M K Choudhary and RNP Choudhary
moisture present in the pores/cracks of the compounds, thus increasing the conductivity
of the ceramic samples in general [25]. Variation of In a with inverse of absolute
temperature at constant electric field (7.87 kv/m"') is shown in Figure 6. The decrease
of resistivity with increasing temperature can be explained on the basis that the insulators
have no free carriers, but due to thermal energy, electrons can be set free from oxygen
ions. Hence, conductivity of SrMo 04 increases due to generation of electrons [25].
However, the compound shows a negative temperature coefficient of resistance (NTCR)
similar to an extrinsic semiconductors above 100°C. As no D-E hystersis loop in wide
temperature range (liquid nitrogen temperature to 3(X)‘^C) was observed, we concluded that
the compound is non ferroelectric, which is very much consistent to our other studies. It is
finally concluded that unlike many molybdates, this compound does not show any
ferroelectric behaviour.
Acknowledgment
The authors wish to thank Sri S Bera and Ms. T Kar for their kind help in some
experimental work.
References
f I J B Wul and L M Goldirun C R Acad Sci. USSR 46 1 23 ( 1 943)
[2] E C Subbarao Ferroelectncs 5 267 (1973) •
[3] K K Deb Fenoelectrics 82 45 ( 1 988)
[4] K S AIcksandary, A T Anistrator. S V Metrikova, P V Klevsov and V N Voronov Phys. Stat. Sol
.67 377 (1981)
[5] K S Alek.<iandary, D H Blat, V I Zinenki, 1 M Iskomcv and A I Kruglink Ferroelearics 54 233
(1984)
[6J S Bera and RNP Choudhary Indian J, Pure Appi Phys. 33 306 (1995)
[7] S Bera and R N P Choudhary Mater Lett. 22 197 (1995)
[8] E T Keve, S C Abrahams, K Nassau and A M Glass Solid State Cnmmun. 8 1517 (1970)
[9] W Buc.s and H Gehrke Z Anorg. Allgem. Chem. 288 307 (1956)
[10] G M Clark and W P Doyle Spectrochim. Acta 22 1441 (1966)
[11] T Janssen and A Janner Adv. Phys. 36 5 1 9 (1987)
[12] AT Moulson and J M Herbert Electroceramics (Materials Properties and Application) (London :
Chapman and Hall) (1985)
[13] S Bera and R N P Choudhary Mater Sci. Utt. 15 251 (1996)
[14] M H Fremcombe Acm Cryst. 13 313 (1960)
[15] K S Singh, Sali and R N P Choudhary Parmana 48 161 (1992)
116] R W G Wyekoff Crystal Structure 3(2) 21 (1964)
[17] L F Johson J. Appl. Phys. 34 897 ( 1963)
1 1 Xj R Loudon Adv. Phys. 13 423 (1964)
jl 9| J P Russell and R Loudon Proc. Phys. Soc. 85 1029 (1965)
Studies of X-rays and electrical properties of SrMo 04
247
[20] J P Russell J. Phys. (Pans) 26 620 ( 1965)
[2 1 ] Powder Diffraction File Set Voi 6-/0 f Revised) Inori^anic Vol. No. PDIS 15 p 404
|22] P Schcrrer Cothn Nachncht 2 98 ( 1 9 1 8)
[23] T Kar and R N P Choudhary Mater. Lett. 32 109 (1997)
[24| R N P Choudhary and N K Mishra Indian J. Pure Appi Phys. 31 945 (1993)
1 25] R C Buchanan Ceramic Materials for Electronics (New York : Marcel Dekker) ( 1 986)
Indian J. Phys. 72A (3), 249-252 (1998)
UP A
- an international journ al
On the structure and phase transition of lanthanum
titanate
H B Lai, V P Srivastava and M A Khan
Department of Physics, University of Gorakhpur,
Gorakhpur 273 009, Uttar Pradesh. India
Received 13 January 1998. accepted 3 February 1998
Abstract : This research note reports the structure and phase transition studies of
lanthanum titanate (LaTi 03 ) through XRD pattern, dielectric and electrical conductivity
measurements It has been found that LaTi 03 has orthorhombic unit cell at room temperature
and satisfies the criterion put forward by Roth for the Perovskite structure. The phase transition
temperature has been found to be ( 1030 ± 10) K.
Keywords : XRD pattern, transition temperature, Perovskile structure
PACS Nos. : 72 80 Jc. 72.90.+y
Most of the compounds with general formula ABO3 have perovskite structure with a
cubic unit cell. The cation A in this structure is coordinated with twelve oxygen ions
and cation B with six oxygen ions. In very early studies, Goldschmidt [1] has put
lorward a criteria for ideal cubic structure in terms of tolerance factor t which is given
by the relation :
2(Rb +/fo)
According to the author and quoted by others [2J, the tolerance factor should lie in the
• dnge 0.8 ^ f < 0.9 for ideal perovskile structure. LaTi03 along with many other
liinthanum compounds with general formula ABO3 satisfy this criteria as evident from
Table 1. However, except LaTi03, all are reported [2] to have orthorhombic unit cell
with parameters ^o, and cq e: given in Table 1 .
© 1998 lACS
250
H B Lai, V P Srivastava and M A Khan
Table 1. Ionic radii, tolerance factor (/) and structural parameters of few LaM 03 type
compounds with orthorhombic unit cell. Radii of La^*^ and 0^~ are 0.1061 nm and 0.1400 nm
respectively
"
i!
Radiu.s
of ionis
(nm)
Tolerance
factor (/)
Unit cell parameter
Reference
%
(nm)
^0
(nm)
^0
(nm)
Mn
0066
0 845
0.5536
0.5726
0 7697
3
Fe
0 067
0.841
0 5556
0 5565
0-7862
2
Cr
0,069
0 833
0.5477
0.5515
0.7755
2
Ti
0 070
0 823
0 5570
0 5796
0,7680
PS
Sc
0.081
0.787
0 5678
0 5787
0.8098
2
PS = Prescni study
LaTiOj is reported [4,5] to have cubic structure with a = 0.392 nm. It is evident
from this table that lower limit of tolerance factor 0,8 is not appropriate for ideal
perovskite structure. It is worth mentioning at this stage that the criteria for different types
of perovskite structure has been dealt in detail by Roth [6] and a summarized result of
the same is presented by Glasso [2]. According to criteria presented in a figure by the
authors [2,6], LaTi 03 should have orthorhombic unit cell at room temperature. To
resolve this anamoly between the reported structure and criteria pul by Roth [6], we have
prepared and studied the structure and phase tran.sition of LaTiO^ by dielectric constant and
electrical conductivity measurement and the results are presented in this note. The starting
materials for the preparation of LaTi 03 were La 2 P 3 (with stated purity of 99.99% from
Rare and Research Chemical, Bombay, India) and Ti02 (stated purity of 99.9% from the
same firm). The two oxides were dried for four hours at =450 K. Then they are mixed in
stoichiometric amount and heated in silica crucible in air at about 1200 K for 48 hours with
one intermediate grinding. The compound is formed according to the following solid state
reaction ;
La2O, + 2Ti02 2 LaTi 03 + 0T.
The loss of the sample after heating was recorded. The loss was well within the range
expected from above equation. The X-i ay diffraction (XRD) pattern of prepared compound
has been recorded using CuK^ radiation (A = 0.15405 nm) and diffraction peaks were
analyzed using standard procedure. All the peaks could be assigned by proper /i, k, I values
(Table 2) as per relation :
dhu = ho[(h/ay-
where a©' sre lattice parameters and a = oo/^o ~ The anal > .si',
shows that the compound has orthorhombic unit cell with a© = 0.5570 nm, bo = 0.579b nm
and cq a 0.7680 nm.
On the structure and phase transition of lanthanum titanate
251
li is normal tendency of the less symmetrical structure to undergo phase transition
and yield more symmetrical structure at higher temperatures. Since LaTiC )3 is orthorhombic
Table 2. Experimental and calculated value.s of and the h, k, I
values for intense peaks in XRD pattern.
^hkl
Experimental
(nm)
Theoretical
(nm)
h
k
/
0.2786
0.2785
2
0
0
0.2440
0.2438
1
2
1
0 2257
0 2255
2
0
2
0.1916
0 1920
0
0
4
0 1812
0.1815
1
2
3
0 1763
0 1763
2
1
0
0 1612
0 1606
3
1
2
0 1449
0 1449
0
4
0
0 1.356
0 1356
0
4
0 1340
0 1337
3
3
0
ai room temperature, it is expected to go to tetragonal or cubic structure at higher
temperatures. If it happens, then we can expect sharp anomalies at transition temperature
in both dielectric constant and electrical conductivity. To sec this, we prepared pressed
pellets ot powdered LaTi 03 , annealed it around 1000 K for few hours and measured its
density. The density of pressed pellets was about 80 percent of the evaluated density using
structure data. Using painted silver and hard platinum electrodes and two-electrode method,
the capacitance and resistance of the pellets were measured at different temperatures
employing LCR Q-meter (Aplab, India). Using these data and dimensions of the pellet,
dielectric constant (K) and electrical conductivity (cj) were calculated at different
icmperatures. The results at higher temperatures are presented in Figures 1 and 2. It is seen
(IO^/tXK"') ->
Figure 1. Variation of logarithm of electrical conductivity (log 0) vs inverse of
absolute temperature (7^S for pressed pellet of LaTiO^.
252
HBLalVP SrivQstavQ andMA Khan
from Figure 1 that (7 drops by a factor of 30 around 1030 K and K vs T plot shows a well-
defined peak at the same temperature. These anamolies are probably due to phase transition
Figure 2. Vanation of dielectnc constant (K) vs absolute temperature (T) for
pressed pellet of LaTiO^
of LaTi 03 around (1030 1 10) K. The detail analysis of a and K data will be presented
elsewhere.
References
[ 1 ] V M Goldschmidt Skrifetes Norshe Vidkenskaps Aknd Oslo I Mat. Naturv Kl No. 8 ( 1 926)
[2] F S Glasso Structure, Properties and Preparation of Perovskite Type Compounds (London Pcrgamon)
(1969) »
[3] R J H Voorhocve, J P Rcmcika, Trinbicic* A S Cooper, F D Disalvo and P K Gallaghar / Solid State
Chem. 14 395 (1975)
[4] , M Kentigian and R Ward J. Am. Chem. Sac. 76 6027 (1954)
[5] W D Johnson and D Sestrich J. Inorg. Nuci Chem. 20 32 (1 961 )
[6] R S Roth J. Res. NBS RP 2736 p 58 (1957)
Indian J. Phys. 72A (3), 253-258 (1998)
UP A
- an intcpnational journaJ
Cylindrically symmetric scalar waves in general
relativity
Shri Ram and S K Tiwari
De|)artment of Applied Mathematics, institute of Technology,
Banoras Hindu University, Varanasi 221 005, India
Received 12 February 7995. accepted II March 7995
Abstract : In this note, exact solutions of Einstein equations with scalar waves
are obtained for the most general cylindrically symmetric space-time which reduce to
essentially static forms. The asymptotic behaviour of the null geodesic near the curvature
singularity of a solution is discussed. The other .solution is found to have no finite curvature
singularity
Keywords : Ein.stein equations with scalar waves, cylindncully symmetric space-time,
exact solutions
PACS No. : 04.20.Jb
General relativity couples gravity with all fields. The study of the exact solutions of
gravity coupled to other fields is important to understand clearly the physical and
mathematical structures of space-times. For many reasons, the study of Einstein equations
m the presence of scalar fields has been an object of special attention and various aspects
of the problem have been investigated by Brahmchary [1], Bergmann and Leipnik [2J.
Buchdahl [3], Janis et al [4], Penny [5], Gautreau [6] and others. Most of the authors have
taken up the problems of interacting gravitational and scalar fields with and without the
rest-mass term. Several physically acceptable scalar-tensor theories of gravitation have been
proposed and are widely studied by many workers. Scalar-tensor theories of gravitation
provide the most natural generalisations of general relativity and thus provide a convenient
set of representations for the observational limits on possible deviations from general
relativity.
The most general spherically symmetric static solution of Einstein equations coupled
with mass-less scalar field was found by Wyman [7], Since then, some authors investigated
nA(3>ij
254
Shri Ram and S K Tiwari
its global properties and a few interesting results were found. Roberts [8] has discussed the
applications of spherically symmetric solutions of the mass-less scalar Einstein equations to
cosmic censorship and has given a non-static solution to the field equations. He has also
constructed the Vaidya form of Wyman solution obeying thq reasonable energy conditions.
Li and Liang [9] have presented the static general solution with plane symmetric scalar
fields and have shown that the singularity in the plane symmetric case is not influenced
essentially by the introduction of the scalar field. Li [10] has presented the general plane
symmetric metric yielded by a scalar wave and concluded that the metric is either static or
spatially homogeneous. He has shown that the Taub Theorem [11] can be generalised to
space-time with a scalar wave. Shri Ram and Singh [12] have derived an exact non-static
scalar wave solution for the cylindrically symmetric Marder [13] metric which give Taub
solution [11] and Li solution [10] in special cases.
In this note, we consider Einstein equations with scalar wave for the most
general cylindrically symmetric metric recently discussed by Banerjee et al [14] in the
investigation of exact gravitational fields due to static and nonsiatic cosmic strings arising
due to the breaking of a global U (\) symmetry. The field equations are completely
integrated and two exact solutions are then presented which reduce to essentially static
form under coordinate transformations. We also discuss the asymptotic behaviour of
the null geodesic near the singularity of one of the solutions. The other solution has no
finite singularity.
Field equations :
The general cylindrically symmetric line element can be written as
^^2 = +dr^) + e^^dz- +W^e-^^de^, (1)
where all of K, U and W are functions of r and / [14]. Setting jc' = r, x^=z, = 0and Jt^ = t,
the non- vanishing components of the Ricci tensor are R| j. R22* ^33> ^44 ^nd R14.
The energy-momentum tensor for a massless scalar field is
Tap = ( 2 )
where the scalar field 0 is the solution of Klein-Gordon equation ;
= 0. (3)
A comma and a semicolon denote ordinary and covariant derivative respectively.
The Einstein equations are
On contraction, the field equations (4) can be written in the form
f<ap * .
( 5 )
Cylindrically symmetric scalar waves in general relativity
255
Because of the syirimetry in the metric (I). ^ is function of r and t. For the line-element (1 ).
the Einstein equations (5) give the following set of equations :
U . W , K , W , U . W .
IV,, W44
IT ~ w ' w ~ w "
■Un + U ^-
y.iv, u,w^
t/.i -u^ +
/filVi 1/|IV| K^W^
+ ^^-2ui = ml.
^ - 21/, I/, =
The Klein-Gordon equation (3) leads to
. . , W'l^i W'4
A linear combii\ation of eqs (6-10) yields
W^- M'l, =0,
f/.tv, U^W,
W„ W44 2/f,W, 2K^W^
W ~ W * W W
A/44 - AT,, + t/„ - 1/44 - Ui + +
21/ - 21/4^
f/.W, 1/4 )V4
w'i4 w'iA /4 'V4/:i ,, „ . .
and - ^ 2l/,t/4 = 8;r0,04.
Here suffixes 1 and 4 are differentiation with respect to r and t respectively.
Solutions of the field equations :
The general solution of ( 1 2) is
W - (^) + WtCt/), where 5 = r + r, 77 = f-r. (17)
256
Shri Ram and S K Tiwari
Let the scalar field ^ be the solution of wave equation, which is referred to as a scalar
wave.
0 = + /l2(T7). (18)
Using (17) and ( 1 8) in (1 1 ) we obtain
dh^ , dw^ dh2 !
~d^ " ~~dv ' di]
where a is an arbitrary constant. In view of these equations, the scalar field 0 becomes
0 = afH-i (4) - WjCtj)} + /», (19)
b being another arbitrary constant. From eqs. (12) and (13). wc can write
U = e log W, 6 = ± 1 (20)
Using (17), (18) and (19) in eq. (15), we obtain
2 ■
(21)
‘ l*i',(5) + Wj(7))]^
A dash denotes ordinary derivative. Eq. (21) has the general solution
K = 4na2H'|(^)tV2(7j) + login'll^) + wj (i))} +«i (4) +«2(^). (22)
where g|((5) and gjlH) are arbitrary functions. Substituting K from (22) into (14) and (16),
wc obtain
+ 47Ca2 [{w, '($)}* ^,(0 + {»'2(77)}^ W2(7))|, (23)
- M'i('i)«2(»?) = - *»’2('7)]
+ 4to^ |{h’, '($)}■ W|(^) - {m'J(JJ)}* «'2(7J)J, (24)
which are equivalent to
g|(^)= jlogwK^) + 2na^{w,(4)}^ + -i-logc, (25)
and SjCT) = ^logM’2(ri) - 2;M2{H-2(r7)}^ + |logC2. (26)
Equations (25) and (26) can be written as
, ( 27 )
( 28 )
Cylindrically symmetric scalar waves in general relativity
257
Case / ; when g a - /
From eqs. (20), (22). (27) and (28), we obtain
^iK^w - ±c^[w, (5) + W2(77)]^ Wj (f)w2(77) (29)
where we take the negative sign if w w J < 0 and c ^ = c , c j is a constant.
The metric of the solution becomes
*2 = + (d^dT])
+ [w,(§) + tVjCTj)]'- dz^ + [w,(i5) + de^. (30)
Using scale transformation.
w,(4)=/? + r and ^i '2 (7j) = /?- r,
(he cylindrically symmetric line element (I) yielded by a scalar wave can be written in the
form
ds^ =c^R^e^'^'^- idR^ -dT^)-^R-^dz^ (31)
The scalar curvature of space-time (31) has the value 327Ui'^ / which tends to
inlinity as ^ 0. Thus /? = 0 is a scalar curvature singularity. Investigating the asymptotic
behaviour of the null geodesic, it is found that the null geodesics approaching /? = 0 in T-R
plane are incomplete.
Case II : when € = + I
From eqs. (20), (22). (27) and (28), we obtain
g2K-iu - i w,'(^)w2 (32)
where we take the negative sign if w w j <0 and =c,C 2 isa constant.
The metric of the solution becomes
ds^ = d^dT}
+ (K-,(5) + vvj(J7)]^<fc2 +<(02. (33)
Using scale transformation,
W\i^) = R+T and w^iTf) - R-T,
the cylindrically symmetric line element (1) yielded by a scalar wave can be written as
ds^ = (^/?2 -JT^ +^02 (34)
The scalar curvature of space-time (34) is 327ia^ which shows that the
metric (34) has no finite singularity.
Hcfercnces
1 1 1 R L Brahmchary Prog.Thtor. Phyx. 23 749 ( r960)
12] O Beigmann and R Leipnik Phyx Rev. 107 1 137 (1937)
258 Shri Ram and S K Tiwari
[3] H A Buchdohl Phys. Rev. 115 1325 (1959)
[4] A Janis. E T Newman and J Winicour Phys. Rev Lett 20 878 (1968)
[5] R Penny Phys Rev. 174 1578 (1968)
[6J R Gautreau Nuovo Ctm B62 360 (1969)
( 7 J M Wyman Phy.^. Rev. D24 839 ( 1 98 1 )
f8) M D Roberts Gen. Ret Grav 17 913 (1985)
(9] Jian-Zeng Li and Can-bin Liang Acta Phys. Sinica 40 643 (1991)
[ 1 0] Jion-Zcng Li / Math. Phys. 33 3506 ( 1 992)
(II) AH Taub Ann. Math 53 472 ( 1 95 1 )
f 1 2 J Shri Ram and J K Singh IL Nuovo Cim. Bill 757 ( 1 996)
mi L Marder Proc. Roy Sac. A244 524 (1958)
[141 A Banerjec, N Baneijee and A A Sen Phys. Rev D53 5508 (1996)
JUNE 1998, VoL 72, No. 3
Review
Identification of astrophysical black holes
Sandip K Chakrabarti
General Physics
On exact solution of anharmonic potentials V(x) =
// = 2,3,4...
M S Ansari and M a Baba
Four element linear array of annular slot antenna under superstrate
cover
Sunil K Khah, Sandhya Gupta and P K S Pourush
Dielectric investigation in binary mixtures involving a nuclear
exiraclant-di-is 9 butyl ketone (DIBK) and nonpolar solvents
S Acharya, S K Dash and B B Swain
Optics & Spectroscopy
Vibialional spectral studies and thermodynamic functions of
4, 6-dihydroxy-5-nitro pyrimidine
B S Yadav, Vipin Kumar, Vir Singh, M K Yadav and
Subhash Chand
Note
Finite element analysis of trapezoidal cross-section lossy waveguide
S B Deshmukh and P B Paul
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Volume 72 A
Number 4
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CONDENSED MATTER PHYSICS
A
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PLASMA PHYSICS
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STATISTICAL PHYSICS, BIOPHYSICS & COMPLEX SYSTEMS
V Daiakrishnan Indian Institute of
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UP SPECIAL ISSUE
STATISTICAL PHYSICS : RECENT TRENDS
Indian Journal of Physics is bringing out a special issue (edited by Professor
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This special issue will explore some not-too-familiar topics in statistical
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ii) Quantum integrabie systems
iii) Quantum phase transitions
iv) Black hole thermodynamics
v) Chern Simons field theory
vi) Turbulence
The contributors are D. S. Ray (lACS), A. Kundu (SINP), B. Chakraborty and
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Indian Journal of Physics A
Vol. 72A, No. 4
July 1998
CONTENTS
Condensed Matter Physics
Crystal growth and characterization of (NH4)iBuCh.2H20
K Byrapha, M a Khanohaswamy and V Skinivasan
The el feet of infrared pulsated laser on the degree of ordering of
eellulosc nitrate
S A Nouh, M M Radwan and A A Bl Hagg
l^uKc method for measurement of thermal conductivity of metals
and alloys at cryogenic lemperalure.s
T K Dly, M K Chahopadhyay and A Kalir Dhami
Study of lorward (C--V ) characteristics oi MIS Sehollky dunles
111 presence ol interlace stales and series resistance
P P Sahay
The el feci of doping on (he microhardness behaviour ol anthracene
Nimisiia Vaidya, J H Yagnik and B S Shah
Study ol bishiuth substitution in cobalt feirite
UuMi M JosHi, Kapil Bhati and H N Panina
IXdecl charactcMzalion ol Sr'’* doped calcium tartrate letrahydrale
crystals
K Suuyanarayana and S M Dharmaprakash
Nuclear Physics
Dynamical short range pion correlation m ultra-relativislic heavy-ion
interaction
Dipak Ghosh, Arciha Dt ii, Md Azizar Rahman, Abdul
Kayum Jai ry, RtNt Chahopadhyay, Sunii. Das. Jayita Ghosh,
Biswanath Biswas, Krishnada.s Purkait and Madhumita Lahiri
Relativity & Cosmology
Harly cosmological models with variable G and zero-rest-mass scalar
lields
Shriram and CP Singh
\Coiu‘:l
Pa^es
259-268
269-279
281-286
287-294
295-300
301-306
307-312
313-321
323-329
on nevi
Matching olFriedmann-Lemaitre-Robertson-Walker and Kantowski-
Sachs Cosmologies
PBorcohain and Mahadkv Patoiri
Note
.Sii'iiciurai and dielectric studies on lanthanum modified
Ba>l.iNb,0,5
K Sambasiva Rao, K Koti.swara Rao, T N V K V Prasad
AND M Rajf-SWAra Rao
331-335
Pa^cs
337-341
Indian J. Phys. 72A (4). 259-268 (1998)
UP A
— an interaational journal
Crystal growth and chsuracterization of
(NH4)3Baa5.2H20
K Byrappa*, M A Khandhaswamy and V Srinivasan
^Department of Geology, University of Mysore, Manasagangotri,
Mysoie-570 006, India
Sri Ramokrishna Mission Vidyalaya College of Arts and Science.
Coimbalore-641 020, India
Received 6 March 1 998. accepted 4 May 1 998
Abstract : (NH4);)BaCl5.2H20 single crystals were grown from aqueous solutions.
The crystals obtained were subjected to a .systematic morphological, X-ray and Ihernuil
analyse.s. The cell parameters are monoclinic, space group, P2]/n. a = l 075 (7), h = 10.828 (8),
r = 6 668 (6) A, p = 91 .20°, Z = 2. V = 492.6 A^
Keywords : Ciysial growth, morphology, X-iay powder diffraction, thermal analy.sis
PACSNos. : 81 IODn,61 10.Nz,8I.70Pg
1. Introduction
A 2 BX 4 (where A = K, NH 4 , Rb, Cs, Na, NCCH^li^; B = Cu, Cd, Co, Zn; X = Cl. Br, I)
compounds represent the largest known group of insulating crystals with structurally
incommensurate phases [ 1 , 2 ], Similarly, A3BX5.2H2O (where A = Na, NH4; B = Ba;
X = Cl, Br) crystals exhibit very unusual physical properties and are closely related to the
A2BX4 group t 3 - 5 ]. These systems have attracted a great deal of attention owing to the
occurrence of varying stoichiometries in them. Although no detailed X-ray single
crystal data is available for these crystals, some structural data for the prototype
compound BaCl2.2H20 is available [ 4 ]. Also some data is available on the
Na^BaClv2H20 [ 5 ]. However, (NH4)!iBaCl5.2H20 has not been studied in the literature.
Here, the authors report the growth of single crystals of (NH4)3BaCl5.2H20 for the
llrst time and carried out their characterization through the morphological studies,
XRD,TGA and DSC.
Author for correfipondence
e 1998 lACS
260
K Byrappa, M A Khandhaswamy and V Srinivasan
2. Crystal growth
(NH4)nBaClv2H20 (ABC) crystals were obtained through slow evaporation of a saturated
aqueous solution. The starting components included 3 moles of NH4CI and 1 mole of BaCl2
and the crystallization reaction occurred as follows :
3NH4CI + BaCl + 2H2O (NH4)3BaCl5.2H2 0 .
Saturated solutions of analytical grade ammonium chloride and t^arium chloride (3 ; 1
molar ratio) were prepared separately using triple distilled water. The two solutions were
mixed thoroughly and filtered. The solution was then poured into a crystallizer shown in
Figure la. The crystallizer was covered by a watch glass and placed on a beaker containing
1 . Watch gla.ss
2. Nylon thread
3 Beaker
4. Saturated solution
5. Seed crystal
Figure 1. Experimental set up for the growth of Na3BuCl5.2H20 .single crystals
about 200 ml of concentrated sulphuric acid. The whole assembly was covered by a large ‘
glass dome in order to protect it from dust and also to provide minimum thermal
oscillations. The crystal growth experiments were carried out with minimum mechanical
shocks. Under such conditions crystallization though slow, yielded transparent, colourless
seed crystals which exhibit platelet habits. Crystallization look place for seven to ten days
either in a neutral medium or acid medium using hydrochloric acid. In the acid medium, the
crystallization process is fast due to common ion effect. The average size of the grown
crystals are of the order 7 x 4 x 2 mm^. In some experiments, the crystals were as big as
20 x 6 x 3 mm^
The crystallization was carried out essentially through spontaneous nucleation. The
spontaneously grown large crystals of (NH4)3BaCl5.2H20 were used as the seed crystals for
the growth of large size single crystals as shown in Figure lb. As the size of the crystals
increased due to increased growth rate, the quality of the crystals slowly decreased.
The authors have made an attempt to grow the (NH4)^BaCl5.2H20 crystals by gel
method and the results were not encouraging. Therefore, the authors paid more attention to
Crystal growth and characterization of (NH 4 ) 3 BaCl^. IHjO 26 1
the solution growth. The crystal growth experiments from solutions in general were carried
out at two different temperatures (27®C and 32®C) and in both cases the experiments
produced crystals of different size, habit and quality. This has been discussed in more detail
under morphology.
The solubility study on (NH4)3BaCl5.2H20 crystals was carried out with varying
temperatures. The Figure 2 shows the solubility curve as a function of temperature for
(NH4)»BaCl5.2H20 crystals in grams per 100 ml of triple distilled water. As is evident from
Figure 2. Solubility curve of (NH4)3BaCl5.2F^O.
Figure 2, the solubility increases with increasing temperature. Higher the temperature,
bigger will be the solubility, and in turn the growth rate increases which reduces the crystal
quality.
3. Morphology
The habit of a crystal is determined by the slowest growing faces having the lowest
surface energy, but it is also apparent that a crystal habit is governed by Kinetics rather than
equilibrium considerations [6]. A number of factors, such as degree of supersaturation, type
of solvent, pH of the mineralizer, etc. effect the habit of a crystal. Kem [7] has shown that
many ionic crystals change their habits when suprsaturation exceeds a certain critical value.
Wells [8] observed that a change in solvent results in a change in crystal habit. Sometimes
ihc pH of the media has a considerable influence on the growth rate of crystals, which
ultimately changes the growth habit [91. Habit modifications are also observed when
significant changes in the growth temperature and occurrence of impurities, because an
increase in temperature increases the growth rates [10]. The most common cause of habit
change is the presence of impurities in the crystallizing solution. It is observed that
even very small traces (0.01%) arc enough to produce significant changes. Therefore, many
262
K Byrappa, M A Khandhaswamy and V Srinivasan
observed crystal habits may be caused by unsuspected impurity effects. This is true with
reference to the (NH 4 hBaCl 5 . 2 H 20 crystals, which show a wide range of morphological
variations not only due to the changes in the growth parameters, hut also due to the
deliberate or accidental entry of impurities. The changes in the growth parameters have
been attributed to the slight temperature fluctuations and growth media.
The morphology of (NH4)jBaCl5.2H20 crystals is very interesting and it was studied
using a phase contrast microscope (Leitz-Laborlux, Germany). Since the experiments have
been carried out at two different temperatures (27®C and 32”C), and also in different growth
media (acid and neutral media), the crystal morphology varies significantly. The crystals
are usually tabular, plate like, rectangular, long thick needles and so on. The most common
faces observed in these crystals are ( l(K)), (010), (001), (01 1), (1 11), (101) and so on. The
overall morphology of the crystals obtained is a typical monoclinic centrosymmetric. The
faces arc very well developed and so also the edges and solid angles. The overall
morphology of these crystals is given in Table 1 .
Table 1. Morphology of (NH4)^BaC!5.2H20 crystals
Growth
temp. (°C)
Common
faces
Growth Prominent
rate face
Crystal
morphology
Surface
morphology
27
(100), ((JOl),
(III). (110),
(010),
y (010) > y ((X)l)
or y(Oll)
or y(lll)
(010)
Long rectangular
platelet [Fig. 3(c)|
surface is iTK>re
or less smooth
12
(100), (001),
(II0),(lll).
(010),
y (100) > y (101)>
y ((H)l) > y (0I0)>
y(lll)
vlOO)
Broad monoclinic
platelets [Fig.s 3(b)
and 3(d)l
spirals, octagonal
interi-upied spirals,
dis.sected di.ssolution
features
The morphology of the crystals obtained at 27°C show long rectangular habit with
the most common faces like (010), (01 1 ), (1 1 1), (001) and so on. The size of the crystals
vary from 20 x 6 x 3 mm^ and even longer. The crystals are highly transparent, vitreous
with very smooth surfaces and well developed faces, edges and solid angles. The growth
along the c-axis is unusually high compared to the a- or /^-crystallographic axes. The
Figures 3(a-d) shows the characteristic photographs of the (NH4)3BaCl5.2H20 crystals
obtained at both 2TC (long rectangular and transparent) and 32'’C (broad jnd semi-
transparent platelets). The crystals obtained at 32®C are fairly bigger, in the sense more
broader and also equi-dimensional in most of the cases. The crystals arc slightly buff white
in colour in some places and colourless in the remaining portions. However, both the
crystals (obtained at 27®C and 32°C) show well developed monoclinic symmetry. The
^hematic diagrams of (NH4)3BaCl5.2H20 crystals are shown in Figure 4. The crystal
drawings were done using the CAMERA-LUCIDA set up, and the actual observed central
distances were used in crystal drawings. As is evident from the Figure 4, the growth
temperature clearly controls the crystal morphology and the growth rate,
Crystal growth and characterization of(NH4)jBaCls.2H-,0
Plate /
Fiijiirc J(a).
FIgurf 3(b).
K Byrappa, M A KhandhasWamy and V Srinivasan
Plate l(ConVd.)
Figure 3 (c).
Figure 3(a-d> Characteristic photographs of (NH4)3BaCJ5.2>^0 crystals.
Crystal growth and characterization ofiNH4}fiaClii,2H20
263
The authors have also studied the surface morphology of these (NH4>^BaCl,v2H20
crystals in order to understand the growth defects and to find out the optimum growth
conditions. The crystals obtained at 27°C show more or less smooth and shining surfaces
without any major morphological features. Whereas, the crystals obtained at 32°C show
32*C
Figure 4. Schematic diagrams of (NH4);3BaCl5.2H20 crystals.
very interesting surface morphological features like interrupted growth spirals, growth
layers, dissolution features. The Figures 5(a-g) shows the characteristic surface
morphological features observed in (NH4)jiBaCl5.2H20 crystals obtained at 32°C. The
Figure 5a represents uniform growth bands in the middle of the crystal on (100) face. The
Figure 5b shows the dissolution features along the growth steps under high magnification.
The step height is relatively moderate. The Figure 5c shows a portion of the growth spiral
and dissolution feature all along the spirals which is shown under high magnification
(Figure 5d). This is also probably the region of impurity concentration. The Figure 5e
shows the presence of small growth hillocks aligned along the (100) face. The Figure 5f
shows the bottom portion of the crystal which is not really smooth, but growth spiral is seen
from the bottom. The nucleus is well in the middle oY the crystal and it is clearly seen in this
picture. The Figure 5g shows most probably an edge dislocation on (010) face. All these
surface morphological features observed in (NH4)5BaCl5.2H20 crystals are actually seen
not in the middle portion of the crystals except for the polygonal shaped growth spiral more
or less in agreement with the symmetry of the face. The slight eccentricity is due to the
anisotropy in the growth environment such as a supersaturation gradient. The
morphological studies on (NH4)3BaG3.2H20 crystals show that the ideal temperature for
the growth of these crystals is around 30®C.
264
K Byrappa, M A Khandhaswamy and V Srinivasan
4. Characterization
The (NH4);)BaCl5.2H20 crystals obtained were characterized using XRD (both powder
X-ray diffraction and single crystals methods were used) and TGA/DSC techniques.
4.1. X-ray diffraction :
The X-ray powder diffraction patterns for (NH4)3BaCl5.2H20 crystals were recorded
using Rich Seifert Unit, Germany, X-ray diffractometer with a monochromatic radiation of
CuK„ (Lambda = 1.5406 A). X -ray powder diffraction studies showed that the resultant
product is a single phase and also a new phase. The powder XRD data is given in Table 2.
Table 2. XRD data for (NH4)^BaCl5.2t^O.
7.78857
00656
Direct
9.86151
01104
Lattice
6.41617
.00440
Parameters
90.00008
.00000
91.71703
.06192
90.00008
00000
Volume
= 492.6 ahe
= 492.8
N
h
k
1
Refi Exp
Pha.se No, I
Ren. Fit
Differ
DK
chi-squ
1
0
1
1
5.3731
5.3764
-.0033
5..3764
.4
2
1
0
1
4.8601
4.8788 '
-.0187
4 8788
13.9
3
1
1
1
4.3779
4.3729
0050
4 3729
1.0
4
-2
1
0
3 6252
3 6207
0045
3.6207
.8
5
_2
0
1
3.3597
y.mi
-01.30
3..3727
6 8
6
-2
1
1
3 1838
3.I9I2
-.0074
3 1912
2.2
7
-1
1
2
2 8831
2.8673
0158
2.8673
10 0
8
0
2
2
2 6914
2.6882
.0032
2.6882
4
9
1
2
2
2 5287
2.5215
mil
2 5215
32
10
3
0
1
2.3953
2.3809
.0144
2.3809
13 0
II
2
3
1
2.3270
2..3233
.0037
2.3233
8
12
-2
2
2
2.2402
2.2384 ,
.0018
2 2384
2
13
1
4
1
2 1962
2.2004
-.0042
2 2004
2.0
14
0
0
3
2.1387
2 1378
0009
2.1378
.1
15
0
3
2.0764
2.0774
-.0010
20774
.1
16
3
0
2
1 9887
1.9883
.0004
1.9883
.0
17
2
5
1
1.6888
1.6908
-0020
1.6908
2
18
-3
1
3
1.6536
16512
0024
1.6512
.2
19
-4
2
2
1..5976
1. 59.56
.0020
1.5956
.2
20
-2
5
2
1.5527
1.5513
.0014
1.5513
.1
21
-2
6
0
1.5134
15141
-.0007
1.5141
.0
22
-4
4
1
1.4953
1.4940
.0013
1.4940
1
23
0
5
3
1.4501
1.4496
.0005
1 4496
.0
24
-4
5
0
1.3850
1.3853
-.0003
1 3853
_ .0
25
-2
6
2
1.3748
1.3754
-.0006
1.3754
.0
26
4
3
3
1.3025
1.3022
.0003
1.3022
. .0
27
-4
6
1
1.2377
1.2368
.0009
1 2368
.0
28
5
2
3
r.2014
1.2035
-.0021
1.2035
.3
29
5
6
1
1.1097
1.1091
0006
1 1091
.0
RefL Exp. - refers w the interplanar spacing values 'd' obtained u.sing Bragg 's angles.
Reft, Fit. - re]er.s to 'd' spacings calculated after having corrected for the intensity of the
Itnes (eliminating the noise and applying a least .squares fit),
DK - refers to tne interplanar spacing for various reflections after applying corrections
for the lines.
Plate U (Cant'd.)
Figure 5(d).
Crystal growth and characterization qf(NH4),BaCly2H20
Plate II (Cant’d.)
Figure 5(f)*
K Byrappa. M A Kiiandhaswamy and V Srinivasan
Plate II (Cant’d.
Ki{<ure 5(g).
Figure 5(fi-g). Characteristic suiface morphological features observed in (NH4)3BaCl5.2H20
crystals obtained at 32°C : (a) uniform growth bands in the middle part of the crystal on (100)
face; (b) dissolution features along the growth steps under high magniricaiion. (c) a portion of
the growth spiral; (d) dissolution feature all along the spirals which is shown under high
magnification; (e) presence of small growth hillocks aligned along the (100) face; (f) bottom
portion of the crystal which is not really smooth, but growth spiral is seen from the bottom and
(g) on edge dislocation on (010) face
265
Crystal growth and characterization of(NH4)jBaCl^,2H20
Therefore, the single crystal X-ray diffraction studies were carried out using Enraf Nonius
CAD4 X-ray diffractometer. The unit cell parameters were found to be a = 7.075 (7) A.
b = 10.828 (8) A, c = 6.668 (6) A and 91.20 (7) from the accurate centered 25
reflections in the 6 range 20 to 30. The space group was found to be P2 1 / n. The number of
molecules in the unit cell was found to he two. A detailed structural refinement work is
under progress for publication elsewhere.
4.2. Tliermogravimetric analysis (TCA ) :
The TGA curves for the (NH4)3BaCl5.2H20 crystals were recorded using Mcttler TA 3000,
and the characteristic curves are shown in Figure 6. It is observed from Figure 6, that there
Iciiipor.iliiiv
Wn'jjiii guin
Figure 6. (a) TG and (b) TGA curves of (Nti4)3HaCl5,2HbO.
was u percentage weight loss of 14.888 when the .sample was heated from 47.5°C to
377. 5°C. The peak temperature was observed at 212.5“C. The molecular weight of
(NH4);iBaCl5.2H20 crystals is 404.8. If the crystal is a dihydrate then the percentage weight
loss should be 8.893 (36 x 100/404.8). Since the actual percentage weight loss is 14.888
and the percentage weight loss due to water of crystallization is 8.893, the remaining 5.995
percent weight loss has to be accounted for. The TGA was carried out by static weight loss
method. A known weight of (NH4),BaClv2H20 crystals was taken in a previously weighed
container. Then the substance was heated around 200°C for half an hour. The weight of the
dehydrated substance was then taken immediately. This procedure was repeated thrice to
get a constant weight of the anhydrous substance. The difference in weight between the
266
K Byrappa, M A Khandhaswamy and V Shnivasan
hydrated substance and the dehydrated anhydrous substance gave the amount of water
present in the crystal. After heating, the transparent crystals became white without losing its
shape and size. The following calculation shows that the percentage weight loss of water is
I4.7H3 which closely resembles that obtained from TGA (14.888). It is observed that in
these crystals there can be only loss of water at about 200°C, by analogy the difference
between the initial and final weights of the (NH 4 ) 3 BaClv 2 H 20 crystals is due to the
dehydration process and not due to the decomposition of the (NH 4 );^BaCls part and this
remains intact during heating around 200X. The remaining 5.995 percent weight loss can
he accounted for the occluded and adsorbed water present in the crystals. Thus the
thermogram predicts the decomposition of water of crystallization as follows ;
(NH4)3BaClv2H20 + 2 H 2 O.
4.3. Dijferential .scanning calorimetry (DSC) :
DSC curves lor (NH4>^BaCl5.2H20 crystals were recorded both at low and high
temperatures using Pcrkin-Elmcr differential scanning calorimeter'2, USA. The
sensitiveness used were between 10 and 5 m cal /.sec. In case of low temperature DSC, the
heating rates employed were 10 K/min. Subambient scans were recorded using liquid
nitrogen as coolant. The instrument was calibrated for low temperature operation using the
standards viz tetrachloride and cyclohexane. Thermal anomaly was found at 275 K in the
heating run. The heat innut to the specimen as a function of temperature is shown in
Figure 7, for the low temperature DSC studies. The thermal anamoly observed at 275 K is
indicative of a phase transition.
gf^owth and characterization of {NH4) fiaCl^.2H20
267
The high temperature DSC study was carried out between 323 K and 873 K. The
sample was analysed with a heat flow rate of 20 K/min. The DSC curve abovt; room
temperature (Figure 8) shows four peaks at 340.2 KI 407.2 K, 460.9 K and 535. 1 K.
Mral flonv
iiV I'lothermnl
leiinH‘*-iUiirc ^
Figure 8. High leniperatuie DSC of (NH 4 )iBan 5
The less pronounced first peak at 340.2 K indicates the loss ol occluded and
adsorbed water in the crystals. The peak a! 407.2 K suggests that one molecule ol water ol
crystallization is lost by breaking the hydrogen bonding and leaving the lattice at this
temperature and the crystal goes from the dihydrate form to the monohydrate lorm. This
involves a phase transition of the following type ;
460 9 K
(NH4),BaCls.2H20 (NH4hBaCl5.H20 + H 2 O.
The fourth less pronounced peak at 535.1 K suggests that the anhydrous
(NH 4 hBaCl 5 compound is stable at least up to this temperature without any decomposition
(NH4CI sublimes at 613 K and boils at 79^ K).
Both the TGA and DSC studies conclude that the (NH4hBaClv2H20 crystals
contain two molecules of water and undergoes mvuiple structural phase transitions.
RcfiTcnccs
1 1 1 K Gesi Ferroele( tries 66 269 ( 1 986)
12) HZ Cummins Plm. Rep 185 21 1 (1990)
13) Z Brosset 4 Allfienm. Chem 235 139(1937)
(4) V M Padmanabhan! W R Basing and H A Levy Ai la Crvsi. B34 2290 (1978)
268
K Byrc^pa, M A KharuUuiswamy and V Srinivasan
[5) S Asolh Bahadur, V Ramakrishnan and R K Riyaram BulL Uaitr. 5ci. 13 161 ( 1970)
[6] J M Gibbs Colleard Wnrkf (London : Longman Green) ( 1 925)
|71 R Kern in Gmwih iif Crptah Vol. 8 ed. N N Shcftal (New York ■ Consultants Bureau) (1969)
|K| A F Wells P/m/. M«/;. 37 184(19461
[9] J W Mullin Tilt /nanural Ui iure on Ciysiallizatum—A Siuih in Molecuhr Enfimeerinn (University
College of London, UK) (1970)
|l()| K Nassau, A. S Cooper, JWShievcr and BE Prescott / Solid Stale Cliein.H2U)(im)
Indian J. Phys. Ilk (4). 269-279 (1998)
UP A
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The effect of infrared pulsated laser on the degree of
ordering of cellulose nitrate
S A Nouh, M M Radwan* and A A El Hagg**
Department of Physics. Faculty of Science, Am Shams University.
Cairo, Egypt
* Department of Physics, Faculty of Engineering in Fayoum,
Cairo University, Egypt
**Faculty of Science (Girls Branch), Al-Azhar University, Cairo, Egypt
Received I August 1997. accepted 5 May 1998
Abstract : The effect of infrared laser pulses on the degree of ordering of cellulose nitrate
(LR-1 15) detector has been investigated X-ray diffraction measurements were carried out on
LR- 1 1 5 solid samples These samples were exposed to laser puLses with different exposure doses
ranging fiom 0.0 to 7 5 j/cm^. The dependence of the integral intensity of the amorphous
regions (/^„,), the integral intensity of the cry.stalline regions index of crystallinity and
the crystallite size (L) on the laser dose was also studied. The absorbance of the LR-1 15 samples
in the infrared range was also investigated The results indicated that a higher degree of ordering
could be obtained when irradiating the LR-1 15 samples by infrared laser pulses up to 7.5 J/ cm^.
Keywords : Infrared pulsated laser, cellulo.se nitrate. X-ray diffraction and infrared spectra.
PACS Nos. : 61,82.Pv, 6l.lO.Nz, 78.30.Jw
1. Introduction
The problem of laser damage attains progressively more and more interest from researches
due to the ever increasing important applications of laser technology. Fleske et al [1]
focused a laser beam on the polymer surface, giving pulses once a second. This allowed
exposure spectra to superimpo.se and permitted a survey of the material surface uniformly.
Ready [2] studied the damage threshold in different transparent solid materials. There was a
difference between the damage threshold at the surface and the damage threshold when the
point was in the interior of the sample. Patel and Baisch [3] studied the effect of incident
laser fluence, laser frequency, and polymer thickness on a single pass cutting speed.
Keiko et al [4] studied the photoablating behaviour of various polymers irradiated by
excimer lasers and YAG lasers. Experiments reveal that a low-damage pattern is obtained
© 1998IACS
270
S A Nouh, M M Radwan and A A El Hagg
with high absorption coefficient. Chmel et al [S] investigated the morphology of laser
damage of polymer films. Bychkov et al [6] studied the laser destruction in polymer
material Kesting et al [7] studied the pulse and time dependent observations of UV laser
induced structures on polymer surfaces. Peterlin and McCrackin [8] showed that the
increase in the crystal fraction decreases its free volume of the amorphous phase in some
semicrystalline polymers. Nielsen [9] pointed oui that polymers are not completely
amorphous but are more or less crystalline. Barakat et a/ [10] applied X-ray diffraction
technique to investigate the effect of irradiation on fibre. Keller [11] gave a survey of
a series of studies on the influence of crystallinity on the radiation induced effects,
Suthar et al [12] performed X-ray diffraction and IR spectroscopy measurements to
study the effect of radiations on the characteristics of polyvinylidene fluoride.
This paper deals with the effect of infrared laser pulses on the X-ray diffraction .
patterns and IR spectra of cellulose nitrate aiming to investigate the induced physical and
chemical changes such as, ordering, disordering and branching. The investigations may
enable one to introduce the basis used in constructing a simple sensor of irradiation.
2. Experimental procedure :
Samples ;
CR-39 is the t»‘ade name of diglycol carbonate. It is a thermo-plastic that combines the
optical properties of glass with mechanical and physical properties superior to other plastic.
The CR-39 sheets used in this study were manufactured by Pershore, LTD England. It is of
density equals 1 .32 gm/ cm'’ and 300 pm thickness.
LR-1 15 is cellulose nitrate manufactured by Kodak Pafhe, France. It consists of a
sensitive cellulose inlrate layer of 12 pm thickness on a 100 pm thick polyester support. Its
densit/is 1.42 gm/cm^.
Makrofol is a polycarbonate foil manufactured by Bayer A G West Germany. It is of
300 pm thickness and density of 1 .23 gm/cm \
Irradiation facilities :
All samples were exposed to laser pulses for different exposure doses at levels between
0.0 and 7.5 J/ cm^, using an infrared pulsated laser tube of 5 Watt power (Model No. SSL3)
USA. The unit is capable of producing 2000 pulse per second with pulse duration 200 nano
seconds at 9040 A. The laser beam were in the form of a circle of 1 .8 cm in dameter, a.id
was focused on the sample surface giving pulses once a second.
The X-ray diffraction measurements were carried out with a Philips Powder
diffractometer Type PW 1 373 goniometer. The diffractometer was equipped with a graphite
monochromator crystal. The wavelength of the X-rays was 1. 5405 A and the diffraction
patterns were recorded in the 2 theta range (4-80) with scanning speed of 2 degrees per
minute.
The infrared measurements were carried out using the Unicam SPIOOO infrared
spectrophotometer which is a double beam, optical null, pr^ision recording instrument.
The effect' of infrared pulsated laser etc
271
This instrument measures in the wave number range 625-3800 cm“*, with wave number
accuracy belter than ±3 cm"' over 625-2000 cm“' and better than ±9 cm~' over 2000-
3800 cm-'.
3. Results and discussions
X-ray diffraction measurements :
Figure I shows the X-ray diffraction patterns for the un irradiated CR-39, Makrofol and
LR-1 15 detectors.
00 77 6A 56 4B 40 31 14 U
2 theto { degree )
Figure 1. X-ray diffraction patterns for umrrudiated CR-39. Makrofol and LR-1 13 detcctoi-s.
From the figure it is clear that the unirradiated CR-39 sample was characterized !*>
the appearance of an amorphous halo extending in the 2 theta range from 12 to 32°.
Also, the unexposed Makrofol sample was characterized by the appearance of an
amorphous halo extending in the 2 theta range 12-24°. This shows that both CR-39
and Makrofol detectors contain major amorphous phase. While, when the unexposed
LR-1 15 sample was examined by X-ray diffraction the X-ray diffraction pattern (Figure I)
272
S A Nouh, M M Radwan and A A El Hagg
amorphous regions in addition ,o diTc Je rfle 7 ^
pl.-= .. ,h. of
pmpcntes of the material the LR IIW . ^ '"'Pot'oo' role in deierminiiig the
-pie. .ere erpo!:: ,o
ranging from 0.0 to 7.5 j/cm^. X-rav diffL ,■ ^ ^ doses
ihcsc samples and the corresponding pattern^aTe
Z«hm... „ a, I aeh ">
The effect of infrared pulsated laser etc
273
reflections (the mass fraction of the crystalline regions) and the integral intensity of the
amorphous halo (mass fraction of the amorphous regions) change with changing the
laser dose. The integral intensity of amorphous regions (/,,„,) and the integral intensity
of crystalline regions (/,,) were calculated and the values obtained are given in Table I and
Table 1. The variation of X-ray parameters with laser exposure dose for LR-1 1 5 detector
Laser dose
(J/cm“)
Integral intensity
of amorphous
regions, (a. u )
Integral inlensuy
of crystalline
regions, (a ii.)
Index o|
crystallinity,
Crystallite
size,
L(A)
00
89.30
46 07
34 0
21.3
0.5
46.65
40 57
44 0
36.7
1 5
43 30
37 60
46.5
40 4
3 0
34.57
32 .30
48 3
44.9
4.5
.32.24
31.90
49 7
50 5
7.5
20 65
21 36
.50 7
-57 8
plotted as a function of laser dose in Figure 3(a,b). From the figures it is clear that the
iniegral intensity of the amorphous regions decreases Nvilh increasing the laser dose up to
7.5 j/cm^ indicating the reduction of the amorjihous phase in the sample. Also from the
figure it is noticed that the integral intensity of the compound crystalline peak decreases
with increasing the laser dose until the sharpness of the crystalline peak has been obtained
ai 7.5 j/cm^ indicating a high degree of ordering.
The interpretation of the reduction in the integrated intensity of the compound
crystalline peak which indicates the decrease in both 1^^ and together with the laser dose
can be as foflows :
For a semi-crystalline sample as prepared (or as deposited) under certain
conditions, nominally, if the sample prepared in its semi-crystalline phase and suffers an
annealing within a certain thermal energy range, its degree of ordering or its degree of
crystallization will be enhanced as a function of that annealing. If this sample contained
traces of an amorphous phase, then with the annealing this phase will disappear
causing an increase in the crystalline phase (indicated by enhanced peak intensities).
Under certain conditions, the crystalline peak (the dominant peak) may decrease due to
(i) presence of an intermediate phase which can volatile with the annealing, (ii) If a
large surface change of the sample arises due to an injection of high dose of destructive
radiation which certainly affects the degree of roughness of the surface beside the
vaporization of surface atoms.
Values of the apparent degree of crystallinity or the index of crytallinity 0)^ were
calculated for the LR-1 15 polymer since the crystalline and amorphous scattering in the
diffraction pattern could be differentiated from each other. The degree of crystallinity was
considered to be the ratio of Che integrated crystalline scattering to the total scattering, both
crystalline and amorphous. The obtained values are also given in Table 1 and plotted as a
function of laser dose in Figure 3(c). The index of crystallinity OJ^. showed an increase on
274
S A Nouh, M M Radwan and A A El Hagg
increasing the laser dose indicating the increase of the mass fraction of the crystalline phase
in the sample.
Lostr dos9(
Figure 3. The dependence of (a) the integral
intensity of the amorphous regions {lam)' (b) the
integral intensity of the crystalline regions {l^r),
(c) index of crystallinity (oi^) and (d) crystallite
size (L) on the loser dose.
Approximate indicative size of the crystallites (L) were calculated by means of the
Scherrer equation [15,16].
0.89A,
AWcos9'
where AW \s the peak's width at the half of maximal intensity and A is the wavelength of
X-rays. The values obtained are also given in Table 1. Figure 3(d) shows the dose
dependence of the crystallite size. From the figure, it is clear that the crystallite size
increases with increasing the laser dose indicating also a high degree of ordering. It is also
The effect of infrared pulsated laser etc
275
noticed that the intensity of the crystalline peaks at 261 = 26, 46.8 and 54 and also the
intensity of the amorphous peak at 20= 23.2 varies with the variation of the laser dose
(Table 2). This means that the mass fraction of the crystalline and amorphous phases
changes with changing laser dose. The interpretation of the above results can be explained
as follows :
Table 2. The variation of peak intensity vi^ith laser dose for LR- 1 1 5 detector.
Non expo.sed
2(T 1
0.5 J/cm^
20“ I
1 5 J/cin^
20“ I
3.0J/cm^
20“ 1
4.5 J/cm^
20“ I
7.5J/cni2
20“ t
23 2
17 9
23.2
8.3
23.2
8 3
23.2
5 9
23 2
45
23.2 3 5
26.0
41.5
26 0
36 1
26 0
35 4
26.0
33.2
26 0
.35 8
26 0 20 4
46 8
0 9
46.8
0.4
46.8
0.25
46 8
00
46.8
0.0
46 8 0.6
54.0
20
54.0
1.0
54.0
1.0
54.0
0 7
54.0
0.6
54.0 0 5
By exposing the sample to laser light, the sample surface rises to the vaporization
temperature and begins to vaporize. The evaporated material will flow away from the centre
oi irradiation due to the thermal gradient leaving behind it a resultant pit. After stopping the
stimuli, the molten material will begin to recrystallize at the colder regions in the
surroundings of the pit. This leads to the growth or multi-layered crystals and different
shaped grains and grain boundaries. Similar effect was observed before [17].
Injrared spectra of LR-l 15 detector :
f igure 4 show.s the infrared spectra of unirradiated and irradiated LR-115 samples.
Tiom the figure it is clear that the half band width V' 1/2 of the infrared absorption bands
were slightly affected, by the laser dose. On this basis and since the intensity of the
absorption band is equal to {k/2) V |/2 log (/q//), the absorbance log (Iq/I) could be taken as
a direct measure of the intensity. Values for the absorbance (/\) and log Uo/l) at the
maximum of the absorption band over the range (0-3800 cm"*) were obtained and are given
in Table 3.
Figure 5 shows the dependence of the absorbance (A) (measured at different
wavenumbers) on the laser dose. It is clear from the figure that the absorbance showed a
linear increase up to a maximum value around 4.5 j/cm^ due to the existence of
amorphous regions followed by a decrease on increasing the laser dose up to 7.5 j/cm^ due
to the high degree of ordering. The interpretation of the above figure can be explained
according to Tobin [18] that a majority of the absorption bands in the infrared spectra of
polymers are associated with both crystalline and amorphous regions. A few absorption
bands, however, are produced by the amorphous regions only, due to the loss of symmetry
by the cooling of the chain in amorphous regions. Also, according to Zbinden and Rudolf
1191 who assigned specific absorption bands as crystalline and amorphous, we may define
the "crystalline band" as one which does not appear in the spectra of the completely
uinorphous polymer and becomes more intense with the increase of the crystalline character
278
S A Nouh, M M Radwan and A A El Hagg
conversely, an “amorphous band" is one which does not appear in the spectra of the
crystalline polymer and become more intense with the increase of the amorphous character.
Under these previous definitions and according to Mostafa [20], it can be pointed out that
ail the bands that increase in intensity with the increase in the amorphous character may be
called amorphous bands. On the other hand, all bands that decrease in intensity with the
decrease in crystalline character may be called crystalline bands. These show that all the
amorphous bands in the LR-1 15 detector tend to be crystalline bands on increasing the
laser dose up to 7.5 J/cm^ (and this agrees with the X-ray diffraction results) due to the
explanation that : by focusing the light of high power pulsed laser on a target surface, the
material surface rises to the vaporization temperature and begins to vaporize. The material
evaporated is partially burnt, and this gives rise to gas bubbles of high pressure and
temperature. The gas pressure produces near the bubbles large stresses and initiates the
development of cracks which proceeds into the target. The cracks become wedged a part by
healed gas. During the course of crack expansion, the hot gas carburizes the crack walls,
enhancing further light absorption.
4. Conclusions
The X-ray diffraction measurements indicated that the exposure of the LR-115
samples to laser pulses, leads to the growth of multi-layered crystals and different
shaped grains and grain boundaries, i.e. the samples tend to be crystalline. This conclusion
was drawn since both the index of crystallinity and crystallite size increased on increasing
the laser dose while the mass fraction of the amorphous phase decreased. The infrared
measurements indicated that the infrared pulsated laser enhances the light. absorption of
the LR-1 15 polymer up to a certain dose (4.5 j/cm^) then it enhances the light transmission
of the polymer due to the high degree of ordering obtained at higher laser doses (4.5-
7.5 j/cm2).
References
[ 1 1 A Felske. W Hagenah and K Laqua Proc. Xil. Coll. Sped Ini. Exeter p 340 (1965)
[2] J Ready Effect of High Power Lcuer Radiation (New York ■ Academic) (1971)
13] R Patel and G Baisch Single Pass Uiser Cutting of Polymers in UA Vol 74 p 282 (1992)
[ 4 ] Keiko Ito, Inoue Masami and Masahra Moriyasu J. Polymer Sci. Tech . 48 725 (1991)
[.5] A Chmel, A Kondvrev, N Leksovskaya, A Radyushin and Yu Shestakov Mater. Lett. 14 94 (1992)
(6] S Bychkov, A Biketov, S Mashakova Pizikai Kfumiya Obrabotki Materilov No. 1 70 (14191)
(7] Resting, Wolfgang. Knittel Dicrk Bahners, Thomas and Schallmeyer Eckhard Appl. Surface Sei,
330(1992)
(8] A Peterlin and F McCrackin J. Polym. Sci. Phys. Edn. 19 1003 (1981)
(91 L Nielsen Mechanical Properties of Polymers and Composites (New York : Marcel-Dekker) Vol 1
(1974)
[101 N Barakat, B Kalifa, F Sharaf and A El-Bahay Egypt J. Phys. 15 237 (1984)
(11) A Keller Developments in Crystalline Polymers. Vol 1 ed. D C Bassett Appl. Sci. (London) p 37 (1982)
[ 1 2J L Surhar Jayant, I Laghar and R Javaid IEEE Trans. Nucl. 5^1. 38 16 ( 1 991 )
The effect of infrared pulsated laser etc
279
1 1 3] H Zachmann. Kunsioff-Hondlmck eds R Vicweg Herau^geber, D Braun Carl Hanser Vering (Munchen)
( 1975 )
f 14| R Miller EmYclopedia of Polymer Science ond Technology Vol. 4 eds H Mark. N Gaylord and N Bikales
(New York ; Wiley-lnierscicnce) (1966)
[151 L Alexander X-ray Diffraction Methods (New York . Wiley Inierscience) ( 1 969)
1 1 6] M Kakudo and N Kasni X-ruy Diffroi fion by Polymers (Tokyo Kodaimha. Amsterdam Elsevier) ( 1972)
[171 M Nagwa MSc Thesis, (Faculty of Science, Ain Shams University, Cairo, Egypt) p 64 ( 1 98 1 )
1 18] M Tobin / Chem. Pfm 23 891 (19.55)
[19] Zbinden and Rudolf /. H. Spei trosvopy ofHi^li Polymers EEE ( 1 964)
(20| A Mosiafa MSi Theus (Faculty of Science, Am Shams University, Cairo. Egypt) (1971)
Indian J. Phys. 72A (4), 281-286 (1998)
UP A
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Pulse method for measurement of thermal
conductivity of metals and alloys at cryogenic
temperatures
T K Dey*, M K Chattopadhyay and A Kaur Dhami
High Temperature Superconductivity Laboratory, Cryogenic Engineering Centre.
Indian Institute of Technology, Kharagpur-721 302, West Bengal, India
Received 17 February J99H, accepted 14 May 1998
Abstract ; A computer controlled experimental facility to measure thermal conductivity
of aerospace metals and alloys at cryogenic temperatures using pulse method has been
described. The experimental set-up has been calibrated using a standard stainless steel 304
sample The reliability of pulse method has been further confirmed by measuring the thermal
conductivity of a copper alloy and an Inconel 718 by both conventional steady state method
as well as by pulse technique The agreement between the measured data obtained by both
these methods has been found to be within -'3%. Advantages of the present technique have
been di.scussed
Keywords ; Thermal conductivity of metals and alloys, pulse technique cryogenic
temperatures
PACSNo, : 72.15.Eb
1. Introduction
In condensed matter physics, measurement of thermal conductivity of solids at low
itMiiperatures is of great interest because it helps one to identify the type of thermal carriers
and their interaction mechanisms 1 1-4], operative at different temperature zones. Besides,
design and development engineers in aerospace industries continue to have urgent need for
thermal property data for new alloys and composites. For most materials, specially
uncommon alloys and new composites, measured values of thermal conductivity at
cryogenic temperatures are not available readily and predictions also can not be made with
adequate confidence. Traditionally, heat conductivity experiment is performed using
sicady-statc technique [5-8]. In this method thermal conductivity (A) is determined by
(’^) For correspondence— e-mail address : tapasdey9hijli.iitkgp.ernet.in
© 1998 1 ACS
282
T K Dey, M K Chattopadhyay and A Kaur Dhami
noting the steady temperature gradient across the sample for a known quantity of heat when
the steady state equilibrium is reached. Steady slate method, in spite of being a straight
forward one, has the main disadvantage of very long waiting times for both temperature
stabilisation, as well as for establishment of a steady state thermal gradient. A typical
estimate by Reese [9] for the time (r) taken for the temperature gradient (AT) to reach
within 1% of its equilibrium value is ;
» = (sec), (1)
71 ^ A.
where, C is the specific heat per unit volume. Thus, a stainless steel sample of -50 mm long
would require an equilibrium time of about I hour at 20 K. It is important to note that long
waiting times also influence precision owing to temporal offset drifts.
Such disadvantages of steady-state method may be eliminated to a large extent by
using Pulse or, Non-steady -state method, in which the bath temperature is allowed to drift
slowly. In the present communication, a brief description of the design and the performance
of the facility for thermal conductivity measurement at cryogenic temperatures by pulse
method is reported. This facility has been developed for the investigation of thermal
conductivity of aerospace alloys between 10 and 300 K.
2. Outline of the pulse technique
In pulse method, as the temperature of the bath is allowed to drift slowly and a periodic
square wave current (period 2r) excites the heater, the system never returns to the steady
Slate. Instead, the temperature (7) of the heat source becomes an oscillating function of
time. Under such conditions, the thermal conductance {K) of the sample may be expressed
in lenris of the pcak-to-pcak amplitude of the signal as 110] :
K =
(4n,„
tank
( 2 )
where, R and Iq arc the heater resistance and the peak current through the heater, r is the
half period of the square wave. Similarly, the heat capacity (C) of the heat source can be
expressed as :
7 Di 2 _
V ) PI)
where (5AT j St) is the time derivative of the peak-tq-peak amplitude of temperature
difference (4T). Thermal conductance (^) of the sample can be determined by solving
equations (2) and (3) by successive iteration. Thermal conductivity (A) is then obtained
from : K (AL/A) where A is the area of cross section of the sample and AL is the
Pulse method for measurement of thermal conductivity of metals etc 283
distance over which the temperature gradient is monitored. Influence of error in C on
may be estimated from ;
C
li is evident from the above that for T > 4C/K, a 100% error in C induces an error of
less than 1% in K. If T » 4CIK, the system reaches the steady stale regime. However, if
r « 4C/K, the measured signal does not correlate with the thermal conductance (^0-
3. Experimental
The present facility for thermal conductivity measurement between 10 and 300 K has been
built using a cryo-refrigerator (APD model 202). It may be noted that the same facility
also enables one to measure the thermal conductivity under conventional steady-state
cmidiiions, with the difference that for pulse method the heating current is pulsed as
square wave with an appropriate lime period. At the two ends of the cylindrical sample
(length ~40 mm, 0 ~4 mm) two small copper electrodes were soldered. One of the copper
blocks was firmly screwed to the second stage (10 K) of the cryocooler with an indium foil
in between m order to ensure excellent thermal contact. The small copper block at the other
end of the sample contained a small healer (50 which was used to generate thermal
LMacheiu (AT) across the sample. A differential Au + 0.07% Fe vs Chromal thermocouple
was employed to monitor AT across the length (AL) of the sample. Absolute temperature (T)
of the sample was monitored using a calibrated Si diode sensor (Lakeshore DRT-470).
The sample was kept enclosed within a radiation shield (connected to the 10 K stage) and
was covered with at least 10 layers of aluminised mylar sheet. The entire assembly was
luiihci surrounded by another copper shield thermally anchored to the first stage (40 K)
(h ilic cryo-refrigerator. Finally, the sample holder and the radiation shields were enclosed
in a stainless steel vacuum shroud. Experiments were performed under a vacuum level of
-I X 10"^ lorr so as to make the heat losses due to gas conduction and convection
negligible.
Figure 1 shows the schematic diagram of experimental set-up. All voltages
were measured by a digital nano voltmeter (Keithley model 181) with a resolution of
10 nV. Pulse heating current of appropriate frequency was generated using a programmable
airient source (Keithley model 220). The sample temperature was drifted at any
desired rate with the help of a programmable temperature controller (Scientific Instruments
model 9600). The entire system was interfaced with a PC 386 for continuous data
acquisition. Typical time for measurement of thermal conductivity of a stainless
^‘cel .sample between 10 and 300 K including the cool down time was about eight
72A(4).5
284
T K Dey, M K Chattopadhyay and A Kaur Dhami
Figure 1. Schematic diagram of the electrical layout and the instrumentation
for the measurement of thermal conductivity of metals and alloys at cryogenic
temperatures by Pulse method.
hours. Maximum uncerlainty eslimated in the measurcmenl of thermal conductivity
(A) was ^8%.
4. Results
The facility described above for pulse method of measL rement of the thermal conductivity •
of metals and alloys has been tested successfully with a standard SS-304 sample. A typical
Figure 2. Typical nature of the time
dependence of the temperature gradient
{AT) across the sample observed in Pulse
method.
plot for the time variation of AT due to square wave current pulse to the sample heater as
the bath temperature is slowly drifted is shown in Figure 2. Measured values of X for
Pulse method for measurement of thermal conductivity of metals etc
285
SS-304 sample between 10 and 300 K is shown in Figure 3 along with the NBS data [11],
It may be seen that the agreement between the two is very satisfactory (maximum deviation
Figure 3. Tempcraiure dependence of
thermal conductivity between 15 and
3(X) K for an SS-304 sample. Serie.s 2
NBS data (A) and Senes 3 measured by
Pulse method (•)
0 100 200 300 400
Tamparature (K)
~±5%). The reliability of the pulse method was further established by measuring A for two
more samples (viz. Copper alloy and Inconel 718). Thermal conductivity of ihese two
samples between 20 and 300 K obtained by pulse method has been cross checked with
the data obtained by conventional steady state method. Figures 4(a)and (b) show the results.
Figure 4. Thermal conductivity as a function of temperature between 1 5 and 300 K measured
by both Pulse method and Steady state method, (a) Copper alloy [Series 2 ; Pulse method (■)
and Scries 3 : Steady state method (a) 1 and (b) Inconel 718 [Series 2 : Pulse method (^) and
Series 3 : Steady state method (•)!.
286
T K Dey, M K Chattopadhyay and A Kaur Dhami
Temperature variation of thermal conductivity between 10 and 300 K for both the samples
measured by steady slate method and by pulse method agrees to better than ±3%.
As noted earlier, the validity of the pulse method for thermal conductivity
measurement depends on the proper choice of the half period (r) of the square wave pulse
applied to the sample heater and the drifting rate of the bath temperature. Principal sources
of error in this measurement arc associated with the measurement of AT, the geometrical
factor (AIJA) and the lime derivative of the peak to peak temperature difference
5. Conclusions
An experimental facility for the measurement of thermal conductivity of aerospace metals
and alloys between 10 and 300 K using pulse technique has been described. Our lest results
on various samples show excellent agreement with those measured by conventional steady
slate method. Pulse method allows faster and accurate measurement of thermal conductivity
of metals and alloys with higher point density and hence could be adopted for routine
measurements.
Acknowledgments
The technical help received from Mr Dilip Kumar Paul of Cryogenic Engineering Centre
during the setting-up of the facility is gratefully acknowledged by the authors.
References
1 1 ) P G Kleinens and L Tewordt /Jcv. Mod Phys. (January) 118(1 964) »
[2] H M Rosenberg Imw Temperature Soitd Stale Physics (U K Oxford University Press) Ch 5 110(1 963)
(3) T K [)cy and K D Chadhuri J Low Temp, Phys. 23 419 (1976)
[41 S D Pcacor, R A Richardson, F Nori and C Uher Phys. Rev B44 9508 (1991)
1 3] B Chanda and T K l>y Sol Stale Commun 89 353 (1994)
[6] G K While Experimental Techniques m Low Temperature Physics 3rd cdn. (Oxford ■ Clarendon)
ChVll 171(1979)
[71 C Uher and A B Kaiser Phys Rev B36 3680 (1987)
[81 W J Hall, R L Powell and H M Rodes Adv. Cryo En^fi. 3 408 (1957)
[9] W Reese J Appl. Phy.c 37 864 ( 1 966)
[10] 0 Maldonado Cryofiemcs 32 908 ( 1 992)
[11] V J Johnson A Compendium of the Properties of Materials at Low Temperatures (Phase I) 60-63
Part II (U.S.A : National Bureau of Standards) 3.301 (Oci., I960)
Indian J. Phys. 72A (4). 287-294 (1998)
UP A
— an intematipnaJ journal
Study of forward (C-V) characteristics of
MIS Schottky diodes in presence of interface
states and series resistance
P P Sahay
Department of Physics, Regional Engineering College.
Silchaj-788 010. India
Rei etved 16 February im, accepted 2R Aprd 199H
Abstract : Forwaid (C-V) churacteristics of MIS Schottky diode in presence of interface
stales have been studied by taking into account the effect of series resistance and using
Shockley-Rcad-Hall statistics Exchange of charge between the metal and the interface state.s is
included in the model it is observed that at a particular density of the interface states and a given
ac signal frequency, the diode capacitance decreases m the presence of a series resistance. In
addition the (C-V) plot exhibits a peak whose value depends on the interface slate density and
the frequency of ac signal as well as the senes resistance
Keyword.s : MIS Schottky diodes, (C-V) characteristics
PACSNos. ; 73.30 +y. 85.30.Kk
1. Introduction
The origin of the excess admittance at forward -biased Schottky diodes is the subject of
a controversy among research workers. Werner and coworkers [1,2] based on their
experiments contend that the excess admittance observed at forward-biased Schottky
diodes is due to imperfect back contacts. They ascribed the capacitance and inductance
to excessive minority-carrier extraction at defective back contacts. On the other hand,
Wu et al [3,4] attributed the excess admittance to the presence of the interface states
ai the boundary of the metal-semiconductor structure. Recently, Chattopadhyay and
Raychaudhuri [5] investigated the frequency dependence of forward (C-V) characteristics
of Schottky barrier diodes considering the series resistance effect. They found that the peak
value of capacitance in (C-V) plot varies with series resistance, interface stale density and
the frequency of ac signal.
© 1998 I ACS
288
P P Sahay
In this paper, forward (C-V) characteristics of MIS Schottky diodes in presence
of interface states and series resistance have been studied using Shockley-Read-
Hall statistics and considering the charge exchange between the metal and the interface
stales.
2. Theoretical approach
2.7. Determination of current density J^c as a function of applied voltage V :
Figure 1 represents the energy band diagram of a forward biased metal /n-type
semiconductor Schottky diode with a thin interfacial layer. Here 0^ is the work function of
Figure 1 . Energy band diagram of a forward
biased metal /n-type semiconductor Schottky
diode with a thin interfacial layer.
the metal, x electron affinity of the semiconductor, ^he semiconductor
surface potential, 5 the thickness of the interfacial layer, A the voltage drop across '
the interfacial layer and V„ the depth of the Fermi level below the conduction band edge
in the bulk semiconductor. and Efp are the respective quasi-Fermi levels for
electrons and holes in the semiconductor at a forward bias voltage V applied to the
diode.
Considering the energy band diagram, the voltage drop across the interfacial layer
can be written as
^ = <l>m-X-Vs-V„-V + lj,Rs, (I)
where is the current across the diode and Rs the series resistance.
The voltage drop across the interfacial layer can also be obtained by using charge
neutrality condition and Gauss law. Thus
4= {Qsc+Q„+Qf], (2)
where is semiconductor space charge density; the interface trapped charge
density and Qp the fixed charge density in the interfacial layer.
Study of forward (C-V) characteristics of MIS Schottky diodes etc
289
Taking the case of the interface state continuum throughout the band gap and
assuming the donor nature of the interface states, the net charge density trapped in the
interface states is given by [6] :
e.,(^ )=</["■[' -fAE,,Vj]D„(E,)dE,. (3)
where Di,{E,) is the interface state density at the energy level V,, the voltage drop across
the semiconductor space charge region at a forward bias voltage V applied to the diode and
/„(£„ VJ, the occupation function of the inlerhicc slates.
The occupation function of the interface state is obtained using the Shockley-
Read-Hall statistics and considering the charge exchanges between metal and interface
stales [7-9]. Thus
fAE,,V,)
", + ypi(E , )
(4)
where and are the quusi-thermal equilibrium densities of electrons and holes
at the semiconductor surface; and p\ arc the densities of electrons and holes if
iheir quasi-Tcrmi levels were coincident with trap energy level £,; / is a parameter
specifying the controllability of minority carriers on the occupancy of the interface
slates
In general, the description of current-voltage characteristics of most Schottky
iliodes is based on thermionic emission theory. Thus assuming inlerfacial layer-
thermionic emission theory [10], the dc current density for these Schottky diodes can be
written as
=4*r2 0„exp
for V
wr
(5)
where A* is the effective Richardson constant, T the absolute temperature and d„ is
transmission coefficient across the intert'acial layer.
The voltage dependence of surface potential y/^ can be obtained numerically from
eqs. (1-3). The values of thus calculated can be used to obtain current density as a
function of applied voltage V.
2.2. AC admittance of the diode \
The ac admittance Y of an MIS Schottky diode is defined as the ratio of total ac current
to the ac voltage 6V. Thus
290
PPSahay
The total ac current across the interfacial layer of an MIS Schottky diode consists of three
current components :
(i) The ac current of the moving electron, given by [2]
kTIg
SWs-
( 7 )
(ii) The displacement current which flows within the space charge region of the
semiconductor due to the change of the electric field, given by
dJ,, =icoC,,6iif,^ ( 8 )
where C\, is the semiconductor space charge capacitance.
(iii) The ac current which flows between the space charge region and the interface
due to the charging and discharging of interface states, given by
SJ„ ={C„ ^iwc„)6yf,, (9)
where and C„ represent the conductance and capacitance associated with the interface
slates.
With these substitutions cq. (6) becomes
ik
K =
kTjq
+ G„ + /a)(C„. +C„)
SV
( 10 )
The expressions for G„ and Q derived by Nicollian and Brews [6] for a MOS structure
with interface slates continuum are given by
G„ = ln(l + a)^T^) (11)
qD ,(E,)
and C,, = " ■ tan-'(Ct>T). (12)
"m
where r is the relaxation time of interface slates and co the angular frequency of the
ac signal.
These expressions may be used to describe the interface state admittance of an MIS
Schottky diode as long as the interface states are in thermal equilibrium with the
semiconductor and do not communicate with the metal [11].
For a MIS diode, the variation of relaxation time T with the applied voltage V is
given by |6J
(,3,
where C7„ is the electron capture cross section of the interface states; v the thermal velocity
of electrons and N,/ the donor concentration in the semiconductor.
Study of forward (C-^V) characteristics of MIS Schottky diodes etc
291
In order to obtain the conductance G and capacitance C of the diode from eq. (10)
one has to express Siffg as a function of SV. Due to the presence of the interfacial layer and
the series resistance, any voltage V applied to the diode is divided across the series
resistance (VJ, the interfacial layer (V,) and the space charge region (V,). Thus for small
incremental change in applied voltage, we can easily write
SV = + SVi + SV^. (14)
But SVj is equal to the incremental change in surface potential and 5Vr = Rs^SJ^c,
where A is the diode area. Hence
SV=S\ir,+SV,-\-R,ASJ^. ( 15 )
Taking the lime derivative of incremental change in voltage drop across the interfacial
layer,
dV,
dt ^ C,[ dt ^ d, \
or
/to5V, = +i(o{C,,+C„)]5w,.
(16)
Substituting the values of SVj and S/ac in eq. (15), we get
5V . .
-= — = a - i(oP,
Sw,
(17)
where
(18a)
and
(18b)
Putting this value of SV j Syf ^ in eq. (10), we get
y =
kTlq
+ G,f + icoiC^^ + )
G + icoC .
a “ iwP
Equating the real and imaginary parts on both sides of eq. (19), we get
G =
and
{w/q
(19)
( 20 )
( 21 )
+ co^p^
These are the required expressions for evaluating the capacitance C and conductance G of
the diode as a function of applied voltage V.
’2A(4)-6
292
P P Sahay
3. DIsciusioii
The study has been carried out on any arbitrary metal /n-type Si Schottky diode where the
meial has the work function 5.0 eV. The parameters used here are = 5.0 eV, % = 4.05 eV,
= IO'‘ciii-3, AV=5x 10" cm-2 £,= 1.12eV. 5= 10 A, 11.9, «; = 3,9, y =0.01,
V = 10’' cm/sec, o„ = 10“" cm^.
The occupancy of an interface state lying within the semiconductor bandgap depends
on the charge exchange between the interface state and the three reservoirs surrounding it,
namely the conduction and valence bands of the semiconductor and the conduction band of
the metal. The charge exchange between the semiconductor conduction or valence bands
and the interface states follows the Shockley-Read-Hall (SRH) theory while the charge
exchange between the interface states and the metal conduction band occurs through direct
tunneling. The occupation function /„ (£;, Vg) of the interface states has been calculated with
the help of eq. (4). The occupation function thus obtained is used to get interface trapped
charge density from eq. (3).
Considering the interfacial layer to be of oxide layer and with = {2q
and Qf~qNp being the density of fixed charges in the oxide layer, the values of have
been calculated for different values of V for a given interface state density. In obtaining the
current density Jdc^ we have used the values of effective transmission coefficient calculated
by Card and Rhoderick [12] for oxide films of thickness from 8 A to 26 A.
Figure 2. Forward (C-V) characteristics of an MIS Schottky diode at
Q) = 2;rx 10^ Hz with interface state density os parameter.
Study of forward fC-VJ characteristics of MIS Schottky diodes etc
293
Figure 2 shows the effect of interface state density on the forward (C-V)
characteristics of the diode at O) s 2^ x 10^ Hz. It is obvious from the figure that the diode
capacitance C increases with the increase of the density of interface states in both situations
(Le., Rg-O and R^ a 10 £2). This is because of the presence of the interface states at the
boundary of the interfacial layer/semicoivductor, which attribute the interface state
admittance thus modifying the diode capacitance C. It may be noted that at a particular
density of the interface states, the value of C decreases in the presence of a series resistance.
This is due to the voltage drop across the series resistance R^ which in turn, increases the
value of SV I and thus decreases the diode admittance according to eq. (10). Further,
in the presence of a series resistance the (C-V) plot exhibits a peak whose V&lue increases
and also shifts towards a lower voltage as the density of the interface states increases. The
capacitance peak in (C-V) plot has been observed in a number of experimental studies on
Schottky diodes [13-15]. However, here the results regarding the capacitance peak position
with the interface state density differ from those obtained by Chattopadhyay and
Raychoudhuri [5]. This discrepancy may be due to the relaxation time dispersion of the
interface states.
Figure 3 represents the frequency dependence of the forward (C-V) characteristics
of the diode at D„ = 5 x 10^^ cm~2 eV"'. It is seen that in both situations {i.e., /?, = 0
and Rs = 10 X2), the diode capacitance C increases in the lower voltage region with the
decrease of the frequency of ac signal. This happens because at lower frequencies, the
interface states respond the ac signal and yield the excess capacitance. However, in higher
voltage region, the capacitance C does not change. This is due to the large relaxation
lime of the interface states lying near the conduction band edge. These inferences are
in consistent with the experimental results observed by Barret and Vapaille [16], and
Singh [17]. ‘
Figure 3. Frequency dependence of the forward
(C-V) characteristics of the diode at D,-, = 5 x 10^^
cm“^ eV"' . Other parametric values are the same as
those used in Figure 2.
Figure 4. Effect of the series resistance on the
forward (C-V) characteristics of an MIS Schotd^
diode at D,, » 5 x 10*^ cm"^ eV“' and » 2« x KT
Hz. Other parametric values are the same as those
used in Figure 2.
294
? P Sahay
Effect of series resistance on the (C-V) characteristics of the diode at D„ a
5 X 10'^ cm’^ eV ' and o) = 2;rx 10^ Hz is shown in Figure 4. It is observed that the
capacitance plot exhibits a peak whose value strongly depends on the values of the series
resistance. As the series resistance increases, the peak value of the capacitance decreases
and also shifts towards a lower voltage. Similar results have been reported by
Chattopadhyay and Raychoudhuri [5] and Venkatesan e/ a/ [18],
References
[ 1 J J tt Wemer, A F J Levi, R T Tung, M Anziowar and M Pinto Phys. Rev. Lett. (lO 53 (1988)
[2] J H Werner in Metallization and Metal -Semiconductor Interfaces cd. I P Batra (New York ; Plenum)
p 235 (1989)
[3] X Wu and E S Yang J. Appl Phys 65 3560 (1989)
[4] X Wu. E S Yang and H L Evans J. Appl Phys. 68 2845 (1990)
[5] P Chattopadhyay and B Raychaudhuri Solid-State Electron 36 605 ( 1 993)
(6! E H Nicollian and J R Brews MOS Physics and Technolofty (New York ; Wiley-lnicrscicnce) Chap 5
p 176 (1982)
[71 L B Freeman and W E Dahike Electron 13 1483 (1970)
[81 P P Sahay and R S Snvasiava Cryst Res. Technol 25 1461 (1990)
[9] P P Sahay Indian J. Phys. 72A 57 ( 1 998)
[10] C Y Wu J. Appl Phys. 51 3786 (1980)
[11] P P Sahay, M Shamsuddin and R S Srivastava Microelectronics J 23 625 (1992)
[12] H C Card and E H Rhoderick Phys. D4 1589(1971)
[13] H L Evans, X Wu. E S Yang and P S Ho 7. Appl Phys. 60 36 1 1 ( J 986)
[14] P Chattopadhyay and B Raychoudhuri Solid-State Electron 35 875 ( 1 992) •
[15] P S Ho, E S Yang. H L Evans and X Wu Phvs. Rev. Utt. 56 177 (1986)
[16] C Barret and A Vapaille Solid-State Electron 18 25 ( 1 975)
[17] A Singh Solid-State Electron 28 223 ( 1 985)
1 1 8] V Venkatesan. K Das, J A von Windheium and M W Geis Appl Phys Lett 63 1065 (1993)
Indian J. Phys. 72A (4), 295-300 (1998)
UP A
— an international journal
The effect of doping on the microhardness
behaviour of anthracene
Nimisha Vaidya, J H Yagnik and* B S Shah
Solid State Physics and Materials Science laboratories, Department of Physics,
Saurashtra University, Rajkot-360 005. India
Received 13 April 1998. accepted 5 May 1998
Abstract : Microindentation hardness studies using the Vickers and Knoop mdenlers were
carried out on single crystals of anthracene doped with carbazole and phenanthrene respectively.
It is observed that the Vickers hardness versus load curve exhibits two peaks at 55 g and 90 g
load.s with hardness values of 5.4 kg/m^ and 7.9 kg/mm^ respectively, for carbazole doped
anthracene, whereas the Knoop hardness variation with load for phenanthrene doped anthracene
is a curve with two peaks at loads 2.5 g and 7.5 g with hardne.ss values 13.5 kg/mm^ and
12 4 kg/mm^ respectively. The Vickers hardness behaviour is explained in terms of the ease of
slip of the (100) plane in comparison to the (201 ) plane in pure anthracene The Knoop
hardness behaviour is explained in terms of the splitting of the (201) fOIOJ type of dislocations
in phenanthrene doped anthracene.
Keywords : Microhardness doping, anthracene
PACS Nos. : 8 1 .40.Np, 62.20 Fe
1. Introduction
Impurities change the physical properties of any substance. The process of intentionally
introducing impurity atoms to obtain a desired change in any physical property such as
conductivity's called doping [1]. A most common example is that of semiconductors like
silicon or germanium in which the extrinsic conductivity can be adjusted over a wide range
by additions of group III and group V compounds.
It has been suggested that impurity atoms are attracted to dislocations [2,3]. At an
edge dislocation, large impurity atoms would be attracted to the expanded region below the
glide plane where there is more room for them. Small substitutional impurity atoms would
he attracted to the region above the glide plane. The segregation of impurities at a
©19981ACS
296
Nimisha Vaidya, J H Yagnik and B S Shah
dislocation, effects the mechanical properties because its movement is hindered or eased by
the impurities. The segregation of impurities at a dislocation will lead to different chemical
properties of a crystal near a dislocation as compared to a normal part of the crystal [4], The
fact that the energy associated with an impurity atom is affected by its proximity to a
dislocation causes the impurity concentration to change in the vicinity of a dislocation
line [5].
Additions of one metal to another result in a steady increase in hardness until a
saturation is obtained and two phased structures are caused due to further increase of the
solute. Various elements with solid solubilities on aluminium have shown a relationship
between hardness and concentration of solute [6].
There does not exist literature on the microhardness behaviour of the addition of an
organic molecular compound with another to the best of the authors' knowledge.
2. Experimental and results
2. /. Crystal growth :
Single crystals of carbazole doped anthracene and phenanthrene doped anthracene were
grown from the melt by the Bridgmann method. The starting material was column
chromatographed, twice vacuum sublimed and zone refined. The material was transferred to
the crystal growth tubes without exposure to the atmosphere [7]. The crystals were cleaved
in the usual manner using a sharp blade. Smooth cleavages were selected after optical
examination. »
2.2. Microhardness :
The crystals were indented on a Carl Zeiss NU 2 Universal Research Microscope. The
indents were made with Vickers and Knoop indenters. A number of indents were made at a
particular load. The average length of diagonals was used in calculating the Vickers
hardness number using the formula :
= 1.8544 X P/d^ (1)
and the long diagonal length was used in calculating the Knoop hardness number using
the formula :
//* = 14228.8 X P/d^, (2)
where P is the applied load in grams and d, the mean diagonal length in micrometers. The
indentation time of 10 s was kept constant as this time was adequate to minimize the
vibration effects on the results. The crystal size was much larger than the indentation size,
thus eliminating the boundary effects on the results. The distance between the indents was
five limes the size of the largest indentation mark. The crystal thickness was relatively
The effect of doping on the mkrohardness behaviour of anthracene 297
large such that the indenter did not sense the lower surface [6]. A number of crystals were
indented.
Figure 1 shows the Vickers hardness variation with load curve for pure anthracene
single crystals [8]. The plot shows two peaks at 30 g and 67 g loads having hardness values
7.2 kg/mm^ and 4.95 kg/mm^ respectively.
Figure 1. The best fit plot of Vickers hardness versus load for pure anthracene
single crystals
Figure 2 is the Knoop hardness versus load plot which also reveals two peaks but at
lower loads, 5 g and 17.5 g [91. These peaks have higher hardness values 13.0 kg/mm^ and
1 1 .4 kg/mm^ respectively.
Figure 2. The best fit curve of Knoop hardness variation with load for pure
anthracene single crystals
Figure 3 shows the result of Vickers hardness studies performed on carbazole doped
anthracene single crystals. The hardness versus load curve reveals two peaks at 55 g and
90 g having hardness values 5.4 kg/ mm^ and 7.9 kg/mm^ respectively.
298
Nimisha Vaidya, J H Ydgnik and B S Shah
Knoop hardness studies were carried out on phenanthrene doped anthracene.
Figure 4 is the curve of hardness variation with load which shows two peaks at loads 2.5 g
and 7.5 g with hardness values 13.5 kg/mm^ and 12.4 kg/mm^ respectively.
Figure 3. 7'hc best fit curve of Vickers hardness variation with load for crystals
of carbazolc doped anthracene
Figure 4. The best Tit plot of Knoop hardness versus load for phenanthrene
doped anthracene single crystals.
3. Discussion
Anthracene crystallizes in the monoclinic structure. The lattice parameters are a = 8.562 A,
b = 6.038 A and c = 1 1 .184 A with fl = 124° 7’. The compound has the structural formula
C 14 H 10 with two molecules per unit cell. The space group is P2^/„ and it cleaves along the
( 001 ) plane [ 10 ],
The effect of doping on the microhardness behaviour of anthracene 299
Phenanthrene crystallizes in the monoclinic structure. Its lattice parameters are
ri = 8.660 A, /? = 11 .500 A and c = 19.240 A with B = 98° 4' with space group The
compound has the structural formula C14H10 with two molecules per unit cell. Phenanthrene
cleaves along the (001) plane [10].
Carbazole belongs to the orthorhombic structure. Us structural formula is C12H9N
and it has four molecules per unit cell. The space group is P2„a„ and it cleaves along the
(010) plane. The lattice parameters are a = 7.772 A, /? = 19.182 A and c = 5.725 A with
a=/J=r=90°[ll].
In case of pure anthracene, the molecules are tightly entangled across the
adjacent (OkO) planes. The first peak in the Vickers hardness versus load plot of
Figure 1 corresponds to slip taking place on (20T) plane and second peak corresponds
10 slip on (100) plane. Thus, on indenting at low loads, dislocations of the type (201)
101 Oj arc generated whereas at higher loads, the (100) fOlO] type of dislocations are
activated.
The plot of Vickers hardness variation with load for carba/.olc doped anthracene
ol Figure 3 show.s that the peak positions have been interchanged as compared to
Figure I. Also they occur at higher loads and the hardness values arc 5.4 kg/mm^ and
7 0 kg/ mnV'^ compare well with the observations on pure anthracene crystals but
interchanged.
The carbazole molecule is very similar to the anthracene molecule and goes in
subsiiiutionally in the lattice of anthracene. This reverses the plot of variation of
hardness with load in the carbazole doped anthracene. The first peak appears at 55 g load
with a hardness value of 5.4 kg/mm^ which compares well with the hardness value of
4 95 kg/mm- at 67 g load in pure anthracene. Both (1(K)) and (201 ) slip planes are facile
hut the (201) plane is more facile since it contains the molecular axis and is the
second most closely packed plane. Slip in the (201) occurs more readily along the [010]
direction. Carbazole dilates the lattice causing the (100) slip plane to be more facile than
the (20T) slip plane. Thus, the movement of dislocations of the type (201) [010] are
rcsinclcd due to the carbazole impurity while movement of dislocations of the (100) [010]
type arc facilitated. The nearly same hardness values at peak positions support these
observations.
The Knpop hardness behaviour of pure and phenanthrene doped anthracene as
shown by Figures 2 and 4 is very similar except that the peaks in the doped crystals appear
ill low loads in comparison to pure anthracene. The hardness values at the peaks are nearly
the same within experimental error. The splitting of the (201) [010] dislocations into
pariials has been suggested in the case of pure anthracene. Similar behaviour is suggested in
ca.se of doped crystals, as phenanthrene substitutes for anthracene in the lattice the
compounds have the same chemical formula except for a change in the shape of the
molecule. Thus, it is concluded that doping pure anthracene crystals with phenanthrene
dilates the lattice causing deformation to take place at much lower loads.
72A(4)-7
300
Nimisha Vaidya. J H Yagnik and B S Shah
4. Conclusions
(1) Anthracene and carbazole doped anthracene belong to the monoclinic system,
having space group P2\/a with two molecules per unit cell. They deform in a similar
manner as seen by the variation of hardness versus load plots of Figures 1 and 3
when indented on (001 ) cleavage surfacc.
(ii) Carbazole is known to go in substitutionally in anthracene causing the lattice to
relax, reversing the behaviour of hardness versus load.
(iii) The movement of dislocations of the type (201) [010] are restricted due to the
carbazole impurity while movement of dislocations of the (100) |010] type arc
facilitated. The hardness values at peak positions support these observations as well
as the peaks appearing at higher loads.
(iv) Knoop hardness studies on pure anthracene show two peaks in hardness versus load
plot. These are due to the (201) [OlO] dislocations splitting into partials.
(v) The Knoop hardness behaviour of pure and phenanthrene doped anthracene is very
similar except that the peaks in the doped crystals appear at low loads as compared
to pure anthracene. The hardness values at the peaks are nearly the same within
experimental error, thus dilating the anthracene lattice.
(vi) Splitting of (201 ) [010] dislocations into partials is also suggested in case of doped
crystals as phenanthrene substitutes for anthracene in the lattice.
References
[11 L V Azarofr and J J Brophy Electronic Processes m Materials (New York McGraw Hill) ( 1 963 ) *
[2] A H Cottrell Effect oj Solute Atoms on the Behaviour of Dislocations, Report of Conferetu e on Strength
of Solids f London Phy.sical Society) ( 1 948)
[31 J S Koehler and F Seitz J Appl. Mech. 14 217 (1947)
[4] J J Gilman Proft Ceramic Sci. ed J E Burke Vol. I (New York : Pergamon) (1961)
[5] J Wecriman and J R Weertman Elementary Dislocation Theory (London : Macmillan) (1964)
[61 B W Mott Micro-indentation Hardness Testinft (London : Butterworths) (1956)
[71 J N Sherwood Fractional Crystallization Vol. 2 cd. M Zief (New York . Dekker) (1969)
[81 R K Marwaha and B S Shah Cryst. Res, Technoi 26 491 (1991)
[91 N Vaidya, M J Joshi, B S Shah and D R Joshi Bull Matter. Sci. 20 333 (1997)
(101 R W G Wyckoff Crystal Structures (New York . Interscience) (1951)
(II] M Kuruhashi, M Fukuyo, A Shimada, A Funisaki and I Nitta Bull. Chem. Soc. Jpn 42B2J74 ( l%9)
Indian J- Phys. 72A(4), 301-306 (1998)
UP A
an intcmationaT jounul
Study of bismuth substitution in cobalt ferrite
Urmi M Joshi, Kapil Bhaii and H N Pandya
Department of Electronics, Suurashtra University,
Rajkol-360 005, India
Rci eived 1 S Dec ember 1 997, accepted 28 April 1 998
Abstract : The bismuth substituted cobalt ferrite, that is, CoBi 2 J:Fe 2 _ 2^04 has been
prepared by ceramic method. The single phase has been confirmed by X-ray diffractograms The
electrical behaviour of the system is studied by the mcasuremenis of clectncal resistivity and
dielectric constant The magnetic behaviour is studied through low field ac susceptibility. The
electrical and magnetic behaviours arc explained on the basis of single domain (SD) to
superparamagnctic (SP) transition
Keywords : Bismuth substituted cobalt, resistivity, susceptibility
PACS No. : 75.50 Gg
1, [nfrnduction
Tlic pure cobalt ferrite has been well studied by many researchers [1-3]. The substituted
(\)Fc 204 has also been studied by various researchers [4-6], Some people have also studied
the mixed cobalt ferrite [7,8]. As far our knowledge goes, there have been no studies on the
bismuth substituted cobalt ferrite. In this paper, we represent the effect of bismuth
substitution in place of Fe in cobalt ferrite. The system CoBi2vFe2-2i04 has been
characterized by X-ray diffraction and found single phase. The resistivity and dielectric
constants have been measured as a function of temperature. The resistivity against
temperature curves exhibit prominent rise after the critical temperatures of the samples.
This is attributed to superparamagnetic (SP) to paramagnetic transition. The susceptibility
measurements indicate the presence of single domain particles.
2. Experimental techniques
All the samples of the system CoBi2jfFe2_2r04 with x = 0.0 to jt = 0.1 in steps of 0.025
were prepared using the standard ceramic method. The stoichiometric proportion of CoO,
Bi20^ and Fe20j of high purity were thoroughly mixed, pelletized and sintered at 950°C
© 1998 lACS
302
l/rmi M Joshi, Kapil Bhatt and H N Pandya
for 12 hours. These samples were reground and refired at 950°C for 12 hours. The X-ray
diffractograms were obtained with the help of Philips (PM 9220) diffractometer using
FeKrt radiation.
The resistivity arid dielectric constant measurements were performed on all the
samples of thickness 4 mm and diameter 10 mm. The Aplab made microprocessor based
LCR bridge was used for above measurements. Before measurements, the faces of the
pellets were carefully polished and rubbed with graphite. The low field ac susceptibility of
powdered samples for all the samples were measured using double coil apparatus (9], from
room temperature to 800 K.
3. Results and discussion
The X-ray dilTraclograms exhibited a well-defined pattern of lines. When these lines were
indexed they indicated a single phase. This is shown in Figure 1. The extra lines for the
Figure 1 . X-ray diffractograms of the Figure 2 . Lattice parameter o vrrmv
system CoBi2^^Fe2-ir04. concentration x.
sample x = 0.100 which is shown with arrow in Figure 1 indicates that for x > 0.100 the
cobalt ferrite does not accommodate bismuth in its cubic phase. The variation of lattice
Study of bismuth substitution in cobalt ferrite
303
constants against concentration is shown in Figure 2. From Figure 2, it is clear that
substitution of bismuth ions slightly but steadily increases the lattice constant. Of course,
this rise is not noteworthy because of the very small concentration of bismuth. The steady
rise can be attributed to the replacement of Fe ions of smaller radius by bismuth ions of
larger radius.
The low field ac susceptibility for all the samples as a function of temperature is
shown in Figure 3, The sample .y = 0.000, that is, cobalt ferrite exhibits a constant rise upto
■ * * • aAl < .
Tmup-IK) r- -- r - ;
Figure 3. Low field ac susceptibility versus Figure 4, Log p versus temperature for the
temperature for x = 0.000. 0.025, 0.050, 0.075 system CoBi2jrFe2-2r04-
and 0.100.
a peak and then decreases. Such behaviour is also observed by Baldhaer al [7], The
constant rise in susceptibility for x = 0.000 is indicative of single domain (SD) particles.
As the temperature increases, single domain particles become superparamagnetic particles
having increased susceptibility. Thus, with rise in temperature, more and more SD particles
304
Urmi M Joshi, Kapil Bhatt and H N Pandya
become SP particles. Before T,, at blocking temperature Tf,, all the SD particles become SP
particles. This is kn(»wn as SD to SP transition. These transition temperatures T/, for all
samples are shown in Table I . The absence of anisotropy peak for x = 0.000 suggests the
absence of multidomain (MD) particles. As the concentration increases, the steady rise in
susceptibility decreases. This shows that addition of bismuth in cobalt ferrite forms a
mixture of SD and MD particles. The critical temperature T^. also decreases as the
concentration increases. This can be attributed to decrease in Fe ions with the bismuth
addition. The transition temperatures 7\ are also shown in Table 1 .
Table 1. Ternperature.s al dip and peak of re.sistivity, blocking temperalures
and critical temperatures for the system of CoBi 2 jrFc 2 _ 2 ,^ 04 .
Concentration
X
Second dip
in
resistivity r^.(K)
Blocking
tempemture
Cntical
temperature
7'c(K)
0000
-
713
803
0 02.*^
-
703
788
0.050
773
693
773
0 075
758
683
758
0.100
743
673
743
The resistivity for all the samples as function of temperature is shown in Figure A
The samples x = 0.000 and x = 0.025 exhibit normal behaviour but the samples x = 0.050,
X = 0.075 and x = 0.100 show two discontinutes at higher temperatures. This suggests that
substitution of bismuth upto certain amount into cobalt ferrite makes the resistivity sensitive
to some kind of transition.
The comparison of these discontinuity temperatures (.sec Table 1) with the blocking
temperature 7/, and the transition temperature 7, of susceptibility suggests that
first discontinuity occurring at lower temperature may be due to SD-SP transition while
the second discontinuity may be due to superparamagnetic (SP) to paramagnetic
transition. The general nature of resistivity is a decrease in resistivity as temperature
increases.
Figure 5 shows the dielectric constant as a function of temperature for all the
samples. The dielectric con.stant for all the samples initially increases quite negligibly; but
after certain higher temperature, it shows prominent rise. It also reveals that dielectric
constant does not get affected due to the SD-SP transition. The critical temperatures of
respective samples are indicated in Figure 5. This suggests that superparamegnctic (SP)
to paramagnetic transition causes a remarkable rise in dielectric constant. It is also
Study of bismuth substitution in cobalt ferrite
305
interesting to note that dielectric constant versus temperature has almost inverse
behaviour compared to resistivity versus temperature behaviour. From this relation, rise in
■ « 0.000
m
m
T, - N3 K •
m
Figure 5. Dielectric constant E
versus temperature for the system
CoBi2;jFc2_2j04
Liu’lcclric constant can also be attributed to rapid decrease in resistivity after critical
temperature.
4. Conclusion
The present study of the system CaBi 2 xFe 2 _ 2 i 04 shows that bismuth can be added only
upto 10% in C 0 FC 2 O 4 . The resistivity becomes sensitive to magnetic transition due to
addition of certain amount of bismuth as the temperature is varied. Dielectric constant is
affected by rapid decrease in resistivity and superparamagnetic (SP) to paramagnetic
transition.
Acknowledgments
The authors are thankful to RSIC, Nagpur for providing XRD facilities, Urmi Joshi is
also thankful to Government of Gujarat for providing financial help in the form of
scholarship.
306
Umi M Joshi, Kapil Bhatt and H N Pandya
References
1 1 ] G H Jonker J. Phys, Chem. Solids 9 165 ( 1959)
[2] G A Sawatzky , F Van der woude and A H Monish J. Appi Phys. 39 1 204 ( 1 968)
[3] G D Reik and J J M Thijsscn Acta Cryst. B24 982 (1968)
[4] J G Na, T D Lee and S J Pask IEEE Trans. Magn. 28 ( 5 ) 2433 ( 1 992)
[51 C M Yagnik and H B Mathur Indian / Pure Appl Phys. 6 21 1 (1968)
[6] B S Trivcdi and R G Kulkami J. Mater. Sci UtL 12 1401 (1993)
[7] G J Baldha, R V Upadhyay and R G Kulkarni Mater. Res. Bull. 21 1051 (1986)
[8] R Satyanarayana and S Ramana Murthy J. Mater. Set. Lett. 4 606 (1985)
[9] C Radhaknshnamurthy, S D Likhitc and P W Sahastrabudhe Proc. Indian Acad. Sci. 87 A 245 ( 1 978)
Indian J. PM- 72A(4), 307-312 (1998)
UP A
— art intematiortal jcwimal
Defect characterization of Sr2+ doped calcium
tartrate tetrahydrate crystals
K Suryanarayana and S M Dharmaprakash
Department of Physics, Mangalore University,
Mangalagangotri-574 199, Karnataka, India
Received 12 February I99H, accepted 2 April 1998
Abstract : The defect content of gel grown Sr^'*' doped calcium tartrate tetrahydrate
single crystals (CST) with molecular formula Cao.g 8 Sro. 12 C 4 H 1 O 6 . 4 H 2 O, was estimated by
dislocation etching. The study revealed the exisiance of dislocation network in the body of the
crystal. CST crystal has only one easy cleavage plane (1 10). The kinetics of etching is studied.
From Arrhenius plots, the activation energy of etching and the pre-exponential factors are
computed, An empirical relation governing the kinetics has been suggested.
Keywords : Defect characterization, etching, calcium tartrate tetrahydrate crystal
PACS Nos. : 6 1 .72.Ff, 8 1 .65.Cf
1. Introduction
An etching technique, in association with optical microscopy, can be used alternatively to
X-ray methods for the detection as well as quantitative and qualitative analysis of defects in
crystalline solids 1 1-5]. The segregation of foreign solute particles during crystal growth
leads to the introduction of defects into the crystal [6,7]. In order to study the effect of
doping of Sr^^ on defect characteristics and to compare the dislocations in pure and doped
crystals, CST single crystals were grown in gels (8). The characterir^ation of CST single
crystals by selective etching and a study of the kinetics of etching arc reported here for the
first lime.
/
2. Experimental
CST single crystals, to be employed for etching studies, were carefully picked up from the
silica gel to avoid any damage during mechanical handling. The crystal morphology was
generally a rhombic octahedron (Figure 1), elongated in the c direction and made up of
72A(4).8
© 1998 1 ACS
308
K Suryanarayana and S M Dharmaprakash
principal faces (110), (010), (Oil) and their symmetry equivalants. The crystals were
cleaved by light pressing with a blade parallel to (1 10) plane which proved to be the only
possible cleavage. A number of analaR grade chemical reagents were examined for possible
use as dislocation etchants. HCl and HNO3 were found suitable etchants for CSt crystals.
Figure 1. Morphology of doped calcium tartrate tetrahydrate single crystal.
Etch pit size was determined by talcing an average of measurements on a number of etch
pits at a constant magnification using a filar micrometer eye piece fitted to the optical
microscope (Leitz-Wetzlar 307-002). In order to ascertain the scope of the etchants used
here in delineating the linear defects existing in the body of the crystal, microscopic
examinations were made of the etched mirrer cleavages and the successively etched faces.
In order to test whether the etch pits are produced at the emergent sites of dislocations,
successive etching was tried with each of the etchants. The etch pattern obtained on the
complementary faces of CST crystal showed one to one corrcspondance of the etch pits on
the two match halves. This indicates that the pits observed are formed at the sites of linear
defects, terminal ends of which lie on both of the match surfaces. The successive etching
resulted in pit widening and deepening for all etchants, thus establishing the reliability of
etchants. Crystals were etched at different temperatures between room temperature anO
50°C. Etch rates for different composition of the etchants was calculated from a number of
measurements of the pit size.
3. Results and discussion
Figures 2 and 3 depict typical etch patterns produced on the habit faces of CST single
crystals by HCl and HNO3 respectively after etching for 10 .secs. It can be seen from these
figures that the etch pit morphology is independent of the nature of etchant used. Some
shallow pits on the etched planes have been observed. Micropits are also found, which
indicate the general dissolution of the surface, because point defects are too sensitive to
etching. Such shallow and micropits formed during etching need not necessarily be related
to the sites of dislocation intersection with the surface. Point defect clusters, impurity
inclusions, surface damage, foreign particles on the surface and other often nontraccable
factors may also lead to the formation of pits on the habit faces. Some of the etch pits on the
surface are not of the same size and depth. The time lag in the formation of pits is
responsible for the non-uniform size of etch pits. When the etchant attacks the dislocation
sites, the pits thus formed will follow the dislocation lines into the body of the crystal. If
the dislocation lines are perpendicular to the face, symmetric pits will be produced [9].
On the other hand, for inclined dislocation lines, asymmetric pits will result. When a series
Defect characterization of St^'^ doped calcium tartrate etc
Figure 2. Etch pattern produced by HNO^ ( 10 sec)
Figure 3. Etch pattern produced by HCI (10 sec)
Defect characterimion ofSt^* doped calcium tartrate etc
309
of dislocations lying in the same slip plane meet a barrier such as a grain boundary, the
dislocation pile-up takes place. The row of etch pits shown in the Figure 3 represents such
an example of pile-up. The morphology and orientation of etch patterns are identical and
mutually inverse. So the etch pattern symmetry on all faces is 1 m [4,10]. This accounts for
the centro symmetric characteristics of CST crystal.
From the distribution of etch pits on the etched surfaces, it is observed that the
dislocation density in CST single crystal is greater than the dislocation density in calcium
tartrate tetrahydrate single crystals. The values of the estimated dislocation densities are of
the order of 9 x 10^ enr^ in CST whereas in calcium tartrate tetrahydrate crystals, they are
6 X 10^ cm”^. The etching experiments revealed that around some foreign particles
incorporated during growth of the crystals, are associated a large number of dislocations.
The presence of such foreign particles may be the chief source of dislocation centres in
doped crystals.
The successive etching experiments reveal that the depth and lateral size of pits
increase with etching time. For quantitative analysis, the pit widths were measured at
different intervals of time. The growth of pits was linearly related with time, revealing
greater etch rate with greater etch concentration, suggesting the consistancy of the rate of
etching.
It was observed that the conccr\tration and temperature of the etchant have
considerable influence on the etch rates. Tuck [11] suggested that the factors contro[ling the
etching rate can be conveniently divided into two main groups : (a) those for which the rate
limiting process is some aspect of chemical reaction and (b) those for which diffusion of
atoms to or from the surface controls the rate. Whether the etching process is chemically
controlled, can be ascertained reliably by determining etch rates as a function of
temperature.
Figure 4. Plot of In^ againRt temperature for HCI.
As a rule, the dissolution process controlled by reaction rate requires an activation
energy in the range I to 3 eV [1 1], while the activation energy of dissolution is limited by
310
K Suryanarayana and S M Dharmaprakash
diffusion change in the interval of 0. 1-0.5 eV [12]. Figures 4 and 5 sho>v Arrehenius plots
of etch rates at different temperatures in the interval of 30 to 50X for different
concentrations of the etchants used. From these plots, values of activation energy and pre-
exponential factors were determined and are presented in Table 1 . [ntereslingly the values
Figure 5. Plot of \nR against temperature for HNOv
of activation energy are independent of acid concentration and lie within the limits of the
reactions in which the diffusion process is predominant.
Table 1. Activation energy E (eV) and pre-exponentiol factors A calculated
from Anchnius plots.
Dislocation
etchants
Etchant
concentration
Activation
energy (cV)
Arrehnius pre-
exponential factor
HCl
0.2 N
0.291
8 65 X 10^
0.4 N
0.289
10.30 X 10^
0.6 N
0.296
1 1.38 X 10^
0.8 N
0.293
12.33 X 10^
1.0 N
0.288
12.97 X 10^
HNO 3
0.2 N
0.369
12.56x10'*
0.4 N
0.366
13.59x10^
06N
0.362
I4.I6X lO'*
0.8 N
0.365
15.78x10^
I.ON
0.360
16.79 X 10^
The acid etchants HCl and NHO 3 react with CST, yielding tartaric acid and
calcium strontium nitrate/chloride. Here, both the reaction products are water soluble. This
reaction is of special interest because the exact reversal of this reaction was employed for
the crystal growth of CST described elsewhere. Hence the etching process is reaction-rate
controlled. No change in morphology and orientation of the pits is observed due to change
Defect characterization ofSi^* doped calcium tartrate etc
311
Figure 6. Plot of \nA against InC for HNO3.
Figure 7. Plot of \nA against InC for HCI.
in temperature. Figures 6 and 7 is the graph of InA against InC, from which A can be
expressed by- the empirical relations :
A= 12.3X10^
and A = 8.6 X 10^
for HNO3 and HCI respectively. This enables us to represent the dissolution of CST crystals
by writing the Arrhenius equation in the form
/? = 12.3 X 10^ exp i-E/kT) for HNO3
/? = 8.6 X 10^ C° exp (-EAD for HCI
where C is the etch concentration.
312
K Suryanarayana and S M Dharmaprakash
4. Conclusions
CST crystal has only one easy cleavage plane (110). The etch pattern obtained on the
complementary faces of CST crystal showed one to one correspondence of the etch pits on
the two match halves. The successive etching resulted in pit widening and deepening, thus
establishing the reliability of HCl and HN 03 as suitable etchants for CST. The morphology
and orientation of etch patterns on opposite surfaces of CST are identical and mutually
inverse thus establishing the point group of CST as nonpolar 222. No change in
morphology and orientation of the etch pits is observed due to change in temperature. The
mechanism of etching of CST in etchants HCl and HNO 3 is reaction-rate controlled.
References
r I ] J J Gilman and W G Johnston J. Appl. Phys. 27 1018 (1956)
f2] V Venkataramonan, G Dhanamj. V K Wadhawan. J N Sherwood and H L Bhat / Crystal Growth 154
92(1995)
[3] F J Rcthinam, D Arivuoli, S Ramasamy and P Ramasamy Mater. Res. Bull. 29 309 (1994)
[4] N Nokatoni Japanese J. Appl. Phys. L1961 30 (19^1)
[51 I Owczarek and K Sangwal J. Mater Sci. Lett. 9 440 (1990)
[6J V B Paritskii, S V Lubenets and V I Startsev Sov. Phys. Solid State 8 976 (1966)
[7] J J Gilman. W G Johnston and G W Sears J. Appl. Phys. 29 747 ( 1 958)
[8] K Suryanarayana and S M Dharmaprakash Cryst. Res. d Tech. 3l K16 (1996)
[9] A R Patel Physica 27 1 097 ( 1 96 1 )
[10] International Table for X-ray Crystallography eds. N F M Henry and K Lousdale (Birmingham
The Ky noch Press) Vol 1 Chap 3 ( 1 969)
[111 B Tuck J. Mater. Sci 10 32 1 ( 1 975)
[12] K Sangwal and S K Arora J. Mater. Sa. 13 1977 (1978)
[ 1 3] Kratkaya Khimicheskaya Entsiklopediya Soviet Entsiklopediya (Moscow) V 5 (1967)
Indian J. Phys. 72A (4), 313-321 (1998)
UP A
- an intematiopal journal
Dynamical short range pion correlation in ultras
relativistic heavy-ion interaction
Dipak Ghosh, Argha Deb, Md Azizar Rahman, Abdul Kayum Jafry,
Rini Chattopadhyay, Sunil Das, Jayita Ghosh, Biswanath Biswas,
Krishnadas Purkait and Madhumita Lahiri
High Energy Physics Division, Depaitment of Physics, Jadavpur University,
Calcutta-700 032, India
Received 13 January ,1998. accepted 7 April 1998
Abstract : The paper presents new data on two- and three-particle pseudo-rapidity
correlation ^iinong showers produced in 0*^-AgBr and ^^S-AgBr interactions at 60A GeV and
200A GeV respectively. The data have been compared with Monte-CarTo simulated values to
look for true dynamical correlation in each case
Keywords : High energy physics, heavy-ion interaction, pion correlation
‘ PACS No. ; 25.70.Pq
1. Introduction
Studies in nuclear matter under extremes of energy and density are gaining momentum
because of the possibility of observing some exotic phenomena. The study of correlation
among the particles produced provides significant features of the nuclear interactions and is
a potential source of information. The correlations can give direct information about the late
stage of the reaction when nuclear matter is highly excited and diffused [1]. Several studies
using well-known two-particle and three-particle correlation functions have been reported
in hadron-hadron [2] and hadron-nucleus [3] collisions. The particles produced in different
types of interaction^ (like hadron-hadron, hadron- nucleus) at high energies seem to be
emitted preferably in a correlated fashion. But it is not possible to say with certainty why
they prefer to do so. While some think that the larger part of the observed correlation
effects, is conditioned by the production of the well-known resonances, hot multi-nucleon
Hreballs or formation of the exotic state of nuclear matter, the quark-gluon plasma, others
observe the experimental data to favour formation of heavier intermediate states,
© 1998 LAGS
314
Dipak Ghosh et al
clusterisation, etc. Moreover, the much-debated intermittency effect is also believed to be a
manifestation of short-range eorrelations [Bose-Einstein correlations, the Hanbury-Brown
Twiss (HBT) effect or the Goldhaber (GGL) effect for identical particles] [4], In this
context therefore, interest in the study of correlation is increasing rapidly. For a better
understanding of correlation effect, it is necessary to investigate data of different projectiles
covering the whole available energy spectrum. However, such studies in nucleus-nucleus
interactions at high energies using different projectiles are rare. We present here some new
data of ^^S-AgBr interaction at 200A GeV and '^O-AgBr interaction at 60A GeV using
some standard techniques to seek for true correlation of non-statistical origin. The
experimental data of two- and three-particle correlation have been compared with the
Monte-Carlo simulated values for the purpose.
2. Experimentation
Stacks of G5 nuclear emulsion plates horizontally exposed to a beam, having an average
beam energy of 200 GeV per nucleon and an beam, having an average beam energy of
60 GeV per nucleon at CERN SPS have been used in this work. Leitz metalloplan
microscopes provided with semi-automatic scanning stage are used to scan the plates, the
scanning being performed by using oil immersion objectives of magnification lOr and 25jr
ocular lenses. The scanning is done by independent observers to increase the scanning
efficiency which turns out to be 98%. The following criteria are adopted to select the
events :
(a) The beam track must not exceed an angle of 3° to the mean beam direction in the
pellicle. .
(b) All the events having interactions within 20 fim from the top or bottom surface of
the pellicle are rejected.
(c) The incident beam tracks are followed in the backward direction to ensure that
events selected do not include interactions from the secondary tracks of other
interactions; the latter events are removed from the sample.
The present analysis is based on the selected 150 primary events of ^^S-AgBr
interactions and 250 primary events of ‘^O-AgBr interactions. All the tracks of the charged
secondaries in these events are classified according to standard emulsion terminology in the
following way :
(i) The target fragments with ionisation > 1.4/o (/q is the plateau ionisation) produce
either black or grey tracks. The black tracks with range < 3 mm represent target
evaporation particles of J9< 0.3, singly or multiply charged particles.
(ii) The grey tracks with a range ^ 3 mm and having velocity 0.7 ^ ^ S 0.3 are mainly
images of fast target protons of the energy range up to 400 MeV.
(iii) The relativistic shower tracks with ionisation < 1.4/o are mainly produced by pions
and are not generally confined within the emulsion pellicle. I^se particles are
believed to carry important information about the nuclear reaction dynamics.
Dynamical short range pion correlation etc
315
(iv) The projectile fragments formed a different class of tracks with constant ionisation,
very long range and small emission angle.
To ensure the target in the emulsion to be Ag/Br, only those events are chosen in
which number of heavily ionizing tracks are greater than eight. The heavily ionizing
particles constituted of types (i) and (ii) belong to the target nucleus, those of type (iv)
belong to the projectile nucleus, and the particles of type (iii) are those produced in the final
state of the interaction. To distinguish between the singly charged produced particles and a
projectile fragment of the same charge, we excluded all the particles falling into the cone of
semi-vertical angle 0^ [5] (0^ = O.TJp^y^, p^^ (GeV/c) is the incident beam momentum per
nucleon) with respect to the projectile direction, from the present analysis.
The spatial angle of emission in the laboratory frame, of all the product particles,
is measured by taking the space coordinates (jc,y,z) of a point on the track, another point on
the incident beam, and of the production point. For measurement, we have used oil
immersion objectives of magnification lOQx and 25x ocular lenses.
3. Method of analysis
i. /. Two-particle correlation :
Generally, the two-particle correlation function is defined as
= o^tid*‘aldy^dyid'^p,^d'^p,2) - {ar^)^
X (d^a/ dy,d^ p„) (d^al dy2d~ p,2), (1)
where v and p, denote the rapidity and transverse momentum of the particles respectively, A
Ks the target mass number and s is the square of the centre-of-mass energy, the subscripts 1
and 2 denote the panicles in the pair considered.
Integrating eq. (1) over/?,,
C2(3',.>2.*.A) = or^(.d^aldy^dy2) - {da I dy^) (da I dy2\ ... (2)
where CT,-' J(d2cr/</y,<<y2 >6'i<^y2 =
trr' I (dal dy)dy = < n, >,
jc2dyydy2 = fr.
h being the multiplicity moment defined as
fi = <n,{n, -l)> - <n, >2.
Now, the two-particle correlation function can be written as
^2 (>1 .>' 2 ) = P 2 (>^1 * 3 ^ 2 ) ” Pi (>'1 )Pi (> 2 ). ( 5 )
where P 2 {yx,y 2 )- ! dy^dy^ andp, (y) = err'da/dy are respectively the two-
and one-particle densities, Oj,, is the total inelastic cross section and d^G/dyxdyi mddo/dy
72A(4).9
316
Dipak Ghosh et al
arc the two and one-particle semi-inclusive distributions respectively. The normalized two-
particle correlation function can be written as
^2(3'i.y2) = p\(y\)p\(yi)‘ (4)
We choose pseudo-rapidity (77) as an approximated variable where 77 s - In tan since
the shower particles are primarily relativistic pions with for most of the pions.
Thus, 77 is closely equal to the rapidity
y=l/21n [(£+?/)/(£ -Pi)].
Here p/ is the longitudinal momentum.
The two-particle coirelation function [6] thus becomes,
CjCtJi.iJj) = a-^{d'^aldr\^dr]i) - (a;^)^(da/ dTjtXda/ dtj^)
= Af2('?i.JJ2)/N-W,(r/,)A',(»J2)//V2, (5)
where N,(77j) is the number of showers with pseudo-rapidity between 77 and rj + dr] ;
^2 (^1 > ^2 ) number of pairs of shower particles with pseudo-rapidity between rji,
7 ]i -I- drji and 772, 772 + drj2. N is the total number of inelastic interactions in the sample. The
normalised two-particle correlation function can be written as
^2(ni.»]2) = a,^(d‘^aldr]\dT\.i)l(daldr]\){daldn.2^) - 1
= N[A/j(f;,,n2)//V,()7,)Wj(J72)] - 1. (6)
The correlation function Rj (77i, 772) is related to the density of emitting sources [7].
3.2. Three-particle correlation : ,
The three-particle correlation function is also defined in a similar way :
= P3(2 i»Z 2*23) + 2p, (Z, )p, (Zj )Pl (^3 )
' P 2 .22 )Pl (Z3 ) - P2 (22 .23 )Pl (2, ) - P2 (23 ,2, )Pi (Z2 ). (7)
while the normalized three-particle correlation function is [8]
^3(21.22.23) = C3(z,,Z2,23)/pi(Zi)P,(Z2)Pi(23). (8)
where the quantities, Pi = \ / CT^J^{da / dz\p2{Z] ,12) = \ I (J^„(d'^a f dzidz2\
p2(Z\ ,Z2^Zj) = 0 / dz^dz2dz2) represent one-, two- and three-
particle densities respectively. For relativistic shower particles, we may take 77 as a variable
i.e., z = 77 ; thus equation (8) becomes
/f3('7|.'72.'?3) = [<Tr'(rf’(T/j7J,rf7J2</n3)
+ 2((Tr„' )Hdaldii,)(da/dTi2'KdoldJii)
- (crr;)Hd^a/dT],dTi2Xda/dT}2)
-(a;^)Hd^aldri2dri2)idcrldTtO -
(d‘^aldr]2di]^ )(daldr]2 )]/ [(ffi,' )’ {daldr]^ )(daldr\2 Xdo/dTjj )]
Dynamical short range pion correlation etc
317
-N/V2(Tj,,nj)/[/V,(r,,)iV,(j7j)l
-yvjV2(n2.»],)/[A^,(i?2)/v,(!7,)]
-yVN2(T?j.r?,)/[W|(/73)/V,(f7|)l + 2. (9)
where Nj(r}\,rf 2 ,T]‘^)\$iht number of triplets of shower particles at r/i , r 72 and 77 ^ .
4. Monte-Carlo simulation
Correlation between the secondary particles produced in high-energy heavy-ion collisions
can be studied by observing pseudo-rapidity (rj) correlation among them. This may arise
Jue to (i) the broad multiplicity distribution, (ii) the dependence of the one-particle
spectrum, do/dJ), on the multiplicity n, and (iii) the non-trivial correlations which occur due
lo kincmatical constraints in the individuals events. To search for the correlation among the
secondary particles in ^^S-AgBr interaction and '^0-AgBr interaction, we have compared
the experimental data with those obtained from the Monte-Carlo method. The simulation is
made using the following assumptions :
0 ) The shower particles are emitted statistically independently;
(ii) The multiplicity distribution in the ensemble of the Monte-Carlo events is the same
as the empirical multiplicity spectrum of the real ensemble;
(in) The one-particle spectrum da/dr}, in the simulated interactions reproduces the
empirical "semi-inclusive" distribution do/dt], with the corresponding for the real
ensemble.
This method has been successfully applied for hadron-nucleus and nucleus-nucleus
inicractions [9, 10, 11]. Gulamov et al [12] compared correlation function calculated from
the inclusive ensembles of random events generated according lo the method adopted here.
The observation of any excess short-range correlation over the Montc-Carlo values
will indicate the presence of dynamical effects which cannot be explained by the
conservation laws. For both two- and three-particle correlations we have compared the
experimental values with the values obtained from Monte-Carlo calculations. The
difference between experimental values R and Monte-Carlo values can be interpreted as
the dynamical surplus Rj which arises due to some kinematics in the reaction process. The
dynamical surplus can be written as
The surplus Rj can be interpreted as a manifestation of dynamical correlation.
Results and discussion
The normalised two-particle correlation function R 2 W 1 . ^2 - ^ 1 ) * ^^e diagonal elements
of the correlation matrix characterising the magnitude of short-range correlation at different
pseudorapidity values for and events are shown in Figures 1(a) and 1(b)
318
Dipak Ghosh et al
respectively. The solid lines in the figures represent values of correlation function due to
Monte-Carlo calculations. Figures 2(a) and 2(b) give the dynamical surplus values in
each ca.sc. The errors shown are only statistical [ 1 3] (the details are given in the appendix).
Figure 1. The normalised two-porticlfc correlation function for different values
of T ) ; (a) for '^O events and (b) for events. The solid curves represenl ihe
Monte-Carlo simulated value.s
16 0
^80
1^
5 00
e
K
-ISO
^ ft I
10 ’ 6 0
n
(a) (b)
Figure 2. The dynamical surplus correlation over the Monte-Carlo background .
(a) for events and (b) for events.
Figures 3(a) and 3(b) represent the variation of normalised three-panicle correlation
function (rji, r)2 = t?i, 7?.^ = 7]|) < the diagonal elements of three-particle correlation
matrix also characterising the indication of short-range correlation of pseudorapidities for
and events. The solid curves show the Monte-Carlo simulated values. The
corresponding dynamical surplus for the three-particle correlation functions are shown in
Figures 4(a) and 4(b) respectively.
One may draw the following inferences from the above analysis :
( 1 ) The two-particle short-range dynamical correlation exists in the targetiragmentation
region t] = 1 for both *^0 and events and in the projectile fragmentation region
(7] s 4 and 5) for '®0 events and ?] = 5 for events.
(2) The three particle dynamical correlations are prominent at 7) = 1 and 5 in case of '^0
events and 7] = 1 , 2 and 5 in case of events.
Finally, one may conclude that both two- and three-particle dynamical correlations
exist among pions produced in '^O-AgBr and ^^S-AgBr interactions. It is also interesting to
Dynamical short range pion correlation etc
319
note thai in case of ^^S-AgBr interaction (heavier projectile with increased energy),
correlation occurs in additional phase space compared to ‘^0-AgBr interaction (lighter
Figure 3. The normalised three -particle correlation function for different
values of rj : (a) for events and (b) for events. The solid curves
represent the Monte-Carlo simulated values.
(a)
Figure 4. The dynamical surplus correlation over the Monte-Carlo background :
(a) for '^events and (b) for events.
projectile at low energy). The data are helpful for an understanding of the physics involved
m ihe particle production in ultra-relativistic heavy-ion interactions.
Acknowledgments
Authors would like to thank professor P L Jain, Buffalo State University, U.S.A., for
providing the exposed and developed emulsion plates. We also gratefully acknowledge the
financial help given by the University Grant Commission (Govt, of India) under their
COSIST programme.
References
[ 1 1 G Giacomelli and M Jacob Pfiys. Rep. 55 I (1979)
[2] F W Bopp Riv. Nuovo. dm. 1 1 (1978)
[3] D Gho.sh, J Roy, K Sengupto. M Ba.su, A Bhattocharya, T Cuhathokurta and S Naha Fhys. Rev. D26 2983
(1982)
[4] R Hanbury-Brown and R Q Twiss Nature 178 1046 (1956); G Goldhober er at Phys, Rev. 120 300
(I960); P L Jain. W M Labuda. Z Alimad and G Pappas Phys. Rev. 8 7 ( 1973)
320
Dipak Ghosh et al
[5] M I Adamovich et al (EMUOI) Phys. Lett. B223 262 (1989)
[6] W R Prasser. L Ingber, C H Mehta, C H Poon, D Silverman, K Stowe, P D Ting and H J Yesian Pev. Mod.
44 284(1972)
[7] C I Kopylov and M I Podgocrtsky Yad Fit. 15 392 (1972); Sov. J. Nucl. Phys. 15 219 (1972): 19 434
(1974); 19 215 (1974); G I Kopylov Phys. Utt. D50 572 (1974)
[8] E M Levin. M G Ryskin and N N Nikolaev Z Phys. C5 285 (1980)
[9] G M Chemov, K G Gulamov. U Gulyamov. S G Nasyrov and N Srechnikova Nucl. Phys. A280 478
(1980)
f 10] D Ghosh. J Roy and R Sengupta Nucl. Phys. A468 719 (1987)
[11] HA Gustafsson, H H Gutbrod, B Kolb, H Lohner. B Ludewight, A M Poskanzer, T Renner, H>Riedcsel,
H G Ritter, A Warwick. F Weik and H Wieman Phys. Rev. Lett. 53 544 (1984); D Ghosh, J Roy and R
Sengupta Z Phys. A327 233 (1987)
[12] KG Gulamov, S A Azimov, A 1 Bondarenko. V I Petrov, R V Buzimatov and N S Scripnik Z Phys.
A280 107 (1977)
[13] W Bell. K Braune, G Claesson. D Drijand, M A Faessler, H G Fischer, H Frehse, R W Frey, S Garpman,
W Geisi. C Gnihn, P Hanke, M Heiden, W Herr. P G Innocenti, T J Ketel, E E Kluge, 1 Lund, G
Momacchi, T Nakada, I Otterlund, B Povh, A Putzer, B Rensch, E Stenlund, T J M Symons, R Szwed, O
Ullaland and M Wunsch Z Phys C22 109 (1984)
Appendix
The calculation of errors :
Experimentally, the two-particle correlation function is calculated as
^(^ 1 .^ 2 )= < nin2 > >1 " for^i’^^2
= <n(n-l)>/<n^ >-l, = ^2-
where ;i| and are the shower multiplicities in a small interval of S„ around 7)] and T] 2 - The
variance in R is given by
- 2(nf/i2)(n,/i2)(n,)(n2)^
- 2(n,nf)(n,)^(n2}(/i,«2) + {«?)(«i«2>^("2>^
+ ("2)(«i«2)^(ni)^ + 2{n,n2)’{«,)(«2)
-(«|"2)^(''i)^(«2)^}|M«i)"(« 2>‘']'' + 0(1/Af2)_
for7I,5tfl2.
and ^[/J] = {n*){n)^ - 4(/i^)(n^}(n) +
+ 2(n^){«>^ -4(«^)^«) + 2(n^)(«>^
forij, =Th.
[>ynamkal short range pion correlation etc
321
N is the total number of inelastic events. 0(1/AP) is a polynomial which is negligible when
calculating the errors, in comparison with the other terms, Similarly, the three-particle
correlation function is experimentally obtained as
= (n(n-l)(n- 2 ))/(n)^
- 3(/i(«-l))/(n}^ + 2 , for tji = J ]2 = 7 ) 3 .
The variance of this quantity is calculated term by term and instead of giving the long
algebraic expression of the net variance, we have computed it and shown the corresponding
errors in the figures.
Indian J. Phys. 72A (4), 323-329 (1998)
UP A
an intcmalional journal
Early cosmological models with variable G and
zero-rest-mass scalar fields
Shriram and C P Singh
Department of Applied Mathematics, Institute of Technology,
Banaras Hindu University. Varunasi-221 005. India
Received 16 December 1997, accepted 26 May 1998
Abstract : Einstein's field equations for zero-curvature Robertson- Walker model of the
universe with variable gravitational ‘constant' G and zero-rcsl-mass scalar Fields are considered
in which the perfect fluid satisfies the ‘gamma-law’ equation of stale /» = (y-Up The y^mdex
describing the material content vanes continuously with cosmological time and this allows a
unified description of the early evolution of universe The solutions of the field equations arc
presented for the inflationary phase and the radiation-dominated phase. Some physical properties
of the cosmological models are also discussed
Keywords : Early universe, scalar fields, cosmological parameters
PACS No. : 98 HO Cq
1. Introduction
In general relativity, the constant of gravity G plays the role of a coupling constant between
geometry and matter in Einstein’s field equations. The value of C has to be constant since
(i-conslancy is in-built as a manifestation of the principle of equivalence. A breakdown
from the principle of equivalence, in any form, would constitute a departure from Einstein’s
general relativity. There are several extensions of Einstein’s theory of gravitation in which
C is taken to vary with cosmic time [1]. The time-dependent G follows as a natural
consequence of Dirac’s large number hypothesis [21. The implication of time-varying G
will become important only at the early stage of the evolution of the universe. It appears
natural to look at this constant as a function of time in an evolving universe. A large body
oi literature can be found on the evolving universe with mailer satisfying the equation of
siaie.p = (y- l)p, 1 < y< 2.
Israelit and Rosen [3] have obtained a singularity-free model of the evolving
universe with matter and studied the transition from the inflationary to radiation-dominated
■’2A(4)-10
© 1998 1 ACS
324
Shfiram and C P Singh
and matter-doniinated periods of the universe by using an equation of state. Recently,
Carvalho [4] has studied a homogeneous and isotropic cosmological model in which the
parameter gamma of ‘gamma-law’ equation of state p = (y- l)p, varies continuously with
cosmic time /. He studied the evolution of the universe as it goes from an inflationary phase
to a radiation-dominated phase.
In this paper, we study the evolution of universe with the zero-curvature Robertson-
Walker models in the presence of zero-rest-mass scalar fields in which the gravitational
parameter G varies with cosmic time t. Solutions are obtained for inflationary phase and
radiation-dominated phase by using the equation of state, suggested by Carvalho [4]. The
physical behaviour of the cosmological solutions are also discussed.
2. Field equations
We consider the homogeneous and isotropic Robertson -Walker line-element
” fir'2- 1
it) + -i- sin^ 6d<l>^ ’ (I)
_\-kr^ J
where R{t) is the scale factor and k is the curvature index which takes values +], 0, -1 lor
the spaces of positive, vanishing and negative curvature respectively.
The Einstein field equations for matter coupled with a zero-resl-mass scalar field are
= -8rtC(f)[7';; +S„], (2)
where g,j is the metric tensor, R,^ the Ricci-tensor, R the scalar curvature, T,j is the energy-
momentum tensor of matter field and S,^ the energy-momentum tensor for a zero-rest-majiK
scalar field given by (51
(3)
where the scalar potential V satisfies
(4)
For a perfect fluid distribution, the energy-momentum tensor Tij is of the form
+ M' = K (-^)
where p is the pressure, p, the matter energy-density and «' the four-velocity vector. A
comma and a semi-colon denotes ordinary and covariant differentiation respeciively.
In comoving coordinates system, the field equations (2) - (5), for the metric (1), lead
to the following equations
^ = -S'fCCOp - G(r)\>2 (6)
P2
3p- + =i7tG(t)p + G{t)V^.
and
(7)
Early cosmological models with variable C etc
325
An overdot denotes differentiation with respect to t. Eqs. (6) and (7) can be rewritten as
I = -^!iC(t)(p + 3p) - |c(f)V2 (g)
and RR + 2(R^ +k) = 4nG(t)(p-p)R^. ( 9 )
Eliniinaiing R from (8) and (9), we gel
^ " f ^G(r)p + iC(t)V2. (,0)
Ei], (4) gives
'ID .
(II)
Eqs. ( 8) and ( 1 0) can be written in terms of Hubble parameter H = r/r as
H + H'^ = - j;rC(n(p + 3p) - |c(f)V2 (12)
and H2 + ^ = |ffC(r)p+ iG(f)V2- (13)
In order to solve the above equations, we assume that the pressure p and energy-
densiiy p are related through the ‘gamma-law’ equation of state
p = (y-\)p, (14)
where the adiabatic parameter / varies continuously with cosmic time during the phase
iiansilion Irdm an inflationary phase to a radiation-dominated phase of the universe.
Carvalho [4] assumed the parameter / of the form
^ 4 +(al2){RIR.y
^ 3 A{R/R.)'^ + {R/R.y
(15)
\^hci c A IS constant and parameter a is related to the power of the cosmic time t during an
inllationary era and lies in the range 0 < a < 1. The function /(/?) is such that when the
scale factor R(t) is less than a certain reference value /?•, we have the inflationary phase
(7< 2/3). As the scale factor increases, /also increases to reach the value 4/3 for R » /?•
and thus wc have the radiation-dominated era.
Substituting the value ofp from (14) into (12), we get
//+H2 = _|;rC(r)(|y-ljp- |c(r)V2' (16)
Eliminating p between eqs. (13) and (16), we obtain
+(|y-l)-j^-(i)'-l)c(/)V'2 =0’
(17)
326
Shriram and C P Sinf*h
To solve eq. (17), wc rewrite it in the form
( 18 )
where a dash (') denotes differentiation with respect to R. For zero curvature Robertson-
Walkcr model (k = 0), eq. (18) lakes the form
(19)
An additional equation relating the time changes of G can he obtained by the Bianchi
identities ~ = 0 = \ which yield
P +
il'") *{<!>* 1 ’^*
(20)
3. Solution of the Held equations
Eq.d 1) has the first integral
\/ = ///?^
where / is the mlcgraiion ct)nslanl. Using eq (21) into (19), we obtain
2^ R l2 ^ I hR^
( 21 )
( 22 )
Eq. (22), involving two arbitrary functions R{t) and G(t), admits solution only if one^ol
these is specified. In most of the variable G cosmologies, G is a decreasing function of
time |6,71. The possibility of an increasing G has also been suggested by Levit |8]
Beesham 19| has discussed the possibility of the creation field with G «= P. Sislcro [10] has
prc.sented exact solutions for zero pressure Robertson- Walker cosmological models with
G R^. For mathematical convenience, wc assume the time-dependent C of the form
Git) = ni{HR^ )^ (23)
where //? being a positive constant. Using eq. (23) into (22), we obtain
"'*[(1 ■ ■£*))""*
where A= l-m is another positive constant. On integration of eq. (24), we get
C
H =
R^\A(R/R.f + (R/R*y
(24)
(25)
where C is the integration constant. U H = H* for R = R*, a relation between C and A can
be written in the form
C = H.\\ R,^.
(26)
Early cosmological models U^ith variable G etc
327
By use of equation (26) into (25), an expression for r in terms of scale factor R can be
written as
5 ^ dR- (27)
During the course of evolution, the deceleration parameter is not constant and its value for
any cosmological time can be calculated from eq. (24) to give
9 = [(3-A)/3](3y/2) + ^-l, (28)
which clearly depends upon R via y.
We solve the eq. (27) for inflationary phase and radiation-dominated phase
separately in the following sections starting with the inflationary phase.
3 J Inflationary phase :
When we consider the inflationary phase (R « /?*). the second term inside the square
bracket on right-hand side of integral (27) dominates over the first term which gives
a phase of power law inflation for 0 < a < 1. The scale factor R for [3fl + (3 -d)k] ^ 0,
is given by
/? = /?*
[3a + (3-a)A]
H.(l + A)<’-*''5f
3/|3fl + (3-«)A)
(29)
The energy-density is given by
P =
(3-A)
8«m
[3o+ (3-a)A]
-|-18/[3£J + (3^)A)
//*(l + /\)^^-^)/3r
(30)
For energy-density to be positive, we must have 0 < A < 3. The solution for pressure is
obtained by using eqs. (14) and (30) with the limiting value y= 2a/3. The Hubble
parameter (A/) and gravitational constant (G) have the expressions :
3
H =
Tf'
where
[3a + (3-a)^]
G — B f (3-A)/[3fl + (3-a)A],
[3a + (3-fl)A]
9m
-Ri
[3a+(3-a)A]^
Using (29) into (21), the scalar potential V is given by
V = /V (A-3)/[3fl + (3-o)A],
(31)
(32)
18/[3«+(3-u)A)
where
[3a + (3-a)A]/
(3-a)(A-3)
[3a-t-(3-a)A]
(33)
1-»/(3u+(3-b)A|
//,(l + A)(’-‘'/’
328
Shriram and C P Singh
Putting the limiting value 2a/3 for inflationary phase in eq. (28)» the asymptotic
value of deceleration parameter in the limit R/R*« 1 , is given by
9= [3(a-l) + (3-a)A]/3. (34)
In order to have expansion, we must have 0 < A < 3 (since, for inflationary phase, the
parameter a must lie in the range 0 ^ a < 1). We observe that the energy-density is a
decreasing function of lime. As r 0, the energy-density as well as pressure become
infinite. Therefore, the model has singularity at r = 0. We see that the gravitational
‘constant’ increases with the age of the Universe which is against to Dirac’s hypothesis [2]
that the gravitational ‘constant’ should decrease with time in the expanding universe. The
scalar potential decreases as time passes.
Using eqs. (30-33), we find that the eq. (20) is identically satisfied.
3.2. Radiation-dominated phase :
When we consider the radiation-dominated phase {R » /?*), the first term inside the square
bracket on right-hand side of the integral (27) dominates over the second term. Therefore,
the solution for scale factor R is given by
R =
(6+ A)
V(64-A)
(35)
The energy-density is given by
P =
(6 + A)
«•(¥)
o-X)/y
(36)
For energy-density to be positive, we must have A< 3. The solution for pressure is obtained
by using eqs. (14) and (36) with the limiting value 7 = 4/3. The solution for Hubble
parameter and gravitational ‘constant’ are respectively given by
and
where
// =
3
(6 + A)
G = Bt /2(3-A)/(6+A),
=
9m
(6 + A)-
-Ri
(6 + A) fl + A^
(3-A)/3
I8/(64-A)
(37)
(38)
The solution of eq. (21 ) for this phase is given by
y = f(X -.3)/(6 + A).
. (6 + A)/ „.J(6 + A) „ n +
"'' = 71337''* — "*1— J
(3- A)/3
-«)/(6+A)
(39)
Early cosmological models with variable G etc
329
Putting the limiting value /= 4/3 for radiation-dominated phase in eq. (28), the
asymptotic value of deceleration parameter in the limit/?//?* » 1, is given by
9=(3 + ;L)/3. (40)
In order to have expansion, we must have 0 < A< 3. The energy-density decreases with
time. As r oo, the energy-density as well as pressure becomes zero and therefore the
model would essentially give an empty universe for large time. The scalar potential
decreases with time and tends to zero as f -4 «.
Using eqs. (36-39), we find that the eq. (20) is identically satisfied.
4. Concluding remarks
We have obtained the solutions for spatially homogeneous and isotropic cosmological
models with zero-curvature in the presence of perfect fluids and zero-rest-mass scalar
fields. A unified description of early evolution of the universe is studied with ‘gamma-law’
equation of state for two different periods where the gravitational ’constant’ is allowed to
depend on cosmic time t. The inflationary phase is obtained according to the value of
parameter a in eq. (15). The model is an expanding one in each phase for 0 < A< 3. We also
observe that cq. (34) reduces to the solution of a pure radiation phase R - (2//*r)'/^ /?• for
A = 0 (see [4]). The solutions obtained in each phase is identically satisfied. The possibility
of an increasing G during the transition period is also discussed. A particular case of
homogeneous and i.sotropic solution corresponds to the dc Sitter phase when A = 3.
Kefercncc-s
1 1 1 S Wesson Cosmoloffy and (leophysics (Oxford/New York . Oxford University Press) Gravity, Parliclex
and Astrophym s (Dordrecht . D Rcidal) (1980)
12 1 PAM Dirac Proc. Roy. Soc (London) A 165 1 1 9 ( 1 938)
131 M Israelii and N Kosen AxtrophyK. J. 342 627 (1989)
1 4 1 J C Carvalho Ini J. Theor. Ph y.r. 35 20 19(1 996)
I I K P Singh. Gulab Singh and Shri Ram Indian J Phys 54B 547 ( 1 980)
[6 1 E B Norman Am. J. Phys 54 317 (1986)
f7J S Weinberg Gravitation and Cosmolony (New York : Wiley)
1 8J L S Levil Lett Nuovo. Oni 29 23 (1980)
[91 A Bcesham Ini J. Theor Phys 25 1295 (1986)
flOJ RobeUe F Sistero Gen. Relativ Gravit. 23 1265 (1991)
Indian J. Phys. 72A (4). 331-335 (1998)
UP A
— an international journal
Matching of Friedmann-Lemaitre-Robertson
Walker and Kantowski-Sachs Cosmologies
P Borgohain and Mahadev Patgiri
Department of Physics. Cotton College,
Guwahati-781 001, India
Ret eived 13 January I99fi, accepted 26 March 199H
Abstract : The matching of Fncdmann-Lemailre-Robertson-Walker space-times onto
Kantowski-Sachs space-times with strings is investigated Employing Darmoft junction
conditions, a spherically symmetric homogeneous anisotropic Kantowski-Sachs metric with
stniigs can be joined smoothly to the present day universe represented by FLRW space-times.
This cosmological model is expected to be an important tool for studying the early stage of the
Keywords : Space-time geometry, strings, Darmois junction conditions, paramctrizalion
FACS No. : 98.80 Mw
I. Introduction
rhe space-time geometry of the present day universe is believed to be described by FLRW
ivpc of metric. But the universe did not have the same type of space-lime geometry just
filler Its birth and has passed through a number of different phases before it reached the
pieseni day form. Different space-lime metrics are developed to describe such different
phases and we have the problem of matching of such metrics which occur during the phase
‘■hange. While the formalism for joining two different space-limes is well developed,
successful examples of its application are very few. The reason is that since the matching of
iwo solutions usually takes place on a surface sharing some of the symmetries, both of the
two matched solutions must come from a restricted subset of all solutions, which is
determined by their shared symmetries — this restriction makes the problem of matching a
dillicult one. The best known examples of matching is probably the matching of FLRW
dust space-times with Schwarzschild interior or exterior spacetimes [1-4]. A second
example is the matching of FLWR metric with the Kasner metric [5]. In this paper, we will
© 1998 lACS
332
P Borgohain and Mahadev Patgiri
present another example of matching of FLRW space-times with Katowski- Sachs space-
time with strings.
2. FLRW metric and Kantowski-Sachs metric with string
The general FLRW metric in its usual spherically symmetric form, can be written as
ds^ =dt^ -R^(t)[dr^ /0-kr^) +r^{de^ +sini 9d(p^)]. (I)
The Kantowski-Sachs metric for spherically symmetric homogeneous anisotropic space-
time in presence of strings is of the form
ds^ =dT^ -b2(r)[d02 +sinZ _fl2(7')^p2 P)
With its solutions for geometric strings [6],
a{T)a(T-T^)-^n
and b(T)a(T-TQ)V\
3. The matching
From now on, we will use the symbols F and to denote indexed quantities associated with
FLRW and Kantowski-Sachs metrics respectively. Hence, the coordinates of the
corresponding metrics can be represented by
X;. = [f,r.6,4«p.b
o.i>= 1.2, 3,4
and Xi =[r,p, ©,<*>],
We 4vill apply the Darmois set of junction conditions since it does not require the use of
the same coordinate systems on both sides of the hyper surface I [7]. The two regions of
space-times are said to match across Z, if the first and the second fundamental forms
calculated in terms of the coordinates on Z, are identical. The first and the second
fundamental forms are expressed as
Ya? = g„ dx'ldu<^ dx> Idu^ , i,j = 1, 2, 3. 4
(4)
a. ^ = 1. 2, 3
and
= {^ij"k -nij)9x‘ tdu" 9xj Idu^ -
(5)
where w" = [m‘ =u,u^ = v, = w] is the coordinate system on the hypersurface and n,
is its unit normal. Let Zbe given by the functions //r [jcJ^ (u“ )] = 0, /* [jcJ^ (m® )] = 0 and
two parametric representations = h‘fr{u^ = h[{u ° ). Then n, can be calculated by
using the relation
«. = f,b I)''",
where i denotes 9/dx‘ .
( 6 )
Matching cf Friedmann-Lemaitre-Rohertson-Walker etc
333
Wc now consider a surface represented by the function /p(xjf) = r-rQ=0 where
ro is a constant and parametrised by jcJ, =G = v,x]r and x); -r-r^.
In K-S frame wc donot know the form of /jf, however, we will use =T = T(u),
xl = 0(m, v), jc J =0 = \sf and J « p = p(u) as its parametrization.
Now the condition Ypafi ^ implies that
1 = {dTIduf -b'^iSeiduY -a^dptdu)^.
(7)
[deidvf = R^rllb\
(8)
R^rllb^ = sin^ 0/sin^ 6,
(9)
b^dBjdu dBjBv = 0.
(10)
From [10] we find that at least one of the terms , dBjdu or dSjdv must vanish.
However, if = 0 or dSj dv = Q, then according to [8] and [9] we have /?(«) = 0 which is
not allowed. Hence, we are left with the only condition dBj du - 0 i.e. 0is a function of v
only. Eqs. (7) to (10) then reduce to
l={dTlduf -a^(dpldu)^, (11)
(deidv)^ = R'^rl /b^ , (12)
and = sin^ 0/sin^ 6. (13)
From (12) and (13), we have 0= 0by adjusting the constant of integration,: Thus (12) and
(13) give us
.R^rllh^ = 1. i.e. Roeb. (14)
Let us now compute the second fundamental forms. The unit normal in the FLRW space-
time can be calculated by using the eq. (6) and also using f f(xp) = r - Tq = 0 , we get
nfi =
A.s can be seen, the normal is space like, Le. = -1. Further, we have =
since dxp /du^ = dr^ jdu^ = 0,
The unit normal in the Kantoswki-Sachs space-lime in presence of string is more
complicated to obtain since we donot know the explicit form of fk(x[) except that it
should not depend on 0 and 0. However, must satisfy the two conditions
= n^pnpi = -I
(15)
and ni^idx‘j^ j = 0.
Thus, we obtain a set of two equations for two unknowns which enable us to derive n^i as a
function of m®. We have
72 A( 4 )- -\±adp ! a dT ! du\. (16)
334
P Borgohain and Mahadev Patgiri
Now differentiating (15) w.r.t. u®, we get
rixid'^x'K j f du^ dx'^r f du° (17)
andfinally, ^ xap ~ ^Khj ^k, I I + n^i d'^x’^ I duP du^ . (18)
FVom eq. (16) and noting that 22 ' 33 ’ 44 non-^ero
Christoffel symbols of interest, the condition fipap ~ ^ ^ ^Kap already satisfied
except for L^ap^ diagonal terms. These three remaining terms are
+ dT/du dp/du + riK, d^T/du^
+ n d^ p 1 du^ = 0,
(19)
(20)
and
■f^A'31
(21)
From cqs. (20) and (21), wc find thatw/^i = 0. Then from (16) we have
^p/dii = 0 .
and from (II),
±du = ±dt (22)
Nt)licc that cq. (19) is automatically satisfied. Now from (14) we get
« = /,/r„ = l/r„(r~r„)2/’ (23)
4, Disciis.sion
From (23) we find that the FLRW region has a scale factor R - l/rg (/ - /o )"^ ' with
consequence that the space-lime is F:instcin-de Sitter type (a trivial displacement in t makes
the argument more evident). Thus, wc would show that the spatially flat Einstein-dc Sitter
space-lime can be joined smoothly to a Kanlowski-Sachs space time with strings. The
presence of the strings in K-S space-time allows the matching of the two space-times
smoothly. Moreover, such a matching can be considered only at the very early stages tif the
universe during which, it is believed the universe passed through a scries of phase
transitions along with spontaneous breaking of symmetry. Such a symmetry breaking may
give rise to topologically stable defects such as appearance of domain walls, strings and
monopolcs. Out of these three only strings can lead to a very interesting cosmological
consequence as can be seen from the following.
Wc have seen that at a surface defined by r = /q = constant and p = constant the two
space-times can be joined smoothly with R - hj - !//■() )'/-^ . This can also be
seen by noting the foims of the two metrics at this surface.
^•^R.RW - dt' -Xj ^dr- r’^(dQ- (24)
and
dsl^ = dT^- -(T'-T’o)**/^ [d9^ +sin2 Gd<P-]-iT (25)
Matching of Friedmann-Lemaitre-Robertson-Walker etc
335
we find from (24) and (25) that the two space time are identified on surfaces r = cons'tanl =:
ro and p = constant if we simply assume Ts r, 0= 0and ^ = 0. It is also interesting to note
ihai since the space times ae matched across surfaces with r = constant and p = constant one
can construct a universe of alternating layers of FLRW and K-S regions. In this scenario,
the thickness of the K-S layers would be decreasing as (r - so that FLRW regions
iTfow with time and at a certain time the K-S region is completely wiped out and the
universe becomes FLRW type.
References
1 1 1 A Friedmann Z Phys 10 377 (1922)
[21 G Lcmaiire Ann. Soc. Sci. Bruxelles lA 53 51 (1933)
[3] H P Robertson Astrophy.^ J. 82 284 (1935)
|4| A G Walker Proc. Ljmdon Math Sac. 42 90 (1936)
|'=i] Charles C Dyer, Sylvie, Landry and Enc G Shaver Phys Rev. D47 4 (1993)
[f)| Subenoy Chakroborty and Ashok Kr. Chakroborty J. Math. Phys. 33(6) (1992)
|7) G Darniois Menwrml des Sciences Mathematiques Fascicule XXV (Gauthcir-Villars, Paris) Chap V
(1927)
Indian J. Phys. 72A (4). 337-341 (1998)
UP A
an international jo urnal
Structural and dielectric studies on lanthanum
modified Ba2LiNb50i5
K Sambasiva Rao, K Koteswara Rao, T N V K V Prasad
and M Rajeswara Rao
Solid State Physics Laboratory, Department of Physics, Andhra University,
Visakhapatnam-530 003, India
Received 5 September 1997, accepted 1 7 April 1998
Abstract : Present paper describes the ceramic preparative conditions and the effect
of lanthanum (La) on structure and dielectric properties of Ba 2 LiNb 50]5 (BLN), The
materials have been sintered at 11S0°C. The low sintering temperature in these compositions
attnbute to the presence of lithium. Unit cell parameters obtained from XRD studies indicate
an orthorhombic structure. Curie temperature of BLN has been found to decrease with
La-content
Keywords : Ba 2 LiNb 50 | 5 , .structure, dielectric properties
PACSNos. : 77.22.-d.61.66.Fx
One of the most important and numerous groups of ferroelectrics is the family of oxygen-
octahedra crystals. The ease of these crystals is a combination of oxygen-octahedra centres
and voids of which other ions are located. One of the family members of oxygen octahedra
ferroelectrics belongs to the distorted potassium tungsten bronze (TB) structure. The
standing representatives of this group are single crystal solid solutions, barium strontium
niobate (SBN), barium sodium niobate (BNN), barium silver niobate (BAN) and barium
lithium niobate (BLN) [1-4]. The greatest interest in these materials are due to their optical
non-linearity and device applications [5-7].
A useful non-linear optical crystal [4], barium lithium niobate, Ba 4 Li 2 Nb]o 03 o
belongs to TB structure with a point group 4 mm. The BLN has phase transition
temperature at 586X. The transition in BLN is first order. It has no microtwinning at room
temperature unlike BNN.
However, studies on ceramic materials of lanthanum doped BLN is still not widely
found. The present communication describes the preparation, characterization and dielectric
studie; on lanthanum (La) doped and-undoped BLN.
© 1998 I ACS
338
K Sambasiva Rao et al
Raw materials used for the preparation of ceramic samples are of reagent grade,
BaCOi, Li 2 C 03 , La20, and Nb 205 . The constituent carbonates and oxides were weighed to
yield the following compositions and mixed well in an agate motor and pestle and calcined
at 875°C for 6 hours. This procedure was repeated twice to give more homogeneous, single
phase materials and then sintered at 1 150°C for 4 hours. The compositions are
Ba4Li2Nb)Q03Q BLN,
Ba^ gLi 2 . |LaQ {NbiQO^Q 0.1 La^BLN,
Ba 3 5 Li 2 , 2 LaQ 2 Nb|Q 03 Q 0.2 La~BLN,
Ba^ 4 Li 2 ^Lhq 3 Nb|QO 30 0.3 La-BLN.
Lattice constants are determined by the powder method on X-ray powder diffractometer,
available at RSIC, Nagpur University, Nagpur, India, using CuK^ radiation. Dielectric
constant has been measured at 1 KHZ using a digital LCR meter type VLCR-6. Silver
paste, fired on the surface of the well sintered ceramic specimens cured at 600°C was used
to form the electrodes.
The compositions are sintered al low temperature due to the presence of lithium and
gives liquid phase sintering. The XRD patterns obtained on BLN have been nicely matched
with JCPD [8]. Figure 1 indicates the XRD pattern on BLN. Also, it is found that XRD
Figure L XRD pattern of Ba4Lt2Nb|o03o-
peaks are single phase belonging to orthorhombic structure [9]. XRD peaks on BLN
have been indexed in Table 1 . It is observed that the values of i/-spacing observed WobJ and
calculated (iicai) are very much closer. It shows that the assignment of Miller indices h, k, I,
values are correct.
The computed lattice parameters are a = 10.194 A, /? = 14.874 A and c ^ 7.928 A,
which agrees well with literature values [8] a = 10.197 A, /? = 14.882 A and c = 7.942 A.
Substitution of La-in BLN does not affect the orthorhombic structure of BLN. TaMc 2
shows the lattice constants of BLN and lanthanum doped BLN.
Structural and dielectric studies etc
Tabk 1. XRD data on BLN.
^obfi
5.0895
^col
5.0970
2
0
0
///o(%)
3
4.1962
4.2044
2
2
0
5
3.9574
3.9640
0
0
2
29
3.7140
3.7187
0
4
0
12
3 5807
3.5856
1
1
2
5
3.4966
3 4982
0
2
2
37
3.3074
3.3127
3
1
0
100
3.1250
3 1291
2
0
2
60
3 0075
3.0621
2
1
2
99
2 9626
2.9625
1
3
2
88
2 8840
2 8842
2
2
2
52
2 8543
2.8558
1
5
0
35
2 8005
2.8029
3
3
0
28
2.7127
2.7127
0
4
2
31
2 5477
2 5485
4
0
0
II
2 4797
2.4791
0
6
0
2
2.41 15
24109
4
2
0
7
2 3922
2.3943
2
4
2
9
2 3178
2 3171
1
5
2
9
2.2888
2.8886
3
3
2
6
2 2389
2.2383
3
5
0
5
2 2279
2.2294
2
6
0
4
2.1438
2,1437
4
0
2
5
2 1187
2.1218
4
1
2
4
2.1031
2 1022
4
4
0
10
2 0797
2 0802
1
7
0
6
2 06(X)
2 0598
4
2
2
5
1.9834
1.9820
0
0
4
20
1 9493
1.9491
3
5
2
19
t.8810
1 88.56
5
3
0
4
1.8593
1 8593
0
8
0
4
1 8430
1 8420
1
7
2
5
1.7984
1.8017
3
7
0
7
l.'^683
1 7615
5
2
2
12
J.7442
1.7467
2
8
0
10
1.7027
1.7028
5
3
2
33
1 7022
1.6990
6
0
0
36
1.6832
1.6833
0
8
2
27
339
72A(4)-il
340
K Sambasm Rao et al
Table 2. XRD and dielectric data.
Umcep^<«A_
b <■
„LN 10 •9'* ’
OiLa-BLN K' ''>87« 1932
02LaBLN 1«1“0
nlUvBLN 1 0182 14 904 2 969
^RT
Dielectnc data
“KtC
7,T
380
nil)
610
6(X)
243
5620
470
478
15^
1607
410
390
185
332^)
--
290
. Af R1 N and ianlhanuiii doped
Vanauon of dielectric (LSI) relat.on
bln have been shown in Figure jie,ecirie constant of oxygen octahedra
'",1 gotilility, when the frequency of the lowest transverse optical
rerroelecincs would go lo tnnm y,
Figure 2. Variation of dielectric
mode goes lo zero.
Also, the behaviour
c„«slanlw..htcmpcral«ofBl.Nan(ll.aBLN
Of Static dielectric constant of ferroelectric mateo
U/nicC lnw
Structural and dielectric studies etc
341
where C is Curie constant. 7\. is the transition temperature. Therefore, as the temperature of
the sample approaches transition temperature 7',., the static dielectric constant goes to a
maximum value. It is evident that a maximum dielectric constants (A/^) ol 1110 has been
observed at 600‘^C in BLN indicating of its transition temperature (T, ). The observed 7\ is
very much close to reported value of 586°C [4|. Also, substitution of La-in BLN affects the
which decreases with increase of La-content from 600 to 290°C The Curie
lemperaliirc obtained on BLN from differential thermal analysis (DTA) peak temperature of
exotherm {Tj) is 610'*C, close to experimental value 6(XLC. The Curie temperatures {T^)
obtained from DTA on 0.1 La-BLN and 0.2 La-BLN arc 470^(' and 410”C. closed to 478
and 390‘’C obtained from dielectric measurements. The room temperature dielectric
constant (A'^y) of La-doped BLN indicates a decrease with increase ot La. But. there is no
systematic variation. This decrease of A'^j may be due to increase in I] Similar behaviour
ha'^ been reported in rare earth modified BNN and BAN ceramics 112,13|. The Curie Weiss
law has been obeyed in all the materials and Curie constant ((’) m each composition has
been computed and found to be of the order of 10^ K, closed to reported value |4|. It has
been observed that the variation of Curie constant with the dopant concentration is very
small 'I'he value C is the evidence that the materials belong to oxygen oclahedra. The
dielectric data is given in Table 2.
It has been concluded that the materials are sintered at a low temperature which may
be due to the picsence of lithium. Substitution ot La affects the '![ ’ AVi and A|^ of BLN but
not stiuciure ol BLN. The value of C. lO*' K indicates that the maiernals belong to oxygen
oc tahedra
UkiiowledKiiicnl
One of the authors, K S Rao is grateful to the Council of Scientific & Industrial Research
(CSIR), New Delhi, India, for their financial support
Ucfcri’iuTS
ilj W A Bonner, J R Caiuthcis and H M O Biyan Afr;f<7 Bull 5 243 ( bl7t))
I : I K R Nciii paonkji , W K Cv\ y and J R Oliver J C ryu Gnm th 84 f)2‘) ( 1 9S7 )
! ^1 Tokuko Sugai and Masannbu Wada Jaimn J Appl PIm 13 K (1474)
!41 Hiio^lii Hiraru), Huiiiihikn Takcir and Shipenao Knide ./pj ./ Appl Pliw 9 580 ( B170)
I'^l I h Cieusic, H J Levinstein, J J Rubin, S Sinpli and L O Van IJileit .A/v>/ /Vos U'ti 11260(1%7)
!«') J t GeusiL. H J Levinstein, S Singh, R (i Smith and L (i Van Uiten Appl Pli\s Lvn 12 306 ( l%K)
i'/J R G Smith, J L Geusic. H J Levinslein. J J Rubin. S Singh and L G Van Uilerl Appl Phw Li ft 12 308
(1%8)
!8| J(TnS27 1215(1977)
V>\ M Mailhe;. ./ Crysi Growth 15 157 (1972)
tO] R M Lyddane. R G Sachs and E Teller Phyy Rtv 59 673 (1941)
1 ■ 1 1 H Frohluh / Itatry oj Dielectru \ (Oxford Clarendon) (1949)
AUGUST 199H, Vol. 72, No. 4
Special Issue on Recent Trends in Statistical Physics
Foreword
J K Bhatiacharji-i-
A slochaslic approach to chaotic diffusion
Bidhan Chandra Bao, Shanta Chaudhuri and Dfh Shankar
Ray
Uiulying structures in quantum intcgrable systems
Anjan Kundu
Quantum phase transition and critical phenomena
A Dum and B K Ciiakrauar ii
Black hole ihcmiodynamics
PMitra
Cliern-Simons theory of quantum Hall effect
V Ravishankak
7’hc problem of turbulence
J K BHA'nACHARJI.L
Finite temperature Field theory
Samir Mai.uk
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ANNOUNCEMENT
20th Bangladesh Science Conference
Date
: September 11-14, 1 998
Theme
Infrastructure and Human Resource
Development in Science and Technology :
Bangladesh Perspective
Venue
Bangladesh University of Engineering &
Technology
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and
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Department of Physics Fax ; 880-2-863046, 880-2-863026
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Engineering & Technology,
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Maiching ol Friedinann-Lcmailrc-Roberlson-Walker and Kantowski-
Sachs Cosinuloj^ics
P BoKCIOMAIN and MaIIADI.V PAltilRI
Note
Siruclural and diclccinc studies on lanthanum modified
Ba2LiNbsO,s
K Samdasiva Rao, K Kivu swara Rao, TN V K V Prasad
AND M RaJESWARA Rao
331-335
Pa^es
337-341
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INDIAN JOURNAL OF PHYSICS
Volume 72 A
Number 5
September 1998
OP
EDITOR-IN-CHIEF St HONORARY SECRETARY
S P Skn Gupta Indian Association for the Cultivation of Science, Calcutta
A K Barua Indian Association for the
Cultivation of Science, Calcutta
S N Behera Institute ofPhysics, Bhubaneswar
D Chakra voRTY Indian Association for the
Cultivationof Science, Calcutta
B G Ghosh Saha Institute of Nuclear
Physics. Calcutta
A
National Physical Laboratory.
New Delhi
C K Majumdar S N Bose National Centre for
Basic Sciences. Calcutta
ESRajagopai. Indian Institute of Science,
Bangalore
CONDENSED MATTER PHYSICS
S K JosHi
NUCLEAR PHYSICS
CVKBaba Tata Institute of Fundamental \ S Ramamviithy Department of Science &
Research. Mumbai Technology. New Delhi
SS Kapoor Bhabha Atomic Research
Centre. Mumbai
PARTICLE PHYSICS
H Banerjee S N Bose National Centre for Prodir Roy
Basic Sciences. Calcutta
D P Roy Tata Institute of Fundamental
Research. Mumbai
Tata Insiitute of Fundamentah
Re.^earch. Mumbai
S Banerji
B K Datta
RELATIVITY & COSMOLOGY
University of Burdwan. NKDadhicii
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Pune
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S M Chitre Tata Institute of Fundamental
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R CowsiK Indian Institute ofAstrcfphysics.
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S K Skn
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'l»is volume.
Indian Journal of Physics A
Vol. 72A, No. 5
September 1998
CONTENTS
Proceedings of Condensed Matter Days — /P97, held at the Department of
Physics, Visva Bharati, Santiniketan, India, during August 29 - 37 , 1997
J'orcword
S K Roy
Quantum magnetism : novel materials and phenomena
Indrani Bosi:
Gas-surface scattering ; A review of quantum statistical approach
S K Roy
An orbital antilerromagnetic state in the extended Hubbard model
BiPLAB CHAm)PAI)HYA\
Idcctronic transport in a randomly amplifying and absorbing chain
Asok K Spn
rKinsporl and Wigner delay time distribution across a random active
medium
Sandppp K Joshi, Ahhuh Kar Gupta and A M Jayannavar
l .aiiice relaxation in substitutional alloys using a Green’s function
S K Das
Semielassical theory for transport properties of hard sphere fluid
BiKI NDRA K SlNdH and SlJRhSH K SiNHA
[ he [iroblem of a composite pic/ocleclric plate transducer
T K Munshi, K K Kundu and R K Mahalanabis
lassical theory for thermodynamics of molecular fluids
Tardn K Dky and Surush K Sinha
Suhiliiy of Ag island films deposited on softened PVP substrates
Manjunatha Pa'itabi and K Mohan Rao
I iK i getics of CO-NO reactions on Pd-Cu alloy particles
Mahesh Menon and Badal C Khanka
Pages
343-350
351-357
359-364
365-369
371-377
379-383
385-389
391-395
397-401
403-406
407-411
[Cant'd an next page]
Pages
Inhomogcneity of vortices in 2d classical XY-model : a microcanonical
Monte Carlo simulation study
S B Ota and Smita Ota
413-416
A new viscous fingering instability : the case of forced motions
perpendicular to the horizontal interface of an immiscible liquid pair
B Roy and M H Engineer
417^20
Energy, Muctualion and the 2d classical XY-model
Smita Ota, S B Ota and M Satapathy
42M25
Phase alternation in liquid crystals with terminal phenyl ring
Jayashre£ Saha and CDMukherjee
427-431
Change in conductivity of CR-39 SSNTD due to particle irradiation
T Phukan, D Kanjilal, T D Goswami and H L Das
433^37
Melastability and hysteresis in random field Ising chains
Prabodh Shukla
439-446
Electron tunneling in heterostruclures under a transverse magnetic field
P K Ghosh and B Mitra
447-454
Slicking of He"^ on graphite and argon surfaces in presence of one
phonon process
GDuttamudi and SKRoy
455^61
Influence of alloy disorder scattering on drift velocity of hot electrons
at low temperature under magnetic quantization in n-Hgo 8 Cdo 2 Te
Chaitali Chakraborty and C K Sarkar
463^67
Proceedings of Condensed Matter Days^l997, held at the Department of Physics,
Visva Bharati, Santiniketan, India, during August 29-31, 1997
Foreword
CONDENSED MATTER DAYS- 1997 was held at the
Department of Physics, Visva Bharati, Santiniketan during
August 29-31, 1997. This meeting was sponsored by S N Bose
National Centre for Basic Sciences, Calcutta; DST, New Delhi;
CSIR, New Delhi; lUC, Indore; SINP, Calcutta; Institute of
Physics, Bhubaneswar; LACS, Calcutta and Visva Bharati,
Santiniketan. This was the fifth meeting in CMDAYS which
attracted nearly sixty front-line workers in Condensed Matter
Physics. Seventy five papers were presented in the conference
inc luding sLx review papers, twenty three oral papers and rest
[)()sler papers. The evening lecture was delivered by Professor
C K Majumdar, Director, S N Bose National Centre for Basic
Sciences, Calcutta. There were extensive discussions on some
recent fields of Condensed Matter Physics. Out of seventy five
jxipers presented some very good papers have been submitted
lor publication in this proceeding. 1 have made all possible
( Iforts to ensure that the standard of the papers meet the
requirement of Indian Journal of Physics. I sincerely thcuik
the authors for their keen Interest. 1 also thank Professor
S P Sengupta. P2ditor. UP and staff editors for giving me full
Ireedom in editing the papers submitted.
S K Roy
Convener. CMDAYS- 1 997
Guest Editor
Indian J. Phys. 72A (5), 343-350 (1998)
UP A
— an mterna(ional joumal
Quantum magnetism : novel materials and
phenomena
Indrani Bose
Depamnenl of Physics, Bose Institute, 93/ 1 . A. P C Road,
Calcutta '700 CKW, India
Abstract : The subject of quantum magnetism has witnessed a iiemenduus surge
in research activity in the last decade Several new materials and phenomena have been
discovered which have made significant additions to our knowledge about magnetic systems
In this review, some of the important developments will tie discussed with approprijiie
examples
Keywords : Quantum antiteriomagnets, spin gap. high 7 ^ cuprates
PACS Nos. : 71 27 fa 71 3*^ ->
1 . Introduction
In the last ten years or so, there has been unprecedented research activity in the area of
magmetisni. Many new materials exhibiting novel phenomena have been discovered. In this
leview. we will discuss some of ihc.se exciting developments. The discovery ol high-
tcinpcralure superconductors m 1986 has given a tremendous boost to research on quantum
aniiterromagncts (AFMs) 1 1 1. The high-T, materials are cuprate systems with a layered
siiuciurc. The common structural ingredient is the copper-oxide plane. All the dominant
electronic and magnetic properties are associated with the plane The plane looks like a
square lattice. The copper tons carrying spin- 1 /2 sit at the lattice sites and the oxygen ions
are on the bonds m between The copper ions interact through AFM supercxchange
uueraction mediated via oxygen ions. The interaction Hamiltonian is the well-known
Heisenberg Hamiltonian given by
H =
- S, are the spins located at the sites / and j, J,^ denotes the strength of the exchange
•nieiaction. Usually, i,j arc ncaresi-ncighhours (NNs) hut further-neighbour interactions are
I99S1ACS
'-AiM-2
.V44
hhlrani Host'
also iniporiani for many real systems. In most materials J,jS arc assumed to be equal to the
value y. The magnitude of spins may be l/2, I, 3/2, 2. ... etc. For positive J, NN spins
favour antiparallcl oncnialion to achieve the lowest energy slate, this is the case of
aniiferromagnetism. For ferromagnetism, J is -ve and the NN spins favour parallel
orientation. For the cuprate systems, J is +ve and S = I /2.
Consider the cuprate system La 2 Cu 04 This system is an AFM and an
insulator. Replacement of La ions by Sr or Ba ions is called doping and v in La 2 .^Sr^Cu 04
IS called the dopant concentration. The effect of doping is to replace some of the Cu
spins in the Cu02 planes by positively charged holes. On doping with a few percent ol
hides, the long range AFM order is rapidly destroyed leaving behind a spin-di.sordercd
stale This has motivated a large number of studies of quantum AFMs with spin-disordered
states as ground stales. Some of these AFMs consliiule a class known as spin-gap (SG)
systems. In Section 2, the SG systems will be introduced In Section 3, doped S(i
anlifcrromagnets will be discus, ^ed. The doped cuprate systems exist in insulating, metallic
and superconducting phases depending on the temperature and dopant conceniralion
The doped systems exhibit strange properties which cannot be explained by conventional
theories. This has motivated the study of other doped quantum AFMs to gam a proper
understanding. Section 4 contains .some more examples of recent developments m the
area of magnetism.
2. Spin-gap antiferromagnets
Recently, several new AFM compounds have been discovcied which exhibit the
phenomenon of SG. Excitations in a magnetic system are ciealed by deviating spins
from their ground stale arrangement The energy E of the excitation is ,i runciion of the
nionienium wave vector A The excitation spectrum is said to he gapless il there is at
least one momentum wave vector at which the excitation energy becomes zero. The
excitation spectrum has a SG if the lowest excitation is separated by an energy gap from
the ground state. The SG occurs naturally in systems with anisotropies of various types
The SG in the new AFM systems, however, has a purely quantum origin and cannot
be ascribed to any anisotropy effect .SG implies the absence of low-energy spin
excitations This is rcmimscciU ol the energy gap in the electronic excitation spectrum ol
a superconductor (SC) 7'he gap opens up due to the formation of bound Cooper pairs
ol electrons in the SC stale For temperature T > the SC gap disappears and the
system becomes a normal metal. In most of the SG systems, the ground state consists ol
singlets (T I - 4Tj ol spins which are spin pairs, the analogues of Cooper pairs.
In the SC ground stale, long range phase coherence is established, all the Cooper pair
wave I unctions have the same phase The singlet ground state of the SG systems also
have long range coherence charactcri.scd by novel order parameters [2|. The ordering is
lost above a temperature 7,. Recently experiments on the cuprate systems show the
Quantum magnetism : novel materials and phenomena
345
evidence of partial spin and charge gaps opening up for T > [3]. The gap has been
designated as the pseudogap and may indicate some kind of pairing without phase
coherence above 7^. In a conventional SC, pair formation and opening up of the energy
gap occur simultaneously at T^.. The effect of doping on the magnitude of the pseudogap
and its evolution to the SO energy gap below 7^. are some of the issues that are yet to
be settled. Study of doped SG AFMS may lead to knowledge about the various
possibilities.
We will now discuss .some AFM spin-gap systems which have been discovered
in the last five years. The most well-known example of a SG system is a Haldane-
gap AFM. These systems are linear chain systems with integer spins. Half-odd integer
spin chains, on the other hand, have a gapless excitation spectrum. The compound
Y^BaNiQs is an example of a 5 = 1 linear chain AFM that can be doped with holes [4].
The ground stale is spin-disordered and can be characterised as a quantum spin
liquid (QSL). The system offers the first example of a doped QSL in I d. The doped
and spin-disordered Cu02 plane of the cuprate systems is an example of a QSL in
d = 2 The holes are introduced by replacing yttrium with calcium. Experimentally,
there is a large reduction in the DC resistivity as the dopant concentration increases
from zero. At the same time, new slates appear within the spin-gap. As in the case of
the Cu 02 plane, the holes move in a background of antifcrromagnctically interacting
spins
The linear chain 5 = l/2 AFM compound CuGcO^ is the first example of an
inorganic compound showing the spin-Peierls (SP) transition [5] This transition is caused
by the coupling of spins to the phonons, the quanta of lattice vibrations, in the system.
Bulow Tsp, the SP transition temperature, the I d lattice distorts bringing successive pairs of
Npins closer. As a result, a gap opens up in the excitation spectrum below Next, we turn
10 the discussion of spin ladders [6|. The spins have magnitude l/2. The simplest spin-
ladder consists of two chains coupled by rungs and interpolates between 1 d and 2 d AFMs.
The Hamiltonian is given by
Wl =^iS5,,S, +7„^S,,S, (2)
mngs chains
where andy^^ are the exchange interactions along the chains and rungs respectively.
The ladder has a gap in the excitation spectrum even in the isotropic coupling limit
>/|] = J i - The ground state consi.sts of singlets along the rungs. An excitation is created by
replacing one of the singlets by a triplet and then letting it propagate. The triplet excitation
spectrum exhibits a gap. A general spin-ladder consists of n chains. One example of such a
^vslcm is Sr„_|Cu„^.i02fl (n = 3, 5, 7, ...) which consists of ladders of (n+\)/2 chains with
frustrated “trellis” coupling between the ladders [7], A ladder with an odd number of chains
lia.s properties similar to that of a single chain, namely, gapless excitation spectrum and a
346
Indrani Bose
power-law decay of the spin-spin correlation function. A ladder with an even number of
chains has a spin-gap and an exponential decay of the spin-spin correlation function.
The significant difference between the properties of odd and even chain ladders has
been verified in a number of experiments [6], The system La444„Cu8+2nOi4+8n a^so has a
ladder-like structure. Another compound of interest is LaCu02.5 [8]. Initial susceptibility
experiments were interpreted as showing a gap in the excitation spectrum but subsequent
/i sr and NMR experiments indicate an AFM transition below 7;^ ~1 10 K. The compound
Cu2(C-sH|2N2)2Cl4 is also an example of a two-chain ladder compound [9]. Magnetic
susceptibility results indicate the presence of weak FM diagonal interactions in the
ladder. The compound A14CU24O41 (A = Ca, Sr, Ba, La, Y) is composed of layers
containing two-chain ladders alternating with layers of CUO2 chains [10]. Spin-gaps
have been seen in the excitation spectra of both the chains and the ladders. A recent
addition to the list of AFM systems exhibiting SG is the compound CaV4Q) [11]. The
lattice structure of this compound corresponds to the l/5-depleted square lattice. In
this lattice, ]/5 of the original lattice sites of the square lattice are missing. The lattice
consists of four-spin plaqucttes connected by bonds. Susceptibility, NMR spin-lattice
relaxation rates and neutron scattering measurements show the existence of a SG in the
excitation spectrum. The spin model on the CaV40^ lattice has been suggested to be in the
Plaquette Resonating Valence Bond (PRVB) phase and includes both NN as well as
further-neighbour interactions [12]. In the PRVB phase, the four spins in each plaquette
is in a RVB spin configuration. The PRVB state is a linear superposition of two singlet
states. In one state, spin singlets form along the two horizontal bonds and in the other
state, the singlets arc along the two vertical bonds. The spin-disordered slate of the
doped CUO2 plane in the high-7^ cuprate systems has earlier been conjectured to be in a
RVB state. CaV40^ provides the first example of a 2^/ AFM system in which the
pos.sibility of the RVB slate is supported by experimental evidence. CaV4CX> is the
member of a class of compounds, CaV„0^„^j (n = 2 - 4) which arc defined on a l/ (n+1 )-
depleted square lattice 113]. The excitation spectrum is gapless (with gap) when n is
odd (even) This behaviour is similar to that found in spin-ladders and half-odd integer
and integer spin chains.
3. Doped spin-gap antiferromagnets
The high-7, cuprate systems have a rich phase diagram as a function of temperature and
dopant concentration [ 1 j. In the undoped state, the cuprate system is an AFM as well as an
insulator where the insulating property is brought about by strong Coulomb correlation. On
the introduction of a few percent of holes, there is an insulator-to-metal (MI) transition. The
underdoped metallic state is characterised by unconventional transport and thermodynamic
properties which cannot be explained by the Fermi liquid theory of conventional metals.
There is a conjecture that AFM spin fluctuations may be responsible for the unusual
Quantum magnetism : novel materials and phenomena
347
properties. There is experimental evidence that short-range AFM correlations persist in the
metallic as well as SC phases. In the SC state of the cuprates, the holes bind in pairs. In a
conventional SC, the bound pairs, the so-called Cooper pairs, consist of electrons rather
than holes. The binding mechanism of the holes in the cuprates is not as yet well-
understood. A class of theories suggest that exchange of AFM spin fluctuations may cause
the binding of holes [14], This is in contrast to the fact that in conventional SCs electrons
bind on exchange of phonons. There has been a large number of studies on quantum AF^s
doped with holes in order to explore various possibilities. Even at the level of a single hole,
one encounters a non-trivial many-body problem, one hole in a background of a large
number of antiferromagnetically interacting spins [15]. Strong correlation demands that no
site is doubly occupied by electrons to minimize the Coulomb repulsion energy. A hole as
soon as it moves, leaves behind its wake a string of wrongly oriented (parallel) spin pairs,
thus raising the energy associated with the AFM exchange interaction. Antiferromagnetism
favours antiparallel spin pairs. Thus there is a competition between kinetic energy lowering
due to hole delocalization and exchange energy minimization. This competition can give
rise to novel types of ordering in the ground state. For example, combined ordering of
charge and spin can occur. This is seen in the hole doped AFM compound La 2 NiQi which
is not a SC. The ordering consists of domains of antiferromagnetically ordered spins
separated by periodically spaced domain walls to which the holes segregate [16].
The problem of doped spin ladders has been addressed in a large number of
theoretical studies^ Dagotio etal [1,6,17] first suggested the possibility of SC in a two-chain
ladder system. Two holes are predominantly on the same rung to minimise the loss in
exchange interaction energy. This gives rise to an effective binding of holes. Bose and
Gayen have constructed a spin-ladder model which includes diagonal hopping and
exchange interaction terms alongwith the corresponding intra-chain and inter-chain terms.
The Hamiltonian describing the system is the well-known t-J Hamiltonian. The kinetic
energy term describes the hopping of holes to sites separated by NN and diagonal distances.
The holes move by displacing spins. The spins interact via Heisenberg AFM exchange
interaction. Bose and Gayen [18-21] have derived a number of exact results (ground
state and excited states) for the cases of zero hole, one hole, two holes as well as more than
two holes. The most important result is that of the binding of a pair of holes [20]. The
effects of both strong correlation and quantum fluctuations have been exactly taken into
account to derive the results. Hiroi and Takano succeeded in doping a spin-ladder
compound Lai_jSrjrCu02.5 [22]. The compound showed a MI transition as x was changed
hui unfortunately no SC was observed. Considerable excitement was created when hole SC
was found in the spin-ladder compound Sro 4 Cai 3 6Cu2404i ^ under a pressure of 3 to 4.5
fiPa [23]. Tc is not high and is of the order of 9 K and 12 K respectively. There is some
experimental evidence to suggest that the spin-gap collapses when SC is stabilised under 29
khar pressure [24]. Studies on this system and its variants are still al an early stage. Doping
348
Indrani Bose
of the ladder systems leads to novel phenomena and poses a number of challenging
problems.
Other examples
The present review has been mainly devoted to the description of SG AFMs and their
connections with high-T^ cuprate systems. A brief discussion of the latter has been
included to explain the current research interest in quantum AFMs. Much of the
knowledge and insight that have been gained so far have expanded the scope and content
of the subject of magnetism. The new magnetic materials that have been discovered
are interesting in their own right apart from their possible relevance in the context of
cuprate systems. Besides SG AFMs, a number of recent discoveries have opened up
new areas of research in magnetism. In the following, we describe some of these
discoveries briefly.
Random 5=1/2 chains are the latest class of novel \d compounds to be
discovered. The compound SriCaPiosIrosO^ can be described as a 5 = l/2 Heisenberg
chain with randomly distributed ferromagnetic (FM) and AFM exchange interaction
bonds [25]. The random spin chain is a quantum mechanical system with disorder. The
surprising experimental result is that at low temperatures, when quantum effects are
supposed to be dominant, the susceptibility behaviour is that of a spin system consisting of
classical free spins. The experimental observation has motivated further studies on
random spin systems. The second example is that of quantum hysteresis in molecular
magnets [26]. Magnetic materials are characterised by hysteresis. Their response to an
increasing magnetic field is not the same as that in a decreasing field. The hysteresis
loop obtained as a smooth shape. Recently, material scientists have fabricated a crystalline
organic compound Mn 12 -acetate consisting of weakly interacting molecules of giant
spin 10. Magnetization measurements made at a temperature below a few degrees
kelvin show a hysteresis loop containing steps. The phenomenon is believed to be caused
by macroscopic quantum tunneling of the magnetic moment associated with the giant spin.
The next example is that of light-induced magnetization in a cobalt-iron cyanide
complex. The system orders magnetically below a critical temperature of 16 K. Sato et al
found an increase in the critical temperature from 16 K to 19 K by shining red light on
the system [27]. On shining with blue light, the enhancement of the magnetization can
be partly removed. Such control over magnetic properties by optical signals may be of
significance in the design of magneto-optical devices. The last example is that of
colossal magnetoresistance (CMR) [28]. Magnetoresistance is the relative change in the
electrical resistance of a material on the application of a magnetic field. All metals show
MR but only a few percent. The phenomenon offers prospects for applications such as
reading heads in hard disk drives and digital videotape recorders. A device whose
conductivity is sensitive to magnetic changes would be ideal for quick conversion of
Quantum mafinetism : novel materials and phenomena
349
magnetically stored information into electrical signals. Some years back, a 220% resistance
change was achieved at 7= 1.5 K in a multilayer of 50 alternating films of iron and
chromium. An even more dramatic effect that has been observed recently is that of CMR.
This has been seen in perovskite magnates of the type Lai.^A^MnO^ where A is a
divalent cation such as an alkaline earth (Sr^^ Ca^^ etc.) or Pb. CMR involves
resistivity changes as large as several thousand percent. The effect can only be seen at low
T and high magnetic fields. Research on the CMR materials has benefilted from the
study of high-7f cuprate systems. The CMR materials also have a rich phase diagram which
has motivated a large number of studies, both theoretical and experimental, to understand
the origin of the various phases. To summarise, we have discussed in this review
various new magnetic phenomena and materials which have led to unprecedented research
activity in the area of magnetism. Many problems still remain to be solved which imply
continuing research activity in the coming years.
References
1 1 1 E Dagotto /?ev Mod Phys 66 763 ( i 994)
[2] I Bose Phv.ucu A186 298 and references therein (1992)
[3] N P Ong Sneru e 273 321 and references therein (1996)
[4] J F DiTusa, S W Cheong, J H Park. G Aeppli. C Broholm and C T Chen Phys, Rev Leu 73 1837
(1994)
|.3J M Hasc. I Terasaki and K Ucliinokura Phys Rev Uu 70 36.31 (1993)
16J E Dagotto and T M Ricc S(ietu f 271 618 and references theii^m (1996)
|7] S Gopalan, T M Rice and M Signst Phy^ Rev B49 8901 ( 19,94)
[K] S Matsumoio. Y Kitaoka. K Ishida, K Asayaim, Z Hiroi, N Kobayashi and M Takano Phvs Rev BS3
11942(1996)
[9] C A Hayward. D Poilblanc and L P Uvy Phys Rev B54 R 12649 (1996)
[ lOJ S A Cancr, B Batlogg, R J Cava, J J Krajewski. W F Peck (Jr ) and T M Rice Phys Rev Lett 77 1378
(1996)
[11] S Taniguchi. T Nishikawa, Y Yasui. Y Kobayashi. M Sato. T Nishioka, M Kontani and K Sano J Phys
Soc.Jpn 64 2738(1995)
[12] A K Ghosh and f Bo.se Phys Rev B55 3613 (1997)
[13] H Koniani. M Zhitomirsky and K Ueda J Phys Soc. Jpn 65 1.366 (1996)
1 14] D J Scalapino Physics Reports 250 329 ( 1993)
[13] I Bose Superconductivity Theoretical and Experimental Effects ed K N Shrivastava (New York ; Nova
Science) p 21 (1993)
1(^1 J M Tranquada, D J Buttrey and V Sachan Phys. Rev B54 12318 (1996)
, I / | E Dagotto. J Riera and D J Scalapino Phys Rev. B45 5744 (1992)
j I K1 1 Bo.se and S Gaycn Phys. Rev B48 10633 (1993)
[ I9| 1 Bose and S Gayen J Phys : Con. Matt 6 L403 (1994)
[20] S Gayen and I Bose J. Phy.x. : Con Matt. 7 3871 (1995)
[21] f Bose and S Gayen Physica B223 & 224 628 (1996)
[22] Z Hiroi and M Takano Nature 377 4 1 (1993)
150 Indrani Bose
123] S Maekawa Scitnce 273 1 5 1 5 ( 1 996)
|24] H Mayafirc. P Auban-Senaer, D J6rotn«. D Poilblanc, C Bonrbonnais. U Ammerahl, G Dhalenne ud
A Rtvcolevschi (prcpnnl)
(251 T N Nguyen, P A Lee and H-Ciur Loye Scitnce 271 489 (1995)
|26] P C E Stamp Name 383 125 (1996), E M Chudnovsky faenre 274 938 (1996)
(27) 0 Sato, T lyoda, A Fujisli’ma and K Hatthimoio Science 272 704 ( 1996)
|28| S Jin, T H Tiefcl, M McCormack, R A Fasinachi, R Ramesh and L H Own Science 264 413 (1994)
Indian J. Phys. 72A (5), 351-357 (1998)
UP A
- an intcmat ional journal
GaS'Surface scattering : A review of quantum
statistical approach
SKRoy
Department of Physics, Visva-Bharati University.
SantinUcetan-73 1 233, India
Abstract : The quantum theory of scattering of gas atoms from solid surfaces is
reviewed with special reference to ^Hc gas scattering from Graphite and Argon surfaces It
has been emphasised that the sticking coefficients and the bound state lifetimes evaluated
from T-matnx formalism are more exact and take into account the problems of overcounting of
scattenng events encountered in usual first order distorted wave Born approximation
(FODWBA).
Keywords : Scattering, 7'-matnx, FODWBA
PACS Nos. : 68.45 Da, 68 35.Md. 82.65.Dp
1. Introduction
The gas-surface scattering refers to processes initiated at a surface and involving atoms of
both the gas and the solid. It is only the scattering of gas atoms which may be studied by
experiments.
Particles adsorbed by weak van-der Waals forces are said to be physically
adsorbed or physisorbed e g, rare gases on metals, alkali halides and on graphite. One of
the most important processes occurring at a gas-solid interface is the capture and
subsequent sticking of incident atoms or molecules. If we want to know how fast and
by what mechanism a gas particle adsorbs on or desorbs from a solid, we have to study
Its kinetics.
There exists in general two types of theories describing the interaction of particles
with a surface those based on classical mechanics for the incoming particle, but
nevertheless allowing quantum effects for the substrate and those based on quantum
mechanics. Rather peculiarly, the quantum description 11-4] predates the classical one
1^.6], As the quantum theory becomes better developed and is able to interpret more and
^ 2 A (^3
© 1998 I ACS
352
SKRoy
more of the forthcoming experimental data, it has become clear that more information on
gas-surface scattering is obtainable from experimental data needing a quantum mechanical
interpretation than from data for which classical interpretation is inadequate.
In this review we look into some theoretical aspects of gas-surface interaction within
the quantum regime from the simple model of gas-solid interaction and discuss some of the
interesting results obtained.
The sticking coefficient is one of the most important but unfortunately a very
controversial parameter in the study of desorption and evaporation as well as atomic beam
scattering. This is particularly true for physisorption where few experimental data are
explained by tob-many and often inconclusive theories. Classical theories obtain a sticking
coefficient of unity as the temperature of the solid approaches zero whereas quantum
mechanical theories yield the zero sticking coefficient at zero substrate temperature
Thermal accommodation and adsorption coefficients of gases have been reviewed
comprehensively by Saxena and Joshi [7].
2. Quantum theory of gas-surface scattering
Almost all the quantum theories dealt in references [1-4] are now understood to he
unsatisfactory as they are based on FODWBA. Gas-surface scattering is too strong to be
correctly described by FODWBA. In FODWBA, the probability of scattering into one or
more final states sometimes exceeds unity (He on graphite). The first-order theories do
not generally conserve the number of scattering particles and is non-unitary. There are
two closely related methods viz. close-coupling formulation (CCF) (8,9] and Cabrera,
Celli, Goodmann and Manson approximation (CCGMA) [10] which are unitary as they are
almost exact and obtained by deriving the exact wave-function results from T-matrix
formalism.
The most suitable formulation for our purposes is the two potential T-matrix
scattering theory due to Gellmann and Goldbergcr [1 1].
2. 1. T-matrix formalism :
The Hamiltonian H of the gas atom plus solid including internal states is
H=Hq^U ( 1 )
where //q is the Hamiltonian for the free gas atom plus the solid and U is the gas-solid
interaction part given by
f;=f/o + f/| (2)
where Vq is the large potential depending only on z and is treated exactly and t/| is the
small potential treated approximately. The transition rale from some initial slate i (of total
energy £,) to some final state/(of total energy Ef) is
( 3 )
Gas-surface scattering : A review of quantum statistical approach
353
A standard result of the r-matrix is
(4)
where is the solution of Schrddinger equation with Hamiltonian Hq, xl the
incoming wave (-) and outgoing wave (+) solutions with Hamiltonian Hq f/o and xt
the solutions of incoming wave (-) and outgoing wave (+) with complete Hamiltonian +
The explicit forms of 'Py and xt
'F, = (l, L,L, )■%«,,,}).'*. -
Xt = (^A^v) ^ |{«nw })«'*" (*,,:-) (5)
where |{/»„„, }) is the solid vibrational eigenfunction of labelled by Ihc phonon
occupation numbers of the normal modes m, e‘^>' ^ is the plane wave gas
atom eigenfunction of Hq (or of A/o + because (7,) depends only on :), lor motion
tangential to the surface, ' is the plane wave gas atom eigenfunction of A/^) for
motion normal to the surface.
Now the 7-malnx equation may be written in terms of reduced /-matrix as [12]
T = tf, +(//2;rp, )S{fs)
iT-.f = |f„ +(l/2;rpj,5(/v)|'
x! =
lor the phase angle =
The reduced /-matrix may be written as
't. = (/M') + +,Ep{f\U\c)l.,
+ £^ (£, -£,+/£)'' (/|[/|/;V*, (7)
where .v, b and c arc scattered, bound and continuum slates respectively. The summation
over continuum slates may be written as the product of summatipiis over phonon slates,
gas atom tangential wave vector stales and gas atom normal wave vector states as the
lollowing
I-III
ki k,.
( 8 )
354
SKRoy
Further, I Pc^^n where is the density of states.
^(£, -£*+.£)-' (£, -£*)-'
h
The integral continuum states (c) does cause a problem and (£, -E^ +iE)"' must be
interpreted as
r f(E,^)dE,, ^ f (E,-E,-it)f(E,,)dE,,
J E, -E, +i£ J (£, -£J2 +£2
( 9 )
-4 pj - iffj /■(£■, J)5(£',
The final result in the exact analysis of the gas atoms is
‘fr = (/I t/k) + I („,, , S t, S ^
+ 1 1 Z P \ (£. - £, )-' p, {/|f/k)t„ (10)
{ ■ ■ ■ ) indicates that only states c satisfying = E, arc considered.
3. Approximations
1 . FODWBA : Retains only the first term on the right-hand-side of equation (10)
= (/I f/U) FODWBA
If (/| fy|i) are small unitary would hold, at least approximately.
2. CCGMA ■ The principal part in equation (10) poses major difficulty in solving the
reduced r-matrix. In CCGMA we selP = 0 i.e. all intermediate continuum states c
which do not conserve energy exactly are neglected and all intermediate bound stales
b are included.
Generally gas-surface scattering is strong and (/|(7|.v) arc large and therefore
FODWBA is generally invalid.
4. Present work
The inadequacy of FODWBA to explain the total inelastic component of the gas-solid
interaction has been discussed in detail in our earlier work [13] with specific examples of
He scattering from graphite and Ar surfaces. The detailed calculations done in this work for
sticking coefficient and bound state lifetimes will not be presented here. Only the results
obtained in this case which highlight the importance of exact 7-matrix calculation for light
particles .scattered with very low energy will be discussed.
Gas-surface scattering : A review of quantum statistical approach 355
For very low energy and light particles, the mechanism is intimately related to the
quantum nature of incoming particles. In all experiments in which He or H 2 arc scattered
from single crystal surfaces both inelastic scattering and selective absorption processes are
present which can have very important effects on elastic scattering. The inelastic scattering
is generally enhanced by selective adsorption resonances so much so that the FODWBA
breaks down. The breakdown of FODWBA is caused by overcounling of the scattering
events that is inherent in the FODWBA. This overcounting has been removed by
renormalisation of the momentum dependent sticking.
5. Discussions
With the above theoretical background, we study the inelastic components in presence of
selective bound state (BS) resonances. We hope that a detailed qualitative discussion can be
made for sticking of light particles on the cold solid surfaces. It has been shown earlier [14]
that the inelastic scattering is greatly enhanced by ^elective adsorption resonances so much
so that the DWBA breaks down. This breakdown caused by the overcounting of the
scattering events can be removed by renormalisation of the momentum dependent sticking
coefficient within the time of interaction with the surface [15]. We have demonstrated in
reference [13] that the problem of overcounting can be tackled by exact T-matrix
calculations without recourse to renormalisation considering the inelastic scattering in
presence of bound state resonances. We agree that the overcounter of the scattering events
can be removed without going beyond one phonon emission/ absorption process. We have
shown in our earlier work [13] that the overcounling of the scattering events can be largely
reduced by considering the inelastic events in presence of bound state resonances even
under DWBA. The results of calculations on slicking coefficients with substrate
lempcraiure using 7-malrix formalism (given in reference 13) are shown in Figures 1-3.
In this work we have used the gas temperature 7’^, = 65 K for He-graphite system and =
55 K lor He-Ar system. The essential feature of this work is ba.sed on the scattering of He"*
from cold graphite and argorv surfaces with extended particle phonon interactions. This
leads to the inelastic scattering in presence of resonant surface bound slates. The inelastic
scattering is however sufficiently weak where only one or two phonons arc created or
destroyed, it is never negligible. The scattering process can rcsonably be described in terms
of how the occupation of the lattice has changed.
It is observed from Figures 1 and 2 that the sticking component is enhanced with
temperature of the substrate thereby indicating the enhancement of the inelastic scattering
in terms of the changes in phonon modes because the inelastic scattering probability reflects
the discrete nature of the phonon exiiation of each mode. It is interesting to note that the
inelastic scattering and hence the slicking coefficient is never greater than unity as is found
in the case of DWBA. Thus within the one-phonon approximation, if the higher-order tenns
are taken into account in the perturbation expansion, as we do in the exact 7-matrix
356
SKRoy
calculation, the overcounting of the scattering events inherent in the Born approximation
can be removed in presence of surface bound state.
Figure 1. Variation of sticking coefficient with solid temperature for He-Ar
system, (Dotted line , results from Reference 15; Sfolid line : result from
f-malrix calculations).
Ts(K)
Figure 2. Variation of sticking coefficient with solid temperature for Hc-
graphiie system (Dotted line results from Reference 15; Solid line . result
from 7'-matnx calculations).
The bound state life times shown in Figure 3 for He-Ar and He-graphite systems
indicate that the decay of the bound states are faster at higher temperatures leading to faster
desorption. The slicking in these systems are more probable upto a temperature of 10 K as
found in experiments.
Gas -surface scattering : A review of quantum statistical approach
357
We thus conclude that the r-matrix formalism which takes into account the higher-
order contribution to sticking coefficient of a phonon-mediated process predicts a non-zero
5 10 15 20 25 30
T.(K)
Figure 3. Bound state life times vs substrate temperature
Liipfure even at higher temperatures to make accurate measurements of sticking coefficients
(.iilficult, the calculations presented in reference [13] suggest that at higher temperature
capture into the physisorption bound state is possible and can be monitored through careful
measurements of specular beam. The results, however, depend on the nature of the
potential, its well depth and the interaction strength.
References
! 1] J M Jack.son Proc Cam. Phil Sac 28 136(1932)
(:i C Zener Phys. Rev 40 1016(1932)
[^] J M Jackson and N F Moll PrtK Roy, Soc A137 703 (1932)
Ml J E Lennerd- Jones and A F Devonshiie Proc Roy Soc. A 156 29 (1936)
[^J G Ichc and P Nozieres J. Physique 37 1313 (1976)
[6] C Coroh, B Roulel and D Sainl-Jamcs Phys. Rev. B18 545 (1976)
[7] S C Saxena and R K Joshi Thermal Accommodation and Adsorption Coefficients oj Gases (McGraw Hill)
(1981)
(81 A T.suchida Surf. Science 14 375 (1969)
191 G Wolken I Chem Phys. 58 3047 ( 1 973)
1*1^1 N Cabrera, V Cclli, F O Goodman and J R Manson Surf Sci. 19 67 (1970)
I'M M Gellmonn and M L Goldberger Phys Rev 91 398 (1953)
1121 F O Goodman and H Y Wachman. Pyruimtcs of Gas-Surface Scattering (New York Academic) 152
(1976)
1 1 3) G Danamudi and S K Roy J. Phys. Condens. Matter 8 8733 ( 1 9%)
1 1 D Stiles and J W Wilkins Phys. Rev. B34 4490 ( 1986)
1 1 51 Z W Gortel and J Szymanski Phys. Lett. AMT 59 (1990)
UP A
— an international journal
An orbital antiferromagnetic state in the extended
Hubbard model
Biplab Chaltopadhyay*
Saha Institute of Nuclear Physics. I/AF Ridhannagar,
Calcutta-700 064, India
9 ^
Abstract : Ground stales of an orbital antdcrromagnetic order, along with the charge
density waves and spin density waves, are considered within the framework ot an extended
Hubbard model The model includes nearest neighbour (-;) and next nearest nt'ighbour (/')
hopping mainx elements as well os on-site {U) and nearest neighbour (VO repulsions between
fermions Ground state phase diagram of the model is calculated within the Hanree-Fock
approximation For t' - 0 . only charge and spin density waves are stable For non-zero r', the
orbital anliferromagnetic order, charactensed by symmetry, is stable over a finite
portion of the phase diagram, which grows in size with inercasing t '
Keywords ; Ground state phase diagram, particle-hole ordering, staggered flux pha,se
l»ACS Nos. ; 7 1 .27 -i-a, 7 1 28 -k 1. 7 1 .35 -fz
The high temperature superconductors (HTS) are doped cuprates of lanthanum, yttrium,
bismuth, thallium etc. and are predominantly characterised by their strange normal state
properties. The parent materials (without doping) are found to be antiferromagnetic
insulators [1] contrary to the conclusions from the band structure calculations [2],
according to which these are metals. This behaviour as well as their strange properties
with doping, are believed to arise due to the strong electronic correlations present. In
addition, properties of these cuprates are found to be dominated by Cu 02 plane.
Hence Hubbard model in two dimension could be an usefull starting point. In fact,
Hubbard model and its different extended or strong coupling versions are extensively
studied in the recent years, in connection with the cuprate superconductors, and
systems with strong correlations in general. Phase diagrams of such models are thus of
inherent interest. In this communication, we study the ground state phase diagram of an
extended Hubbard model focussing on the stability of an orbital antiferromagnelic state.
email ; biplab^cmp.saha.emet.in
© 1998 I ACS
'I2A (5)-4
360
tfipiao {^naiiuf^uunj>i*j
The Hamiltonian of the extended Hubbard model on a square lattice, is given by
H = Ho + Hi,
k.a
and H, = Y S S
I i.a.a'
where ^ (r ^ ^ ) creates (annihilates) a fermion with momentum k and spin O’, n, are
number operators, fj. is the chemical potential and = ±aJc, ± ay are nearest neighbour
lattice vectors with a being the lattice constant. The interactions are on-site ([/) and nearest
neighbour (VO repulsions and independent of lattice sites. The band dispersion, comprising
of nearest neighbour (-r) and next nearest neighbour (t') hopping matrix elements, is
given by = -r(cos(A:^<3) + cos(^'^.f/)) r'cos(A,fl) cos(/:^fl). For t'= 0 and at half
filling, the Fermi surface (FS) is perfectly nested where Q = (Ti, ;r) is the
nesting vector. With the nested FS and for = 0, the ground slate Hartree-Fock, particle-
hole ordering is a two sublaltice spin density wave (SOW) for small H Detailed
random phase approximation (RPA) studies [3] of the collective excitation spectrum
have shown that this is the case even at large U. Introduction of V induces a charge
ordering in the form of charge density waves (CDW) [4], which is energetically favoured
loSDWfor \/> f//4.
It was first noted by Halperin and Rice |5] that, particle-hole ordering can
produce ground states other than CDW and SDW. One such is an orbital antiferromagnetic
(OAh) .state. The OAF stale arises out of a if-wavc parlicic-hole pairing, that gives rise
to circulating charge currents in the square lattice with the orientation of the circulation
being opposite in the neighbouring elementary plaqucttes 16]. This is, thus similar to
the staggered flux phase studied earlier (7J as a possible ground slate of the t J model.
The £/-wave particle-hole pairing, involved in the formation of the OAF stale, is interesting
in the light of a pseudo-gap found in the underdoped cuprates by pholoemission
experiments [8,9], where the gap function has a d^ 2_^2 symmetry. Within the extended
Hubbard model, OAF stale was discussed earlier [10,1 1] giving vent to the speculations
that this might be stable compared to CDW and SDW ground stales if the nesting of FvS
is removed [6]. However, no attempt has been made to establish its stability compared
to the CDW and SDW, which is the principal focus of the present communication. We do
a Hariree-Fock analysis of the ground stales of different particle-hole orderings such
as CDW, SDW and OAF, and obtain the ground state phase diagram for the half filled
band [12].
Mean-field decoupling of tbe Hamiltonian in eqn. (1) enables one to have the CDW
order parameter (in the real space) of the form
{"-.(j) = 7 + fcosCg.R, )
( 2 )
if»c cAicnuea nuDoara model
361
and the SDW state (considering az-polarized state) is defined as
= I + (3)
where p amd m are the amplitudes of the induced charge and spin density modulation
respectively and CT= +1 (-1) for up (down) spins. The OAF slate has non-zero intersitc
averages
= igcos(Q.Ri) (4a)
‘"'<1 {‘^la<^,±ya) = -<gcos(fi./f,) (4b)
Substitution of the mean field values from eqns. (2-4), yields a mean-field decoupled
Hamiltonian from eqn. (1) as
RBI
^ MF “ k Q ~ k^Q,a^ k■*^Q.a\
k.a
RBZ
i,a./9
The reduced Brillouin zone (RBZ) is due to the modulation by Q, which effects a folding up
ot the original Brillouin zone. Here A '‘^ ^ (k) is the generalized order parameter with 5 = 1 ,
2 and 3 corresponding to CDW, SDW and OAF slates respectively. Thus,
for the CDW Stale (6a)
lor the z-polarizcd SDW state ( 6 b)
^\pik) = iS ~co^k^.)(-2V)g for the OAF stale (6c)
where O'" is a Pauli matrix. The constant part (in eqn. (5)) for different states are, Xj =
(2V'- Ui4)p^N, X 2 = {lHA)m^N and X 3 = 4Vg^N, where N is the total number of sites. In eqn.
(3) we suppress writing the factor (U/2 + 4V) for each state and absorb it in the chemical
potential. In X^ also, we ignore the contribution - (U/4 + 2V)N for each state, which shifts
the ground-stale energies by equal amount.
The mean field Hamiltonian of eqn. (5) is diagonalized by a canonical
transformation. The folding of the Brillouin zone results in the formation of two bands
el*\k) = (rj* -n) + yjel
(7a)
and
«!.■’(*) = (n* -^l) - yje] -I- Aj(k'
(7b)
'vherc Ai(k) = (U/2 - 4V0p, ^(it) = - (U/2)m and A^(k) = - 2Vg(cos(k/2) - cos(k,a)). It is
^^•ear that CDW (4i(^)) and SDW (A 2 (k)) slates are isotropic ( 5 - wave) whereas the OAF
362
blpiao L^naiiopuanyuy
slate (zl^CA)) has order parameter of d^ 2 ,y 2 symmetry. Here rj* =:(^^ =
(hi ~ '?ii 4 e 2. For /' = 0, one has rjf, = 0, enforcing the nesting of the Fermi surface. This
also implies that /J=0 and the lower band el~^ is completely full whereas the upper band
' is completely empty. However for nonzero t\ 7]/^ ^ 0, nesting of the Fermi surface is
removed at half filling, and the overlapping of bands is possible.
The sclf-consisleni equations for the chemical potential p, and the order parameters
are
RBZ
I = (8a)
and
J_
r,
(8b)
where F] = 4V - U12, r 2 = t//2, F^ = VI2 and 0)^ h the symmetry factor of the order
parameters with tuj =1 (for SDW and CDW) and (ol =lcos{k ^a)-cos{k ^.a)]^
(for OAF).
Expression for the ground state energy (per site) turns out to be
RB7
E' = ^ + j^'^[e['\k)0(-e\-\k)) + f !'^*(*)0(-f ]*' (*))] (9)
k
Self consistent equations (8) are solved numerically to obtain the values of p and gap
function A,{k), which are then used to evaluate the ground state energies £, in eqn. (9). In
Figure 1. The mean-field phase diagram of the extended Hubbard model, as
obtained by compan.son of the ground .state energies of the states considered
The region of stability of the OAF slate grows with increasuig t'/t ratio, as i.s
evident in the figure The phase boundaries for small values of Uft and V/t are
not shown since numencal difficulties prevented an accurate determination of
their positions
Figure 1 we present the ground state phase diagram in the (U/t, V/t) plane, for different
values of parameter t'/t. For /' = 0, the CDW and SDW states are only stable and are
separated by a phase boundary at (/ = 4V (not shown). Any nonzero r* results in the
destruction of nesting of the FS and thereby helps to stabilize the OAF state at weak
coupling.
The gap functions in the CDW, SOW and OAF states are proportional to 4V- t//2,
UH and 21^ respectively. Thus the OAF slate is expected to be energetically close to the
other two stales near the t/ = 4V line. In fact, for the perfectly nested FS (r' = 0), the OAF
state becomes degenerate with the CDW and SDW at weak couplings on the (7 = 4^ line
(13]. At large (/, the minimization of the double occupancy in the ground stale is achieved
by the SDW ordering with m -> 1, whereas for large V the energy cost associated with
having electrons on neighbouring sites could be avoided through a CDW ordering with
p -> 1. Thus the OAF state is expected to arise only at weak coupling and close to the
U = 4 V line. Although a physical interpretation of the fact that, introduction of f stabilizes
the OAF state, is not possible at this stage, our results suggest that the OAF state is more
robust for the case of a non-nested Fermi surface. This is interesting since real materials
often have non-nested Fermi surface.
for the OAF stale
along the symmetry directions of reduced Bnllouin zone, .shown explicitly
in the inset. The solid curve corresponds to the conduction band while
the dashed curve i.s for the valence band. The parameters arc t'/i = 0.1.
U/t - I ..*> and V/f = 0.37 for which the OAF state is stable
To understand the material characteristic of the OAF state, we plot in Figure 2, the
dispersion e|[’^ande[^^ (for the OAF stale) as a function of k, for nonzero f'/t, with U/t and
Vlt chosen to ensure the stability of the OAF slate. The overlapping of bands implies that
ihc valence band is not completely full and the conduction band is not completely empty
iind the density of states at Fermi surface is found to be suppressed but nonzero. The OAF
state IS therefore a poor metallic state with a pseudogap at the Fermi surface.
It should be memtioned here that, a circulating spin current along the elementary
plaquettes of the square lattice, could produce a spin nematic (SN) order [11,14]. Within the
364
Uipiao \^nuiiuf/uunjrv»ji
Hartree-Fock approximation, the SN state is degenerate with the OAF state and has
indentical bands, although physically these slates are quite different. The OAF
state originates out of circulating charge currents which produce staggered orbital magnetic
moments in the elementary plaqucttes, whereas the SN state is due to the circulating spin
currents. Thus, though a magnetic neutron scattering experiment could differentiate
between OAF and SN states, they will appear identical to any experiment probing only their
band structures or single particle properties.
To summarize, we have obtained the ground state (zero temperature) phase diagram
of the extended Hubbard model on a square lattice at half-filling showing relative stability
of the competing orders of CDW, SDW and OAF states. We find that for nonzero values of
t\ the OAF state is favourable to the SDW and CDW slates, over a finite range of
parameters in the weak coupling region. The size of this region, where OAF state is stable,
increases with increasing t\
Acknowledgment
The author thanks Dattu Gaitonde and Atin Das for useful discussions.
References
[1 ] R L Greene et al Solid State Commun. 63 379 (1987); D Vaknin ei al Phys. Rev Lett 58 2802 (1987),
J H Brewer et al Phys Rev Lett 60 1073(1988)
[2] L F Mattheiss et al Phys Rev. Lett. 58 1028 (1987). J Yu et al ibid 58 1035 (1987). S Massidda et al
Phys Un A122 198(1987)
[3] J R Schneffer, X G Wen and S C Zhang Phys Rev. B39 1 1663 (1989)
[4] R Micna.s, J Ranninger and S Robaszkiewicz Rev. Mod. Phys. 62 1 13 (1990)
[5J B 1 Halpenn and T M Rice Solid State Physics Vol. 21 cd. F Scilz, D Turnbull and H Ehrenrcich
(New York Academic Press) ( 1 968)
[6] C Nayak and F Wilczck Prepnnt, cond-iiiat/95 10132
[7] I Affleck and B Marston Phys Rev. B37 3774 (1988)
[8] D S Marshall et al Phys. Rev. Lett. 76 4841 (1996); A G Loesen et al Science 273. 325 (1996)
[9] H Ding et al Nature 382 51 (1996); Preprint, cond-inal/961 1 194
1 1 0] H J Shulz Phys. Rev . B39 2940 (1989)
[11] A A Nersesyan and G E Vachnadze J. Low Temp Phys. 77 293 (1989); A A Nersesyan, G 1 Japandze
and 1 G Kimeridze J. Phys. Cond. Matt. 3 3353 (1991)
[12] B Chattopadhyay and D M Gaitonde Phys Rev. BS5 15364 (1997)
[13] B (2hinopadhyay (unpublished)
[14] L P Gorkov and A Sokol Phys. Rev. Utt. 69 2586 (1992)
UP A
— an international journal
Electronic transport in a randomly amplifying and
absorbing chain
Asok K Sen*
LTP Division, Saha Institute of Nuclear Physics,
1/AF, Bidhannagar, Calcutta-700 064, India
Abstract : We study localization properties of a one-dimensional disordered system
characterized by a random non-hermitian hamiltonian where both the; randomness and the
non-hermiticity arise in the local site-potential, its real part being ordered (fixed) and a
random imaginary part implying the presence of either a random absorption or amplification
at each site. The transmittance (forward scattenng) decays exponentially in either case
In contrast to the disorder in the real part of the potential (Anderson localization), the
transmittance with the disordered imaginary pan may decay slower than that in the case
of ordered imaginary pan
Keywords : Localization, transmittance
PACS Nos, : 05 40 +j, 42 25 Bs. 74 55 Jv. 72. l5.Rn
The study of the spectra of systems with non-hermitian hamiltonians and of the
inicrfercnce of waves multiply scattered from such a system of scatterers have of late
become very fashionable. The physical reason for such a description lies in the fact that
the scattering in any real medium is never perfectly elastic and that in many cases the
deviation from perfectly clastic scattering may be described, for example, by absorption
through other inelastic channels or by amplification due to enhancement of the wave-
amplitude (e.g., population inversion in an active medium) of incident particles or
waves. We are interested in the class of non-hermitian hamiltonians in which the non-
herimiicity is in the local part (typically in one-body potentials) [1-15], It is well-known
that an imaginary term in the local part of the hamiltonian behaves like a source or a
^mk (depending on the sign). It may be noted that this type of complex potentials, called
optical potentials, have been extensively studied for isolated atoms in nuclear physics,
l^or obvious reasons, a medium having scattering potentials with positive imaginary
e mail address : Qsok(9hp2iiaha.ernet.in
© 1998 lACS
366
Asok K Sen
part 7] (sink) at each site is called an absorbing medium and a medium with negative
7] (source) an amplifying medium.
In a disordered chain with ramdom but real-valued site-potentials, almost all the
states are exponentially localized and hence an incident wave (~e'^ propagating in the
positive jc-direction is completely backscattered due to the well-known localization effects
[16]. While a purely ordered chain with fixed absorbing site-potentials (sinks for particles)
leads to an exponential decay of the transmittance (forward-scattering), one naively expects
that the transmittance would increase indefinitely if each of the fixed imaginary site-
potentials is amplifying (source of particles). Interestingly, it was shown by the author [7]
both analytically and numerically that the transmittance asymptotically (in the large length
limit) decays exponentially in both the cases and that the asymptotic decay constants are
identical for an absorbing and an amplifying chain with the same magnitude (1 77 1) for the
strength of the non-hermitian term. This somewhat surprising duality between the
amplifying and the absorbing (ordered) cases was confirmed' later by Paasschens et al [8]
for a classical Helmholtz equation describing propagation of radiation (light) through a
medium with a complex dielectric constant. While the above duality was originally [7]
obtained for a tight binding hamiltonian, recently we [9] observed the same generic
behaviour for an ordered Schrddinger hamiltonian as well. Generically, the transmittance
decays monotonically with length for an absorbing chain, while it increases in an
oscillatory fashion for an amplifying chain upto a length determined by I 77 I, beyond
which the transmittance decays exponentially. The study of disorder in all the works
considered so far has been constrained to the real part of the potential (dielectric
i
constant, in the classical case). In this work, we generalize over our work in [7]
and consider the effects of randomness in the amplification/absorption (imaginary disorder)
at each site.
We consider a quantum chain of N lattice points (lattice constant unity), represented
by the standard single band, tight binding equation :
+C„4l)- (1)
To calculate transport, we consider the open quantum system which consists of the
above chain coupled to the external world (two reservoirs at very slightly different
electrochemical potentials) with two identical semi-infinite perfect leads on either
side. Here E is the fermionic energy, V is iht constant nearest neighbour hopping term
which is the same in both the leads and the sample, £„ is the site-energy, and c„ is the
site amplitude at the n-ih site. Without any loss of generality, we choose = 0 in the
leads and V' = 1 to set the energy scale. Inside the sample, we choose + ie, where
both the real and the imaginary parts could be random [i = V^). For the purpose of this
work, there is no disorder in the real part and we take for simplicity £r = 0. The imaginary
part £, has the form /2], where the constant part 7] may be
either positive or negative or zero and W/ is the width of the uniform random distribution
. >,* I* luituumiy umpujying and absorbing chain 367
in £,, The complex transmission amplitude in the ordered case ( W, = 0) was calculated in [7]
to be
'*1 f-r _ g-'* j g y ^ ^g»*._g-tt jg-it(L+2)
_ ggiAji-g-yZ. ’ ^ ^
where c = -l)^, J = -l)^, (3)
and the decay' length \/\y\ = and the wave- vector are given by
£ = 2cosik = (eJ' +e“i') cosifc,, (4)
and r/ = (el' ) sinfcj. (5)
The transmittance or the two-probe conductance T ~ g 2 = obtained Irom the
eq. (2) is found to decay monotonically (exponentially) towards zero for a set of absorbers
(rj > 0). But, for a set of ampliBers (7} < 0), g 2 increases first to a high value but eventually
(for large L) decays as
Disorder in the real part (e^) in ID is known to give rise to an exponential
decay [16]. Let us consider the case of a disorder in the imaginary term of the potential
with Tj = 0, but W, ^ 0. It may be noted that in this case (for a long enough chain), about
half of the sites act as absorbers (77 > 0) and about half as amplifiers (Tj < 0). Then a^
discussed above, the net contribution to the transmittance from all the sites would
essentially be decaying with a superposition of various decay constants. The modes with the
fastest decay rales will possibly dominate the net transmittance. When tj ^ 0, ilu
distribution of all the decay constants will be asymmetric, and the net decay constant
L
Figure 1. The variation of the logonthmic transmittance as a function of L in
units of the lattice constant for various combinations of n = 0. W, := 0.3. There
is no disorder in the teal part of the site energy. The pure absorbing/amplifying
case means that rj = ± 0.01, W, = 0. the symmetric disordered case means that
Tj. = 0, W, = 0.3; and the asymmetric absorbing/amplifying case imply
disordered cases with r] = * 0.0 1, W, = 0 3. Note that the transmittance decays
faster in the pure cases than in the disordered ones.
^xpecied to be somewhat different from the 77 = 0 case. In Figure I, wc have sbo'\ m die
various cases with T] =4).0l and W, = 0 3. We find that the pure imaginary case . // '
(.'i)-5
368
Asok K Sen
W, - 0) gives rise to a decay length 1/2/= 100 as obtained from the equations above. For
the symmetrically disordered case (ry = 0, W,- = 0.3), the decay length is about 440. Clearly
the latter decay length in the disordered case is much larger than the same for the pure
imaginary case (somewhat counter-intuitive). Finally for the asymmetrically disordered
case (T} = 0.01, W, = 0.3), the decay length is about 120 which is in between the two
extreme cases.
Figure 2. The same as in Figure 1. but for different combinations of T) =
± 0.01, W, = 0.7. Again, there is no disorder in the real part of the site energy.
For these parameters, the transmittance decays faster for the disordered ca.ses
than in the pure ones
In the Figure 2, we have considered another situation with the same ry = 0.01 but a
different disorder W, = 0.7. For the symmetrical disorder case (ry = 0, IV, = 0.7), the decay •
length is about 85 which is smaller than that in the pure imaginary case. This is what one
normally expects to be the role played by disorder (disorder in the real part). Further, in
contrast to that of Figure 1, the transmittance decays faster in the asymmetrically disordered
cases (7y = ± 0.01, W, = 0.7) than in the symmetrically disordered case (ry = 0, W, = 0.7), the
decay length in the former case being about 80.
To summarise, we have studied the transmittance through a ID chain with randomly
amplifying and/or absorbing site-potentials at each site. We find that in contrast to the real
disordered case, the decay of the transmittance (exponential localization) may be faster in
the disordered case. This in particular implies that the scattering from the disorder of this
type may never be incoherent. Thus there is- no cut-off length scale for a crossover from
localized to diffusive behaviour even when the chain consists of both amplifying and
absorbing potentials.
Acknowledgments
The author would like to thank the organisers of the CM Days 97, and the warm hospitality
of the Department of Physics, Vishwa-Bharati University, Santiniketan, during the progress
of this workshop.
Electronic transport in a randomly amplifying and absorbing chain
369
Refcrencei
[1] S John Phys, Rev. Lett. 53 2169 (1984)
[2] A Z Genack Phys. Rev. Lett. 58 2043 (1986); A Z Gcnack and Garcia Phys Rev. Lett. 66 2064 (1991),
N M Lawandy, R M Balachandran, A wS L Gomes and E Sauvin Nature 368 436 (1994); D S Wicrsmu,
M P van Albada and A Ugcndijk Phys. Rev. Utt. 75 1739 (1995)
[3] R L Weaver Phys. Rev. B47 1077 (1993)
[4] A Rubio and N Kumar Phys. Rev. B47 ,2420 (1993); P Pradhan and N Kumar Phys Rev B50 9644
(1994)
[5J V Freilikher. M Pustilnik and I Yurkevich Phys. Rev. Lett. 73 810 (1994)
[6] A Kar Gupta and A M Jayannavar Phys. Rev. B52 4156 (1995)
[7] A K Sen Mod Phys. Utt. BIO 125 (1996); A K Sen ICTP Preprint No. IC/95/391 (1995)
[81 J C J Paasschens, T Sh Misirpashaev and C W J Beenakker Phys Rev. B54 11 887 ( 1 996)
[9] N Zckn, H Bahlouli and A K Sen ICTP Preprint No. IC/97/J3I (1997), Preprint Cond Mat /97 10173
(submitted for publication)
1 1 01 Z Q Zhang Phys. Rev B52 7960 (1995)
1 1 1 J C W J Beenakker, J C J Paasschens and P W Brouwer Phys Rev Lett 76 1 368 ( 1996)
(121 V Freilikher, M Pustilnik and I Yurkevich Preprint Cond. Mat./9605090
(131 T Sh Misirpashaev, J C J Paasschens and C W j Beenakker Physica A236 1 89 (1997)
(14] M Yosefin Europhys. Utt. 25 675 (1994)
(15] V Freilikher and M Pustilnik Phys Rev B55 653 (1997)
(161 A Lee and T V Ramaknshnan Rev Mod Ph\s 57 287 (1985), see also Statterinfi and hualnuiion r//
UVnej in Random Medio cd P Shensi (Sinjzaport* World Scientific) ( 1990)
(171 ICTP Preprint No IC/97/U0 (1997): Pnpnnt Cond Mat/97l0.m
Indian J. Phys. 72A (5), 37 1-377 (1998)
UP A
— an inieinaiional )oumal
Transport and Wigner delay time distribution across
a random active medium
vSancIcep K Joshi*, Abhijil Kar Gupta* and A M Jayannavar*
Inslitulc of Physics, Sachivalaya Marg. Bhubaneswar-T.Sl OD.S. India
' Insiiiutc of Maiheiiiuiical Sciences. Taramani. Chcnnai-60() 1 13, India
Ab.slract ; Wc siiuly ihe wave propagation through a single-channel (single-mode)
coherently amplifying disoidcred medium A new crossovei length scale is introduced in the
legmie ol stioiig disoidei and weak amplification Wc show that in an active medium rellcctancc
aiises due to .synergetic eftcet of localization and coherent amphricalioii Oiii study reveals that
I he tail ot the Wigner delay lime distribution from a disordered passive medium exhibits a
universality in the sense that it is independent of the nature ot disorder
Keywords : Disorder, amphncation. localization, delay time
PAC\S Nos. : 73 23 Ps. 42 25 Bs, 7 1 55 J
Wave (Propagation in passive disordered media continues lo be a subjcci of greal
inieresl |1| Simple models of wave/parliele moving m a random jioleniial can be used
lo describe such variety of phenomena as Anderson locali/ation, photon or light
localization in a random dielectric medium [2], .sound propagation in inhornogenous
media, eU\ These waves, though qualitatively diflcrent. obey the Helmholtz equation
111 appropriate limit. The common oficrative Icatiirc is the interference and diffraction
ol W.iVCS.
In recent years the subjeel of wave propagation in an active lantlom medium, i.c., m
Ihe presence of amplifiealion/absorption, has ailracied considerable attention 13-51 Light
wave propagation through a spatially random but laser-active (amplilying) dielectric
medium is an cxccclicnl laboratory for studying the interplay between disorder-induced
iocali/alion (Anderson localization) and coherent ampli heal ion. To describe the
•miplil Kaiion/absorpiion complex potentials arc uscil leading to rion-Hcrmitian
h-miilionians and hence non-eonservation of particle number. It is worthwhile to note
ih>ii ilic temporal coherence of wave Ls preserved in spile of the amplificaiion/ahsorpiion In
« lyysiACS
372
Sandeep K Jos hi, Abhijit Kar Gupta and A M Jayannavar
present work we will be concerned with two aspects of the transport through one-
dimensionul disordered systems, namely, the statistics of transmission and reflection in
presence of coherent amplification and the universality of the tail of the Wigner delay time
distribution.
The dual role played by an imaginary potential as an amplifier/absorber and as a
reflector has been emphasized in Ref. [6]. Using duality relations it has been shown that the
amplification suppresses the transmittance in the large length limit just as much as
absorption does irrespective of the strength of the disorder [7]. Even though the
transmittance decreases exponentially in the asymptotic limit, the transmission coefficient
(/) is a non-scif-averaging quantity but with a finite well-defined average value [5]. This is
in contradiction with the naive expectation of (f) being infinite owing to the contribution
from the resonant states. However, the fact that even for the case of no disorder (all states
resonant) asymptotically t 0 clarifies this ambiguity [5]. There exists a crossover length
Lj, below which the amplification enhances transmission and above which the amplification
reduces the transmission which vanishes exponentially in the L — > limit. The length
was shown [8] to behave like 1 / , where W is the strength of disorder and 7] is the
strength of amplification. This .suggests that as W -^0, would tend to infinity. This is in
contradiction with the analytical result which clearly shows that is finite and non-zero
even for 0 case. Evidently the result - I / W^frj is valid only in certain region of the
parameter space. To investigate this, we consider the following single-band light-binding
Hamiltonian to model the motion of a qausi-particle moving on a lattice [4,5 J ;
»
V is the off-diagonal matrix clement connecting nearest neighbors .separated by a lattice
spacing a (taken to be unity throughout) and ln> is the non-degenerate Wannicr orbital
associated with site n, where - if] is the site energy. The real part of the site
energy e„ being random repre.sents static disorder and e„ at different sites are assumed to be
uncorrclaied random variables distributed uniformly iP{£n) = IW over the range - W/2 to
W/2. We have taken imaginary part of the site energy 7] to be spatially uniform positive
variable for amplification. Since all the relevant energies can be scaled by V, we can set V
to unity. The lasing medium consisting of N sites (n = I to N) is embedded in a perfect
infinite lattice with all site energies taken (o be zero. To calculate the transmission and
icRcction coefficients we use the well known transfer-matrix method, and the details arc
described in Ref. 14,5].
In our studies wc have set the energy of the incident particle at £ = 0, i.e., at a
inidband energy. Any olher value for the incident energy does not affect the physics of the
problem. In calculating average values m all cases wc have taken 10,000 realizations ol
landoni site energies The strength of the disorder and the amplification arc .scaled with
respect to V. i.c., IV (= \t7V) and i)IV\ The length L= Ua.
Transport and Wigner delay time distribution etc
373
In Figure 1 we have plotted <lnt> against L for ordered lasing medium (W = 0, tj =
0.01 ), disordered passive medium (W = 1.0, Tj = 0) and disordered active medium (1V= 1 .0,
7] = 0.01). The present study is restricted to the parameter space of 7] and W such that rf «
1 .0 and W> 1,0. We notice that for an ordered lasing medium, the transmittance is larger
L
Figure 1. Vuiiaiioii ol <lnt> with L The new length scale which arises for
1 / « 1 (J IS shown by a vertical dotted line The inset shows the vanation of
with Tj tor W -- I 0 The nuiiiencal fit shown by the thick line indicates that
.scales as in this regime
than one. We have taken our range of L upto 300. For a disordered active medium ( W = 1 .0,
I] = 0.01), we notice that the transmittance is always less than one and monotonically
decreasing. Initially, upto certain length, the average transmittance is, however, larger than
ihai in the disordered passive medium (W= 1.0, 7] = 0). This arises due to the combination
of lasing with disorder. In the asymptotic regime transmittance of a lasing random medium
lalls below that in the passive medium with same disorder strength. This follows from the
enhanced localization effect due to the presence of both disorder and amplification together,
i c , ^ < I where ^ is the localization length in the presence of both disorder and
amplification and / is the localization length due to disorder alone. It is clear from the figure
ihai <lnt> does not exhibit any maxima and hence the question of does not arise. We
noiicc, however, from the figure that for random active medium initially <Jnt> decreases
with a well defined slope and in the large length limit <tnt> decreases with a different slope
icorresponding to liKalization length ^). Thus wc can define a length scale 4 (as indicated
m ihc figure) at which there is a cross-over from the initial slope to the asymptotic slope. In
inset of Figure I we have shown the dependence of 4 on 77. Numerical fit shows that 4
''^ales as I / ^ , as we expect 4 r) 0. As one decreases 7], the absolute value
374 Sandeep K Joshi, Abhijit Kar Gupta and AM Jayannavar
of initial slope increases and that of the asymptotic one decreases. Simultaneously, the
cross-over length increases. In the rj 0 limit both initial as well as asymptotic slopes
become identical.
0.0 50.0 1 00.0 150 0 200.0
L
Figure 2. Variation of <lnr> with L for values of W indicated in the figure The
two length scales Li(W0 and Lq{W) associated with the reflectance are shown
with arrows.
We would now like to understand the role of interplay between Anderson
localization and coherent amplification in enhancement of the reflection. In Figure 2 we
plot <lnr> as a function of the length L for a fixed value of amplification strength rj = 0. 1
and for various values of the disorder strength Was indicated in the figure. In the absence
of disorder (W = 0) as one varies length, initially the reflectance increases to a very large .
value through large oscillations and after exhibiting a maximum again through oscillations,
it eventually saturates to a finite (large) value. In the presence of disorder one can readily
notice that initially <lnr> increases and has a magnitude larger than that for 1V= 0 case and
asymptotically beyond a disorder dependent length scale Z.|(W), it saturates to a value
which is smaller than that for a W= 0 case. The saturation value of <lnr> decreases as one
increases the disorder as a result of localization induced by combined effect of disorder and
amplification. Below the length scale LifW) we identify another disorder dependent length
scale L 2 (W). Above L 2 (but smaller than L,) further increase in disorder suppresses the
reflectance whereas below it enhances the reflectance. The length scale being much
smaller than the localization length I for the passive medium, increase in disorder causes
multiple reflections in a sample of size smaller than Li and due to the increase in delay time
we get enhanced hack reflection. Beyond L 2 due to disorder induced localization delay time
decreases and as a consequence we obtain reduced reflecUnce.
We now dissertate on the issue of the universality of the tail of the distribution of
Wigner delay lime of a passive one-dimensional random medium. The delay time in the
scattering process is generally taken to be related to the duration of a collision event or time
spent by the particle in the region of interaction. The delay time statistics is intimately
connected with the dynamic admittance of microstructures. For a single channel the
Transport and Wigner delay time distribution etc
375
distribution of the delay time for a disordered semi-infinite sample has been obtained earlier
by using the invariant imbedding approach [9]. The stationary distribution P,.(t) for the
dimensionless delay time ris given by
^(T) =
(gXKt2 - 1 )( 1 +“ t 2 )’
( 2 )
where A is proportional to the disorder induced localization length and the most probable
value of T occurs at T^ax f T'he long lime tail of the above distribution scales as I/t^.
The average value of Tis logarithmically divergent indicating the possibility of the particle
traversing the infinite sample before being totally reflected, due to the resonances. If the
disordered region is semi-infinite, the reflection coefficient will be unity, and the complex
reflection amplitude will have the form R = If the wave packet is incident on the
disordered sample it will not be immediately reflected back into the lead region, but will be
delayed by time proportional to T = hdO/dE. This energy dependent random time delay
leads to a non-cancellation of the instantaneous currents at the surface involving the
incident and reflected particles. This is expected to lead to a low temperature \/f type noise
that should be universal [9]. A very recent study based on analytical work found the delay
time distribution in the one-channel case to be universal (especially the long time tail is
independent of the nature of disorder) [10]. We would like to examine this through our
study. We would like to emphasize that, in order to obtain P^iT) (eqn. 2) earlier studies
invoke several approximations such as the random phase approximation (RPA), which is
only valid in the small disorder regime and moreover, the correlation between the phase and
the delay time neglected.
In order to calculate the reflection amplitude we use the same model as described
above except for the disorder distribution. We consider three kinds of disorder where the
site energies £„ are assumed uncorrelated random variables having distributions which
arc uniform (P(£„) = l/W), Gaussian (P(£„ ) « e"*"- ) and exponential (P(£„ )
oc ). The transfer-matrix method [4,5] is used to calculate the reflection amplitude
r(£) = and its phase 6(£) at two values of incident energy E^Eq± dE. The delay
time is then calculated using the definition T = hdO/dE. Throughout our following
discussion we consider the delay time tin a dimensionless form by multiplying it with V
and we set ft = m = 1 . In view of the fact that the value of the incident energy Eq will not
change the physics of the problem, in the following we choose Eq = 0 and dE = 26E =
0.002. In calculating the stationary distribution of delay time we take at least 10*
realizations of a disordered sample of length (L) equal to 8 times the localization length {^),
where the localization length is calculated by a standard prescription [4,5].
In Figure 3(a) and (b) we show the numerical data (thin line) for the stationary
distribution Pg{T) of the delay lime Tfor weak disorder (W = 0.5) and strong disorder (W =
20) respectively. The thick line in the figure is the numerical fit obtained by using the
( 5)-6
376
Sandeep K Joshi, Abhijit Kar Gupta and A M Jayannavar
expression for P^it) given in eqh. 2. We see that the fit is fairly good even for strong
disorder ( W = 2.0) for which the stationary distribution of the phase of the reflected wave,
Figure 3. The stationary distribution of delay time P^it) for (a) weak disorder {W = 0.5)
and (b) strong disorder (W = 2.0).
shows (inset of Figure 3(b)) two distinct peaks indicating the failure of the RPA in
this regime.
We now look at the tail of the delay distribution and its universality for the three
kinds of disorder beyond RPA. Since the origin of the tail is due to the appearance ot
I
Figure 4. The plot of tail of P,(r) for the case of uniform (U), Gaussian (G) and
exponential (E) disorder The disorder strength in all the three coses is W = 1 0 The
plots have been shifted on the K-axis to avoid overlap which would obscure the details.
resonant realizations which are independent of strength and the type of disorder, we expect
that the tail distribution would be universal beyond RPA. In Figure 4 we plot the tail
Transport and Wigner delay time distribution etc
377
distribution of Pg(T) for uniform, Gaussian and exponential disorder characterized by the
strength Wb 1.0. The numerical, least-square fit for the expression a / to the long-time
tail dau gives all the cases. The^ values of exponent for the different kinds
of disorder and different strengths of disorder are summarized in Table 1. For the value
Table 1. The values of exponent obtained by leait-square Tit for the
expression a/t^ to the data for different kinds of disorder and different
strengths of disorder.
Kind of disorder ptwW^l.O /}forW=L5
Uniform (U) 1.979 2.006
Gaussian (G) 2.047 1 987
Exponential (E) 2.024 1.961
W B 1 .0, we are in a regime beyond RPA as can be seen from the non-uniformity of the
stationary distribution Pg(9) of the phase of the reflected wave shown in the inset of the
Figure 4. For the stronger disorder case of Wb 1.5 also we obtain the value of exponent p
to be 2. Therefore, our numerical simulation results suggest the existence of universality in
the long time tail distribution.
RcTcreiiccs
[II P A Lee and T V Ramakiishnan Rev. Mod. Phys. 57 287 (1985)
[2] S John in Scattering and Localization oj Waves in Random Media ed Ping Sheng (Singapore : World
Scientific) ( 1990)
[3] P Pradhan and N Kumar Phys. Rev. B50 9644 ( 1994)
[4] Abhijit Kar Gupta and A M Jayannavar Phys. Rev. B52 4156 (1995)
[5] Sandeep K Joshi and A M Jayannavar Phys Rev B56 12038 (1997)
[6] A M Jayannavar Phys Rev. B49 14718 (1994)
[7] C W J Beenakker et al Phys. Rev. Lett. 76 1368 (1996)
[8] Z Q Zhang Phys. Rev. B52 7960 (1995)
[9] AM Jayannavar et al Z Phys. B75 77 ( 1 989)
110] A Comtet and C Texicr / Phys. A30 8017 (1997)
111) S K Joshi, A K Gupta and A M Jayannavar Cond Mat/97 1 225 1
Indian J. Phys. 72A(5), 37^-383 (1998)
UP A
— an intemational journal
Lattice relaxation in substitutional alloys using a
Green's function
SKDas
Azad Physics Centre, Department of Physics. Maulana Azad College,
8, Rafi Ahmed Kidwai Road, Calcutta-700 OB, India
Abstract : We calculate nearest neighbour relaxation in dilute substitutional alloys
Au-Cu, Cu-Au, Cu-Ni and Ni-Cu using a lattice static Green's Function and the Morse potential.
Distant neighbour relaxation is calculated by invoking a continuum approximation Using the
above relaxation, we calculate volume changes in the above alloys. It is observed that the simple
model predicts values which are in reasonably good agreement with the expenmental values in
most cases. But a major discrepancy is found to occur in one case when gold is substituted in
copper. Possible reason for the discrepancy is discussed.
Keywords : Lattice relaxation, substitutional alloy. Green's function and Morse potential.
PACSNos. : 61 72.Ji.«l 72.Ss
1. Introduction
In studies of point defects, lattice relaxation plays an important role. Estimation of above
lattice relaxation using a method based on the first principles is very difficult, and a huge
computational time and effort are necessary for the purpose. We suggest an alternative and
simlificd approach based on a Green's function to determine lattice relaxation. The method
IS very powerful and it can be applied to alloys with a finite concentration of defects with a
suitable modification using Huang's idea [1]. Our work along this line on (K-Cs) alloy is in
progress and will be reported elsewhere, Datta Roy (Paul) and Sengupta [2] employed the
Green's function method to study the variation of nearest neighbour separation with
concentration in alkali halides and they have got good agreement between the calculated
and experimental values. The Green's function approach has been discussed in detail by
Caldwell and Klein [3] and by Tewary [4]. In studies of point defects, it is usual to divide
ihc crystal into two regions. Region I consists of the immediate neighbourhood of the
point defect, and this region is treated in details, atomistically. The remaining portion of the
© 1998 lACS
380
SKDas
crystal is the region II. Using the continuum approximation for the region n, one puts the
relaxation of an ion at a distance r from the defect as K/t^, where AT is a constant which we
call the defect strength constant. For the region I, relaxation, of nearest neighbour to
defect is written as u„„ = (Jr,, where r, is the nearest neighbour distance of the host crystal
and (J is a parameter determined by the Green's function discussed in the next section'. The
procedure of determination of K is also discussed in the next section. Utilising the above
relaxations, we calculate volume changes in dilute substitutional alloys, Au-Cu, Cu-Au,
Cu-Ni and Ni-Cu.
2. Theory
For a monatomic crystal, the relaxation of an ion in the /-th cell along the a (= x.y,z)
direction is given by
i/fn = y g (^
[a) Air.fi
(I)
/ /',
where the lattice static Green's function ^ J is defined by
/)) " ”*"(o 3 ■
X exp[(^.(r, -r;)]/to2(9,j), ( 2 )
where m is the mass of an ion of the pure crystal, 0 is a square x3N) force constant
matrix of the pure crystal, N the number of cells in the crystal, b(q, a, j) the eigenvectors
and CO (q,j)y the eigenfrequencies and 7 = 1 to 3. We have replaced cc^iqj) by an average •
<( 0 ^> according to Einstein approximation.
in eq. ( 1 ) is the additional force
experienced by an A ion in the /' cell along the P (= x,y.z) direction due to the substitution
by a foreign B ion at the origin. From equations (1) and (2) we can find the displacement
^(a) crystal provided the phonon spectrum of the perfect lattice is
known. In the present paper, we have replaced (0^(q,j) by an average Einstein frequency
defined by
(aj2} = (l/3yV)I^^ (3)
We have used Morse potential for the purpose of calculation of <(ij^> and
two-body central potential 0 (r|,)
(a) 2 ) =
+ '(% )]•
For/c.c. crystal and interaction upto the second nearest neighbour (s.n.n.), we can write
( 4 )
Lattice relaxation in substitutional alloys using a Green's function
381
where 2r^ is the lattice parameter, r^ and r^ are respectively the nearest neighbour (n.n) and
second nearest neighbour distances. Hence, relaxation of the A ion at the n.n site of the
defect B ion at the origin is
The Morse potential function is usually written in the form
= i>[exp(-2a(r,y -Tq)) - 2exp(-a(ry -ro))], (7)
where D, a and tq are potential parameters. We shall, however, use a more convenient but
equivalent form given by
0(r,^) = D(exp(~2a(r,^ - a)) - exp(-Q:(ry -cr))), (8)
where D = 4D,a = a and a = Cq - (9)
The potential parameters are determined by using cohesive energy, equilibrium and
bulk modulus. These parameters along with the data used for their evaluation are listed in
Table 1 . If the potential parameters for A -Type atoms are and Ca and ag and Og
Table 1. Parameters of the Morse potential for pure metals and data used for their evaluation.
The harmonic lattice parameter is denoted by d dis calculated by using harmonic density p and
atomic weight Coliesive energy is denoted by U The harmonic bulk modulus is denoted by /I.
Metal
P
tg cm"^)
[9]
d
(Kr® cm)
U[\0]
(I0-*2
erg/atom)
^[9]
(l0-»2
dyne/cm^)
D
(10-” erg)
a
(10* cm"')
''0
(10"* cm)
Au
\9 55\
4.0600
6.0552
1.814
89181
1.6300
2.9214
Cu
9 083
3.5950
5.6067
1 433
7 7988
1 3980
2.6230
Ni
9 020
3.5096
7 1044
1.888
9 8500
1 4100
2.5629
are the corresponding parameters for the B-type atoms, then the parameters for the AB
interaction are determined from the inteipolation formulae [5], [6]
Dab = (10)
= 0.5(a^ +agy (11)
The parameter r^Ag is related to the parameter Gab by
^OAB ~ 2 / ( 13 )
The expression for the volume change can be obtioned from Eshelby's continuum theory of
elasticity [7] and is given by (for/.c.c. crystal)
= 4V2»*^
( 14 )
382
SKDas
where C is the fractional concentration of the impurity ion. The parameter ^ determines the
nearest neighbour relaxation and is determined by the equation
u(llO) = (15)
The defect strength contant K is determined by matching the nearest neighbour relaxation
with the macroscopic relaxation in the rest of the crystal. The relevant equation is
K = ku„„r} = k^rl (16)
where the merging parameter k is taken to be unity following Brauer's [8] assumption.
Hence the equation (14) reduces to
= 4V2.I.
Considering the image term 17], th
(17)
if -
3(1- V)
(I+v) ’
(18)
where vis the Poisson's ratio. Taking v= 1/3 which is the case for almost all metals, we get
1
C V/
= 6V2;r<J.
(19)
3. Results and discussion
From the Table 2, we note that except Cu-Au alloy, agreement between the theoretical anth
experimental values is reasonably good. For Cu-Au alloy the large discrepancy (about 60%)
Tabic 2. Evaluated volume changes loi several alloys and comparison with
experimental results.
Alloy
{MO {AVIV)
(Calculated)
{MO {AVIV)
(Experimental)
Ref
AU'Cu
-0.0091
-0.2426
-0.2687
111]
Cu-Au
0 0276
0 7357
0.4266
fill
Cu-Ni
-0.0024
-0.0640
-0.0900
[11]
Ni-Cu
0.0021
00570
0 0700
[11]
between theory and experiment is due to the large value of the merging parameter k
determined by Brauer's approximation. Actually the value of k should be somewhat smaller
than unity. This point is discussed in our paper [5].
Acknowledgment
I am grateful to Dr. (Mrs) S Dutia Roy of Bethune College, Calcutta, for helping me in the
formulation part of the problem.
Lattice relaxation in substitutional alloys using a Green's function
383
References
[ I ] K Hiuing Proc. Roy. Soc . A190 1 02 ( 1 947)
[2] S Datta Roy (Paul) and S Sengupta Rhys Stat. Sol. (b) 162 89 (1990)
[3] R F Caldwell and M V Klein Phys. Rev. 158 851 (1967)
[4] V K Tewaiy Adv. Phys. 22 757 ( 1 973)
[5] S K Dos, D Roy uid S Sengupta J. Phys. 7 5 (1977)
[6] A Soran Indian. J. Phys 37 49 1 ( 1 963)
[7] J D E<ihelby Solid Stale Physics 3 cd. F Seitz and D Turnbull (New York : Academic) (1956)
[8] P Braucr Z Natutf. 7a 372 ( 1 952)
[9] G Simmons and H Wang Temperature Variations of Elastic Constants and Calculated Agurenate
Properties (Cambridge, Mass. ; MIT Press) (1971)
[10] C Kiitel Introduction to Solid State Physics (London : Wiley) 78 1 80 (1968)
[11] W B Pearson A Handbook of Lattice Spacings and Structures of Metals and Alloys (London . Pergamon)
(1958)
72A (5) 7
Indian J. Phys. 72A (5), 385-389 (1998)
UP A
- an micrnational journal
Semiclassical theory for transport properties of hard
sphere fluid
Bircndfii K Singh and Suresh K Sinha*
Departincnl of Physics. L S College. B B A Bihai Univcisiiy,
Mu7.afrarpur-842 0()l. India
Abstract : The slatisdcal mechanical theory is ties eloped to esiimaic the tiuanium
coireclions to the transport properties (TP's) of the semiclassical hard sphere iSCHS) fluid in
terms of a classical hard sphere (CHS) fluid of piopcrly chosen hard sphere diameter The
eKplicit cKpressions for the shear viscosity and thermal conductivity of the SCHS are given The
numerical results arc discussed The theory is further applied to Nc. where the agreement with
the experiment is good at low temperature
Keywords : Transpon properties, shear viscosity, thermal conductivity, semiclassical fluid
I* ACS No. ; 6I20P
1. Introduction
The transport properties (TP's) of hard sphere fluid aroused considerable interest in recent
veais [1,2]. Considerable progress has been made in recent years in understanding the TP's
ol ihc classical hard sphere fluid |l,2|. However, our understanding of quaniuin fluids of
hard spheres is less .satisfactory [3].
In this paper we investigate the quantum corrections to the TP's such as the
shcat viscosity and thermal conductivity of dense fluid of hard spheres in the
semiclassical limit i.e. at high temperature. The exchange effect is not considered in the
pieseni paper.
2. Ba.sic theory
Wc consider the semiclassical fluid of hard sphere molecules of diameter a. The quantum
etlccis modify the hard sphere diameter [4J. However, the structure of a dense semiclassical
hard sphere (SCHS) fluid is very similar to that of the classical hard sphere (CHS) fluid of
For correspundcncc . Rutnani Mohan Garden, Kalambagh Road Chowk.
Muzaffarpur-842 002, India
© 1998 I ACS
3K6
Birendra K Singh and Suresh K Sinha
the properly choosen hard sphere diameter d. The TP's of the SCHS fluid may be evaluated
through the TP's of the CHS fluid.
3. Effective hard sphere diameter
The second virial coefficient B and equation of stale pp/p of the SCHS fluid, correct to the
first order quantum correction, are given by [5]
B = {2;r(T’ /3)|1 + (3/2V2 )(A/ct)] (1)
and PP/ p = H\ + ri+rj^ - ri)^\
+ 3V2(A/(THT)(l + r}-(l/2)t}2)/(l-Tj)M ( 2 )
where A is the thermal wave length and rj = npaV6 is the packing fraction,
Jn order to determine the effective diameter d of the equivalent CHS molecule,
we consider the second virial coefficient and equation of state of the CHS fluid. They are
given by
B = iTtd^ /3 (3)
and PP^P = \ + 4r]jgid) (4)
where g{d) is the radial distribution function (RDF) of the CHS fluid at the contact and Jh =
npd-/t = Nvhere d'^= die.
Equating eqs. (1) and (3), the effective hard sphere diameter (EHSD) d is
expressed by
d- = [l + (3/2V2 )(A/ct)]‘^’ - l + (l/2V2)(A/cr) (5)’
Thus the quantum effects for the hard sphere fluid is taken into account by replacing the
actual diameter (7 by an effective diameter of 1 + (1/2V2) (A/o)]. This is in accordance with
the result found previously [4J.
Similarly equating eqs, (2) and (4), the RDF gid) of the CHS fluid is given by
g(d) =^‘((T)[l + (3/2V2)a(A/cr)][l + (3/2V2)(A/CT]“' (6)
where g^(a) is the RDF of the CHS fluid of the diameter Oat the contact and given by (6].
g^ia) = (l-r//2)(l-77)-\ (7)
and a IS the correction coefficient
a = {\+T}-T]^ /2)/(l-r]/2)-'(l-r/)-' (8)
4. Transport properties of semidassical hard sphere fluid
We employ the revised Enskog theory (RET) of Beijener and Ernst [ 1 ] to estimate the shear
viscosity p and thermal conductivity K of the CHS fluid. They arc given by [1]
H = «(d)-'[l + (4/5)(4n.,«(d)) + 0.7615 (4n.,«(d))^]/io
(9)
Semiclassical theory for transport properties of hard sphere fluid
387
K = «(ri)->[l + (6/5)(4j)./«(r/)) + 0.7575(47;,«(r/))^ JaTo
(10)
where
Po = (S/lOiiirf^xiwnlkn'/J » /ij jd'^
(11)
Ko = OSklM7td'^)(jtkTlmyi^ =
(12)
with
A/J = (5/16w2)(2«m*T)'/2
(13)
ATJ = (75klMna^){iikTlmyi'‘
(14)
Here fjQ&nd are, respectively, the shear viscosity and thermal conductivity of the ideal
classical gas. m is the mass of a molecule and 7 is the absolute temperature.
With the help of eqs. (5) and (6), eq. (9) can be expressed as
M* = +(l/2V2)//;(A/(T)][l+(3/2V2)a(A/(j)]'' (15)
where = ^lc/^io =«‘^(O)‘'[i + (4/5)(4i7^'^((7)) + 0.7615(47)«‘^(<T))^](16)
IS the shear viscosity of the CHS fluid and
+ 3a#‘ (a)-'[(4/5)(47ji‘((T)) + 1.5230(4?j^‘ (cr))^| (17)
IS ihc first order quantum correction coefficient to it.
Similarly from cq. (10), we obtain an expression forK
K' ^ K/K^ = [/:; +(l/2V2)/(,-(A/<T)][l + (3/2V2)a(A/(T)]'' (18)
where AC* = s' (CT)-' |l + (6/5)(4n«‘ (CT)) + 0 7575(4T)g‘ ((T))^ j (19)
IS the thermal conductivity of the CHS fluid and
k; =/(.* +3a«'((j)-'|(6/5)(4r/g'(cr))+l.5150(4r)^'(CT))^] (20)
IS the first order quantum correction coefficient to it.
We have calculated the shear viscosity /i* (using eqs. (9) and (15)) and thermal
LDiuluciivity IC (using eqs. (10) and (18)) fora range of packing fraction r]at Vo' = 0 and
12
8
4
n
0 0.1 0.2 0.3 0.4 O.S
Figure 1. The shear viiscosity p* of che hard sphere os a function ofn at A/cr s 0 and 0. 1 .
388
Birendra K Singh and Suresh K Sinha
0. 1 . A/a = 0 corresponds to the classical values. These values are shown in Figures I
and 2 as a function of rj. The values of // and K* obtained under different
approximations are comparable at low t] and begin to deviate with increase of Tj.
The quantum effects decrease the values at low value of rj (rj ^ 0.10) while increase
them for 77 >0.15.
0 0.1 0.2 0.3 0.4 O.S
V
Figure 2. Thermal conductivicy X* of the hard sphere as a function of rj at A/ (7= 0
and 0. 1 .
5. Transport properties of real fluids
This theory can be applied to estimate the TFs of real fluids such as Ne whose molecules
interact via the Lennard- Jones (12-6) potential. For such a system. A* is the
quantum parameters.
No experimental results are available for dense semiclassical fluids. In order to test
the accuiacy of our theory, we apply it to calculate /i and K of dilute Ne gas and compare
with the experimental data [3] as well as those obtained previously by us [7]. The
agreement with the experiment is good at low temperature and decrease with increase of
temperature. On the other hand, the previous results [7] are good at high temperature and
deviate with decrease of temperature. Thus these two methods are complimentary to each
other.
Table 1. Shear viscosity /i and thermal conductivity A for Ne.
H X 10^ (g. cm"^sec."')
Kx 10 ’ (Cal. cm-'
' scc.“* dcg“'
f(k)
Present
theory
Previous
theory
Expt.
m
Present
theory
Previous
theory
Expt.
80
1280
1366
1198
90.2
505
562
489
120
1596
1803
1646
m2
914
1102
1092
160
1968
2108
2026
mi
1077
1341
1357
200
2099
2425
2376
Semiclassical theory for transport properties of hard sphere fluid
389
6. Summary
The purpose of the present paper is to develop a theory for quantum corrections to the TP's
of the SCHS fluid using the EHSD method. This theory is applied to estimate the TP's of
Ne. The agreement is good.
Acknowledgment
We acknowledge the financial support of the University Grants Commission, New Delhi.
References
[ I ] H Van Beijeren and M H Emsl Phystca 68 437 (1973)
[2] J R Dorfman and H Van Beijeren Statistical Mechanics Pert B ediied by B J Berne (New York : Plenum)
(1977)
[3] J 0 Hiarschfelder, C F Cuiti.ss and R B Bird Molecular Theory of Gases and Liquids (New York - John
Wiley) (1954)
[4] W G Gibson MaL Phys. 30 1 3 ( 1 975)
[5] Y Singh and S K Sinha Phys. Rep 79 213 (1981)
[61 N F Camhan and K E Siarling J. Chem. Phys. 53 600 (1970)
[7] B K Singh and S K Sinha Proc. C M DAYS - 96. Indian J. Phys. 71A 285 (1997)
Indian J. Phys. 72A (5), 391-395 (1998)
UP A
— an inlenuuional jou rnal
The problem of a composite piezoelectric plate
transducer
T K Munshi
Department of Physics, Kharagpur College, P.O. Kharagpur,
Dist. Midnapore, West Bengal-721 305, India
K K Kundu
Department of Physics, City College, Calcutta'700 009, India
and
R K Mahalanabis
Department of Mathematics, Jadavpur University, Calcutta-700 032, India
Abstract : An attempt has been made to investigate the mechanical disturbance of a
composite piezoelectnc plate transducer executing vibration in the thickness mode which is
taken along A^-axis. The portion of the thickness x s 0 to x = 1 is excited electromechanically and
ai the end x = 0, is applied an impulsive voltage input. The problem involves of interaction of
, two fields, vi:., electrical and mechanical. The method of laplace transform has been used to find
the disturbances and for small lime scale ranging the nature of the disturbances is found to be
linear in nature and it is of the order of 10“^ cm
Keywords : Piezoclectncity, plate transducer, mechanical disturbance.
PACS Nos. ; 77 65 -j. 77 70.-»-a, 77 65 .Dq
1. Introduction
The Studies in ihe disturbances of a piezoelectric material from the stand point of mechanics
ol connnuou.s media have been initialed by [1-4]. These types of problems are very much
inicresling due to their various practical applications in different branch of science and
lechnology 15-7 1. The problems of a composite piezoelectric slab form a very fascinating
branch in the theory of piczoelectricty. Researchers 13,6>8] have investigated the
disturbance in the piezoelectric slab sandwiiched between different medium under variety
ol excitations. Such type of problems are important in view of their direct applications to
practical problems in which conversions of electro-mechanical energies. are involved like
inicrophone, underwater signaling [7,8], etc.
72A(5).g
© 1998 lACS
392
T K Munshi, K K Kundu and R K Mahalanabis
2. The problem and the fundamental equations
We consider a piezoelectric plate transducer executing vibration in the thickness mode
which is the mode generally used [8,9] in the generation of ultrasonic waves. Let the
thickness direction of transducer be taken in the jc-axis and let its extremities be = 0 to
x-X. The portion of the thickness jc = 0 to x s 1 is excited electromechanically. To the end
jc = 0 is applied an impulsive voltage input V given by
V = Vo5(0 (1)
where S{t) is the Dirac delta function and Vq is a constant. Obviously, this constitutes
one of the types of a 'composite' transducer [5,6]. The object of this paper is to
investigate the nature of the mechanical response owing to the voltage input given
by eq. (1).
The mechanical displacement ^ in the x-direction satisfies the equation of motion
= dTjdx (2)
where p is the density of the material and is the stress.
The constitutive relations [1-3] arc
r, = c,,5, + (3)
P, = (4)
where 5| , £] and P] are respectively the components of strain, electric intensity and
polarization and C|], e\ \ and A:|| are the elastic, piezoelectric and susceptibility coefficients*
respectively.
From cqs. {2)-{4) we get
pd^^ldt^ = {c|, -ef, lkit)d'^^ldx^ +«ii j dx (5)
where S\ = 9^/dx. In accordance with our assumption we consider the polarization gradient
of the form
dP^tdx = PQSinax, <o>0 (6)
where Pq is a constant. Eq. (5) becomes
i92$/i9x2 + e„/(c,,ik„-e2)PoSintur = l/v^ (7)
where represent the velocity of propagation in the transducer.
Writing £, = dV/dx wc get from eqs. (5) and (6)
dV/dx = 1/^,1 (Po^rsincof-e,! ^^/<?x)
Solving eq. (7) after taking Laplace transform, we get
$ = /texp^^'*' + £exp + Poe,, /pk,,(»/(p 2 + 6)2 )p 2 (g)
where A, £ are constants and p is the Laplace transform parameter.
The problem of a composite piezoelectric plate transducer
393
To ascertain the constants A and B we must enumerate the boundary conditions of
the problem. The most general type [7] of the problem can be thought of as consisting of a
transducer of impedance ^ situated between the two systems of impedance Z] and Z 2 . The
conditions of continuity of the displacements at the extremeties jc = 0 and x = X as well as at
X s 1 when formulated give rise to
i)
at X = 0,
(^1 )o
= (l)o;
ii)
at X = 1 ,
(f.')l
= (hr.
iii)
at X = X,
(?2)x
= ih),-.
(9)
where the suffixes 1 and 2 denote the entities of the materials at x = 0, x = X respectively
and we write
I, = A, expP'^"« -I-
I’j = A 2 expP''"2 + flj exp-^^'^^ (10)
I' = A'expP^'‘' + B'exp’^'^'' + Po^n^/P^n (P^
To simplify the calculations we consider the transducers to be rigidly backed [10] at the
extremely x = X so that A2 = P2 = = 0 16]. We get
= V/Zj/D[(exp<’''‘-fl 3 / 0 jp 2 + 03 ){Zexpi’('- 2 '>'''
-exp-P’>'' + \c 2 e^^p + aJ)/Z^} + (exp - 03/02^2+0,)
X {exp*"'" - Zexp»’<^-2')''’ - »?lc204 (^ + 03 )/Z* }]
+ 0, ipVle^p'^ + 0j (11)
where D is the material constant and
z= z,-z;/z,+z;,z* = c,(z, +z;). r? = z,+z,/z, -z,,
Z, =pvS= KZ/v(c„-«?,/*,',). C, = ZJZ,-Z,
04 = l-cxpZi’<^-')'''/cxp<><*-W'.03 = 2Z;/1C204 ,
01 = eii/*ii.fl2 = = «n/p*ii
and C2 = v(Zj -Zj )pfc|| /(C||fcii -ej*| ).
Now Laplace inversion of the expression eq. (11) is too much cumbersome, so, to get
an approximate value we have taken the recourse of asymptotic expansion for small
and large values of time [9] to get an idea about the nature of displacement at the
pointxs 1.
394
T K Munshi, K K Kundu and R K Mahalanabis
2. 1 . Calculation of displacement for small values of time
Substituting the value V from eq. (I) we obtain approximate displacement at the point
jc = 1 and is given by
^ - [2(n-i)cjXej/0j
(X-w/vM - {2c,Z, -(i? + l)z;}2/v]x+ 2(77-l)Z;f
- {2c,Z. +(n-l)z;}2x/v + 3f/2 (12)
where ^5 = Cj(Z^ + Z'},
2. 2. Calculation of displacement for large values of time
After a lot of calculation we obtain the displacement for large values of time as
^ Vo//(/)/pe„D (13)
where H{t) is the Heaviside unit function. Since D is the material constant contains 2, X, p
and ^ 11 , the mechanical displacement for the particular case can be obtained easily by
assigning suitable values to Z, X and Vq.
3. Discussions,
For the purpose of numerical calculations, we take the following standard numerical values
of the material constants for quartz [2, 1 1],
p= 2.65 gm/cm^/cii =- 1.2 x lO^en =0.513 x 10*\c|| = 86.74 >< 10’^ dyn/cm^*^
1 ^ = (86.74 X 10'V2.65)''2cm/sec.
To facilitate numerical computations, the values of Z;, Z^., Z', X, Pq ^ have
been chosen suitably [7,9} as Z| = 2, ^ = 1, Z' = .5, X = 10 cm, Po = 1, tu = 1-57 and
Vo = 3CX)v.
Figure 1. Variation of mechanical
disturbance with time.
The numerical values of the mechanical displacement corresponding to small values
of r for X = 2 cm has been shown in Fig. I . It is found that the nature of the disturbances is
linear in nature and it is of the order of 10"^ cm.
The problem of a composite piezoelectric plate transducer
395
References
[ 1 1 W P Mason Piezoelectric Crystals and their Applications to Ultrasonics (New York , D Van Nostrand)
p 84 (1950)
2] W G Cady Piezoelectricity (New York . McGraw-Hill) (1959)
(31 M Redwood J Acousl. Soc. Am 33 527 (1961)
[4j R D Mindlin on the Equation of Motion of Piezoelectric Crystals; Problems on Continuum Mechanics;
Muskhelishvih Aniv. Volume (S.I.A.M Philadelphia, Pcnsylvania) p 282 (1961)
[5] S K Chaneijee Pev. Poum. Phys. 16 113 (1970)
1 6] T K Munshi, K K Kundu and R K Mahalanabis J. Acousl. Soc. Am. 96 2836 (1994)
[7] V 1 Alshits el al Wave Motion (Netherlands) 19 1 13 (1994)
[8] T Musha J Phys Soc Japan 18 1 326 ( 1 963)
[9] LD Ivanov 5m' Phys Crystallogr (USA) 36 466 (1991)
(101 S N Kunikina. N P Kazakov and K G Yurchenko Tech Phys. (USA) 38 50 (1993)
(Ml D F Gibbs Ferroelectrics (UK) 82 1 33 ( 1 988)
/n</ttwy.M)is.72A (5). 397-401 (1998)
UP A
- an initniMioMl j ournal
Semiclassical theory for thermodynamics of
molecular fluids
Tarun K Dcy
Department of Physics, Government Mahila Inter College,
Pumia>854 301, Bihar. India
and
Suresh K Sinha*
Department of Physics, L. S. College, B. B. A. Bihar University.
Muzaffarpur-842 001, Bihar. India
Abstract : Using the 'preaveraged' pair potential method, we derive an effective
Lennard Jones (EU) (12-6) potential for the semiclassical molecular Huid, which includes
the influence of the angle-dependent parts of potential and quantum effects through the
expressions of the effective molecular diameter Of and well-depth ej. We employ this, theory to
calculate the critical point location, surface tension and thermodynamic behaviour along the
liquid vapour coexistence curve of N 2 and 02- In all these cases the agreement with the
•experiment is good.
Keywords : Semiclassical fluid, effective pair potential, critical constants
PACSNos. : 65.50.+m,05.70.Ce.3l 15.Gy
1. Introduction
Aim of ihe present paper is to compute the thermodynamic properties of molecular fluids in
the semiclassical limit. To deal with the problem one may use the 'preaveraged' potential
method, which is an extension of the method employed for the classical molecular
fluid [1,2].
In the present paper, we employ the 'prcaveraged' potential method to derive
effective pair potential for the semiclassical molecular fluids. This effective pair potential is
used to calculate the thermodynamic properties of molecular fluids such as N 2 and O 2 ,
treating semiclassically.
For correspondence : Romani Mohan Garden, Kalambag Road Chowk,
Muzaffarpur-842 (X)2, Bihar, India
(g) 1998 lACS
398
Tarun K Dey and Suresh K Sinha
2. Theoretical basis
We consider a molecular fluid of diatomic molecules interacting via pair potential of the
form
w(rca,a)2) = + u^(rQ)^(02) . (1)
where uoir) is a spherically symmetric central potential and Ug is the angle-dependent
part of pair interaction. Here r = I ri -r 2 1 is the separation between molecules 1 and 2, and
( 0 , represents the orientation of the molecules i. For central potential, we take the Lennard-
Jones (U) (12-b) potential
ML/(r) = 4 e[((T/r)'2 - (CT/r)*], (2)
where e and a are, respectively, the well depth and molecular diameter. For angle-
dependent interaction, we take
Ma =^QQ +Wd.s, (3)
where uqq is the interaction between the permanent quadrupole moment of the molecules,
and U|n is the interaction of the induced quadrupole moment in one molecule with the
permanent quadrupole moments in the other molecule and is the anisotropic dispersion
forces of molecules. We use the explicit angle-dependent form of interaction [3] in the
present calculation.
The two body Slater sum W 2 (r cUiOJi) of the molecular fluid may be used to define
the effective ’preaveraged' pair potential 'Fir). For thermodynamic properties, Fir) m
defined by the relation
jexp[-)3'#'(r)]</r = dr, ' (4)
where p = and ( represents an unweighed average over the molecular
orientations to, and (O^.
In the semiclassical limit, where the quantum effects are small and treated as a
correction, the Slater sum is expanded about the Boltzmann factor [3]. Substituting the
expansion of the two-body Slater sum [3] and eq. (1) in eq. (4), and expanding and
integrating over the angles, we obtain an expression for the effective 'preaveraged' pair
potential for the semiclassical molecular fluid.
V/(r) = 4c([(l + A-,2)(t7/r)'2 / r)'*]
-[(a/r)‘ + (Lt+AgKa/r)^ + A, 0 «J/r)'»]). (5)
The coefneients L„ and A„ are given in terms of the reduced quantities T‘ =kTle.
a'=a!a\ Q’^=Q^lea\ A‘=hla(mey'^ and S'=hl(ley'^ as Lj =
(5/8;r2T•)A•^ L,4= (W! Ak'^T')A‘^
399
Semiclassical theory for thermodynamics of molecular fluids
A8= (3l4)a*Q*K
A^q = (7/20T‘)(Q*2)2 ^ pi%n^T)a^Q*^A*^
- (53 1
>\,2 = -(9J5T*)K^(]-h].9K^) - (7/247t^T*^)(Q*^)^A*^.
Here A* and S* are quantum parameters (Here m is mass and / is the moment of inertia of
a molecule). In the classical limit, A* = 5* = 0 in the expressions of and i4„,.
Eq. (5) can be expressed in the LJ (12-6) potential form by simply replacing
cr — > Or (T* ,A* ,5* ) and e^e-p (T* ,A* ,5 * ) in eq. (2).
As the quantum effects are largely determined from the hard-core [4], we
approximate r / (T » 1 in Z.^. On the other hand we approximate r / cj = / cr = 2 [5J in
Then eq. (5) can be expressed in the U (12-6) potential form
V(r) = 4e,. [((Jr /r)'^ - ( 0 ^ lr)0].
(6)
where
a'' s a-r lo =
(7a)
s €j / € — (1+j4j2+Z.|4)F^,
(7b)
F = (l+La+A82-''3+A,o2-2'3)/(H-/4|j +L,^).
(7c)
Thus the effective 'preaveraged' pair potential is expressed as the effective LJ (ELJ) (12-6)
potential form in terms of effective well depth ej (T* ,A* ,S* ) and molecular diameter
Cj iT\A* ). Then the system can be treated as the classical LJ (12-6) system.
In the. following sections, we apply our theory to investigate the thermodynamic
properties of N 2 and O 2 . The force and quantum parameters employed previously by Singh
and Sinha [6] are used here. We have investigated the properties with and without the
dispersion interaction (the results are not reported here). On the basis of this study, we
have neglected in case of O 2 in the present investigation.
3. Critical point location
In this section, we employ our theory to study the critical temperature T^, critical volume
V' and critical pressure for the semiclassical molecular (SCM) fluid as well as classical
molecular (CM) fluid. For the classical U (12-6) fluid they are given by
T*skTj€ =1.26, (8a)
VlsVJNa^ =3.1, (8b)
Pi = P^M^f € =0.117. (8c)
In order to obtain the critical constants for the molecular fluid, we replace and
(Jr in eq. (8) and write
r; = 1.26 e", (9a)
72A (5).9
400 farm K Dey and Suresh K Sinha
v;'= 3.1 <t^3_ (9b)
P* = 0.117 e'' /cr-'’ (9c)
Eq. (9a) may be solved by the interative process. Knowing T' , one may obtain V*
andP;.
Table 1. Critical constants of N 2 and P 2 -
System
r,(K)
Vj.(cm3)
Pf. (atm)
PcV^/RTc
N2
SCM
126.86
89.16
33.61
0 288
CM
127 70
88.76
33.99
0.288
Expt.
126 10
90.10
33.50
0 292
02
Theory
153.94
72.84
49.86
0.288
CM
154.68
72.59
50 27
0 288
Expt.
154.40
74.40
49 70
0.292
The results of N 2 and O 2 obtained under the classical and semiclasical limits are
compared with the experimental data [7] in Table 1. The agreement is good. From the
study, we find that the quantum effects decrease the values of and while increase the
value of Vc-
4. Liquid-vapour coexistence curve
In this section, we apply our theory to study the behaviour of the molecular fluid on the
liquid- vapour coexistence curve. The behaviour of the classical LJ (12-6) fluid may be
described by the following equations [8]
v; /Vf- = 1 + (3/4)(l-r* ITl) + (7/4)(l-7* (10)
v; / v'* = 1 -I- (3/4)(i-r* /r!) - (7/4)(i-r* (ii)
where ^|* , V*andV* are the reduced liquid, vapour and critical molar volume. Here T* =
1.26 and V* = 3.1 . These equations show the behaviour of V* / V* for 7* / 7‘* ^ 1.
Figure 1. The comparison of theory
with experiment for the liquid-vapour
coexistence curve of N2-
Semiclassical theory for thermodynamics of molecular fluids
401
For the SCM (or CM) fluid, where € is replaced by € 7 - and o by Oj, eqs. (10) and
( 11 ) can be expressed as
v;/v; = ((T;^/(TMMi+( 3 / 4 )(i-r*/i. 26 )
+ (7/4)(l-r*/1.26)i'M (12)
= (cy;/(TMHi+(3/4)(i-r*/i.26)
-(7/4)(l-r /1.26)‘/3] (13)
where is the value of G'^ at T* = 7* , Then the reduced density is p* s V*"* .
The values of density p obtained under the semiclassical and classical limits for N 2
are demonstrated in Figure 1 as a function of T $ 7* along with the experimental results
|9]. The agreement is good. The quantum effects decrease the values of p\ while increase
the values of p^.
5. Concluding remarks
The effective pair potential is expressed in the EU (12-6) potential form by replacing
iT\A\5*) and G-^ G^iT* ,A\5*). Then the system can be treated as the
classical LJ ( 12 - 6 ) system. The ELJ (12-6) potential is employed to estimate the
theiTnodynamic propenics of N 2 and O 2 over a wide range of temperature and density. In all
these cases, the agreement with the experimental data is good.
Acknowledgment
Wc acknowledge the financial support of the University Grants Commission, New Delhi.
RpftTcnces
1 1] A K Singh and S K Sinha Phys Rev. A30 107g (1984), Phys Rev. A35 295 (1987)
f21 G Sicll. J C Rasaioli and H Narang Mol. Phys. tl 1393 (1974)
m K P Shukla, L Pandey and Y Singh J. Phys. C12 4151 (1979)
(4] M H Kalos, D Levesque and L Vcrict Phys. Rev. A9 2178 (1974)
I Om Singh and A W Joshi Pranuma 15 487 (1980)
[6] A K Singh and S K Sinha Mol Phys. 61 923 (1987)
1 2] JO Hirschftldcr, C F Curtis and R B Bird Molecular Theory of Gases and Liquids (New York ■ John
Wiley) (1954)
\^] R A Young Phys. Rev. 23 1498 (1981)
l‘^l F Dm Thermodynamic Function for Gases Vols. 1 to 3 (London : Butteworth) (4961 )
Indum J. Phys. 72A (5). 403-406 (1998)
UP A
— an intemational journ al
Stability of Ag island films deposited on softened
PVP substrates
Manjunatha Pattabi and K Mohan Rao
Department of Materials Science, Mangalore University.
Mangalagangotri-574 199, India
Abstract : The results of the aging studies carried out on Island silver films deposited on
Poly(2-Vinylpyndine) (PVP) coated glass substrates held at temperatures above the glass
transition temperature of PVP are reported in this article The instability or aging of island silver
films reduced considerably on softened PVP substrates compared to films on rigid substrates.
This is attributed to the formation of sub-surface particulate structure, which is confirmed by
X-ray Photoelectron Spectroscopy (XPS)
Keywords : Island films, PVP. aging. XPS
PACS Nos. : 68 55.-a, 73.61.Al. 8l.l5.Ef
1. Introduction
A discontinuous metal film deposited on a dielectric substrate may be regarded as a system
of randomly distributed metallic and dielectric regions. These films have attractive
electrical properties which can be exploited for device applications like high sensitivity
strain gauges, temperature sensors etc. But, the main hurdle is their temporal instability or
aging, which manifests itself as an irreversible resistance increase with time, even in
vacuum. Aging is attributed to the mobility of islands followed by coalescence, leading to
an increased inter-island spacing [1].
It is reported that vacuum deposition of materials such as Se, Sn, In etc. onto
softened polymer, results in the formation of sub-surface particulate structure [2]. The
morphology of such sub-surface structure is dependent on deposition factors and polymer
metal interaction [2,3], The studies on the electrical properties of island films deposited on
softened substrates are sparse. The dispersion of very small particles can be obtained in a
PVP matrix due to the interaction of lone pair from nitrogen atom in PVP with silver [3].
Therefore, one can expect that silver island films deposited on softened PVP substrates
© 1998 lACS
404
Manjunatha Pattahi and K Mohan Rao
would result in reduced aging. In this paper we present the results of studies carried out on
the aging of silver island films deposited on PVP substrates, held at a temperature much
above the glass transition temperature.
2. Experimental details
Silver (purity better than 99.99%) films of various thicknesses were evaporated onto PVP
coated glass substrates held at 425 K and 455 K in a vacuum of 8 x 10"® torr. The film
dimensions were 1 cm x 1 cm. A Chromel Alumel thermocouple was used to measure the
substrate temperature. A quartz crystal monitor was used to measure the deposition rate as
well as the thickness of the material deposited. The deposition rate was 0.4 nm/s for all the
films. The film resistance was measured using a Keithley DMM 2001. XPS was used to
determine the formation of sub-surface structure.
3. Results and discussion
Figure 1 shows the variation of normalized resistance with time for various film thicknesses
at 425 K arid 455 K. An aging curve for the film depc^sited on PVP at room temperature is
Time (min)
Figure 1. Variation of normalised resistance with time for .silver films deposited on PVP
substrates.
Stability ofAg island films deposited on softened PVP substrates
405
also given in the same figure for comparison. The variation of resistance on a softened
substrate is considerably reduced when compared to a film on a rigid substrate. Figures 2
Figure 2. C Is core level XPS
spectrum at two different ETO
As for Ag on PVP Continuous
1100-75“, filled circlcs-45®
and 3 show the C Is and Ag 3d XPS spectra for a typical silver film deposited on PVP held
at 425 K at two electron take off angles (ETOA defined as the angle between electron
emission and surface parallel) of 75° and 45°.
Figure 3. Ag 3d core level XPS spectrum at two different ETOAs for Ag on PVP. Filled
squareS'7S°, filled circle5-45“
The mobility coalescence model predicts the aging rate to increase with an increase
in mobility. The mobility of the islands on the substrate surface is an activated process and
406
Manjunatha Pattabi and K Mohan Rao
ai higher temperature a higher aging rate is expected. But, in the case of silver deposited on
PVP held at 425 K and 455 K, the aging is considerably less when compared to a film
deposited on PVP at room temperature (Figure 1). This deviation can be readily understood
if one assumes that the silver islands are formed beneath the PVP surface. The silver islands
inside the polymer would have much lower mobility due to the polymer viscosity resulting
m a reduced aging rale. Copper island films deposited on softened polymethylmethacrylate
(PMMA) exhibited higher aging rale where only a surface deposit was formed [4]. Silver is
known to form a sub-surface structure [2] and therefore, the reduced aging of silver clearly
indicates the formation of a sub-surface structure. Further, silver deposited on rigid PVP
(substrate held at room temperature) shows film continuity at a thickness of 25 nm,
while films on softened PVP substrates have resistances in the range of megaohms even
for a thickness of 200 nm. This is possible only when silver is dispersed inside the PVP
matrix.
Angle dependent XPS studies is an useful technique for studying depth profiles. This
IS made .possible by the small electron inelastic scattering lengths in condensed matter
(typically 2-5 nm). The depth sensitivity of the spectroscopy can be changed by varying
the ETOA. The C 1 s signal at two different ETOAs show little change (Figure 2) indicating
that carbon is homogeneously distributed within the surface region [5]. Considerable
attenuation of the Ag 3d signal is observed at a lower ETOA. This implies that the silver is
buried beneath a layer of PVP [5], thus confirming the formation of a sub-surface structure.
Detailed analysis of the XPS studies will be published elsewhere.
4. Conclusions *
The aging of silver films deposited on softened PVP is very much less compared to Ag
films on a rigid substrate indicating the formation of a sub-surface discontinuous film
structure. The XPS studies at two different hlOAs confirm the formation of a sub-surface
silver film.
Acknowledgment
The authors thank the DST, Govt, of India, for the research grant.
References
[ 1 1 J G Skofronick and W B Phillip.s J. Appl. Phys. 38 479 1 (1967)
12] G J Covacs, P .S Vincett, C Trumblay and A L Pund.sak Thin Solid Films 101 21 (1983)
[-3] Martin S Kunz, Kenneth R Shull and Andrew J Kcllock J. Appl. Phys. 72 4458 (1992)
[4] Manjunatha Pattabi, M S Murali Sastry and V Sivararnakri.'Jhnan J Appl Phys 64 437 (1988)
[.*'] C S Fadicy Progress in Solid State Chemistry eds J McCaldin and G Somorjai, (New York . Pergamon)
p 265 (1 976)
Indian J. Phys. 72A (5), 407-41 1 (1998)
UP A
— an international journal
Energetics of CO-NO reactions on Pd-Cu alloy
particles
Mahesh Mcnon and Badal C Khanra
Condensed Matter Physics Group, Saha Institute of Nuclear Physics.
1 /AF Salt Lake, Calcutta-700 091, India
Abstract : The bond-order conservation- Morse potential model (BOCMP) has been
used to study the CO-NO reaction on Pd-Cu alloy particles having total number of atoms
per particle in the range of 200-1300. Monte Carlo simulation has been performed to find
the surface composition of the particles. Cu has been found lo segregate to the surface for
particles of all sizes — the extent of segregation slowly increasing with particle size Activation
energy analysis shows that CO 2 formation is the rate-limiting step for the overall CO-NO
reaction. The most active sites are found to be the three-fold hollow adsorption sites with
three Cu nearest neighbours, and the adsorption sites with two Cu atoms and one Pd atom as
nearest neighbours.
Keywords : Adsorption, (BOCMP) model, segregation, activation energy. CO oxidation,
NO reduction
PACSNos. ; 68 10.Jy,82.65Jv
1. Introduction
Auiomobile exhaust gases have a composition of approximately 15% carbon monoxide
(CO) and 10% nitrogen oxides (NOj^). —the rest consisting of a large number of unburni
hydrocarbons and some sulphur dioxide etc [1-3]. Efficient pollution control would mean
almost total conversion of CO (by oxidation) to CO 2 ; reduction of NO to N 2 and oxidation
of the hydrocarbons to CO 2 and H 2 O. Intensive research over last two decades on suitable
catalysts for simultaneous oxidation of CO and hydrocarbons on one hand and the reduction
of NO on the other has led to the development of a near-ideal catalyst, namely, Pt-Rh/ ceria
11-3]. However, in view of the excessive use of Rh and Pt in the auto-catalytic converters
the world stock of Rh and Pt is fast dwindling and it is important therefore to find some
alternative but cost-effective catalysts. Pd is a good oxidation catalyst and Cu is a good NO
reduction catalyst since Cu dissociates NO very efficiently. It is the purpose of this present
work to study the adsorption, segregation and catalytic properties of these metals and their
alloys for CO oxidation and NO reduction.
© 1998IACS
72A(5)-10
408
Mahesh Menon and Badal C Khanra
A full investigation on the CO-NO reaction on a catalyst surface would.require the
knowledge of the possible reaction steps, the heat of adsorption of the reactapts like CO,
NO, O, CO 2 etc., activation energy for dissociation of NO, activation energy for
recombination of CO and O to form CO 2 and the activation energy for formation of N 2 . In
addition, for supported bimetallics one should also have a knowledge of the surface
composition of the catalyst particles. In section 2 we present the results for surface
composition of Pd-Cu particles obtained by Monte-Carlo simulation. In section 3 we
discuss the possible reaction steps and the rate-limiting step on the basis of heat of
adsorption and the activation energies for various steps. These are done on the basis of
bond-order conservation model of Shustorovich [4]. In section 4 we calculate the activity of
the Pd-Cu bimetallics as a function of the particle size. Also we discuss the role of local
surface order on the activity.
2. Segregation in Pd-Cu bimetallic particles
The bimetallic particles, generally used as catalysts, have diameters in the 2 nm-3 nm range
having the number of atoms per particle in the range of 200-5000. It is a general property
of the bimetallic alloys that atoms of one constituent may preferentially enrich the surface.
We use here the theoretical Monte-Carlo technique to calculate this surface composition of
Pd-Cu particles. For simplicity we consider the fee cubo-octahedron shaped particles. This
is because, thermodynamically for particles of 2 nm-5 nm size cubo-octahedron is the most
stable geometry. The details of the Monte-Carlo technique used in this calculation is^
described elsewhere [5]. Essentially, the method relies on finding the most stable
configuration energy with respect to switching the A and B atoms of an alloy A^B. For the
bond energy between two nearest neighbour atoms j and it with the coordination n and m
respectively we use the formula
Ejt=[o>j,/Z] + [£[(«)/«] + [£t(m)//7i]
where j, k = A or B atom and cUy* = 0 if; ^ k. The first term, known as the interchange
energy is obtained from the molar excess heat of mixing [5]; the second and third terms
denote the cohesive energy per bond of the j-ih atom having n coordination and the k-ih
atom having m coordination respectively. For this calculation the interchange energy is
found to be -0.0197 eV. The cohesive energy per bond is calculated from the surface-
modified pair potential formula is given in Ref. [6].
The results presented here correspond to the Pd 5 oCu 5 o composition in the bulk. We
have calculated the surface composition of particles with 201, 586 and 1289 atoms. The
dispersion (D) and the fraction of surface sites covered by Pd and Cu atoms are presented in
Table l.Njis the total number of atoms in a particle. is the surface concentration. It may
be noted that for 50% Cu concentration in the bulk the surface concentration of Cu in the
surface is much higher for all the particles. The surface composition as obtained by MC
Energetics of CO-NO reactions on Pd-Cu alloy particles
409
simulation for a typical 586-atom particle is shown in Figure 1 . The shaded atoms are Cu
atoms. It may be noticed that the Cu atoms occupy the corner and edge sites of the particle.
Table 1. Site statistics of fee cubo-octahedron Pd-Cu particles
201
586
1289
D
0.6
0.46
0 37
X, (Pd)
0 35
0.335
031
Xv(Cu)
0 65
0 665
0 69
Figure 1. Surface composition of 586-
atom Pd-Cu cluster
The Monte-Carlo simulation also gives the average number of surface Pd (and Cu)
neighbour per surface Pd (and Cu) atom. These numbers are useful to study the role of
surface ordering in catalytic activity.
3. Surface reactions and activation barriers
The overall reaction step may be given as
CO^+NOj^COi+jNj (1)
where the suffix ‘s' denotes the components in the adsorbed phase. But in reality it is the
iniermcdiale steps which are important and should be considered seriously to find the rate-
limiting step. The intermediate steps are the following :
CO^ — > COj,
(2)
NO^ -4 NO,,
(3)
NO, N, + 0 „
(4)
CO2,
(5)
N, + N, N2.
(6)
NO, + N, -► N2O,,
(7)
NA -►.N2 + 0,.
(8)
410
Mahesh Menon and Badal C Khanra
The expressions (2) and (3) denote the adsorption of the molecules from the gas
phase. On the basis of available experimental adsorption energies and the bond-order
conservation — Morse potential model [4] the activation energy for various steps may be
found. The rale of adsorption from gas phase depends on the sticking coefficients of the
relevant molecules. But, once the molecules are adsorbed further reactions take place with
activation energy as given in Table 2. It may be mentioned here that since experimental
adsorption energies for particles are rare we calculate first the activation energies for single
crystal (111) surface of the metals. For particles with number of atoms larger than 2(X) the
average adsorption energy of atoms and molecules and the activation energies for various
steps may then be calculated by a statistical analysis [7].
Table 2. Activation energy (in k cals/mole).
Metal
Pd (111)
cudii)
NO, ->Nj + 0,
9 1
1.0
COy + 0 , — > CO 2
24
20
N, + N, N2
43
25
NO, + N, N 2 O,
25.7
12 5
N2q,,-^N2 + 0,
-VC
-ve
From the results presented in Table 2 the following conclusions are drawn. It may be
noted that Cu dissociates NO very fast. Formation of N 2 from two adsorbed N atom^»
requires higher activation energy than formation of N 2 O which subsequently dis.sociales
spontaneously into N 2 and O. However, in view of the alloy segregation property since
65-70% surface sites are occupied by Cu atoms surface properties of Cu will control the
overall surface reactions. In this respect since CO 2 formation on Cu bas higher activation
energy, this CO 2 formation is most likely to be the rate-limiting step. For Pd sites also the
activation barrier for CO 2 formation is close to the activation barrier for N 2 O formation.
Thus in all likelihood for the Pd-Cu single crystal alloy CO 2 formation from adsorbed
CO and O is the rate-limiting step. From the energetics of adsorption on particles (since
adsorption energy varies very slowly with for Nj- > 200), the rate-limiting step is found
to be the same for Pd-Cu particles.
4. Activity of Pd-Cu bimetallic particles
The activity of Pd-Cu particles for the CO oxidation reaction is expressed as
‘a’ = ^/lp,(X,)exp(-£,7/fr) (9)
where A is a constant depending on the collision frequency of the gas-solid system
p, {X is a steric factor and is a function of the surface geometry of the system. It
denotes the probability to find a chemisorbed bond with i Pd and 3-i Cu nearest
neighbours.
411
Energetics of CO-NO reactions on Pd-Cu alloy policies
Usually, p, (XJ is given by the binomial distribulion
p,(Xj = [3!//!(3-0!]x;,(I-Xj-'"' (10)
The number 3 comes into picture since the adatoms are assumed to occupy the centre
positions (sites with three-fold symmetry in the (! 1 1) surface of /re lattice). The calculated
activity for CO oxidation is shown in Figure 2. The activity increases linearly with increase
Total
^■0(3Cu n.n)
^ l (2 Cu and 1 Pd n.n )
_t- - — l^^i:^y,ind-2Pd n.n)
300 SoTT m ■
Figure 2. Activity of Ptl-Cu allojr for CO oxidation and contribution from
different i values (Ecjn 9)
111 the size of the particles. This is because, Cu segregates increasingly with particle size to
the surface; and Cu has a lower activation barrier for CO oxidation than Pd. Therefore, with
increase in particle size the activity increases. The main contribution to the activity comes
lioiii adsorption sites with 3 Cu sites as nearest neighbours and adsorption sites with 2 Cu
and one Pd atom as nearest neighbours.
Acknowledgment
Mahesh Menon thanks Council of Scientific & Industrial Research for the award of a senior
rc.scarch fellowship to work on the project.
r
Ki’fcrence.s
1 1 1 K C Taylor Oitalwx . Scicncf and Technolof^y eds. J R Anderson and M Boudarl (Berlin Springer
Verlag) Vol 5 p 1 19 (1984)
121 KCTaylor Catal. Rev.-Sa. % 35457(1993)
l-^l K C Taylor Cuuilysis and Auwmotive Pollution Control eds. A Crucq and A Frcnnct (Am.sterdam .
Elsevier) p 97 (1987)
14 ] E Shustorovich and A T Bell Surf Sci 289 127 (1993)
f‘'l J L Roussel, B C Khanra, A M Cadroi, F J Cadelc Sanios Aires, A J Renouprez and M Pellaiin Surf Sa.
352-354 583(1996)
(N J K Sirohl and T S King J Catal. 118 53 (1989)
1^1 R Van Hardeveld and F Hartog Surf. Sd. 15 189 (1969)
Indian J. Phys. 72A (5), 413-416 (1998)
UP A
— Ml iniemational jour nal
Inhomogeneity of vortices in 2d classical XY-model :
a microcanonical Monte Carlo simulation study
S B Ota and Smita Ota
Institute of Physics, Sachivalaya Marg,
Bhubaneswar-751 005, India
Abstract : The extended 2d classical XY-model has been studied using microcanonical
Monte Carlo simulations. Simulations have been carried out on 30 x 30 spin system on a square
lattice. We find that the maximum inhomogeneity of vortex distribution occurs at a temperature
which coincides with the position of specific heat peak in the Kosterlitz-Thouless (KT) case and
in the coexistence region in the first order case. The inhomogeneity is found to be more in the
KT case as compared to the first order one.
Keywords : XY-model. vortices, microcanonical
PACS Nos. ; 75. 10 Hk. 02.70.Lq, 05.20.-y, 05.70.Fh
Study of the two dimensional (2d) classical XY-model has unfolded several interesting
physical properties of 2d system [1] and still demands further investigations. The 2d
XY-model has been considered in the literature to understand the high temperature
superconductors (HTSC), which is however not completely successful [2]. The
experimental data can lead to interesting information on HTSC if one knows the right
extension of the XY-model that explains them [3]. Thus, there is a need to understand the
nature of the vortex-driven transition in 2d XY-model. The lack of long range order,
the presence of topological defects called vortices, and the Kosterlitz-Thouless (KT)
transition are some of its notable properties known as yet [4-21]. The two types of
excitations that dominate at low temperatures are spin waves and vortices. The low-
temperature phase has only bound vortex-antivortex pairs and the KT transition is
associated with the unbinding of the vortex-antivortex pairs. Investigations have been
carried out for the possibility of a first order transition in this model, without disturbing
the essential symmetry. Domany, Schick and Swendsen [22] suggested that by sufficiently
reducing the width of the nearest neighbour interaction potential in this model a first
order transition can be observed. The first order transition is understood to result as due to
<E) 1998 lACS
/ of vortices. In this paper, wc report the
Me classical 2d XY-system, which has sofar
id classical XY-model is given by :
y{(e.-9j)/2)\ ( 1 )
al meaning. For = 1; the Hamiltonian reduces to the
/lich admits the KT transition. By increasing the value of
ide narrower and for > 10 the transition becomes first-
Wfc ^ canonical MC simulations on a square lattice having 30 x 30
spins 1 23,24). \Vv periodic boundary conditions and have calculated system
temperature (7), vorticity, magnetization square and their respective standard deviations at
each given total energy (E). We used 1 x 10^ MCSS for equlibration and 1x10'’ MCSS for
averaging. The accuracy of the mean value of the physical quantities was estimated by
performing block averages consisting of 5 x 10^ MCSS each and then finding the standard
deviation of block averages. The KT transition temperature occurs at 0.9 and the maximum
of temperature dependence of specific heat occurs at 1 .09.
We analysed the Monte Carlo configurations evolved during the simulation to study
the inhomogeneity of the vortices across the transition. To this end we estimated t^ie
average number (K„(r)) of vorlices/antivorlices at a distance r from any given vortex. From
this we obtained the number of positive (negative) vortices (V^)) surrounding a
positive vortex within a di.stance of ^^2a and 2^/2a {a is the lattice spacing). Forp" = 1. ii
is seen that when determined over a distance of , decreases with lemperaiurc
initially upto T= 1.1 and then increases with further increase in temperature. The decrease
of with temperature is not observed, when determined over a larger distance of 2^f2a
Whereas, shows a steady increase with increase in temperature. On the other hand, lor
p^ = 50, both and arc seen to increase with increase in energy even when
determined over a smaller distance of ^!2a [25].
We next determined the net vortex charge (V^,) within a distance of V2fl and
lyflM from a vortex as functions of T (or E). Figure 1(a) shows the temperature
dependence of for p^ = I , and comments on some features of this graph are in order.
Firstly, goes through a maximum. Secondly, the magnitude of is reduced as the
distance increases. Finally, the maximum occurs at T = 1.1, which corresponds to the
specific heat peak. Similar graph is shown in Figure 1(b), forp^ = 50. The behaviour is
qualitatively similar to the case for p^ = 1, except for the following differences. The
magnitude of is comparatively smaller. The maximum of occurs in the coexistence
region of the first-order transition. The vortices arc therefore, not distributed
homogeneously in the lattice. Examination of the configurations revealed the presence of
Inhomogeneity of vortices in 2d classical XY-model etc
415
i^ortex clusters in the critical region, which has also been pointed out by Toboknik
and Chester [13].
Figure 1. (a) The temperature dependence of net voitex charge in a radius
of yjla (+) and l-fla (x) for = I. (b) The energy dependence of net
vortex charge in a radius of Via (-f) and 2 Via (x) for = SO. The data
represents averages over 1 x 10^ MCSS.
We have not yet come across an explanation of the observed inhomogeneity of
vortices. However, certain features can be understood as follows. In the low temperature
(energy) insulating phase the vortices are bound tightly which results in a small value of
In the high temperature (energy) Debye-Hiickel regime, is also small due to the
presence of large number of free charges in the liquid phase. We speculate that clusters of
vortices in the critical region are responsible for the peak in The difference between
= 50 and 1 cases can be attributed td the change in interaction ’^m ln(r la) io r la as
changes from 1 to 50 [3].
In conclusion, we have studied the vmtices in Che classical 2d Xy^modcl undergoing
KT and first-order transition. We have reported a subtle aspect of the vortices^ that is the
mhoniogeneity of vortex distribution.
Acknowledgments
SBO acknowledges discussions with Dr. A Baiimgartner. The authors thank Dr. V C Sahni
lor helpful suggestions on the manuscript. So acknowledges the Council of Scientific &
Industrial Research, India, for financial assistance.
References
[1 1 For a review, see M N Barber Phys Rep , (Ne(herland.s) 59 376 ( 1 980)
[2] L J de Jcwigh Solid State Commun. 70 955 (1989)
[^1 F Mila Phys. Rev. B47 442 (1993)
14] N D Mermin and H Wagner Phys. Rev. Utt. 17 1 133 (1966)
151 VLBai!aMkii5w.Phy5./£r/»32 493(IOTO^
1^1 J M KoenhMtz md D J Thonleu J. Phys C9 L134 (t972>; / Phys. CIO 1 181 (1973); J M Kortcrlttz
' Fftyj.C7 1046(1974)
^2A(5)-11
416
S B Ota ar^ Smita Ota
[71 F J Wegner Z Phys. 206 465 ( 1967)
[8] V L Berezinskii Sov. Phys. JETP 34 610 (1971)
[9] J Zittaitz Z Phys. 23B 55. 63 (1976)
[10] J V Jose, L P Kadonoff, S Kirkpatrick and D J Nelson Phys. Rev. B16 1217 (1977)
[11) C Kawabata and K Binder Solid State Commun. 22 70S ( 1 977)
[ 12] S Miyashita, H Nishimori, A Kuroda and M Suzuki Prog. Th£or. Phys. 60 1669 (1978)
[13] J Tobochnik and G V Chester Phys. Rev. B20 3761 (1979)
[14] W J Shugard et al Phys. Rev. B21 5209 (1980)
[15] J E Van Himbergen and S Chakravarty Phys. Rev. B23 359 (1984)
[16] H Betsuyaku Physica A106 3 1 1 ( 1 98 1 )
[17] J F Fernandes, M F Ferreira and J Stankiewicz Phys. Rev. B34 292 ( 1 986)
[18] R Gupta, J Delapp, G G Batrouni, G C Fox. C F Raillie tind J Apostolakis Phys. Rev. Lett. 61 1996 ( I9KK)
[19] R Gupta and C F Baillie Phy.^. Rev. B45 2883 (1992)
[ 20] U Wolff Nucl. Phys. B322 759 ( 1 989)
[21] J Kogut and J Polonyi Nucl. Phys . B265 [FS 1 5] 3 1 3 ( 1 986)
[22] E Domany, M Schick and R W Swendscn Phys. Rev. Lett. 52 1535 ( 1984)
[23] M Creutz P/iv.v. Rev. Utt. 50 1411 (1983)
[24] S Ota, S B Ota and M Fdhnle J Phys. . Condens. Matter 4 541 1 (1992)
[25] The energy is used instead of temperature to present the points clearly in the coexistence region of the
first order iransilion.
Indian J. Phys. 72A (5), 417-420 (1998)
UP A
- an international journal
A new viscous Angering instability : the case of forced
motions perpendicular to the horizontal interface of
an immiscible liquid pair
B Roy and M H Engineer
Department of Physics. Bose Institute. Calcutta-700 009, India
Abstract : We report the discovery of a new, three dimensional instability in pairs of
flowing immiscible liquids. A travelling ultrasonic wave sent along the axis of a vertical tube
containing a pair of liquids sets up steady, circulating flows in both liquids. If the wave
propagates from the less viscous member of the pair to the more viscous one the interface
changes shape with the strength of the drive. First, a pronounced upward bulge develops. At a
critical drive strength, a long finger of the less viscous fluid tunnels into the more viscous one.
The phenomenon is os universal os the famous two dimensional viscous fingering instability
discovered by Saffman and Taylor [Proc. Roy. Soc. (Lxindon) A245 312 (1958)] .
Keywords : Interfacial instability, interfacial npples, viscous fingering
PACS Nos. : 47 20.Gv, 47 35.+i. 47.55.Hd
1. Iniroduction
Inierfacial waves generated by instabilities inherent in stratified flows of immiscible
Huids have been discussed by several authors [1-3]. A striking example of the effects of
Huid mechanical non-linearity on such waves was reported by Roy et at [4]^ and
Chatterjee et at [5]. There the instability occurs when two immiscible fluids kept
in a container are forced acoustically to move parallel to their horizontal resting
mterface. In the present paper we report an even more striking manifestation of
non-linearity when stratified liquids are acoustically driven perpendicular to their
resting interface.
Experimental details
A cylindrical glass tube, of inner diameter 3.42 cm and length 1 7 cm, was held with its axis
vertical. A gold plated, X-cut Quartz transducer (Valpey Fisher Division. USA),
1998 lAPS
418
B Roy and M H Engineer
mounted axially at the base of the tube, was made to oscillate at around 5.0 MHz, using a
tunable RF oscillator. The oscillating quartz plate sets up an ultrasonic wave which
propagates inb the liquids in the glass tube; as is well-known, such waves generate
hydrodynamic flows in liquids [6] via the so-called quartz wind [see also 4,5]. The upper
end of the glass container could be kept open or fitted with an acoustic terminator —
sometimes a polythene disc terminator of diameter 3.4 cm and length 0.6 cm was used, and
sometimes a carefully machined bakelite cone of length 5 cm. The heavier, and less
viscous, liquid was first poured into the tube; thereafter the remaining space was filled, with
the lighter and more viscous liquid, l/se of this method allowed us to easily establish the
necessary flow pattern while avoiding turbulence.
Experiments arc reported here for two different oil-water systems : (i) the less
viscous and denser liquid was water and the more viscous one was a mixture of castor oil
and chloroform. The latter's density could easily be made very close to that of water by
changing the proportions of its constituents; (ii) the less viscous and denser member was a
mixture of carbon tetrachloride and petroleum ether and water was the other member of the
pair. In both systems, the interfacial tension was lowered to a value of about 2 dyne/cm by
mixing a small amount (0.2 %) of Triton X-100 (Fluke Chemie) in the distilled water. A
negligible amount of water soluble (but oil insoluble) dye was used for clear identification
of the liquids and their interface.
Generally we work with density differences of the order of 0.(X)5 gm/cc and arrange
for the less viscous liquid to be in contact with the driving ultrasonic U'ansducer. We have
observed that the behaviour reported is insensitive to whether the less viscous fluid is the
denser or the lighter member of the pair.
3. Observations and results
The measured values of the densities, viscosities, ultrasonic velocities and the interfacial
tensions of the liquids used are given in Table 1. The motion of fine sus[)ended particles, at
Table 1. Measured values of parameters for liquids used.
Liquids
Density
gm/cc
Viscosity
poise
Interfacial tension
dyne/cm
(between watei/oil mix.)
Water
1.01813
0.00882
Pet. ether-
CCI4 mixture
1.02404
0.0044
2
Water
1.01813
0.00882
Castor oil-
chlorofomi
mixture
1.01108
4.42
2
A new viscous fingering instability ; the case of forced motions etc
Plate /
TRANSDUCER
ATTACHED
AT THE BASE
I 1
0 1 cm
Figure 2. Viscous fingering inslubiliiy with overall flow structure in
water/castor oil-chloroform mixture system.
A new viscous fingering instability : the case of forced motions etc
Plate n
transducer
attached
AT THE BASE
• ~\
0 1 cm
*’‘;^ure 3.
^.iicY/casior
Observed viscous fingering instability at the interface of
OiUchlorofornn mivrun*
A new
etc
PUtte III
^ANSDUCEr
r
0
n
Icm
A new viscous fingerinfi instability : the case of forced motions etc
419
various mean flow velocities, is schematically shown in Figure l(a-c) indicating that the
steady velocity field is mostly circulatory on both sides of the interface. When the liquids
Figure 1. Schemalic diagram of the fluid flow profile at different mean
flow velocities.
are at rest, the interface is horizontal; the slight curvature at the glass walls arising from
intcrfacial tension. The shape of the interface was observed to change markedly with
change in driving acoustic power. In both oil-water systems, we examined the flow stability
as a function of the strength of the driving acoustic power by observing the behaviour of the
interface. The shapes remain unchanged for very small driving strengths, even though both
fluids do flow. Eventually, beyond some critical flow strength, the less viscous liquid
abruptly tunnels into the more viscous one and a transition takes place to a new state of
steady flow. In the new state, the interface deforms strongly into an inverted funnel whose
stem size is strongly system dependent. The tip of the stem breaks up into droplets of less
viscous liquid at higher flow strengths.
(i) The observed instability of the interface in case of water/ castor oil-chloroform
system is shown in Figure 2. The inverted funnel with thin stem can be seen at the
centre. Figure 3 shows the overall nature of the flow as also the viscous finger.
420
B Roy and M H Engineer
(ii) The instability observed in case of the water/petroleum ether-CCl 4 mixture system is
shown in Figure 4. In this system the oil mixture is the less viscous liquid and it has
been made denser than water. Since the flow was driven from oil to water it was
expected that the less viscous oil mixture would finger into more viscous water,
which indeed happened in our experiment. The shape of the interface in this second
case differs in detail from that observed in (i). However, the fingering phenomenon
is present though the finger is both thicker and unsteady. Occasionally droplets of oil
break off from the tip of the finger and fall back into the oil. All these observations
are yet to be explained theoretically.
4. Conclusions
Interfacial instabilities in the form of inverted funnels have been observed in both the
oil/water systems studied. In the well-known viscous fingering phenomenon [7], the
growing finger points from the less to the more viscous liquid; the tip of the inverted funnel
has exactly the same property in our experiments i.e., only less viscous liquids can tunnel
into more viscous ones. Accordingly, the present study reveals that the viscous fingering
instability is possible in fully three dimensional flows as well, contrary to the existing belief
in the fluid mechanics community.
References
[ 1 ] S Chandrasekhar Hydrodynamic and Hydromaffnetic Stability (New York : Dover) (1961)
[2] S A Thorpe J. Fluid Mech. 39 25 (1969)
[3] S A Thorpe J. Geophys. Res . 92 523 1 ( 1 987)
[4] B Roy, B K Chattei 3 ee, M H Engineer and Pradip Roy Physica A186 250 ( 1992)
[5] B K Chacieijee, M H Engineer, B Roy and Pradip Roy J. Fluid Mech. 248 663 (1993)
[6J J E Piercy and J Lamb Proc. Roy. Soc. (London) A226 43 ( 1 954)
[7] P G Saffman and S G Taylor Proc Roy. SfH. (London) A245 312(1 958)
Indian J.Pkys.T2\ (5), 42\-^25 (J998)
UP A
- an intemauo nal journal
Energy, fluctuation and the 2d classical XY-model
Smita Ota and S B Ota
Institute of Physics, Sochivaluya Marg. Bhubaneswar-751 005, India
and
M Satapathy
Deportment of Physics, Utkal University, Bhubaneswar'75 1 004, India
Abstract : General analytical expressions on the fluctuation of the demon and system
energy and the relationship between them have been established for microcanonical Monte Carlo
simulations of systems with continuous symmetry These hav^ been computationally verified for
the 2d classical XY-model We suggest an alternative equilibration check and demonstrate that
the system energy distnbution is a Boltzmannian.
Keywords : Monte Carlo simulations, 2d XY-model, fluctuation
PACS Nos. : 75 1 0 Hk, 02.70 Lq. 05.70 Fh
Computer simulation has become very powerful and inevitable branch in theoretical physics
of laie. Use of different techniques for the study of model systems has increased its domain
in us procedural prescriptions in simulations. One such prescription in this branch is the
microcanonical Monte Carlo algorithm [1]. This technique interpolates between the
Metropolis e( al algorithm [2] and microcanonical formulation. Here, each one of the
accessible microstates of the ensemble consisting of the spin system of interest alongwilh
the extra degree of freedom (the demon) is equally probable. The microcanonical MC
algorithm thus simulates the sum :
to generate a sequence of spin configurations k via the Markovian process where E is
conserved. The system passes through a sequence of configurations with energy E^(k)
having demon energy £,/ l.i the phase space in a hopefully crgodic manner with the help of
the demon. Although the composite system consisting of the demon and the system has
niicrostates with constant total energy, individually the constituent parts suffer from
n actuation with respect to their energy.
‘©I9981ACS
422
T Phukan, D Kanjilal, T D Goswami and H L Das
in equilibrium, has the following distribution :
P(E,i) -exp(£,, /t^r)
Here T is the temperature of the spin system, kg is set to be equal to unity
()1 £j. when continuous, leads to the following analytical result :
<£,:;>/r" = «!
where n is an integer.
The above equation and the constraint in cq. ( 1 ) lead to
5E" = ^(-l)'rt !/(«-/) I' !£'''-''5£;, (4)
t^2
where / is an integer.
To verify eqs. 3 and 4 computationally, we consider a classical 2d XY-model with
225 and 900 spins witnessing the generalized potential [4],
w= 27^[l - cos2/-({fl, -e,)/2)| (5)
(> /)
where, all notations used have their usual meaning [31, This Hamiltonian reduces to the
usual classical 2d XY-sysiem for = I, undergoing the Kosterlilz-Thouless transition and
withp^ = 50 the system undergoes first order transition A single demon as the temperature*
controller and a square periodic bound lattice are the specifications for the system under
study using the microcanonical Monte Carlo simulation technique. The simulation proceeds
as has been described in one of our earlier papers [3]. The system is allowed to
equilibrate for 1 x lO*' MCSS and the averaging of the physical quantities has been done
for 1 X 10^ MCSS forp2 = i and 50.
Figure 1 depicts the comparison of the equilibration of {M^) with that of (EJ )
for /f = 2. 3 and 4 (for = 1, £ = 519.3 (£ corresponds to a value close to the KT
transition temperature) ( l.a) and for p^ = 50, £ = 1620.0 (a value in the coexistence region
of a first order transition) (l.b)) with 4 x 10*^ MCSS. The rate of equilibration of
was found to gel reduced when the change in ) between the initial stage and the
final stage was large. For p^ = 50, the change in ) is approximately the same for
£ = 2160 and 3060 and in this case the rate of equilibration of (M^ ) is fast when
(E^i) = 2(£= 3060), the height of the potential well. This is attributed to the
reduced width of the potential well for p^ = 50. The system was found to be trapped
quite often in meiasiable states for p" = 50, when the simulation was started with
higher system energy. The mciastable state consists of large regions of aligned spins.
This situation can be circumvated by initially aligning a portion of the lattice and then
( 2 )
. This distribution
(3)
Energy, fluctuation and the 2d classical XY-model
423
starting the simulation. This essentially reduces the volume of the boundary between
the regions of aligned spins. In this situation, (EjJ >/7'' serves as an alternative check
of equilibration.
Figure 1. 2 X (fj > //I'T'* , for /> = 2 (O).
3 (+) and 4 (x) and 10 x N~^ < > (□).
V (A), 3/2 X < > (*) and 2 x < M > (0) as
functions of MCSS for 30 x 30 spin
system. Each data point represents the
average over the conngurations up to a
given MCSS. The continuous line is a
guide to the eye. The initial configuration
were with all spins parallel to each other
(a) = I, £ = .‘>19.3, T = 0.918 and V is
scaled to V/2, (b) = 30, E = 1620.0,
T= 1 .0 1 and V is scaled to 1^/20.
Table 1 depicts )/7'" for n = 2, 3 and 4 and for the spin system with 225
and 900 spins. This estimates 5E" using eq. (4). Here, we observe that, (EJ)/?*" is
close to n ! and deviation from n ! value increases as n increases for a given system size
and is more for smaller system size. 6Ej is observed to be a straight line parallel
to X<axis at y' s I which is contrary to the conventional nature (a peak at a usual
transition temperature). We also find that dE^ jT’^ = 1. From the simulation result,
we infer that the mean square fluctuation of the demon and the system energy are the
same and it is found that the system energy distribution is reflected through the demon
energy distribution. The standard deviation of various quantities are given within bracket
in the table.
72A(5)-12
424
Smita Ota, SB Ota and M Satapatky
In conclusion, we obtained a simple relationship between the fluctuations of the
system- and the demon-energy for systems with continuous symmetry analytically in
a microcanqnical framework and demonstrated it for the case of the classical 2d XY-model.
Tabk Lvalues of £. rand <£”)/r” with - 1 and 50.
Synem
size
1
50
£
60.7
129.8
310.3
67.5
405
742.5
T
0.4986
0.9222
1.7552
0.5280
1.0286
1.6170
(±0.0041) (±0.0098) (±0.0241) (±0.0065)
1 (±0.0123)
(±0.0216)
ISxlJ {E^)/T^
1.979
1.988
1.984
1.989
2.002
1.978
1
0.074'
1 1
^+0.097^
) 1
0.122 >
1 (
0.103 >
1 (
0.104'!
1 (
■♦‘O.lll'l
1
^-0.071^
1 1
0.094^
) 1
0.116;
1 1
0.098;
1 1
0.100 J
1 1
0.106 J
5.821
5.901
5.834
5.892
6.032
5.802
U o.4ir
1 1
0 . 535 '
1 1
0.646'
1 1
0.529^
) (
'+ 0.558'
1 (
^+0.569'
1
1
0.394^
I 1
0-506^
1 1
^-0.601^
) 1
0.496^
I 1
. 0-524^
1 1
,-0.531,
1
(Ej)lr*
22.66
23.22
22.57
23.13
24.24
22.46
3.79^
('+4.16')
f+3.40'1
('.+ 3.67')
('+3.58')
U 2 . 71 J
U 3 . 51 J
i-3.78j
i-3.12j
1 3.38J
1-3.26J
E
243.0
519.3
1241.1
270
1620
2970
T
0.4986
0.9165
1.7815
0.5275
±.0175
1.6442
(±0.0056) <± 0.0120) (±0.0267) (±0.0029) (±0.0164) (± 0.0241 ]r
30x30 {EI)/T^
2.003
2.003
1.994
2.001
2.005
1.987
(
'+0.098^
1 (
'+ 0.106']
1 j
^+0.12n
0.055'
r+ 0.141'
) (
'+ 0.126''
1
^-0.094^
1 1
0.101 J
> 1
,- 0 . 115 ]
0.054^
1- 0-133,
) 1
,-0.120;
(eI)It^
6.011
6.017
5.939
5.996
6.0S6
5.875
(
+ 0.538''
1 (
"+0.575^
1 j
0.635'
1 1
^+0.348'
) (
'+ 0.727'
1 1
'■+ 0.620'
1
1
^-0.507;
1 1
^-0.538^
1 1
,-0.588,
I 1
^-0.338,
) 1
,-0.66»,
1 1
0.575,
)
24.15
24.08
23.43
23.82
24.52
22.88
3.88A
('+4.06')
U 4.20'j
('+2.64')
r+ 4.70'!
('+3.81')
1-3-58J
l-3.70j
i-3.78j
i-2.54j
i-4.2lj
I- 3 . 45 J
Although, this equation is independent of the order of transition and the number of extra
degrees of freedom, we undertook the present study with a single demon. We prescribe
EJ /T" = n ! as an alternative equilibration check in a microcanonical framework for
continuous systems in specific circumstances. We also observe that the system energy
distribution is a Boltzmannian when the system is controlled by a single demon.
AckBowkdgiiieiii
SO acknowledges Council of Scientific & Industrial Research, India, for financial assistance.
Energy, fluctuation and the 2d classicai XY-mbdel
42S
RfffefCOOM
[1] M Otutz Fhys. Rev, Lett. 50 141 1 (1983)
t2] N Metropolis, A Rosenbluth, M Rosenbluth. A Teller and E Teller I Ckm. Phyt 21 1017
(1953)
[3] S Oto, S B Ota and M Flhnle J. Phys. Condens. Matter 4 541 1 (1992)
[4] E Domany, M Schick and R H Swendaen Phys. Rev. Un. S3 1535 (1984)
Indian J. Phys. 7IA (5), 427-431 (1998)
UP A
— an inlemaliCMl joamal
Phase alternation in liquid crystals with terminal
phenyl ring
Jayashree Saha' and C D Mukherjee^*
'S. N. Bose National Center for Basic Sciences. Block JD,
Sector 111, Salt Lake, Calcutta-700 091, India
^Saha Institute of Nuclear Physics, 1/AF Bidhannagar,
Calcutta-700 064, India
Abstract : A mean-field model for the phase alternation between homologues. as
observed in the case of some liquid crystalline homologous series with a terminal phenyl ring, is
presented considering the anisotropic interaction of the rigid part and bulky ring of a molecule in
the field of other molecules. Bui the chain interaction is ignored though the chain conformations
influence the relative configuration of the other parts Numerical calculations were done to
reproduce the phase diagrams of the phase alternation for the first five members of the
homologous series Q>-phenylalkyl-4-p-phenylbenzylidene which is in good agreement with the
experimental results
Keywords : Terminal phenyl ring, phase alternations, mean-field theory
PACSNos, *: 61.30. 64 .70.M
1. Introduction
Unusually pronounced even-odd effect in nematic-isotropic transitions has been observed in
liquid crystalline compounds with terminal phenyl ring or Q>-phenylalkyl cinamates [1].
These types of series also show the very interesting behaviour of phase among the
homologues. The member of the series with odd number of the methylene units (N) in the
flexible alkyl chain has both smectic and nematic phases but the same series with even N
has only smectic phase. As a result an alternation of phase occurs [1], because nematic
properties are extinguished for odd number of methylene units in the flexible alkyl chain. In
an earlier work [2], we formulated a mean-field model by incorporating the effects of the
chain conformation explicitly which directly influence the relative configuration of both
parts (viz., central rigid part and the terminal phenyl ring) and as a result the longitudinal
polarizability of the molecules is changed. This model has made it possible for us to
Conespondence electronic address ; chandi9hpl .sahB.emet.in
©1998 lACS
428
Jayashree Saha and C D Mukherjee
reproduce the unusually pronounced even-odd effect in N - I transition. In this
communication we present an extension of the earlier model to the smectic phase by
introducing a pure translational McMillan parameter [3]. Conformational and dispersive
energies for the rigid part and the terminal ring of a molecule in the field of the other
molecules are considered. Numerical calculations have been done on the homologous series
fl>-phenylalkyl-4-p-phenyl benzylidene for the first five member of the series. The
calculation reproduces the phase alternation behaviour which is observed in the
experimental phase diagram of that series.
2. Method
Following reference 2, we consider the mean-field experienced by a molecule to consist of
three parts — one for the rigid core (E^), the other for the end phenyl ring (£^) and the
conformation energy of the chain segment (Econf)- For simplification, we ignored the
contribution of chain part to the mean-field as it is small compared to the other contribution.
Before writing the energy expression let us state the definitions of the order parameters to
be used.
Orientational order parameter for the rigid parts
rj, = (Pj(cosflj). (1)
where is the angle between the rigid part and the mean-field direction. < ^ means a
statistical average.
The phenyl ring order can be defined as
TJ* = (P,(cos0,m}. (2)
where 6^, is the polar angle between the phenyl ring and the direction of the field. 0 is the
angle of rotation of the ring about the rigid part.
The translational order parameter
and
( 6 )
Phase alternation in liquid crystals with terminal phenyl ring
429
In the above expressions the suffixes 'a' and 'b' have been introduced to denote entities
pertaining to rigid and phenyl parts of a molecule respectively. and Cb(N) are the
respective volume fractions of the N-ih member of the homologous series and Vf, are the
molecular volumes of the two basic components. The volumes of the different parts are
estimated from the table values for molecular weight and density data of the sample.
Vifh and are the coupling constants for the interaction mean field with the first suffix
representing the component molecule that experiences the mean field and the latter suffix
indicating the mean field producing agent. By analogy with the result for the binary-
mixtures of nematogenic molecules [4], we have
= IVoaVbk
Therefore, only two of the coupling constants remain as adjnstable parameters, which are
fixed from the nematic-isotropic transition temperature of two homologues, as explained in
reference 2. The parameter S is the relative coupling strength of the Kobayashi two-particle
potential [S] which is constant for a homologous series and also temperature independent.
The value of this parameter is obtained by using the value of the smectic to
nematic/isotropic transition temperature of the homologue. Here we assume that the
correlation between the orientation order and translation order is very weak. Hence the
mixed parameter term in the model potential is neglected. U is the internal energy of i-th
segment of a chain and ^ represents any of the three rotation isomer states namely r(lrans),
g^(gauche). The value of these states are given in reference [6]. The values of the order
parameters at a particular temperature T can be obtained by full self-consistent solutions of
the following equations :
' Vo = 4 - Y {'p2(cose,)txp[E/kT]d{cos9)d<l>dz (7)
Hi, = y, f f f /’2(cose|,(^))exp[£/fcr]<i(cose)d^(fe (8)
£‘=os[^)exp[£/tr]d(cose)rf0rf2 (9)
nllconf
where Z„ is the partition function of the system in ordered phase at a particular temperature
T and it is given by.
Zo
exp [E/ kT]d {cos 0)d(l>dz
For each temperature, the self-consistent solution of the order parameters is found and the
stable solution is picked up corresponding to the minimum of the Helmholtz free energy per
particle. The expression for this free energy is given as,
^ + (2n„ Vt+ST^) + Vkh vl ]
-kT\n[ZJZi]
where Zj is the partition function in isotropic phase.
( 10 )
430
Jayashree Saha and C D Mukherjee
3. Result and discussion
The calculated transition temperatures of the homologous series a)-phenyl alkyl-4-p-
phenyl- benzylidene are compared with the experimentally observed value in Figure 1.
The estimated volumes are, = 433.91 V/, = 108.48 A^, together with the volume
for the each chain segment 27 A^. The values of the coupling constants and
are 2059950.0, 22162.85 and 213668.82 respectively in C G S unit. The value of 5 is
0.37. It is to be noted that the ratio of the strength parameters are constant throughout
the nematic phase and independent of the number of the homologue whose value is
9.64.
Figure 1 . Plot of tiansition tempciaiuie
{T) agam.st number oT methylene graups
(yV) 111 the chain.
In conclusion, we would like to mention that this is possibly the first calculation
where the generally observed feature of the cinnamate homologous series, namely the phase
alteration is well reproduced. The calculated transition points are seem to be only about
10 - 15 K off the experimental values. This small deviation may be due to the sieric effect
which has not been considered directly into our model potential expression. As the
molecules are packed in a layer in the smectic phase there is considerable lateral sieric
repulsion between the molecules [7], It is quite likely that those conformations for which
the chain segments deviate much from the molecular axis are suppressed dde to the lateral
repulsion by the neighbouring molecules. Further efforts should be made for better
agreement with experiments.
Acknowledgments
We are grateful to Prof. M K Roy and Prof. M Saha for their valuable comments and
suggestions. One of us (J Saha) wishes to thank the Council of Scientific and Industrial
Research, India for the Research Associateship offered to her.
Phase alternation in liquid crystals with terminal phenyl ring
431
References
in GW Gray and K J Harison Mol. Cryst. Liq. Crysr. 13 37 ( 197 1 ); D Coates and G W Gray J. Physique,
Colloque Cl C365(1973)
[2] C D Mukheijee, T R Bose, D Ghosh, M K Roy and M Saha Mol. Crysi. Liq. Crysi 140 205 ( 1986)
[3] W L McMillan Phys. Rev. A6 936 ( 1 972)
{4J R L Humphries, P G James and G R Luckhurst Sympositum of the Faraday S’w. 5 1 07 ( 1 97 1 )
(5] K Kobayashi and Oyo Butsuri 40 532 ( 1 97 1 )
(6] J Saha, B Nandi. C D Mukheijee and M Saha Mol. Cryst. Liq. Cryst. 214 23 (1992)
[71 M Nakagawa and T Akahone J. Phys. Soc. Japan 53 1951 (1984)
^2A(5).13
Indian J. Phys. 72A (5), 433-437 (1998)
UP A
- an inicma tional jou rnal
Change in conductivity of CR-39 SSNTD due to
particle irradiation
T Phukan, D Kanjilal*, T D Goswami and H L Das
Department of Physics. Gauhati University. Guwahati-781 014,
Assam, India
^Nuclear Science Ccntic, New Delhi-1 10 067, India
Abstract : The electrical conductivity in gamma iriradiated CR-39 SSNTD increase
substantially compared to pnsiine samples On Si^'*' irradiation in pristine samples the
conductivity decreases below the resolution ol the present measurement. When the gamma pre-
irradiated samples are subjected to Si®'*’ irradiation (fluencc = 5 x 10*^ to 5 x lo’ ’ particle.s/ cm^)
conductivity reveals an increasins tendency With the increase of Si®'*’ fluence, the conduction
activation energy in these ( 7 + Si®"*) citposcd samples changes towards the activation energy of
the only y irradiated samples Tltcsc results are explained on the basis of dipolar orientation ol
the polymer chains.
Keywords : SSNTD, CR-39 polymer, clectncal conductivity
PACS Nos. : 29 40.0x, 61 SO.Ed, 72 HO.Le
1. introduction
C'R-39 (allyl diglucol carbonate) is a polymeric track detector used in different fields of
nuclear radiation detection for its high sensitivity. Various efforts have been made to study
iis track formation mechanism so as to exploit its full potentialities for a sensitive track
ciciccior in diverse fields of applications. From the ion explosion spike model [I] it is
Miggcsied that the physical parameters like carrier density and mobility, dielectric constant
I'h of a material are mainly responsible for its track recording character. It has also been
icportcd lhai gamma-irradiation can be used in polymeric materials to change or modify the
near surface characteristics of a bulk polymer [2]. So an investigation has been carried out
U) study the changes in conductivity in CR-39 SSNTD due to Si®’*' irradiation in pristine
CR-39 and pre-exposed 50-gray gamma- irradiated samples.
2. Experimental details
CR-39 (Pershore Mouldings Ltd. U.K.) sheets of 250 pm thickness were cut' in the
dimension 0.9 cm x 0.9 cm. These samples were chemically cleaned properly and allowed
lo dry in room-temperature. The cleaned samples were exposed to gamma irradiation of 50
© 1998 I ACS
434
T Phukan, D Kanjilal, T D Goswami and H L Das
gray dose. This was done with the help of a phoenix telecobalt unit*. The average energy of
the gamma-rays was 1.25 MeV. The irradiation was carried out at room temperature. These
samples were then exposed to 100 MeV Si^ radiation in the material science scattering
chamber of the 15 UD Pelletron at Nuclear Science Centre, New-Delhi.** The fluences used
ranged from 5x10^ cm"^ to 10‘^ cm*^. Beam size was adjusted to 1 cm x 1 cm and the
pressure during irradiation was -3 x 10~^ torr. The temperature during irradiation was
maintained at 85 ± 2 K with the help of a LN 2 cold-finger arrangement.
The resistivity of the sample; vas measured by an high impedance (10'^ Q or
higher) ECIL electrometer amplifier. Ag electrodes of dimension 0.3 cm x 0.5 cm were
vacuum evaporated on the two sides of CR-39 samples with the help of a conventional
Hindhivac coating unit. As a result an Ag-CR-39-Ag sandwich type cell structure was
obtained. The experimental sample assembly connected with electrical and thermocouple
leads was kept inside A B-34 ground-glass jacket with the provision of continuos
evacuation through a stop-cocked side-tube. The experimental observations were carried
out inside a Faraday cage to avoid external noise and pick-up.
3. Results and discussion
Figure I depicts the 1-V characteristics of the samples. There is no deviation from linearity
and hence the conductivity processes involved must be of bulk conductivity type and not of
Figure 1. I V.V V curves of CR-39 at different conditions (Dots - experimental points)
(Line - fitted curves)
Change in conductivity of CR-39 SSNTD due to particle irradiation 435
due to the electrode contact controlled type. It may be noted that the current increased
with the increase of temperature for the same applied voltage in both the pristine and
irradiated states. In the present study, the samples are polymer dielectric materials and thus
have the possibility for dipolar orientation. Increase in conductivity in case of gamma-
irradiated can be attributed to the fact that the dipolar orientation due to the applied field
is reduced [3].
Flaurc 2. In y (Am"^) vs I000/T(ir*) for pristine CR-39, (Dote - experimental points)
(Line • fitted curves).
The temperature dependence of conductivity is depicted in Figure 2 and Figure 3
for pristine and irradiated samples respectively. The activation energies have been calculated
from these plots. It is observed that in the fresh sample there two regions of activation
energies while in the irradiated samples there are three activation regions. It may be noted
that the conductions in region A and B are due to carrier excitation to unlocalised and
localised states respectively. Region C may be attributed to carrier hopping transport. If the
436
T Phukan, D Kanjilal, T D Goswami and H L Das
density of defect states is high, then process B will not dominate in any temperature range
and a direct transition from A to C will result which is the case of the pristine sample [4],
1000 /T(K"' )
Figure 3. In J (Am”^) vs 10(X)/T(K“*) graphs of irradiated Qt-39. (Dots - experimental
points) (Line - fitted curves).
The corresponding computer fit equations in each region are also given in the figures. The
activation energies are given in Table 1. From these data it may be noted that the activation
Table 1. Activation energies for fresh and irradiated samplcB.
Sample type
Activation
(region A)
Energies
(region B)
(region C)
Pristine
1.02 eV
1.76 cV
Gamma (SO gray)
irradiated
2.0ScV
0.936 eV
Gamma (SO gray)
and Si (S x 10^
0.964 eV
0.684 eV
0.171 cV
Gamma (SO gray)
and Si (10*')
1.01 eV
0.688 eV
0.192eV
Gamma (SO gray)
and Si (Sxio")
1.05 eV
0.724 eV
0.284 eV
Change in conductivity ofCR~39 SSNTD due to particle irradiation
437
energies decrease in general in irradiated samples compared to fresh samples. For values of
activation energy lower than 0.2 eV. the conduction may be due to an electronic mechanism
and for values more than 0.6 eV it could be either electronic or ionic [5]. From the observed
values of activation energies it is seen that the conduction may be electronic or ionic in
nature.
Acknowledgments
This facility has been provided by the B Borooah Cancer Institute, Guwahati. We highly
acknowledge their kind help.
We heartily acknowledge the Nuclear Science Centre, New Delhi, for the various
facilities including financial sponsorship provided to us in this respect.
Rercrences
[1] R L Fleischer, P B Price and R M Walker Nuclear Tracks in Solids : Principles and Applications
(Berkeley : University of California Press) (1976)
[2] L C^lcagno, G Compagini and G Foti Structural Modification of Polymer Film by Ion Irradiation. Nuci
Inst, and Meths. in Physics Research B65 413 (1992)
[3] M El Shahawy, A Hussein and A Tawansi CR-39 as a Gamma Dosimeter : Dielectric and Infrared
Studies, Journal of Material Sc. 27 6605 (1992)
[4] N F Mon and E A Davis Electronic Processes in Non-crystalline Materials (Oxford : Clarendon Press)
(1971)
[5] B Bhattachaijee, H L Das and T D Goswami DC Conductivity of Cellulose Nitrate Particle Track
Detector. Radiation Measurements Vol. 23 (No. 1) p 231 ( 1994)
Indian J. Phys. 72A (5). 439-446 (1998)
UP A
— an international journal
Metastability and hysteresis in random field Isine
chains
Prabodh Shukla
Department of Physics. North Eastern Hill University,
Shillong'-793 022, India
Abstract : Zero-temperature non-equilibnum dynamics of one dimensional random field
Ising models is analysed for metastable states and disorder-dnven hysteresis. Ferromagnetic as
well as anti -ferromagnetic cases are considered. In the ferromagnetic case, we obtain an exact
expression for the hysteresis loop in the zero-frequency limit. In the anti -ferromagnetic case, an
exact solution of the problem is not possible (so far). Some interesting aspects of the anti-
ferromagnetic dynamics arc discussed. Its relationship with the dynamics of an ANNNI chain is
also examined.
Keywords : Ising model, metastablc states, disorder-dnven hysteresis
PACS Nos. : 75.60.E. 75 70.K
1. Introduction
The random field Ising model (RFIM) has played an important role in understanding
disordered systems. It first came into prominence around 1975, when Imry and Ma [1]
argued that Ising magnets with a quenched random field were incapable of sustaining a long
range order below two dimensions. It was an appealing argument, and a kind of a clear
statement which was lacking in the context of other prominent models of quenched
disorder, e.g. the Sherrington-Kirkpatric (random bond) model of a spinglass [2]. Thus
several people were attracted to the study of the equilibrium statistical mechanics of the
random field Ising model. Soon a controversy was generated. Dimensional reduction
arguments based on field theoretic methods showed that the lower critical dimensionality of
RHM was three rather than two as predicted by Imry and Ma. It took some years to resolve
that the application of the dimensional reduction method in this context was unjustified
because it necessarily assumed the existence of a unique solution of the field equations. It
was shown that the field equations for systems with quenched disorder have a large number
72A (5)-U
© 1998 lACS
440
Pndfodh Shukla
of so/uiions (mctastable states). Due to the presence of numerous metastaWe states in the
system, the numerical simulations too proved rather difficult and inconclusive, and the
initial enthusiasm for the model faded in due course.
Some years later, interest in RFIM revived for the same reason it had faded earlier.
Its richness in metastable states was a deterent in the study of its equilibrium properties, but
made it a good model for the study of nonequilibrium phenomena in glassy and complex
systems. These systems are characterised by extremely slow relaxation, and history
dependent effects which arise from the presence of several metastable states in the system.
There are two broad time scales; (i) the lifetime of the metastablc state (T]). and the
transition time between neighbouring metastable states (T 2 ). Generally, « X\, and thermal
excitations are too weak to push the system from one metastable state to another over
practical time scales. However, a sufficiently strong external force can easily achieve this.
A cyclic driving force takes the system through a hysteresis loop. The loop shows that the
system can rest in two different states for the same value of the external parameters
depending upon the history of the system. This is a non-equilibrium effect, and one can
argue that in the limit of the frequency of the driving force going to zero, the area of the
loop will also go to zero. This is fine, but in several systems hysteresis loops are observed
even at driving frequencies of the order of 10^ Hz (corresponding to periods of several
days). These loops show no sign of disappearing over time scales which test the patience of
the experimentalist. Thus, for practical purposes, we need a theory for this nonequilibrium
phenomena.
Recently, Sethna et al [3] used the RFIM to study hysteresis and other related
phenomena such as the return point memory effect, and the Barkhausen noise. Hysteresis is
a kinetic phenomenon, and therefore one needs to put in a dynamics in the model. Sethna
et al employed the zero-temperature Glauber dynamics of Ising spins. It showed remarkable
success in reproducing the observed features of hysteresis and other phenomena mentioned
above. The success of the Sethna model is not unreasonable. It is a minimal model which
takes into account the most important aspects of hysteresis. The zero-temperature dynamics
effectively sets Ti =«», and T 2 = 0. This is a reasonable approximation at finite temperatures
on laboratory time scales. Although the dynamics is deterministic, there is a stochastic
aspect to it coming from the randomness of the quenched field. The metastable states of the
RFIM become fixed points (stable states) under the zero- temperature dynamics. This
simplifies their numerical as well as analytic characterization. However, the model retains
the key features of the original problem. There is a broad distribution of energy barriers
between nearby stable states. When the system is driven by a smoothly increasing applied
field, it jumps from a stable state to a nearby stable state of lower energy when the applied
field crosses the barrier between the two states. As the barriers are random variables, the
trajectory of the system is not smooth. On a microscopic scale, it consists of irregular jumps
in the magnetization (Barkhausen noise).
Experiments show that there is a broad distribution of the size of the magnetization
jumps. Averaged over the entire hysteresis loop, jump distribution shows power laws over
Metastahility and hysteresis in random field Ising chains
441
several decades (usually three). This has lead to suggestions that there is a self-organized
criticality in the system. The Sethna model provides a framework for examining this
question. Although it does not appear to support self-organised criticality in the system, but
there is a “plain old critical point” on each half of the hysteresis loop. At this point, the
magnetization jumps show true power laws. The critical region appears to be rather broad.
Thus approximate power laws are expected over a wide sector of the hysteresis loop. The
extensive study of the Sethna model is based on numerical simulations of the model, and its
analysis in the mean field approximation. We have initiated a modest effort to solve the
Sethna model exactly in one dimension, and also on Bethe lattices to clarify its critical
behaviour. Here, we limit ourselves to the one dimensional case. Although one dimension is
definitely below the lower critical dimension of the random Ising model, but the model
shows interesting and non-trivial non-equilibrium phenomena. In fact there is nothing very
one dimensional about the hysteresis loop in the one dimensional model. It looks
qualitatively similar to the one in three dimensions. The analysis of the one dimensional
model serves to illustrate the basic method which can be applied to Bethe lattices as well.
As we shall see below, there are several questions which cannot be answered (so far) even
in the one dimensional case. We hope that readers may be persuaded to investigate these
questions.
2. The model
Consider a one dimensional Ising model with spins {s, = ±l ), nearest neighbour interaction
7, a uniform external field h, and a quenched random field h, at each site i drawn from a
continuous probability distribution p(/7,).
The effective field seen by a spin s, is given by :
h = + h, + h (1)
The energy of spin s, is equal to The zero-temperature relaxational dynamics of the
system attempts to lower the total energy of the system by flipping each spin which is not
aligned in the direction of the local field at its site. It updates spins according to the rule,
s, = sign(/, ) (2)
"he relaxational dynamics is an iterative process. If lowering the energy of a spin increases
the energy of one of its neighbours, then that neighbour is updated at the next step. After
a number of steps, the dynamics converges to a stable configuration where each spin
satisfies equation (2).
The total energy of the system is given by,
H = - - h^Si (3)
i.y I I
A Slate satisfying equation (2) is a local minimum of the energy of the system. It may be
possible to obtain states of lower energy by flipping pairs or larger clusters of spins
together, but these states are outside the scope of the dynamics considered here.
442
Prabodh Shukla
The locally stable slate obtained by our dynamics depends on the history of the
system. For example, two initial states, one with all spins down, and the other with all spins
up yield different stable states at the same applied field h. We focus on the lower half of the
hysteresis loop. In the following, we outline a method to calculate the magnetization per
spin m(h) in a field h starting from a saturated state (m = -1) at /i = -op. The magnetization
in the upper half is related to m{h) by the symmetry m{h) = -m{-h). At present we are not
able to calculate the magnetization m(h) for an arbitrary initial state.
3. Ferromagnetic interactions
The model described above possesses two important properties if the nearest neighbour
interactions are ferromagnetic (J > 0). These properties are :
1 . The stable slate does not depend upon the trajectory of the applied field from /i = -«
to /t, as long as it remains everywhere bounded below h.
2. The stable state does not depend upon the order in which the spins are updated
during the relaxational process.
The above properties greatly simplify the analysis of the model. Suppose we wish to
calculate m(h) starting from m(-oo) = -1. In view of the first properly, we do not have to
worry about the detailed trajectory of the applied field if it was raised from /! = -«> to its
present value h slowly. We can start with the initial state with all spins down, and relax it
directly in field h.
To calculate m(/i), we have to calculate the probability that a spin at an arbitixiry
lattice site O is up in the relaxed state at field h. This calculation is performed in two steps.
In the first step, the spin at the site O is kept down, but all other spins on the lattice
are relaxed. The spin at O is connected to two semi-infinite lattices, and spins on each half-
lattice can be relaxed independently of the other half-lattice. We focus on one half-lattice,
say the one on the left of O. Consider a long chain of N spins extending to the left of the site
0. Number its sites by n = 1,2, . . ., N - 1 , N, N + 1 ; n = N + 1 denoting the site O. We relax
the spins on this chain in the following order. Spin at site 1 is relaxed first, then at site 2,
and so on. Relaxing a spin means checking the local field on that spin, and if it is positive,
to turn the spin up. The spin at site n = 1 has only one neighbour which is necessarily down
(because it is not relaxed so far). Thus the local field at the end spin is /, = V + /i] + /i, and
we turn it up if /| is positive. Next we relax the spin at site n = 2 but keeping the spin at
n = 3 down. If spin at site 2 turns up during the relaxation, we re-examine site 1 to see if it
would turn up as well. Similarly if a spin at site (n) turns up, spin at site (n - 1) is re-
examined, if this turns up then the spin at (n - 2) is re-examined, and so on till we come to a
site where the spin is either already up, or it is down and remains down even after its right
neighbour has turned up.
The advantage of choosing the above order for relaxing the spins is that we can write
a recursion relation for the probability that a spin is up at site (n), given that the spin at site
(n + 1) is down. This probability becomes independent of n if n » 1, i.e. if one is sufficiently
Metastability and hysteresis in random field Ising chains
443
far from the end of the chain. Let P*(h) be the conditional probability that a randomly
chosen spin at site n is up, given that its nearest neighbour at site n + 1 is down (not relaxed
yet), but the spin itself and all spins to its left are relaxed. We obtain,
PHh) = p,(h)P^-Hh) + PoWll-P’^-'m (4)
Here p^^W is the probability that the local field at a site is positive if m of its nearest
neighbors are up, (m = 0, 1 , 2).
Pm(.h) =
p(h, )dhi
The probability that the local field on a boundary site is positive is
(5)
p'(/i)= r p(h,)dh,
( 6 )
Using the above initial condition we can determine all P" recursively for n > 1 . For large n,
tends to a fixed point given by the self consistent equation,
P^ih) = p^{h)P\h) + Pom\-P*m (7)
The second step is to relax the spin at site O. Its two nearest neighbors have been relaxed,
and each of these is up independently with probability P^(h). Thus the probability that the
spin at site O is up is given by,
p{/i) = Pj(/,)(P‘]2 + 2p|(/i)P* 11- P*1 + Pod (8)
We obtain,
P(h)
= Po
1-P? +P 0 P 2
l-(Pi -Po)
(9)
The magnetisation per spin (on the lower hysteresis loop) is given by
m{h) = 2p{h) - 1 (10)
The above results were derived in reference [4] by an alternate method, and checked
numerically by Monte Carlo simulations. These results have been extended to Bethc lattices
as well [5]. Somewhat surprisingly, the behaviour on a Bethe lattices with coordination
number three is similar to the one dimensional case, but behavior on lattices of higher
coordinalion number is qualitatively different. We refer the reader to reference [5) for
details.
4. Anti-ferromagnetic interactions
The anti-ferromagnetic chain is described by a negative J(J < 0). In this case the two
properties of the ferromagnetic model listed at the beginning of the preceding section are
lost. Therefore the method developed there is no longer useful. We describe briefly a
typical numerical simulation. For simplicity, consider a flat and bounded distribution of
quenched fields in the range -A^h,^ A. Start with a sufficiently large and negative applied
lield such that all spins arc down { 5 , = -1} initially, and raise the field slowly. The anti-
444
Prabodh Shukla
ferromagnetic interaction does not like the adjacent spins to be aligned in parallel, and
therefore the applied field has to be more negative than U -A\o keep all the spins down.
At, ^ = 27 - 4 the first spin flips up. On raising the applied field further, more spins flip up,
and the magnetization rises to a value equal to -exp(-2) at ^ = 27 + A The magnetization
remains fixed at this value (first plateau) upto h - -A. Further increase in the applied field
from -A to 4 increases the magnetization continuously to a value which is about 10%
lower than exp(-2). It remains fixed at this value (second plateau) upto h = -27 -A. Further
increase in the applied field cause the remaining spins to turn up gradually, and at /i = -27
+ we get m = 1. This completes the lower half of the hysteresis loop in an increasing
applied field. The upper half loop lies very close to the lower half, and therefore the area of
the hysteresis loop is very small. These features of the zero-temperature anti-ferromagnetic
dynamics are easy to understand. Limitation of space does not allow us to go into the details
here. We refer the reader to reference 16] for an approximate analysis of the numerical
results. An exact analysis has not been possible so far, but we are working on it.
The key to understand the anti-ferromagnetic dynamics is to note that when a spin
flips up, an adjacent spin, if it was down initially, is stabilized in its down position.
Therefore a spin flipping up as a result of increased applied field does not give rise to the
possibility of an avalanche. It may cause a neighboring up spin to flip back down, but it can
not change the stale of spins beyond the nearest neighbor. In other words, a microscopic
increment in the applied field never causes more than two spins to flip. The two-flips arc
relatively uncommon (less than 4% approximately). In the majority of cases spins arc
turned up one at a lime. The smallness of the two-flip effect is responsible for the smallness
of the hy.steresis in the simulations.
5. Random ANNNl chain
ANNNI (axial next nearest neighbor Ising) chain 17] is described by the hamiltonian,
I I
Here J^ and J 2 are competing interactions (/| > 0, and 72 < 0, or 7 1 <0, and Jj < 0). li
supports a rich and complex short-range structure, and has been studied extensively in the
context of spatially modulated periodic structures in magnetic and other systems. The phase
diagrams of the ANNNI chain obtained from a dynamic criterion often show considerable
differences from those obtained from purely energetic considerations. Numerical studies
indicate that the most stable slates of the system (the true equilibrium states) are not
necessarily the most probable states of the system. Some issues in this context can be
clarified by the study mentioned in the preceding section. Defining new Ising spins a, =
Hx can be transformed into the form
H 2 = - Ji'Zo. - J2'Zo.o,,, (' 2 )
I i
Hamiltonian H 2 is similar to the one studied in the previous section, and gives us an
occasion to comment on the effects of quenched randomness on the non-equilibrium
Metastability and hysteresis in random field Ising chains
445
dynamics of the ANNNI chain. Let =7] -k-h^ where hj is a quenched random variable
with zero mean value. We can make contact with the equilibrium states of the non-random
chain in the limit ^ ♦ 0. An equilibrium state is determined by energetic considerations
alone; it is the global minimum of energy. The dynamically stable states (which may
correspond to the meta-stable states at finite temperatures) are the local minima of energy.
We wish to compare the nature of ordering in the two sets of the states.
It is useful to recall the equilibrium results for the non-random ANNNI chain.
Consider Hamiltonian H\ with Jj < 0, and J 2 < 0. The zero-temperature ground state is
ferromagnetic if 7| < 2/2. and an anti-phase state (two spins up followed by two spins down
and so on) if Jy > anti-phase state can be seen easily with the help of the
transformed Hamiltonian For 7) = 272, ^ ground state is not very discriminating with
respect to any particular 16ng-range order. It is infinitely degenerate with any sequence of
it^'bands (k adjacent identically oriented spins, terminated at both ends by oppositely
oriented spins) having the same energy. The degeneracy goes up as a Fibonacci serips, and
scales as d^, where N is the number of spins in the chain, and d = (V5 + 1) / 2. The two spin
correlation function averaged over the degenerate states can be obtained
analytically, and decays exponentially with an oscillatory modulation. At finite
temperatures, there arc two qualitatively different regimes. For 7i < IkIi, where ic is a
temperature-dependent parameter, the correlations decay exponentially without an
oscillatory modulation, For 7| > 2rc/2, the exponential decay of correlations is spatially
modulated by a multiplicative factor of the form cos qr where q varies with K as well as
temperature.
Coming to the random ANNNI chain, we see that in the region 7| < 2/2 - d (region
A), the dynamically stable state is a ferromagnetic state with all (7 spins down. In the region
2/2 - ^ ^ 7) ^ -2/2 + d (region B), the system settles into a state of an arbitrary sequence of
^-bands. The number and the structure of the dynamically stable states in region B is the
same as that of the equilibrium states of the non-random chain at 7) = 2/2. In the region,
2J2-A^Jy^-A (region C), the dynamically stable states are the jammed states (the
slates on the plateaus mentioned in the preceding section). The jammed states occur over
a large region (A can be arbitrarily small), and have a certain universality in the sense
that they can be characterized by a common property (no more than two consecutive
spins are parallel) independently of the parameters of the system over a wide range of the
parameters.
In regions A and B, the dynamically stable states have the same structure as the
equilibrium states. In region C, however, the dynamics leads to jammed states, while
energetic considerations yield the anti-phase state with perfect long-range order. Thus the
non-equilibrium dynamical effects are most striking in region C The jammed states have a
random distribution of energies, but are statistically similar in structure. The structure factor
of these "glassy" states can be calculated analytically. There is no true long range order in
tbe jammed states, but large sections of the jammed chain can show periodic structures
which are quite similar to the anti-phase state.
446
Prabodh Shukla
6. Concluding remarks
There is an obvious . scarcity of exact results in the field of random systems and non^
equilibrium statistic^ mechanics. We have described a method which provides an exact
result in one dimensibli for the zero-temperature non-equilibrium dynamics of the random
field Ising model with ferromagnetic interactions. The method can be generalised to a Bethe
lattice. Hysteresis loops as well as avalanche distributions (Barkhausen noise) can be
obtained exactly [8]. So far we have been unable to solve the problem of anti-ferromagnets
exactly, but work is in progress in this direction.
References
[1] Y Imry and S K Ma Phys. Rev. Lett. 35 1399 (1975); For a more modem reference, see T Nattermann,
cond-mat 9705259
[2] D Sherrington and S Kirkpatrick Phys. Rev. Lett. 35 1 972 ( 1975)
[3] J P Sethna, K Dahmen, S Kartha, J A Krumhansl, B W Roberts and J D Shore Phys. Rev. Lett. 70 3347
(1993); 0 Perkovic, K Dahmen and J P Sethna Phys. Rev. Lett. 75 4528 (1995)
[4] Exact .wtution of zero-temperature hysteresis in a ferromagnetic Ising chain with quenched random fields
P Shukla Physica A233 235 (1996)
[5] Zero-temperature hysteresis in the random field Ising model on a Bethe lattice D^pak Dhar, Prabodh
Shukla and James P Sethna J. Phys. A30 5259 (1997)
[6] Zero-temperature hysteresis in an anti-ferromagnetic Ising chain with quenched random fields P Shukla
P/i>jicoA233 242 (1996)
[7] W Selke Physics Reports 170 213 (1988); See also, W Selke in Phase Transitions and Critical
Phenomena Vol IS eds C Domb and J L Lebowitz (London - Academic Press) and references therein
(1992)
[8] To be published (collaborative work)
Indian J. Phys. 72A (5), 447^54 (1998)
UP A
— an international journa l
Electron tunneling in heterostructures under a
transverse magnetic field
P K Ghosh and B Mitra
Department of Physics, Visva-Bharoti University. Sanliniketan-73 1 235.
West Bengal, India
Abstract ; The transfer matrix formalism is used to study the electron tunneling in
semiconductor heterostructures in the presence of a transverse magnetic field. The transntission
coefficients for heterostructures where tte barriers are arranged in a manner either periodic or
quasiperiodic are calculated. In a quostperiodic heterostructure, the group of resonant peaks is
depressed relative to the resonant peaks in a periodic heterostructure. The magnetic field
produces a shift of the transmission coefficient to a higher energy value and, when the field
increases, the peaks in the group of resonances are depressed progressively and finally disappear
in a stronger magnetic field.
Keywords : Transmission, heierosinictures, transverse magnetic field
PACSNos. : 73.20 Dx. 73 40.Lq
1. Introduction
Recent advances in submicrometer physics have made possible the fabrication of low-
tlimcnsional electronic systems (Roukes et al 1989). This has naturally stimulated interest
in their physical properties, especially those related to transport phenomena. There have
been numerous studies, both experimental and theoretical, devoted to the physics of
transport in semiconductor heterostructures under a variety of conditions related with
temperature, electric and magnetic fields, dimension, arrangement and many-body
interactions (BUttiker 1988, Landauer 1989, Harris et al 1989). In particular, electron
tunneling through a heterostructure in a transverse magnetic field has been studied
extensively (Ando 1981, Xia and Fan 1989, Helm et al 1989, Cruz et al 1990, Zaslavsky et
at 1990, Curry et al 1990).
Hung and Wu (1992) have considered the GaAs/Alj^Gai.^As heieroslructure
and obtained the energy levels and electron tunneling in such a heterostructure under an
^2A(5)-15
© 1998 1 ACS
448
P K Ghosh and B Mitra
in-plane magnetic field. At the same time, transmission through a one-dimensional (ID)
quasiperiodic system has attracted considerable attention (Wurtz et al 1988, Avishai and
Berend 1990, 1991). Singh et al (1992) have made a comparative study of electron
tunneling in periodic and quasiperiodic superlattice systems. Recently, Chen et al (1994)
studied the electron tunneling in the semiconductor quantum-wire superlattice with
randomly distributed layer thicknesses.
In this article we study the transport properties of a semiconductor heterostructure in
a transverse magnetic field. We make quantum-mechanical calculation of the electron
tunneling in heterostructures under a transverse magnetic field. To calculate the
transmission coefficient we solve the Schrbdinger equation in one cell and then by the
successive multiplications of the transfer matrices we obtain transmission and reflection
amplitudes for the whole structure. We calculate the transmission coefficient for a
heterostructure where the barriers are arranged in a manner either periodic or quasiperiodic
and compare the results for these two cases.
The remaining part of the paper is organized as follows. Section 2 introduces the
heterostructure under study and contains the theoretical formalism used in our calculation.
In section 3 we give the results with detailed discussions. Section 4 is a summary.
2. Theoretical formalism
Here we consider semiconductor heterostructures in which each building block consists of
double layers. We further assume that the first (second) layers are constituted by the same
semiconductor material e.g. by GaAs (Al/Jaj.j^s). In the presence of a transverse
magnetic field, the Hamiltonian can be written as
H= j^[pl+{Py+eA)^ +pI] + U(x), (1)
where m* is the electron effective mass and the building blocks are assumed to be arranged
along the x direction. For the gauge A associated with the magnetic field, it can be written
as A = (0, 0, 0) for jt < 0, A = (0, Bx, 0) for 0 < x < L, and A = (0, BL, 0) for x > U where L
is the size of the system in the x direction. Substituting the wave function
9'(x,y,z) =
into the Schrodinger equation H'P = E% one obtains the eigenvalue equation
AL
2m*
dx^ 1
ft J J
<p(x)-^U{x)(p(x) =
hH}]
E *
2m* J
<P{x)
( 2 )
(3)
In what follows, we use the Kronig-Penney model to characterize the potential U(x), ie. the
potential is assigned as constants 0 and V within the first and second layers (corresponding
to the well and barrier) of each building block, respectively.
Electron tunneling in heterostructures etc
449
We divide the i-lh well (barrier) into M(N) slabs and treat the gauge A within
every single slab as a constant vector (Taylor 1977). For explicitness, in the ;-th slab of ihe
i-th well, (Xi /M , jc, +(;>l)fl, /M)J = 0. I, 2, M-1, where a, is the width of
the i-th well, the term + {eBlh)x^ in eq. (3) is approximated by [iky + {eBjh) x
(X, + ;a, /Af)]^ and within this slab eq. (3) then becomes
h'^ d^y(x)
2m* dx"^ ^
which has the plane-wave solution
xe (x,+ ja, /M, x, + {j + \)aJM).
where is given by
(5)
2m‘E
■k^
*. +
( 6 )
Withia the >-th slab of the I'-th barrier, (jt, +a,- + jbj /N,Xj +<!,• + (y + l)hj /N),
; = 0, 1, 2, A/,-1, where b, is the width of the I’-th barrier, eq. (3) is approximated by
^2
<P(x)
d'^9(x) ^
2m* dx^ 2m*
. eB( .bA
= \E-V-
2m*
<p{x)
(7)
with the plane- wave solution
where
(jc) = C . ~ IN)) ^ (* ” /A>) ^
Jte (x/ +a, +jbilN, x, +a, +(; + l)h, /N),
=
2m*(£-V)
[*v+f(^. + “<+^77]
i/2
( 8 )
(9)
Alter the above approximation, the gauge A within the heterostructure is replaced by a
stair-step vector potential. When the number of the slabs in eachrW'ell (barrier) is
sufficiently large, eqs. (4) and (7) will accurately characterize the behaviour of the electron
in the heterostructure.
450
P K Ghosh and B Mitra
From the wave-function-matching conditions at the boundaries of the slabs along the
X direction, a set of coupled equations linking the amplitudes ( Cq, ) and { C, j, Z)g } can
be derived :
1+-^
k. i ^
1-T-^
'^I.J+1 J
g-ik,ja, fM
( C. .■ ^
J
( 10 )
; = 0, 1,2 M-2,
i f I + 1 e “< / " i f 1 - , -‘‘..--I -I / "
\ ^/.o J 'I ^-.0 J
•f,
A “ ^..0 J
r.r a:.. ^
D.
1,7 + 1
'"c.+i.o
l^f + 1.0 J
'I ^/.o
f K \
1 I iiL
( 11 )
('C,, ^
( 12 )
y = 0. 1.2 N-2
\ + bJN iL_ *, /N
^ * 1 + 1,0 j * 1 + 1.0 J
\ _ V -ifij.-. if 1 + V -«..»-i ^ /'v
1, * 1 + 1,0 J * 1 + 1.0 j ^
f ^l.N-
D,,n.
. (13)
By successive multiplications of the transfer matrices given in eqs. (10-13), we are able to
to obtain the transfer matrix (Af = (my) linking the amplitudes of the wave functions at the
left and right ends of the structure, and finally calculate the transmission coefficient by the
following equation :
fm^^ mi2Vl
V"*2i "*22 Hr
(14)
where r(r) is the amplitude of the reflected (transmitted) plane wave at the left (right) end of
the structure. Here, the amplitude of the plane wave incident to the structure is chosen to be I .
From eq. (14) it follows that the reflection coefficient is /? = |r|^ ^
"22
transmission coefficient is thus given byr=l-/? = 5- ^ according to the law of
”72
probability conservation.
Electron tunneling in heterostructures etc
451
3. Numerical calculations
We first consider periodic heterostructures in which two wells with widths Ol and are at
their left and right ends, and the widths of barriers and other wells take values a and b,
respectively. In our numerical calculations, we use dimensionless quantities, i.e., the energy
and magnetic field are in units of h^tfllrnti^ and h/eti^ respectively. When the transverse
magnetic field is applied, we divide every well (barrier) into M(N) = 100 slabs in the
numerical calculations. In Figure 1 we present the transmission coefficcient for the double-
barrier case in which oi^Or^ 0.25, a = 0.5, b = 1 , and V= 1; the magnetic field is chosen
to be B = 0, 0.02, 0.06 and 0.1, corresponding to the solid, dashed, dotted and dash-dotted
curves, respectively. Also, ky and are both taken to be zero. From Figure 1 one sees that
Figure 1. Transmission coefriciem for a double-bomcr .structure, where ai = aR-
0.25, a = 0.5, b = I and V = 1 and = 0 (solid curve), 0.02 (dashed curve), 0.06
(dotted curve) and 0. 1 (dash-dotted curve). Also ky and are both taken t&be zero
only one resonant peak occurs in the considered energy range and the peak shifts rightward
as the magnetic field increases. This observation matches the results obtained by Hung and
Wu (1992) for the double-barrier heterostruclurc. In Figures 2(a)-2(d) the transmission
coefficient is calculated for a periodic heterostructure with five barriers, in which the
parameters are chosen to be the same as in Figure 1 and B = 0, 0.04, 0.07 and 0.1,
respectively. It can be seen that with the increase of the magnetic field, the resonant-peak
group shows an overall rightward shift and the resonant peaks are depressed. Particularly,
when the magnetic field is strong enough, a given peak can even be completely depressed.
We have also calculated the transmission coefficient for a periodic heterostructure with 13
harriers (sec Figure 3), where the parameters arc the same as in Figure 1 and B « 0 and
02 . Also, it can be seen that there exists apparent depression of the resonant peaks as
induced by the applied maganetic field.
Finally, we study a heterostructure with the barriers arranged in a quasiperiodic
tanner. The parameters of the structure are chosen to be * ag = 0.25, fl = 0.5 and V= 1,
Figure 2. Transmission coefficient for a periodic structure with five barriers, where the
parameters arc the same as in Figure I and F = (a) 0, (b) 0.04, (c) 0.07 and (d) 0. 1
The barriers are arranged according to the construction rule for the Fibonacci
sequence (Kohmolo et al 1987) ; Sf+i = {5/, 5/_j) with / ^ I and the initial conditions Sq =
(5) and 5] = (/\ }. For this construction rule, the number of letters A and B in 5/ obeys the
recursion relation F/+| = F/ + f/_i with Fq = Fj = 1. Figures 4(a) and 4(b) show Ihe
transmission coeliicient for a quasiperiodic heterostructurc with Fg = 13 barriers, where the
magnetic field is chosen to be 5 = 0 and 0.02, respectively. In the absence of the magnetic
field, the group of resonances is not as high as in the periodic case [comparing Figure 4(a)
with Figure 3(a)]. This overall depression of the resonant peaks is due to the quasiperiodic
order existing in the heterostructure. When the magnetic field is applied, the transmission
coefficient shifts rightward and the field-induced depression of the resonant peaks also
occurs.
4. Summary
In summary, we have studied the effect of a transverse magnetic field on the electronic
transmission in semiconductor hetcrostructures. The transfer matrix approach is employed.
Eiwgy
Figure 4, Transmission coefficient for a quosiperiodic structure with 13 barriers, where the
parameters are the same as in Figure I but the width of each barrier takes either 1.2 or =
0.8, and B » (a) 0 and (b> 0.02.
454
P K Ghosh and B Mitra
The heterostructures under study have the barriers arranged in a manner either periodic or
quasiperiodic. We have compared the results for quasiperiodic system with those of the
periodic system both in the presence or absence of a magnetic field. We have employed a
plane>wave transfer matrix formalism in this work, while the parabolic-cylinder-function
transfer matrix was used by Hung and Wu (1992). Our approach is more efficient and less
cumbersome, and the results become accurate when each of the wells and barriers is divided
into a large number of slabs. For a periodic double-barrier hetcrostructure there is a single
resonant peak [in the energy range considered], while there is a number of peaks in a
group for a heterostructure with more barriers. The presence of the magnetic field results in
an overall rightward shift of the resonant peak group. With increase of the magnetic field,
the resonant peaks are depressed progressively and are totally depressed in a strong
magnetic field. For the quasiperiodic hetcrostructure, the group of resonances is depressed
relative to the resonant peaks in the periodic case. This is due to the quasiperiodic order
existing in the hetcrostructure. With the application of the magnetic field, the transmission
cocITicicnt shifts to a higher energy value and the resonant peaks are depressed.
Acknowledgments
P.K.G. and B.M. were supported by the Department of Atomic Energy, Government of
India and Visva-Bharati University.
References
[ I ] T Ando J. Pfiys. Soc. Jpn 50 2978 (1981)
[2] Y Avishai and D Bcrcnd P/iyv. Rev B41 5492 (1990)
13| Y Avishai and D Bcrend Phys Rev B43 6873 (1991)
141 M Buuiker/flAf y. Res Dev 32 3l7(t988)
[5] X Chen, S Xiong and G Wang Phys Rev B49 14736 (1994)
[6] H Cruz, A Hernandez-Cabrera and P Aceituno / Phys. ■ Condens. Matier 2 8053 (1990)
[7] L A Curry, A Celeste, B Goutiers, E Kanz and J Portal SuperlaU. Microstruct. 7 415 (1990)
[8] J J Harris, J A Pals and R Woitzer Rep Prog. Phys. 52 12 17 ( 1 989)
[9] M Helm, F M Pecters, P England, J R Hayes and E Colas Phys Rev. B39 3427 (1989)
[10] KM Hung and G Y Wu Phys. Rev. B45 3461 (1992)
1 1 1 1 M Kohmolo, B Sutherland and C Tang Phys. Rev. B35 1020 (1987)
[121 R Landauer J. Phys. : Condens. Matter 1 8099 (1989)
[131 ML Roukes et al Science and Engineering of one- and zero-dimensional Conductors edited by S P
Beaumont and C M Sotomayor-Torres (New York : Plenum) (1989)
[14] M Singh, Z C Tao and B Y Tong Phys. Status Soltdi B172 583 ( 1 992^
[15] PL Taylor P/iys. /?ev. B15 3558 (1977)
[16] D Wiinz, M P Socrensen and T Schneider Helv Phys. Acta 61 345 (1988)
[17] J B Xia and W J Fan Phys. Rev. B40 8508 ( 1989)
[18] A Za.siav.sky, Y Yuan, P U, D C Tsui, M Santos and M Shayegan Phys. Rev. B42 1374 (1990)
Indian J. Phys. 72A (5), 455-461 (19^8)
UP A
— an international journal
Sticking of He^ on graphite and argon surfaces in
presence of one phonon process
G Duttamudi and S K Roy
Department of Physics. Visva-Bharali University, Santinikctan-731 235,
India
Abstract : The sticking coefficient of He^ gas particles on to the surfaces of graphite and
argon is evaluated using the Greens function method. We explicitely look into the variation of
sticking coefficient with the incident energy of gas particles from exact scattering T-Matrix of
the system Contrary to the classical prediction we report a distribution of sticking coefficient
with incident gas particle energy. This distribution is obtained for the sticking coefficients
calculated in presence of bound state resonance and the phonon emission/absorption The exact
time evaluation of the incident particles shows the expected nature and values of sticking
coefficients of He^ on graphite and argon surfaces.
Keywords : Sticking coefficient. T-Matrix. Greens function
PACS Nos. : 68.45.Da, 68.35.Md. 82.63 Dp
1. Introduction
The kinetics of adsorption and desorption of atoms physisorbed on solid surfaces have been
reviewed experimentally and theoretically in recent few years [1-3], Most of them tackled
ihc problem by First Order Distorted Wave Born Approximation (FODWBA). In our
previous work [4] we have shown that in the phonon assisted scattering of a gas-solid
mieraclion the lower order DWBA may not he adequate to explain the total inelsatic
vuinponeni of the gas solid interaction.
On the other hand the mechanism behind the sticking in the quantum regime
''i understood from the observed sticking coefficient (SC) and the elastic scattering
probabilities from the cold surfaces. A recent paper by Z W Gorlel et al [5] shows that
may lead to a value greater than one by FODWBA. They removed this difficulty
linding renormalised SC, specially for Hc-Ar and He-Graphiie systems. Our aim is to
ihe SC by removing the overcounting of the scattering channels and to lake into
account all the inelastic components and the scattering channels. Because of low
© 1998 1 ACS
456
G Duttamudi and S K Roy
inelastic components and the scattering channels. Because of low energy incoming
particles, the inelastic process becomes more important and responsible for sticking.
We therefore concentrate on phonon mediated physisorption of gas particles at
normal incidence for low coverage at localised adsorption sites assuming that interaction
between the adsorption sites is negligible.
We have proposed a Hamiltonian in terms of localised and phonon basis and
changed the phonon basis into the localised basis by a canonical transformation. Hence we
obtain the general theory for the temperature dependent bound state energy for the adsorbed
system. The theory developed can then be used to calculate the sticking and inelastically
scattered intensities for different bound states of the systems.
2. Theoretical model
The model Hamiltonian for the gas-solid system with localised and nonlocalised basis
may be written as [6]
H = (1)
where is the Hamiltonian of the non interacting gas system in a box of length L For just
one shallow BS the three dimensional theory can be reduced to one dimensional theory and
by introducing the second quantized creation/annihilation operators in the slate
\k > the first part of the Hamiltonian becomes
( 2 )
where £i^ = ^ ^ /c ^ / 2m is the kinetic energy of free gas particle.
The second part of Hamiltonian (1) is that of the solid which in the harmonic
approximation may be written as
with I bp, is the creation/annihilation operator of longitudinal acoustic phonons ol
frequency (Op in the absence of gas.
The third term in equation (1) gives the gas-solid interaction. It consists of two parts,
a static and a dynamic i.e. //‘J and respectively. Using the creation/annihilation
operators the static part of the Hamiltonian becomes
where Eg is the eigen value of the free particle state and BS energy £„ with n * 0, 1.
2,3....
Sticking of He^ on gnyykite and argon surfaces etc
457
In fact the phonon-mediated gas solid interaction is accounted for by the dynamic
part of the Hamiltonian which in the lowest order harmonic approximation is given by
^dyn - p {^p'^^pY’q*
Q>Q-P P
where for local surface potential we have
Here 0^(x)'s are the eigen function of denoted by the eigen value equation
Now with the help of the above equations, the Hamiltonian H in equation (1) takes
the form,
q p q>q-p
( 8 )
P
With solid particle mass and the number of particles normalised in a box of length 'L.
Now by using the similarity transformation we reduce the above Hamiltonian to [6]
r .r^ c
where ='S^-
We have solved this Hamiltonian by T-Matrix formalism. We take the static surface
potential as
VqW = (;o(exp-2y<^-^o)_ 2exp-y<"-'oM. (10)
Now in order to obtain the Dyson equation for scattering T-Matrix, we write the single
panicle Green’s function as
The Green's function may , be written in the form of Dyson equation using above
Hamiltonian and taking the fourier transformation as
C«(£) = G*(0) + Cj(0)7TGt„(£),
( 12 )
458
G Duttamudi and S K Roy
which on iteration becomes
where
and
G^iE) = Co(£) + Go(E)TGo(El
1 InAt
P
( ! ! \
~ ^tf-p p ~ ^q-p P ,
( 13 )
(14)
Here 'q' is the momentum of the gas particle in the localised state and iq-p) is that in the
BS. E is the effective final energy with transformed BS energy is due to gas solid
interaction and the continuum state energy.
The relative gas atom occupation number in the substrate maintained at substrate
temperature and at gas temperature is
'• (15)
with, il'® chemical potential of the gas which for He gas
is taken here as 144°K [7]. ^
Now while evaluating the P)\^ we consider the dimensionless parameters
as 18)
2mUo ^ H^.-l
ft 2 '
2mcOj
~hP~
; ^ ^0 ~ Xo-^o
and the normalised BS wave function as (x) = V7/« (<^ )
f„(4) = (2(To)^- r'-''2>(25„)pY”)
exp(-aoe-‘4-4o))e-s.(«-{o)i,“- (2(7of'^-^»)), O^*)
where S„ = (To withR = 0, 1, 2 ... andL^^’ (u) is a Laguerre Polynomial.
The continuum state wave functions of momentum 'q' normalised in a box of length
1' ; (- 1 < X < L\ are given by 0, (x) = (2L)'''^ f(T\\^)\r\ = ql y ax\A
f(m^) =
r(i/2-(To-«'n)
r(2iTi)
1^(1 / 2 - <Jo + i»?, 1 + 2«»J, 2 <To e * ),
(17)
Sticking of He^ on graphite and argon surfaces etc
459
where }iKa,b,z) is a confluent hypergeometric function that vanishes at z Again while
evaluating the sums we have to invoke the thermodynamic limit Le. -> (L/ /r)lQ dit
and perform the sums over phonon states for a Debye model Le.
3N
X -4 ( — 0 . This leads to a real and a imaginary part of the T-Matrix.
The transition probability from the initial continuum to a final BS under the
emission of phonon may be given by
axp
k-p
5(£t ft Wo). (18)
Hence the transition probability from continuum gas state of the momentum k to all
the As is,
(19)
n
So the sticking coefficient for a particle of momentum k normalised by the flux of incoming
particle tL defined by tL = is
5^ = (20)
Now we define the total sticking coefficient S as the average of 5* over the spectrum of
thermal flux of incoming panicles as [9]
I*
nh^
2mK^T^
m
(21)
where the is the incoming current density of particles of momentum fik, and n is the
normal to the solid surface of area A.
With = exp(/3^/i - ) we get the expression for sticking coefficient as
S = 16
nh'^
C^|icVI[F(Jt)]^
[(G(r-S2))2 + {nl2(r-Sl)]^]N„txp{pgti)e(x-r)
( 22 )
With
= [ dyF{y)
Jo
(23)
F(x) =
5inh(2ffVjc)
sinh^(*Vjt)+cosh*(OTTo )
|r(l/2 + ffo+«VIl*
U + «Jo-«-l/2)2p,
(24)
460
G Duttamudi and S K Roy
Gix) = r + -
JC-J..
-In
r-(x-sl +j2)
r + (jr-j2 +j2)
f
(r-x)
(25)
(26)
3. Results and discussion
The essential feature of this work is based on the calculations of Sc of He'* on cold graphite
and solid argon surfaces with extended particle phonon interactions giving rise to the
inelastic scattering in presence of resonant surface BS. Although the inelastic scattering is
sufficiently weak in this case where only one or two phonons are created or destroyed, it is
never negligible.
Figure 1. The variation of slicking coefficient with incident energy ‘
(i) Dashed line for He-Ar and (ii) .solid line for Hc-graphite .system.
The variation of sticking coefficients for Hc-graphite and He-Ar systems with
incident He gas particle energies has been shown in Figure 1. Clearly the slicking
coefficients for both the cases show broad peaks at intermediate gas particle energies. This
is in contrast with the classical prediction but is in good agreement with the experimental
results confirming the fact that the experimental predictions for low energy particles need lo
be done quantum mechanically. The figure also indicates that for both the systems the
slicking coefficients are less than unity. This suggests that the present T-matrix calculation.s
of slicking co-efficients also lakes care of the problem of ovcrcounting of scattering events
as was encountered in DWBA calculations. However at higher incident energies the nature
of the slicking curve is that of the classical predictions confirming the fact that the particles
at higher energies will be bounced back thereby reducing the sticking.
Sticking ofHe^ on graphite and argon surfaces etc
461
Acknowledgment
The authors are thankful to the Department of Atomic Energy, Govt, of India for financial
support.
References
[1] H Schlichtinge/<i/P/(y.f. Rev. Ler/. 60 2515(1988)
[2] G Armand and J R Man.son Phys. Rev B43 14371 (1991)
f31 M D Slilc.s and J W Wilkins Php. Rev. Uti 54 595 (1985)
[4] G Duttamudi and S K Roy J. Phys. C8 8733 (1996)
[5] Z W Gortcl and J Szymansky Phys. Lett. A147 1 59 (1990)
[6] G Duttamudi and S K Roy Indtan J. Phys. 70A 709 (19%)
[7] MW Cole, D R Frankl and D L Goodstein Rev. Mod. Phys. 53 1 99 ( 1 98 1 )
[8] Z W Gorlel, H J Kreuzer and R Teshima Phys. Rev. B22 5655 (1980)
[9] M D Stiles and J W Wilkins Phys. Rev. B34 4490 (1986)
Indian J. Phys. 72A (5), 463-467 (1998)
UP A
— an internatio nal journal
Influence of alloy disorder scattering on drift velocity
of hot electrons at low temperature under magnetic
quantization in n-Hgo.8Cdo.2Te
Chaitali Chakraborty
Department of Electronic.^ and Telecommunication Engineenng,
Jadavpur University, Calcutta'700 032, India
and
C K Sarkar
Department of Physics, B. E College (D U ).
Shibpur, Howrah-? 1 1 103, India
Abstract ; The drift velocity of hot electrons in n-Hgo.8Cdo.2Te has been calculated in the
presence of parallel electric and quantizing magnetic fields at low temperatures. The low
temperature scattering mechanisms such as acoustic phonon scattering via deformation potential,
piezoelectric coupling, ionized impurity and alloy disorder .scattering are considered. The effect
of high electric field leading to a disturbance in the phonon distribution has also been
incorporated. The effect of alloy disorder .scattering on the drift velocity has been analysed for
equilibrium and disturbed phonon distributions.
Keywords : Hot electron, dnft velocity, alloy disorder scatienng
PACSNo. ; 72.I0.-<1
1. Introduction
Mercury Cadmium Telluride is one of the important materials widely investigated by
many workers due to various reasons. Firstly, the material offers an excellent choice
lor an infrared detector useful for operating in atmospheric window region. Secondly
It being a narrow-gap semiconductor with small effective mass, magnetic quantization
condition can easily he achieved in this material with reasonably low magnetic
field.
Transport properties in n-Hgo.gCdo zTe at low temperatures are governed by acoustic
phonon and alloy disorder scattering. The recent analysis of mobility in the extreme
quantum limit (EQL) [1] shows that the alloy disorder scattering is one of the dominant
mechanism in determining the mobility at low temperatures.
© 1998 lACS
72a (.5). 17
464
Chaitali Chakraborty and C K Sarkar
Furthermore the hot electron transport is also an important aspect to be investigated
as far as device applications are concerned. In n-Hgo,gCdo. 2 Te, high field transport ie.
energy loss rate and drift velocity of hot electrons have been investigated theoretically
and experimentally by many authors [2-5]. However, the effect of alloy disorder scattering
on the high field quantum magnetotransport properties has not being investigated in
detail. It will be interesting to study the influence of alloy disorder scattering on drift
velocilf'of hot electrons in n-HgCdTe in the presence of a quantizing magnetic field.
Such investigation is useful in understanding the specific role of the alloy disorder
scattering and its influence on high field transport in the presence of quantizing
magnetic field which is quite different from hot electron transport without magnetic
field.
In the present paper, the drift velocity has been investigated under high field
condition in the presence of a quantizing magnetic field (longitudinal configuration). The
dominant scattering mechanisms are acoustic phonpn via deformation potential and
piezoelectric coupling, ionized impurity scattering and alloy disorder scattering.
Furthermore, the equilibrium phonon distribution function which obeys Bose-Einslein
distribution is assumed to be disturbed due to the presence of high electric field. This is
because the carriers supply the energy obtained from the applied electric field to the
phonons at much higher rate compared to the rate at which phonons lo.se excess energy to
the thermal bath and it has a feedback effect causing a change in the energy loss rale of the
hot electrons. In calculating the drift velocity, cairiers are confined to the lowest Landau
level (EQL) and obey displaced Maxwellian distributions [6]. The nonparabolicity of the
band structure, modified free carriers screening due to high magnetic field and
nonequipartilion of phonons are also considered.
Finally, the influence of alloy disorder scattering on the longitudinal drift velocity of
hot electrons are being examined for equilibrium and disturbed nonequilibrium phonon
distributions.
2. Theory
Assuming that the electric field is applied to a semiconductor with a nonparabolic band
structure [7] in the same direction as in the high magnetic field B and taking a
Maxwell Boltzman distribution for carriers occupying the lowest Landau level
characterised by an electron temperature f,, the drift velocity of electron can be obtained
from the relation
Vj =ne ( 1 )
At the steady condition, energy loss rate can be written as a function of electric
field as
(dE/dT) = v,HE
( 2 )
Influence of alloy disorder scattering on drift velocity etc
465
Again the mobility is given by
n = e(T)/m‘ (3)
where < T> is the average momentum relaxation time.
The momentum relaxation limes needed in the equation (3) for computing the
drift velocity are taken for elastic phonon scattering, piezoelectric coupling, ionized
impurity and alloy disorder scattering. Then the combined effects of relaxation times
can be expressed as
1/ T = 1/ T„ + 1/ Tp, + 1/ i- 1/ Ti^ (4)
The explicit T's afe obtained from [8] including the magnetic field dependent
screening, non equipartition of phonon and Landau level broadening.
Furthermore, the average energy loss rate has been obtained assuming elastic
acoustic phonon via deformation potential and piezo electric coupling which have the
dominant loss scattering mechanism at low temperature. The alloy disorder scattering being
an elastic scattering does not affect the energy loss rate while its contributions to the
combined relaxation time is quite important. Also the calculation of energy loss rate
includes magnetic field dependent screening and Landau level broadening and non
equipartition of phonons [3].
In obtaining the drift velocity we consider (a) the phonon distribution which is
independent of magnetic field given by Bose Einstein distribution, (b) phonon distribution
which is given by the following rate equation [3]
{dN,/dT)^ ={N,-No)/r^ (5)
The ultimate phonon distribution as a function of electric field can be obtained by
the solution of the rate equation (5).
Finally, the drift velocity for cases (a) and (b) with the inclusion of all scattering
mechanisms are discussed.
3. Results and discussions
The longitudinal drift velocities of hot electrons in n-Hgo 8 Cdo. 2 Te are calculated as
a function of electron temperatures ranging from 12 K to 40 K at a lattice temperature
Tl = 10 K and magnetic field B = 4T. Variation of drift velocity with electron temperature
studied f6r all scattering mechanisms. A comparison of the results both for equilibrium
and nonequilibrium phonon distributions with phonon life times Tp = 100 ns and ip =
1000 ns is also done here.
Phonon life time at low temperature is assumed to be governed by the phonon
l>oundary scattering which is given by T = //v where I is the dimension of the sample and v
IS the acoustic velocity. The other nonelectronic phonon mechanisms responsible for
466
Chaitali Chakraborty and C K Sarkar
phonon annihilation is phonon-phonon etc. These effects are dominant at high temperature
only [9,10].
Figure 1 shows that drift velocity increases with electron temperature, but the rate
of increase slows down in the higher electron temperature. It is seen that the inclusion of
Figure 1. Variaiion of drift velocity of hot
electron in n-Hgo.gCdo. 2 Te for different
scattering mechani.sms as a function of
electroi\ temperature. The upper curve for
the acoustic phonon via deformation
potential and piezoelectric coupling, the
middle curve for ionized impurity .scattering
and the lower curve for alloy disorder
scattering.
10 20 30 40
Electron Temperature T«(K)
ionized impurity scattering decreases the drift velocity value but the alloy scattering reduces
the magnitude of drift velocity quite significantly.
Figure 2. Variation of drift velocity of hoi
electron in n-Hgo.gCd() 2 Te for all scattering
mechanisms as a function of electron
temperature for equilibrium (solid) and
nonequilibrium phonon disiiibuiimiN
(dashed curve for tp = \0O n.s and dotted
curve for Tp = I (XX) ns),
10 20 30 40
Electron Temperature T«(K)
Now the energy loss rate at Icnv temperature due to acoustic and piezoelectric
phonon scattering increases with electron temperature and so drift velocity also increases
with electron temperature. But when ionized impurity and alloy disorder scattering
are considered, the drift velocity decreases because of the enhancement of scattering rate.
The low temperature high field drift velocity is primarily determined by the momentum
relaxation time of nonphonon type of scattering such as alloy disorder scattering
because the effect of electron temperature on drift velocity is not so significant due to
small energy loss rate when acoustic phonon scattering is considered. As a result alloy
scattering plays dominant role at low temperature and decrease the value of drift velocity
significantly.
Influence of allay disorder scattering on drift velocity etc
467
In Figure 2, a nonequilibrium phonon effect is considere4. The energy loss rate
process which is due to acoustic and piezoelectric scattering, the nonphonon ionized
impurity and alloy disorder scattering is not affected by the non equilibrium phonon
distributions. It is clear from the nature of the graph that the value of drift velocity for non
equilibrium phonon distributions is lower than that of equilibrium phonons. Actually the
inclusion of non equilibrium phonons slows down the cooling processes due to reabsorption
of phonons emitted by hqt electrons [3]. This process may be considered as a feedback
process which leads to decrease in energy loss rate and as a result value of drift velocity is
also decreased for nonequilibrium phonon distributions.
Acknowledgment
This work is financially supported by the UGC and the CSIR, India.
References
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Dr. A Barman
Dr. Arani Chakraborty
Dr. A Mishra
Dr. Anindya Sarkar
Dr. Amit Chatterjee
Prof. A Rahaman
Dr. (Mss.) Ashmita Sengupta
Prof. Asok Sen
Dr. Asil Kumar Kar
Prof- B C Khanra
Dr. Bidisha Nandi
Prof, B K Chakraborti
Dr. Biplab Chattopadhyay
Prof. Bikash Gupta
Dr. B Sanyal
Dr. B Sundaravel
Dr. B Roy
Prof. C. K. Majumder
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Dr. Goutam Dutta Mudi
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Dr. Jayashrec Saha
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Prof. K Mohan Rao
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Prof. N C Misra
Dr. Parangama Sen
Dr. M Ghosh
Dr. P A Sreeram
Prof. P Shukla
Prof P K Ghosh
Prof. R C Patnaik
Prof. Ranjit Pal
Dr. Ranjana Roy
Prof. R Ranganalhan
Prof. S M Bhattacharya
Dr. S Bhattacharya
Dr. S Biswas
Dr. S N Choudhari
Dr, S B Ota
Dr. Sanjoy Mukherjee
Dr. S Kumar
Dr. S. K De
Prof. S K Ghatak
Dr. S K Joshi
Dr. Sourav Banerjee
Dr. Soma Dey
Dr. S. K. Das
Dr. S K Ghosh
Dr. T K Dcy
Dr. Smila Ota
Dr. T Purohit
Prof. S K Sinha
Dr. T P Sinha
Prof. S K Roy
Dr. T K Munshi
Dr. Sudhakar Yadagadda
Dr. U Dey
OCTOBER 1998, Vol. 72, No. 5
Review
Scattering of electrons and photons by atoms and ions
S N Tiwary
Astrophysics, Atmospheric & Space Physics
A total technique for tropospheric communication performance
estimation
S K Sarkar
Atomic & Molecular Physics
Kq L'/Ka X-ray satellite intensity ratio in phosphorus excited by
photons
B Malukarjuna Rao, B Seetharami Reddy, K Premachand,
ML N Raju, K Parthasaradhi, M V R Murti and P Suresh
General Physks
Dielectric relaxation of some diol/alcohol mixtures in different
solvents
Azima L G Saad, Adel H Shaftk and Faika F Hanna
On Hal plate collector- new approach
M K El-Adawi
Plasma Physks
Diagnostic study of N2 laser produced plasma on indium target
Smita Tulapurkar, a G Bidve, S S Patil and
Sharada Itagi
Static pair correlation function of electrons around an infinite mass
positively charged impurity in one and two component classical and
quantum rare hot plasmas
S P Tewari, Kakou Bera and Jyoti Sood
^influence of Hall effect on thermosolutal instability of acomposite
rotating plasma with finite Larmor radius
M Vasiu and a Marcu
/ Cant'd, on next page ]
Notes
OH (8,3) band emission from different excitation mechanisms
S K Midya, P K Jana and S K Mondal
On the dynamical origin of H and Ps binding
S Bhattacharyya and B Talukdar
Nonlinear propagation of dust-acoustic waves in a magnetized
dusty plasma
A A Mamun and M N Alam
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The Comprehensive List of Conference Proceedings for 1 995-97 is also
available on the lUPAP Web site at
http 7/ww.physics.uinanitoba.ca/IUPAP/confproc.html
iiihomogcneity of vortices in 2d classical XY-modcl : a microcanon ical
Monte Carlo simulation study
S B Ota ano Smita Ota
A new VISCOUS fingering instability : the case of forced motions
perpendicular to the horizontal interface of an immiscible liquid pair
B Roy and M H Engineer
Energy, lluctuation and the 2d classical XY-model
Smita Ota, S B Ota and M Sai apathy
Phase alternation in liquid crystals with terminal phenyl ring
Jayashrll Saha and C D Mukherjli:
Change in conductivity of CR-39 SSNTD due to particle irradiation
T Phukan, D Kanjilal, T D Goswami and H L Das
Meiastabilily and hysteresis m random field Ising chains
Pkamodh Shuki.a
Electron tunneling in hctcrostructurcs under a transverse magnetic field
P K Ghosh and B Mitra
Slicking of on graphite and argon surlaccs in presence of one
phonon process
G Dutiamhdi and S K Roy
lutluence of alloy disorder scattering on drift velocity ol hot electrons
ill low icmpcralure under rnagnelic qiianti/.aiion in n-HgonCdo iTc
Chaitai.i Chakramorty and .C K Sarkar
Pages
413-416
417-420
421-425
427-431
433^37
439-446
447-454
455-461
463-467
^ ued by Bishnupada Chowdhury at Print Home, 209A, Bidhan Sarani, Calcutta 700 006
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Indian Journal of Physics A
Vol.72A, No. 6
November 1998
CONTENTS
Proceedings of the XII DAE Symposium on High Energy Physics held at
the Department of Physics, Gauhati University, Guwahati’78] 014, India,
during December 26, 1996 - January 1, 1997
Part 1
Pages
f'oicward
D K CHOUDHURV
(i)-iiii)
Standard Model
PiCL ision icsis oi lhe Standard Model : Present status
G \V \ \M Bh A ri ACH ARY YA
469-477
Heyond Standard Model
Suiius ol wcak-scale supersymmetry
f^KOMiK Roy
479-4*^4
Stains 111 supersymmetric grand unified theories
B Ananphanarayan and P Minkowski
495-502
Collider Physics
Results from LEP 1
S N Ganc.uli
503-514
Plivsiis at LHP 200
A Gurtu
515-532
New physics at eV colliders
Saurabh D Rindani
533-545
QCD and structure functions
Structure functions — iielecied topics
D K Choudhury
547-566
Nuclear structure functions
D Inoumathi
567-578
H eavy flavour physics
Heavy flavour weak decays
R C Verma
579-600
\Coiu'd on ne.M
Paf>es
Heavy ion physics and quark gluon plasma
Quark gluon plasma— current status of properties and signals 601-619
C P Singh
Formal field theory
Blackhole evaporation - stress tensor approach 621-634
K D Kkori
Light-front QCD : present status 635-640
A Harindranath
Methods of thermal field theory 641-66 1
S Mallik
Quantum intcgrable systems : basic concepts and brief overview 663-677
Anian Kundu
Summary talks
Perspectives in high energy physics 679-6K7
G Rajaskkaran
Lxperimenul summary — XII DAE HEP symposium, Giiwahaii, 1997
SuNANDA Bani.rji:c
689-700
Proceedings of the XII DAE Symposium on High Energy Physics held at the
Department of Physics, Gauhati University, Guwahati‘781 014, India, during
December 26, 1996 - January 1, 1997 Part I
Foreword
The Twelfth High Energy Physics Symposium was held In the
Department of Physics, Gauhati University during December 26. 1996
to January 1. 1997 under the auspices of the Department of Atomic
Energy, Government of India.
The symposium was inaugurated by Professor K M Pathak.
Vice Chancellor, Tezpur University. The inaugural function was
presided over by Professor H L Duorah, Vice Chancellor. Gauhati
University. Professor J C Patl of University of Maryland delivered the
keynote address as the distlngulsed guest of honour. Earlier, Professor
S Jols, Dean, Faculty of Sciences, Gauhati University welcomed
the delegates. The Inaugural function ended with votes of thanks by
Professor S A S Ahmed, the secretary of the symposium.
The history of the DAE symposia now spans more than two
decades. The first DAE symposium of High Energy Physics was held in
the year 1972 In Mumbai. Since then, this symposium has become a
major biennial event of high energy physics community of India with
venues shifting to different centers of the country : Santiniketan (1974),
Bhubaneswar (1976), Jaipur (1978), Cochin (1980), Mysore (1982),
Jammu (1984), Calcutta (1986), Chennai (1988), Mumbai (1992),
Santiniketan (1994) and now Guwahati (1996).
The first request to hold the DAE Symposium on High
Energy Physics in Gauhati University came to me from late Professor
P K Malhotra in 1990. But due to time constraints and lack of
infrastructural facilities, we refrained from such a venture at that time.
Only in 1996, we took courage to hold the Xllth symposium, even
though, our infrastructural facilities eind financial capability have not
improved in ahy significant way.
Total 150 delegates from different parts of India and abroad
participated in the symposium. It was a wonderful experience for the
teachers and student community of the university to be with such a
Jistinguished international gathering of scientists.
The symposium consisted of 21 planery talks and 22 invited talks
of 25 minutes duration in the parallel sessions. As has been customary,
a booklet containing the abstracts of the contributed papers (total 128)
was published and distributed to each participant during the registration.
Of the total 45 invited talks including the key note address and
conference summaries presented at the symposium, only 27 talks could
be included in this volume as the manuscripts of others were not
received in spite of our best efforts.
A special feature of the present symposium is the introduction of
the convenerships in the contributory parallel sessions. The abstracts
of the contributed papers were first addressed to the respective conveners
of the parallel sessions for their erudite comments and approvals.
On the very first evening of the symposium, it was decided in a
meeting of the participants that the thirteenth symposium in High
Energy Physics will be held in Punjab University, Chandigarh in 1998.
Hallway through the symposium on a Sunday, a one day break
was arranged during which, most of the participants enjoyed a cruise
on the mighty Brahmaputra. Scientific deliberations in the symposium
got accompanied by the artistic expression with an evening devoted to
a cultural programme of classical and folk music and dances.
In this s 3 miposlum. another evening was devoted to condole the
sudden passing away of Professor Abdus Salam, the Nobel Prize winner
of Physics (1979), the founder Director of International Center for
Theoretical Physics, Trieste, Italy and an outstanding leader of
development of science in the third world. The symposium also felicitated
Professor G Rajasekaran of Institute of Mathematical Sciences, Chennai
for his three decades of dedicated contributions to High Energy Physics
in India and Professor G C Deka of Guwahatl for his leadership in
experimental High Energy Physics in North East India.
The symposium was organised with major funding from the
Department of Atomic Energy, Government of India and we are thankful
to Dr Z Chidambaram, Chairman, Atomic Energy Commission and
Professor S S Jha, Chairman, BRNS for the same. ASTEC and DSTE.
Government of Assam need special mention for their generous grant for
the symposium. We are also thankful to DST, New Delhi, lUCAA, Pune,
TIER, Mumbai. INSA, New Delhi, CSIR, New Delhi, lACS, Calcutta,
Tezpur University, Tezpur, Manipur University, Imphal, Nagaland
University, Kohlma and Dibrugarh University, Dibrugarh for financial
support for the symposium, besides the host Gauhatl University. It Is
also a pleasure to acknowledge support from Oil India, Dullajan, IOC,
Dlgbol, Noomallgarh Refinery Limited, Gauhatl Refinery, Coal India
Limited, Assam Carbon Limited, Assam Electronics Development
Corporation, Assam Industrial Development Corporation, Sanghi Textiles
Limited. Down Town Hospital Limited, State Bank of India, Gauhati
University Branch, Assam Petrochemical Limited, Namrup, Assam Gas
Company, Dullajan and Decision Computers, Guwahati.
We would like to record our appreciation for the untiring efforts of
the members of the local organising committee and the efforts of our
student volunteers and research scholars, which made the symposium
a great success. We thank Gauhati University authorities for their all
round cooperation. We gratefully acknowledge the efforts of the
members of the Gauhati University Computer Center for ensuring smooth
e-mail service during the symposium. We are specially thankful to
Dr. N N Singh and Dr. J K Sarma for providing assistance towards
preparation of the booklet of the abstracts and Ms. Ranjita Deka for
her invaluable help in publication of the souvenir.
We are grateful to the editorial board of Indian Journal of
Physics, particularly to Professor S P Sengupta, Editor in Chief and
Dr. (Mrs.) K K Datta, Assistant Secretary, for agreeing to publish the
Proceedings as a supplementary issue.
D K Choudhury
Convener
Indian J. Phys. TIK (6), 469-477 ( 1998)
UP A
— un international loumal
Precision tests of the Standard Model : Present status
Gautam Bhattacharyya
Diportimenio di Fisica, Universiih di Pisa and INFN.
Sezione di Pisa, 1-56126 Pisa. Italy*
Abstract : Electroweak data from LEP and SLC as well as data from TEVATRON
(CDF/DO) have established the credentials of the Gla.show-Weinbcrg-Salam nu)del (the so called
Stondard Model) at such a level that there is no other competitive model for the purpose of
describing physics at the 100 GeV scale In this talk. I review the status of the Stundaid Model
by comparing precision data with precision calculations,
Kejrwords : Standard model. LEP. Tevairon
PACS Nos. 12 lO.Dm. 14 70.Hp
1. Introduction
During the 6 years (Summer 1989 - Summer 1995) of running on and around VJ - . ihc
4 experiments (ALEPH, OPAL, DELPHI and L3) of the Large Electron Positron Collider
(LEP) at CERN have in total collected nearly 20 millon Z-events. The analyses of those
data have led to an unquestionable superiority of the Standard Model (SM) over any others
at -100 GeV scale. Just immediately after LEP started running, the situation in August
1989, as regards the key quantities of interest, was the following [I] : = 91.120(160)
GeV, m, = 130(50) GeV, sin^6^ff = 0.23300(230) and a^mj) = 0.1 10(10). Their present
values are [2] : = 91.1867(20) GeV, m, = 173(5) GeV (including CDF/DO), sin^^,, =
0.23152(23) (LEP + SLD Average) and = 0.1 19(4) (World Average). The progress
IS overwhelming! Remarkable is the fact that the measurement uncertainties of the
electroweak observables have now been brought down toper mille level [3]. The CDF and
OO Collaborations of the Permilab pp collider TEVATRON have in the mean lime
succeeded in finding the top quark [4]. The targets of these machines, when they started,
: (i) perform precision tests of the SM at a few per mille accuracy, (ii) count the
number of light generations, (iii) search for the lop quark, (iv) search for the Higgs and
^ddres.s from I January 1998 : Ibeory Group, Saha Institute of Nuclear Physics.
I/AF Bidhan Nogor, Catcutto-700 064, India
© 1998 I ACS
470 Gautam Bhattacharyya
(V ) l(x>k tor new resonances (if we were very lucky!). Although the first three objectives
have been successfully met (a fourth light neutrino is now 9 1 a away!), the Higgs boson still
eludes detection and no new resonances have been found. The theoretical uncertainties are
at present at the same level. The theoretical uncertainties in the SM, for given m//and
ofsOM/). stem mainly from the uncertainties associated with the hadronic contribution to the
photon vacuum polarization [a’Hmz) = flr'(O) (l-&af) * 128.89 ± 0.09, where the
evaluation of the light quark content yields 5Q(had) = 0.0280 ± O.QOOl (5]], which is
included in the RG running of the electromagnetic fine-structure constant 0(0) a(m 2 ).
This propagates as a permille error in the final predictions. Full one-loop and leading two-
loop corrections are now available, but neglecting higher order effects manifest mildly
through renormalization scheme dependence. Genuine weak loop effects (0(G^m,^)) are
now tested at the 5cT level and the present precision is high enough to sense the quantum
corrections of the Higgs boson mass.
2. Physics at LEP
2. L Principal observables :
During 1989-95 LEP has operated on and around theZ-peak with an integrated luminosity
of 160 p/r' and there is a collection of -5 million visible Z-dccay events per experiment.
The principal measurements at LEP have been ;
• Cross section (J{e^e~ ff\ vs ^fs (where V7 * niz and a few around m^). The
peak cross section is given by = (12 ). where and F , are partial
widths of the Z in the channels e and /and is the total width (half width at half
maxima at the Breit-Wigner resonance).
• Partial widths = TfZ -> j/) - (v^ -t-a J )(l + /4;r )/6;rV2 ,
where the vector and axial vector couplings of the Z to the fermion /are given
^ - ^Qf ) and Of = VpLi = I leptons and =
3( I + a V ) / ;r+. . . ) for quarks. The couplings have been dressed with improved
Born-approximations : their meaning, particularly the implication of p-parameter
and how and why the effective weak angle (0en) differs from its tree level value, will
be clear shortly.
• Forward-backward asymmetry s (cjf - (T£j )/ (a^ + / 4, where the
suffixes F and B correspond to the forward and backward hemispheres, and Af-
IvjOj I (vj ). In a purely parity-conserving interaction, the number of particles
thrown in the forward and backward hemispheres would have been the same; a non-
zero Ayg indicates an interference between the vector and axial vector couplings.
• Average T-polarization Pj --A^.
In SLC (the SLAC Linear e^e- Collider operating on theZ-peak with a total luminosity of
5 p/r' upto 1996 and with an average electron polarization of 80%), observables related to
polarized beam are :
Precision tests of the Standard Model : Present status
471
• Left-right asymmetry ■ (CT^ - + (T/, ) =■ - A, .
• Left-right forward-backward asymmetry / 4 .
2. 2. Renormalization procedure and radiative corrections :
To have a feeling why radiative corrections became necessary not long after LEP
started running, let us look back to the situation in Summer 1992 [6] : the measured =
-0.0362!58o 3^ when compared with its tree level SM prediction = -0.5 + 2 sin^
9\\ = -0.076 (sin^^v obtained from = 7ta{0) / sin^ 0vv ^os^ 0^ ), showed a
1 3 CT discrepancy, inevitably calling for the necessity of dressing the Born-level prediction
with radiative corrections. However, just the consideration of running c«(0) -> oK^ni/} and
extracting sin^0 (to replace sin^6lv in the expression of v/) from cos-0 sin-0 = tea
On /)! enabled one to obtain v/ = -0.037, Le. within I cTof its experimental value at
that period. The essential [wint is that it was possible to establish a significant consistency
between data and predictions just by considering the running of a and it was only much
laici, with a significantly more data, that the weak loop effects were felt. To understand the
essential features of the renormalization procedure, let us follow the on-shell scheme
and the readers are referred to two excellent reports 17,8] for details. The steps arc the
lollowing : (i) write the bare Lagrangian and scale the fields and coupling constants by
S(niic constant a priori arbitrary parameters called ‘the renormalization constants’ (these are
usually denoted by c and 0, J^nd gj respectively); (ii) select
renormalization input parameters— usually these are the best measured experimental
(|uantities-^m this case,
• cr '(())= 137.0359895(61),
• 1.16639(2) X It) ‘^GeV--,
• m- =91.1867 ± 0.0020 GeV;
un) impose renormalization conditions (sec below) and (iv) extract those effects that cannot
he absorbed during renormalization — these parametrize the effects of radiative corrections.
The renormalization conditions are ;
• The masses are defined as the pole positions of thex:orresponding propagators. Thus
■* - -
for a vector boson ^(W, Zy), = where Lyy denotes a renormalized
two-point (self-energy) function between V and V'.
• The residue of the photon propagator at = 0 is unity (QED demands it), i.e.
f 'yy = 0, where a prime on a L denotes its derivative w.r.t, q^.
• There is no photon-Z mixing at q^ = 0, ie. Zyiiq^ =0) =0 (QED is thus not
contaminated by Z).
• The photon-clectron-eleciron vertex at = 0 with electrons in their mass shell
472
Gautam Bhattacharyya
All the renormalization constants have by now been used. The net effects of
renormalization then manifest through :
• a{q^) = aiO)/ il + ReZ'yyiq^)) : this way a(0) - (137.0)“' aCmJ )
- (128.9)-'.
• Residue of the Z propagator at the Z-pole is not unity (^ ^ = ) ^ oj and
this gives rise to a non-trivial wave function renormalization on an on-shell Z
(decaying to ff) line. This leads to the celebrated p-parameter : p = (1 - Ar) / (1 +
^hcre the muon decay radiative correction Ar (which is indeed a
charged-current radiative correction) enters into the game when we use obtained
from /i-decay, in the neutral current decay (Z decay) formula.
• Non-zero photon-Z mixing at q^ =m (q^ = m 2 ) ^ 0). This modifies
sin^^to sin^Qff.
2.3. Parametrization of radiative corrections :
2.3. f. Oblique parameters
Wc should first note that not all renormalization constants could be absorbed in the
redefinition of parameters. Those unabsorbed ones cast observable impact. We first
concentrate on universal corrections, i.e. those which originate from the renormalization of
vector boson two-point functions and thus do not depend on external fermion lines. How
many independent parameters carry the observable effects ? Essentially there are four types
of two-point functions, namely, I yy(q^), Ey^iq^ ), ) and ). Jbere
are 2 relevant energy scales at which measurements arc made : = 0 and q^ ~ ’
Hence there are eight such parameters. QED demands (see the renormalization conditions) ;
(0) = 0 and (0) = 0. Out of tht other six, three are absorbed in the
renormalization of the input parameters a, and Hence the remaining three (in fact
three linearly independent combinations of those two-point functions) will have observable
effects. These are usually parametrized by 5, T and*G, the so called ‘oblique parameters’,
defined below [9]' :
• 5= l6nm-^[X„(0)-X3y(/nJ)],
• r=4;rmz^ [X,,{0)-X„(0)]/sin2 e^T.
• U= i67an^ [Xii (m^ )-Xi, (0)] - I6»nz^ [X^j (m| )-X33(0)].
In the above definitions, I have used the ( 1 , 2, 3, Y] basis of the SU(2) ® U(I ) gauge theory
rather than the (y, Z W*| basis. The T parameter is related to p by Ap * p- I = aT. The
' I have adapted (he definitionti used by Bhaitacharyya. Baneijee and Roy in [9]. A linear -expansion of Ihe
form ^ ) = £(0) + 2 £ '(O) yields the definitions in Pcskin and Takeuchi (9]. All these definitions assume
Ihe vacuum polarization dominance of radiative corrections. A slightly more general parametrization. using (be £/
vuiiables, has been used by Altarelli and Barbieri [9].
Precision tests of the Standard Model : Present status
473
leading SM contribution to Ap is quadratic in top mass and logarithmic in Higgs mass and
is given by [10] :
7'^'^ / rtil )-(w| -m )ln(m^ )]/ n.
Ai ibis point, it is worth pointing why 7^*^ is quadratic in m^. The effect, as is evident
Irom the definition of T. is generated by self-energies of massive vector bosons. Since the
longitudinal components of those gauge bosons are essentially Higgs scalars, each vertex
of u self-energy diagram (with top-quark floating inside the loop) picks up one power of
/;/, and hence the quadratic dependence. The Higgs mass appears logarithmically due
to Veltman screening’. The parameter T. as matter of fact, captures the effect of
custodial SU(2)’ breaking. To appreciate this point, let us consider the SM scalar
poieniial ^(0) = -m ‘ (0 ' 0) + A(0 ^ 0)^ /6. Before the spontaneous symmetry breaking,
mere is a global 0(4) symmetry in the potential which is broken to 0(3) once the symmetry
IS broken in one direction. This 0(3) is equivalent to a global SU(2) which we call
custodial SU(2)\ It is precisely because of this custodial SU(2) that even after
Spontaneous symmetry breaking Once we apply this global
symmetry to the Yukawa sector as well, we realise that this custodial SU(2) is broken by
(i) fermion mass splitting in a weak doublet (this effect is quadratic and m, naturally
ilominaics) and (ii) hypercharge mixing (proportional to (m^ ) that multiplies In
/////). Ap also parametrizes the quantum correction in photon-Z mixing at q- = in
the following way :
cos- sin ‘
• sm- -sin- 6-^ , " . , ” -Ap,
cos - -sin -
where sin- is determined through cos" 0sin- 9-Ka{m2)j^2G^m\
S and U are sensitive only to logarithmic effects and their SM expressions are not
displayed. It is worth noting that 5 is sensitive to non-decoupling physics. Even a
degenerate chiral fermion generation (which does not contribute to T), even in the infinite
mass limit, contributes to 5 and its contribution is estimated to be
where is colour and r corresponds to the third component of the weak isospin of the
Icli- and right-handed component of the fermion i. The contribution of a heavy degenerate
4ih generation or a mirror family is 2/2n « 0.21 and the fitted value of the new physics
contribution to S (5'^'^ = -0. 19 ± 0. 16^5*7 [II]) allows not more than one such
generation at 2<7. (^D-like Technicolour models generally yield large positive S and are
excluded |9]. However, the walking Technicolour models survive as they contribute to 5
^’nly by small amount (even negatively) [12].
474
Gautam Bhattacharyya
2.3.2. Ibb -vertex correction :
Non-universal vertex corrections are generally not important except in one situation,
namely, the 2bb -vertex correction. The W^-boson and top-quark mediated triangle graphs
induce a sizable correction to the Zbb -vertex. Since the contribution comes from a chiral
fermion inside the loop, it is non-decoupling and since the longitudinal W couples to a
fermion with a strength proportional to its mass, the effect is quadratic in m^. Moreover,
there is no CKM suppression as - 1. The effect is parametrized by a shift of the vector
and axial couplings of Z with 6-quark compared to those with ^-quark ;
• Vh(a^) = v,,(aj)-l9AV'i /60, where
• -{a! K)\m} ! m\ + (13/6) In /m| )].
A noteworthy point is that , on account of its negligible m^y-dependence, allows an
indirect measurement of without any need of specifying m// (not so is the situation
with Ap!).
2.4. (nil) Crisis! Is it over ?
When most of the measurements at LEP were agreeing so well (perhaps loo well!!)
with their SM predictions. /?/,(= T ^ continued to stay a few sigma above and
Rt (- / ^had ) ^ sigma below from their respective SM predictions. The fact that all
4 LEP-groups were reporting the same trend amused the physics community for more than
a year leading to lots of speculations (.sometimes wild!) for physics beyond the SM. First
we note that for m, = 1 80 GeV, R^^ =0.2158 and = 0. 1 72 . The experimental values
reported at Beijing 1995 1 1 3] were :
• = 0.2219 ± 0.0017 (3. 7a above SM!).
• = 0.1543 ± 0.0074 (2. 5a below SM!).
In fact, the crisis was just not a Rf, -Rf. crisis, it was rather a crisis! Strictly
speaking, there was an a 5 anomaly too— while ocs{mz) from LEP was pointing towards a
central value 0.120, its measurement from scaling violation in deep inelastic scattering was
showing a central value 0.1 12 (although these two measurements had an overlap within
1 a). Notice that at LEP, one way to measure is from
• /?/ = (r,„d/r,) = (r«'*/r,)[i + a5(mz)/jr +
where is the weak part of the hadronic width. Notice that if one could add a few
MeV to (due to a possible positive interference from some new physics contributing
to 6-quark partial width), one would not only push up the theoretical prediction for
W/, making it more comfortable with data, cc^niz) measured the above way would be
drifted down closer to the value obtained from scaling violation. Thus
of the crisis could, in principle, be solved in one stroke! However, as I mentioned before.
Precision tests of the Standard Model : Present status
475
It was a ihrce-prong crisis, and any attempt to subtract out the required McV from
f wcak ^ problem, could only push lurlher away from that
obtained from scaling violation. How reliable were tho.sc data ? Did they survive the test
01 lime '! No! The situation look a dramatic turn in Warsaw 1996. The reported numbers
there 1 14| :
= 0.2179 ± 0.0012(1. 8(7 above SM!) and
R"^^ = 0.1715 ± 0.(X)56(on the dot!!!),
not only amu.sed even more, but puzzled the community this lime as to how all 4 LEP
Collaborations could simultaneously change their numbers in the same directions and that
UK) by such a large extent! What happened
2 4. 1 Wliaf happened to R^ ?
The data allow a direct measurement of (T, )Pic-^ X, )Br(X, ), where X^ is a
hai iiied hadron (D^\ D-, . K'Tt. .). R, has gone up because [15,161 :
• More data have been included.
• Decrease in Br (D*' -4 X ;r) from (4.01 ± 0.14)^;^ to (3 83 ± 0.12)% (ARGUS
input).
t iX'crease in Pi}i — ♦ tc). which is a backgound.
• New techniques have been employed (t- .if “slow pion lag”) at LEP to measure
P\l -> Br (D > Ttiy^) and this has also gone down.
2 4 2 0 lull happened to X/, ?
I<i, ha.s gone JoH7i mainly because 1 15.16] :
• More data have been included.
• Primary vertex for each event has been reconstructed scpcraiely in two hemispheres
leading to small hemisphere correlations (ALEPH).
• Belter understanding of the charm sector has been made possible.
• Sexeral mutually exclusive tags have been used, which reduces the systematic
eiiors,
Also the tt^^-anomaly had gone away at the same time. The different measurements of
have now come to a much better agreement than before.
The present experimental situation is the following |2|
• R""^ = 0.2170 ± 0.(KH)9,
• = 0 17.34 ± 0.(K)4H,
• (/.,\niy) ^ 0.1 19 ± ().(K)4.
conclusion is ; the Rf^R, -CTs-t/#!/) crisis seems to he over ’
476
Gautam Bhattacharyya
3. Summary
Here I list the main results ;
• Number of light neutrinos = Ny=r,„,/r5“ =(ri„, /T, )(r, /T,, )*“ =2.993
± 0.01 1 . Indeed Ty is an SM input in this determination. Splitting the ratio of partial
widths into two such factors (as shown) reduces the theoretical uncertainties. A
fourth light neutrino is ruled out by 91a!
• The mass of the Z-boson, m 2 = 91.1867 ± 0.0020 GeV, has been measured with a
precision -2 x lO^'^.
• The world average of the top mass m, is 173.1 ± 5.4 GeV (Direct search ai
TEVATRON dominates).
• Fitted value (LEP) of m/y = 115;^^ GeV, which implies < 420 GeV (959}
CL).
• The world average of the W-mass is given by, ntw = 80.43 ± 0.08 GeV. By the ciul
of LEP2 and TEVATRON Run 2, the error is expected to be reduced to 30
40 MeV.
• Partial widths of z are measured at a per mille level. The forward>backwaul
a.symmetrics arc measured at a few per mif level.
• The effective weak mixing angle has been measured to a great accuracy :
sin^ 0,fy( LEP) = 0.23 199 ±0.00028 and sin^ (SLD) = 0.23055 ±0.(K)04 1
These should he compared with sin ^ = 0. 23 157.
• Number of extra heavy chiral generation could almo.sl be one (from 5-paramcier)
My conclusion : Order reigns in electroweak physics !
Acknowledgments
I thank Professor Dilip Choudhury for the invitation to give this talk in the XII Hf-P
Symposium in Guwahali.
References
1 1 1 G Allarelli Proceedings of the liUerriattoruil Symposium on Lepion and Phoion /nieroi innis, Sfiin/nnl
(1989)
(21 D Ward Pienarv Talk or the EPS Meeting. (Jerusalem. August 1997)
1.^1 S N Ganguli Indian J. Phys. 72A .*>27 ( 1998); A Guilu ibid S.19 (1998)
[4) The CDF ColUihoraiion. F Abe et a) Phys. Rev. Left. lA 2626 ( I99.S), The DO Collohoivnon. S Abaclu
ei lit Phys Rev. Lett. 74 2632 (199.*>)
|‘'j .S Eidelman and F Jegerlehnci Z Phws. C67 .S85 (199.*)), H Burkhardi and B Pictivyk Pli\''> /<■'
IW56 39K(I99.S)
161 See lor example V Novikov. L Okun and M Vysmsky Nud. Phvs B397 .^3 (1993)
17) M Bohni. H Spiesbcrgcr and W Hollik Fans. Phvs 34 687 1 1986)
Precision tests of the Standard Model : Present status
All
|8] W HoIIik Forts Hhys. 38 165 ( 1990)
[91 M Peskin and T Takeuchi Phys Rev. Lett 65 964 (1990). Ptiys Rev. D46 381 (1991). G Altarelli
and R Barbieri Phys. Lett. B253 161 (1990): D Kennedy and P Langackcr P/n\v Rev Lett 65 2967
(1990); W Marciano and J Rosner Phys Rev Lett.iS 2963 (1990): Eiratum ihicl 68 H98 (1992).
G Bhatiacharyya. S Banerjce and P Roy Phys. Rev. D45 729 (1992); Enaiuin ihtd. D46 3215 ( 1992)
1 101 M Vellinan Nucl. Phys. U123 89 (1997)
1 1 1 1 P Langackci' and J Eiier hcp-ph/9703428
1 1 2| For a review, sec K Line Pnneedin^s oj ICHEP‘94, Glasgow, 1994 (hep-pli/9409304)
I Ml LLP Liet troweok Working Croup Report, prepnnt CERN-PPE/95-172
[141 ILP Llei troweak Working Group Report, prepnnl CERN-PPIiy96*183
I I S| M Dcinartau hep-ph/961]019
I ihj G Allarclh hcp-ph/9611239
Indian J. Phys. 72A (6), 479-494 (1998)
UP A
— an inl emational journal
Status of weak-scale supersymmetry
Probir Roy
Tata Institute of Fundamental Research. Homi Bhahha Knad.
Colaba, Mumbai-400 OO.S
V
Abstract : This article includes discussions on ;
(i) Standard Model and motivation for supersymmetry,
(ii) Supersymmetry and MSS M.
(lii) CMSSM and the mass spectra of sparticles,
(iv) Experimental constraints on CMSSM parameters and
(V) Conclusions
Keywords ; Supersymmetry, Higgs, CMSSM
FACS Nos. ; 14 80 Ly . 1 4 80 Gt
1, Standard Model and motivation for supersymmetry
SupcrNyrnmciry, as a generalized spacetime invariance under which fermions and bosons
Hailstorm into each other, is undoubtedly a beautiful idea. But why should particle
physicists look for ii — especially at or slightly above the weak scale? The answer is that
solily broken supersymmetry with an intra-supermuliiplet mass breaking < 0 (TeV) can
aiic (he ^Standard Mcxlel (SM) of particle physics of a serious theoretical dericicncy. viz. the
ladiaiive instability of the Higgs mass.
Bcloic I elaborate on the last point, let me first briefly review the impressive
c\pcrimenlal succes.ses 11] that SM has had — if only to underscore the absence of any
phenomenological need at present to go beyond it. Table 1 below contains a ”pull-plol”.
Viinous measurables (on the Z-peak) of the standard electroweak theory have been listed in
il'e tnsi column. The second column contains the measured values (combining SLD and
1 IP numbers) with la errors. The "pull", defined as the deviation (with sign) of the central
from the theoretical prediction divided by the laerror in the measurement, is given in
the third column. The fourth column displays the same information geometrically in terms
hori/onial bars drawn in units of a.
© 1998 lACS
480
Probir Roy
In Table 1 there are seventeen data items and one fitting parameter, namely the Higgs
massm//, ;^/(dcgTte of freedom) being 18.5/15 in the fit, Tlie best fit for the latter is [1]
m„ = 149:^jf GeV,
to be contrasted with the latest result mu > 70.5 GeV from direct search experiments
at LEP. One would readily agree that the data represent an outstanding success of the EW
Table 1. Pull-plot for elect roweak measurables on the Z-peak
-3 - 2-10 12 3
MzfGeV)
91 186.3 ± 00020
.17
PzlGeVl
2.4946 ± 0.0027
05
Ohjdr
41 508 ± 0.056
97
R,
20.754 ± 0.057
- 22
Rp
20 796 ± 0.040
.73
Rt
20 814 ± 0.055
.00
a""'
0 0160 ± 0.0024
32
^ih
0.0162 ± 0 0013
.74
^Ih
0 0201 ± 0.0018
2 70
0 1401 ± 0 0067
-..37
Ae
0 1382 ± 0 0076
-..57
Rb
0.2179 ± 0 0012
1 70
Rc
0 1715 ± 0(K).56
- 14
().h
^ih
0 0979^1 0 0023
- 87
a""
^Ih
0.0733 ± 0.0049
43
sin^ 5'
0.2320 ± 0.0010
-.09
1/a
128.894 ± 0.090
-25
electrOweak sector of SM, though mild doubts can be entertained regarding the
forward-backward asymmetry at the Zand the Z-dccay branching fraction into bl7 . Turning
to QCD [2], Figure 1 shows a "besi fit” plot of the QCD fine structure coupling evolving
via the renormalization group equation as a function of the energy scale Q. Given the
large number of different determinations at different scales, one would call the agreement
quite impressive.
In the light of such an outstanding experimental success, any theoretical motivation
for going beyond SM needs to be compelling. Such a motivation was indeed put forth by
'i Hoofl in 1980 by showing the radiative instability of the Higgs mass : a feature of SM
Status of weak-scale supersymmetry
481
known as the naturalness problem. Already, at the 1-loop level, the presence of a quadratic
divergence in the summed diagrams of Figure 2 implies the following fact. If there arc
Q/[GeVl
Figure 1. Evolution of with Q QCD theory vt expenment.
unknown superheavy fields al some high scale M {e.g. Planck scale Mpi), SM has to he
viewed as a residual theory at low energies after these superheavy fields have been
iniegralcd out. However, the latter procedure makes the finite mass of the electroweak
Higgs shift quadratically to that high scale Af. An unnatural amount of fine tuning is needed
Higgs
gouge boson
— fermion
Figure 2. I -loop coniribution.s to (he Higgs mass.
order by radiative order between the Higgs mass and self-coupling parameters in the
Lagrangian to keep within an electroweak range. Generally, one can represent the effect
of integrating the superheavy fields as ;
{^Hgb, ) = . (I )
where (^jight ) remains the effective Lagrangian for the residual theory. In (1 ) is the
>ile dimension of Che operator 0^ in the operator product expansion of the RHS. The
482
Probir Roy
problem Wtlh the Higgs mass term is that the leading value of d„ in the RHS of ( 1 ) is 2 so
that M comes in as and the high scale contribution evidently does not become smaller as
M increases. The fine tuning would be to adjust the corresponding coefficient c„ to zero.
Th^e have traditionally been two types of suggestions for the way out : (I) the
strong coupling (new structure) option and (2) the weak coupling (new symmetry) option.
(I) is presently disfavoured by precision tests in which SM has performed very well. In
particular, the clectroweak oblique parameters S, T, U (which vanish for SM) arc now
experimentally know to be |I1 -().04^J ,-(). and 0.07 ± 0.42 respectively. These
arc gcncrically expected to be 0(1) in option (1). A similar negative conclusion regarding
this option follows from rather strong upper limits which exist on any llavor-changing weak
neutral current. In option (2) .supersymmetry is perceived to be the new desired symmetry.
Here quadratic divergences of fermionic and bosonic loops cancel with opposite signs and
any radiative shift in the Higgs mass .squared gets controlled by the .squared mass-difference
between particles and their superpartner sparticlcs within the same supermultiplet. ^So long
as the latter is < 0 (TeV^), there is no problem. Referring back to ( 1 ), the coefficient c„ of
0„ with dn = 2 IS naturally made to vanish by supersymmetry.
The power of supersymmetry can best be understood from a simple toy model :
scalar electrodynamics. The mass of the .scalar field in this theory is unprotected against
large radiative corrections by any symmetry and suffers from the naturalness problem
owing to a quadratic divergence present already at the 1-loop level. The fermion mass in
.spinor electrodynamics, on the other hand, is protected by chiral symmetry and does not
have this problem — the corresponding loop divergences being logarithmic In
supersymmetric quantum electrodynamics (SQED). the mass of the scalar is equal to the
mass of its partner fermion by supersymmetry and hence gets protected. Thus SQED, unlike
scalar QED, is a natural theory. Moreover, this feature persists even with supersymnuMiv
breaking so long as the latter is done by soft terms {i.e. of scale dimensions le.ss than four)
2. Supersymmetry and MSSM
The supersymmetry idea, originally due to Golfand and Likhimunn |31, was developed
lurther by Akulov and Volkov and more specifically in the context of quantum f ield ihcoi v
by Wess and Zumino as well by Salam and Strathdec. It postulates the existence of particle-
sparticle supcrmultipleis with the superparlners differing in spin by 1/2 unit. Thus i\
supersymmetric theory contains supcrmultipleis with spins 0 and MX (e.g. quarks and
squarks or electrons and selectrons or Higgs bosons and higgsinos) as well as those with
spins I and 1/2 {e.g. photon and pholino or gluons and gluinos or fVs and winos or Z and
zino etc.). Neutral higgsinos mix with the /.ino and the pholino into four physical
neulralinos. while charged higgsinos and winos mix into two pairs of physical charginos
The new particles (called sparticlcs) become nccc.ssary since csiahli.shed quantum numbers
forbid one to make supcrmultipleis out of the known fermions and bosons. In the local
version of supersymmetry there is also the supermultiplet comprising the spin 2 graviton
and the spin 3/2 gravitino.
Status ofweak-scale supersymmetry
483
The zoo of sparticleSt as well as the symbols for themselves and their superfields,
appears in Table 2 below while Figure 3 graphically shows how different particles and
sparticles are denoted by characteristic lines in Feynman diagrams. These sparlicles should,
in general, have masses characterized by the intra-supermultiplet splitting scale M, where
Afiv < Af, ^ 0 (TeV). In particular, if all extra particles — necessitated by a supersymmetric
extension of the Standard Model — are heavier than 200 GeV, supersymmetry will decouple
[4] at presently available energies. Residually left will be the Standard Model in its pristine
form with a somewhat light Higgs particles.
Table 2. Zoo of sparticles.
Name
Symbol
sleptons
h.it
(selectron, smuon, suu)
P L.R • )
squariu
(j-up. i-down, j-charm.
(“iL.il • • ^L.R '
r-stnuige. «op, sbottom)
^LR ^LR )
gluino
X
charginos
^1*2
ncutrolinoi
gravitino
G
The minimal supersymmetric extension of the Standard Model, i.e. the one with the
minimum number of extra particles is called the Minimal Supersymmetric Standard Model
,.L fO ••zr*-..
quack, Itpfon r.S.W.Z Hlgq. grovltpo
• •O-- ►••••• ••'5 ' ■
»qu«rk,ileffep qouflno iilwt»no grovnino
figive 3. Legend for describing panicles and sparticles in Feynman diagrams.
MSSM [5]. Its Spectrum consists of the particles of SM — with a minimally extended Higgs
sector— and their partner sparticles. For panicles, the only new feature, as already
mentioned, is that — in place of one — there are five physical Higgs scalars (a charged pair
two CF-even neutrals — the lighter h and the heavier H — as well as one CP-odd neinral
A) emerging from two Higgs doublets which occur here instead of one as in SM. Tlie ratio
= (VEV of the neutral Higgs field which couples to up-type fermions) + (VEV of that doing
so with down-type ones) is called tan p,
^2A(6)4
4S4
Prohir Roy
The generic sparticle is expected to be heavier than the corresponding particle by an
amount CKM^\ though the mass ordering could get reversed for the top + stop, IV
chargino and Z + neutralino systems. By assumption, MSSM has a built-in conservation
law : that of the multiplicative quantum number /^-parity Rp ^ ^
baryon no., L = lepton no. and s s spin, which is positive for particles and negative for
sparticles. This implies an absolute stability for the lightest sparticle (LSP : a candidate for
cold dark matter in cosmology), usually taken to be the lowest-mass neutralino Xw ^be
LSP, being extremely weakly interacting, escapes through the detectors without leaving any
visible trace. The production of sparticle pairs in collider experiments and the consequent
decay of each of them is characterized by missing transverse energy E-p signatures. One
other consequence of /?p-conservaiion is the prevention of catastrophic proton decay
processes such as p — > e"*" ;r® which could otherwise proceed with lifetimes ~10"® s, instead
of >1(>’^ yrs as dictated by experiment.
The Lagrangian density of MSSM contains the supersymmetrized minimal
extension of that for SM plus the most generally allowed soft supersymmetry breaking
(SSB) terms
^ SM ^ MSSM = ^ SSM + ^ SSB
An attractive feature of MSSM, following from the above, is that couplings among particles
and sparticles arc simply related by supersymmetry. Some of the vertices, related in these
way, are shown below in Figure 4. Note that, in any vertex, sparticles always appear in
pairs owing to the constraint of -conservation.
fermion fermion sfermion
Figure 4. MSMM veniees gen^riied by supenymnwtiy ftom (hose of SM.
Status of weak-scale supersymmetry 4S5
More quantitatively, the superfield content of the model in the matter sector, written
in a transparent notation (i s i, 2. 3 is a generation index), is ;
1
•^L =
.Qi.LVW, =
[//fj ^1
[ 4 }
(2)
The corresponding superpotential (with /^-parity assumed conserved) is :
with ^'s as Yukawa couplings.
The scalar potential can be derived from (3). Writing 0^ for a generic scalar field and
incorporating the soft supersymmetry breaking terms, we have
>'-1
j
9W
+ D - terms
•J
+A/)A/)^t + AEXgei.h^Cff '^Bph\ ./12 +/f,C.}. (4)
In |41 the third RHS term includes mfh^h^ +m|/i 2 ^2 ^ vanishing mi 2 where /i) 2
refers to the scalar component of the superfield H 12 . Also, V| 2 = (^n) tan/3 =
V 2 / v'l . The physical fields can be expressed in terms of the superfield components given
above. For instance, the field for the lightest neutral scalar is /i = V2(Re/i2 -V 2 )cosa
-V2(Re/i®. -V, )sina, where a is an angle which enters via mixing. The orthogonal
heavier combination is H = ^f2(Rth2 - V 2 )sina + V2(Re/i° “V|)cosa while A equals
V2(Im/i2 cosfi- tmli,^ sin/3). The partners of the CKM matrices in the scalar sector are
assumed to posaess safety properties which suppress dangerous flavor-changing neutral
current processes that could emerge from [3].
At the tree level itself one has several mass relations.
m^ = + A/J, , (5a)
(5b)
i — (5c)
Icoslfil
• (co 62^)“* (mf P-mf cos^ P) -
3B/i ® (mj* -ffij ) ian2P + sin 2)8 , (5e)
m\ ^ + m^ + 2/x^.
(5f)
486
Probir Roy
On including I -loop quantum corrections in the leading log approximation, the upper bound
on the squared mass of h reads (Fi 2 are the two physical squarks, assumed to weigh more
than the lop) [6] :
Ml < Af I cos^ 2^ +
3a
EM
m.
In-
2;rsin^ Af m
(130 GcV)2
( 6 )
This is a ‘'killing prediction" of MSSM.
The renormalization group evolution of the three gauge couplings g^^ (a = 1,2, 3)
with the energy scale Q are quite different for SM and for MSSM, as shown [7] for
a~^ s ^ng~^ in Figures 5a and 5b. The low energy values of the couplings are now known
(*) (b)
Figure 5. RGE of the gauge couplings in (a) SM and (b) MSSM.
rather accurately and have been used as inputs in these curves. For MSSM the couplings do
unify at A^gut - 2 x 10'^ GeV, while for SM they do not. In Figure 5b M^ has been chosen
to be - 1 TeV, but the broad features of the figure do not change when M^ is varied between
100 GeV and 1 TeV. Earlier, when the low energy data were not as precise, SM was
compatible with minimal grand unification at -10‘^ GeV with just a desert in between.
Such is no longer the case. This change is illustrated dramatically in the measured values
and errors of the sine squared of the Weinberg angle, as shown in Figure 6 for various years
starting in 1975. Clearly, grand unified theories, without supersymmetry and basing
themselves only on SM at low energies, are ruled out now.
3. CMSSM and the mass spectra of sparticles
Though MSSM is the simplest supersymmetric extension of SM, it introduces 31 new
parameters in addition to those of SM. That makes MSSM not very easily testable in term.s
of predictions that can be pinned down, the predicted upper bound on the lightest Higgs
mas.s [6] being an exception. From a phenomenological standpoint, a more popular version
Status of weak'Scate supersymmetry
Plate I
Figure 6. Chronologically progicssivc icduclion ol eirors in ihc Mieasurcincnls ol mu*-
Status ofweak-scale supersymmetry
487
is the supcrgravity-constrained [8] MSSM or CMSSM which has only 4 extra parameters
plus a sign and hence many dfiniiive predictions — especially on the mass spectra of
sparlicles — that can be tested.
CMSSM has the same Lagrangian density as MSSM. But it is characterized by
several simplifying extra assumptions. All of these pertain to boundary conditions
(inspired by supergravity theories) imposed on various parameters at the unification
scale Mx ~2 X lO'^GeV. Specifically, all supersymmetry-breaking scalar (gaugino) masses
lire assumed to be universal and equal to one mass rrjo (Afi/ 2 ). Squared masses of the
Higgs at the unification scale have the additional contribution jU^ where p is the
supersymmetric Higgsino mass parameter in the MSSM superpoteniial in [3]. Another
.issurnption is that all supersymmetry-breaking trilincar couplings in [4] are taken
to be equal {= >4o). Here and M \/2 are supposedly of the order of the gravitino mass
'^'hich sets the scale of Mg. Now mo, M |/2 and tan p (plus the sign of p) can
be chosen to be the four parameters of CMSSM, or could be traded for one of the
I’irsl two.
The CMSSM boundary conditions at My imply
(7)
Tin ning to gaugino masses M, (i = nonabelian gauge group index) and considering I -loop
RCif effects, one can write — with a,, as the unified fine structure coupling —
M,(Q) =
( 8 )
For the t/(l)K case, with the standard deFinition of Y, there is an extra factor of 5/3 in
the RHS. It turns out that Mi(Mz) = 0.41 Af |/2 and M 2 (Mz) = 0.84 M \/2 with a mild
C dependence in Afj 2 . However, the situation is quite different for My The physical on-
shcll gluino mass m- is given by [9]
M,{Q)
1 +
«5(6)
4;r
- 15 - 18ln
M^(Q)
Q
-^I'dxxln
jcmJ +(l-x)m| -x{\-x)Ml
(9)
Jiid is independent of Q. For M^siOA TeV and m- = 1 TeV, the difference between m.
Jnd Mu {M^) can be as much as 30%.
The spectrum of the remaining sparticles can be parametrized, after accounting for
•ctiormalization group evolution, as follows [10] :
m? = mi + 0.15M?/, -
o * 0 1/2
sin 2
(10a)
488
Probir Roy
mj^ = ml + 0.52A#,% " ~
ml = ml+ 0.52M2j + |d; (10c)
"'|t« = '"o + iO OT + C-)Mli2 + Isin^ 9y/Di (lOj)
•^\lR = “o + (0.02 + CpW,% - |sin2 e^D-, (lOe)
m}^^ = ml + (0.47 + C- )A/,% + - jsin^ 0*, jo (lOf)
'"lu = '”0 + (0.47 + C.)A/,% - - |sinJ 0^ jo. (lOg)
ffi, M. ^ fti rn, rn. M
a • ’g \ *i •
Figure 7. Ranges of some spaitide masses.
Here C-, = \[a](m^) j a](M x) - 1] and D - Ml cos^ fi while we have / = e. jU, =
u, c and qi = d, s, b. For stops and staus, considerable lefMight mixing is anticipated. The
corresponding mass-squared matrices are given by
+0.35D -m, (A, +/iCOt^)
-^ 0.160 j
(lla)
Status ofweak'Scale supersymmetry
489
? +ml-0,27D -m^iA^+ptanP)]
^ ' ( 11 b)
" +^tani3) m? +m^ -0.23D
Wc should remark here that arguments exist [11] why tan p should lie between 1 and
ni, /ni,, .
A sample scatter plot of the ranges [12] of some characteristic masses in the
model — showing the extent of variation in the parameter space — is shown in Figure 7. One
should also mention that five squarks (i.e. all except the stop) need to be taken as nearly
mass-degenerate in order to avoid an unacceptable FCNC-induced mixing. This
Lould be a problem in Figure 7 [12] which has a rather large bi - mass splitting. A
similar argument vis-a-vis the FCNC-induccd p ey decay requires the near mass-
degeneracy of all sleptons except r.
4. Experimental constraints on CMSSM parameters*
In I his section, I concentrate on zones in the CMSSM parameter space that can be
excluded by use of results from completed or currently running experiments. Some of the
constraints, discussed below, involve data from the SLD e'^e~ annihilation experiment
,it Stanford and the pp collision experiments at the Fcrmilab Tevatron. However, the
large majority of them follow from measurements made at the CERN LEP experiments
(1 will exclude from this talk direct mass limits on squarks and gluinos since those will
he covered by D P Roy). The LEP experiments, so far, have an analyzed data sample
ol more than 20 million Z-peak events at LEP I plus nearly 20 pb~^ of data in LEP 1.5
ut e*e CM energies E^m of 130, 136 and 140 GeV and also about 50 pb~^ of data at
161 GeV.
Let us first state some results in the slepton sector. Sleptons, if accessible in
energy, can be pair-produced at LEP. Their characteristic decays with £7 signatures have
been looked for. For the right selectron e^, the lower mass bound [13] is m-^ > 75 GeV
wiili the assumption that the mass difference between and the LSP exceeds
.LS GcV. The latter caveat is necessary in the light of the iff -decay signatures which
have been sought in obtaining this bound. For instance, if this mass-difference is
taken to exceed only 3 GeV, the said lower mass-bound reduces to 58 GeV. For
smuons and staus, the lower mass bounds, with the former assumption, the lower
mass bounds (with the former condition) are somewhat weaker, being 55 GeV and 50 GeV
'cspcciively, since they get pair-produced only by j-channel processes whereas
^electron pair-production has both j- and r-channel contributions. If all sleptons are
mass-degenerate and weighs less than 30 GcV, then the slepton lower mass-bound
76 GeV.
Tills IS US of the summer of 1 997.
490
Prohir Roy
We turn next to light stops. The physical candidates are F, 2 with
F, = cosdF^ +sin0Fyj, (12a)
ti = -sin fiF^^ +cos0F^ (12b)
and f, being lighter. The search process looks for the production e^e~ F|F,*, followed
by the decay F, so that the final configuration ccEj ■ The exclusion zones on the
- F, mass plot are shown [14] in Figure 8 for 0 = 0 and Q^k/1 along with the regions
excluded by previous LEP 1 and D^experimeriis.
95 % Exclusion limit:
Figure 8. Exclusion zones on the - t| mass plot for extreme values of B
Coming to the gaugino-higgsino sector now, let us talk spefifically about charginos
and neutralinos. Exclusion zones [15] in various mass plots, i.e. vs 1^0 =e + /5 + T),
X^ vs lanp aind x^ vsv ff are shown in Figures 9(a-c) with labels specifying input
assumptions. The chargino has been taken to decay by X- ) r ■ ^
distinction has been made between assuming the former^ to decay through
the process • Assuming that the wq parameter is large (> 500 GeV) and that
the M |/2 parameter is bounded from above by 1 TeV, the following lower mass bounds
ha\e been obtained [15] : > 24.6 MeV, > 32.2 GeV, ni: >91.1 GeV,
/n > 103.7 GeV, /?/ -- > 73.6 GeV and /w-. > 96.2 GeV.
X\ X:
Coming finally to Higgs scalars, the lightest CP-even supersymmetric Higgs h
as well as the CP-odd A have been searched for in the Bjorken process e*e~ Z
hZ* — > bhl(q)iiq), while both have been sought in LEP 1.5 and LEP 2 in the associated
processes Z* — >Z/i— ► l(q)l(q)bb and €~ — >Z* — >Z4— > bbbb^ bbtT, The
Status of wrak-scale siipersymnietry
491
Liincni lower limils arc /;>/,> 62.5 GeV, > 62.5 GeV for all values of lan Sirongcr
Iowlm limils arc available lor specific assumed values of lan p. In particular, the exclusion
/one 111 the lan p v\ m,, plane is shown in Figure 10.
95% C.L Excluded region in MSSM
No iquirK miiing, a 174 G«W. a 1 T*V.
M a -150 C«v, 30 < M« < lOCOGdV
30 40 SO U 70 U M 100 110 120
(GeV)
FiHure lOt Exclusion zone m the (on P vs M/, plane.
^2A(6)-5
492
Pmhir Roy
Rciurnini! lo ihe parameters of CMSSM, one can choose five independent
parameters (/% tan ^and p). This is tantamount to covering all sfermions (but not
Status ofweak-scale supersymmetry
493
(Itir ffk) “ Figure 1 la and I lb. Furthermore, we can
compare SM and MSSM fits to the data. The SM fit of Table 1 may be compared with
a corresponding “pull-plot” in the CMSSM case shown in Table 3 for tan 1 .6. The ratio
;^/(degree of freedom) now is 16.1/12, so that one cannot say that CMSSM i.s doing
significantly better than SM.
5. Conclusions
Wc can summarize as follows.
(i) Stability considerations of the SM Higgs provide the strongest motivation for ncar-
weak-scale supersymmetry.
(iij The nature of explicit soft supersymmetry-breaking terms in the low-energy
effective Lagrangian is sensitive to input assumptions about high-scale boundary
conditions.
(fii) CMSSM, a well-posed theoretical model, is open to challenge from immediate as
well as forthcoming experiments.
(IV) The parameter space of CMSSM is getting increasingly restricted as more and more
data pour in.
(V) There is a distinct possibility that supersymmetry in nature is decoupled with all
sparticles lying near or above I TcV.
Acknowledgments
1 am grateful to Sunanda Banerjec, Manual Dress and Gobinda Majumdar for helpful
discussions. 1 thank Dilip K Choudhury for making this symposium a success.
Ki'rcrL'iice.s
1 1 1 UP Collulwration Report CERN-PPE/96-183
|2| M .Schnielling Proc. 28ih Inti Ctmf. High Energy Phyxic\ (Warsaw, 1996) p 91
1^1 Yu A Golfand and E P Likhtmann JETP Lett 13 323 (1971); D V Volkov and V P Akulov Phyx Lett
46B 109 (1973); J Wess and B Zumino Niu l. Phvs B70 .39 (1974), A Salam and J Slraihdee Nucl PIm.
1180 317(1974)
(41 H E Haber Proc. Phystcxfnwt Planck Scale tv Electroweak Scale (Warsaw. 1994) p 49
l^il HE Haber and C L Kane PItys. Hep. 117 75 (1985)
I (>) HE Haber and R Hempfling Phys. Rev. Lett. 66 1 8 1 5 ( 199 1 ); J Ellis, G Ridolfl and F Zwirner Phyx. Lett.
B257 83 ( 1991); Y Okala, M Yamaguchi and T Yanagida Phys Utt. B262 54 (1991 )
|7| U Anialdi etal, Phys. Rev. 036 1385 (1987)
|H] For reviews, see H P Nilles Phys. Rep. 110 I (1984): P Nutli, R Amowilt and A Chamseddine Applied
N^I Supergravity (Singapore : World Scientific) (1984); M Drees and S P Martin in Electroweak
Svmnieiry Breaking and New Physics at the TeV Scale cds T Barklow, S Daw.son. H Haber and J Siegrisr
(Singapore : World Scientific)
S P Marlin and M T Vaughn Phys. Utt. fi318 33 1 (1993)
494
ProbirRoy
lioj
im
M Dress and S P Martin Jfr/. [8J
L E Ibanez and G Ross in Perspfrnvfs on Physft\^ edCL Kane (Singapore ; World
p239
1 1 2j JL Feng, N Polonski and S Thomas P/ns. Len B370 W ( 1 996)
I l.y B flaratc e/ n/A/ep/i Cofldhnmn Report CEKN^PPE/97-056
[141 A Dc Min P/iysws from the Pliimk Scale to the Electroweak Scale [Delphi CoUahoraimi Talk Givlh m
the 3rd Warsaw Workshop)
[I5J S Banerjec Private Conununn atm { lioin Ihe U Collaboration)
/>/.y5.72A (6), 495-502 (1998)
Indian J-
UP A
an intem ational journal
Status of supersymmetric grand unified theories*
B Ananthanarayan
Centre for Theoretical Studies. Indian Institute of Science.
Bangalote'560012. India
and
P Minkowslti
Institut fiir Thcorelische Physik. Universitdt Bern,
5 Sidlerstrassc. CH-3012. Bern. Swiu^jrland
Abstract : We begin with a brief discussion of the building blocks of supcrsymmeinc
grand unified theories. We recall some of the compelling theoretical reasons for viewing
supersymmetric grand unification as an attractive avenue for physics beyond the standard model.
This is followed by a discussion of some of the circumstantial evidence for these ideas.
Keywords : Supersymmetry, grand unified theones, status
PACSNos. ; 14 80 Ly. 12.10 Dm
1. Introduction
The standard model of the strong and electro-weak interactions is based on a Lagrangian
field theory of quark, lepionic, scalar Higgs and gauge bosonic degrees of freedom [1,2].
Central to the standard model are the principles of gauge invariance and its spontaneous
symmetry hreakdwon via the Higgs mechanism. The standard model predicts the existence
of a scalar Higgs particle which is the remnant of the Higgs mechanism by which the gauge
bosons of the broken generators of SUi2)i xt/(l)y become massive when the gauge
symmelry is broken down to the residual The mass of the yet to be discovered
Higgs boson is not fixed therein, but is bounded from below from present day experiments
and from above by requirements of vacuum stability.
Indeed, if the standard model were to be vindicated by the discovery of the Higgs
giand unification appears to be a path to go beyond the energies where the standard model
is the correct theory, while continuing to be based on these principles. There would then be
unification scale Mg - 10‘^ GeV suggested by gauge coupling unification, above which
the Memory of Prof. Abdufi Solam.
® 1998 1 ACS
496
B Ananthanarayan and P Minkowski
physics would be described by a grand unified theory [3] based on a gauge group G. Such a
theory would then make a whole host of predictions and simplifications of our
understanding of fundamental phenomena. A compelling goal of theoretical physics is to
replace what are the engineering aspects of the standard model by a fundamental theory; for
example arbitrary parameters of the standard model, hitherto Fixed by experiment, would
then be explained as consequences of a unified and symmetric structure. Furthermore,
within grand unified theories, one uncovers highly desirable properties such as anomaly
free representations of certain grand unified groups. One expects the unification of hitherto
unrelated quantum numbers such as baryon and lepton numbers. These in turn imply
concrete low-energy predictions which can be confronted by experimental and/or
observational information.
The presence of disparate scales in the theory, Mq and the weak scale M^ - 174
GeV, expected to be separated by more than ten orders of magnitude, would render the
mass of the Higgs scalar of the electro- weak model - M^, unnatural-natural. Should the
Higgs scalar be elementary, then one manner in which it would remain naturally at the
weak scale is due to cancellation of divergences as in supersymmetric unified models [4,5].
Supersymmetry [6] is the only symmetry that has non-trivial commutation relations with
the generators of the Lorentz group and is a fermionic object that interchanges bosonic and
fermionic degrees of freedom. Although supersymmetry docs not appear to be manilesi, it
could be broken softly while preserving all the desirable properties of supersymmetric
theories. Models with softly broken susy are popular and significant experimental effort
will be made to test the predictions of these models. •
The final frontier that still remains to be explored is a framework within which a
consistent incorporation of the gravitational interactions is successful. Whereas it has not
been possible to replace the Einstein theory by a quantum version due to bad ultra-violet
behaviour, supergraviiy possesses improved ultra-violet properties [4J. String theories I?)
often contain supergravity in their low energy spectrum and as a result supersymmetric
unification is a favored candidate for these reasons as well.
Other significant avenues exist for the exploration of these theories. Note, for
instance, non-perturbative aspects of the theory such as the possibility of finding
topological defects at the time of .spontaneous symmetry breakdown when combined with
standard big bang cosmology imply .specific constraints on grand unified models. Examples
of such defects are monopoles, cosmic strings and domain walls.
The task of this talk is to briefly summarize the building blocks of supersymmetric
grand unification and recall the main circumstantial evidence for the program. The most
significant advance from the experimental direction has come with the precision
measurements of the gauge coupling constants at the 2? factory LEP [8] (and SLC) and the
discovery of the top-quark at the Tcvairon [9] by the CDF collaboration and confirmed by
the DO collaboration. These advances place significant constraints on .scenarios ot
unification for a start. More spectacular is the fact that certain scenarios of unification
Status of supersymmetric grand unified theories
497
predicted that the top-quark mass would have to be sufficiently large and roughly in the
range where it has been found. Note that combinations of theoretical tools such as the
requirement of infra-red fixed point structure of Yukawa couplings as well as fmiteness
also accommodate top-quark masses in this range. Challenges today lie in spotting the first
traces of the supersymmetric partners of the known particles, e.g. figuring out search
strategies for these for future collider experiments as well as at non-accelerator
experiments.
2. Spontaneously broken gauge theories
Whereas the gauge invariance of the standard model rests on the gauge group SU{3)c x
SU{2)i xf/(l)y, with the quark, lepton [matter] fields and Higgs fields transforming in a
specific manner under the gauge group, at low energies, the 5t/(2)^ x is
spontaneously broken to the ^(Oeir subgroup at the weak scale via the Higgs mechanism.
The result is that three of the gauge bosons, tV* and Z® pick up masses at the weak scale as
does the neutral Higgs scalar. The fermions become massive through the Yukawa couplings
lo the scalars since the vacuum expectation value < 0 0 .
It is possible lo envisage u scenario wherein this is embedded in a larger group C,
which would be the basis of the gauge invariance of a theory manifest above a unification
scale Aff;. below which it would be spontaneously broken via the Higgs and possibly some
()lhcr mechanism to a sub-group large enough to contain the standard model (in a multi-step
scenario), which would then be further broken down lo the standard model gauge group at
various stages.
Circumstantial evidence for this, is found from the renormalization group evolution
of the gauge coupling constants of the standard model gauge group, which appears to bring
them all together at a large scale Mq - 10'^ GeV when the normalization on the hyper-
( luirge coupling constant as required by grand unification is imposed.
Indeed, the arrival at the structure of fundamental interactions from renormalization
group How has a predecessor in the example of asymptotic freedom in deep inelastic
scattering experiments and thus gauge coupling unification is an extremely encouraging
sign that grand unified theories are the right step for a theory of fundamental interactions,
liarliest examples of grand unification were provided by those based on the groups 5(7(4)
x5(7(2)x5(7(2),5(7(5) and 50(10).
Grand unification, of course, implies more than just the coming together of the
gauge coupling constants. One would be gratified if it were possible to unify the particle
content of the theory as well. Indeed, simplifying features of grand unification include
embedding several of the matter fields of the standard model into irreducible
lepresentations of the underlying gauge group. That such an embedding should at all be
possible is an astonishing property of grand unified theory : furthermore, it has the capacity
explain the charge ratios for the elementary fermions in terms of simple group theory. It
turns out that 50(10) [10], for instance, still remains one of the most elegant unification
m
B Ananthanarayan and P Minkowski
groups, with an entire standard model family and a right handed neutrino accommodated in
a single 16 dimensional representation.
Whereas grand unified theories are based on local Lagrangian field theories
possessing symmetries, it is then important to address the question of anomalies in such
theories. Compelling theoretical reasons for viewing grand unification as a consistent road
to physics beyond the standard model include the fact that several grand unified groups
ensure the vanishing of anomalies of gauge currents from the very nature of their
representations; e.g., for any irreducible representation of 50(10), TiiY) and Tr(Y^) vanish
automatically [3], where Y is the hypercharge generator. Thus anomaly cancellation which
may appear somewhat mysterious in the standard model is natural in grand unification; it
may be worth noting that while the structure of the strong interaction was arrived at through
the analysis of the anomaly in -^2y [2], the structure of theories beyond the standard
model may also be uncovered from such considerations of anomalies. In addition, global
anomalies arc related to the centre of the gauge group : Z 2 in the case of 5C/(2), 50(10) and
Zi in
In turn, processes involving transitions from one set of matter fields to another
predicting, say the decay of the proton at measurable rates are intrinsic features of
unification. The continued failure of the proton to decay within present day expieriments in
turn implies constraints on scenarios of grand unification [11].
Note that whereas in the standard model, the field content forbids a Dirac mass for
'.he neutrinos since the right handed neutrino is absent and Majorana mass is forbidden by
the conservation of lepton number. In grand unified models, neither of these principles is
respected and a wide variety of possibilities exists for the generation of neutrino masses.
However, far from being arbitrary, it should be possible to uncover information regarding
the structure of unified theories from accurate determination of small and eventually
large neutrino masses and mixing angles, viz., neutrino masses may be viewed as bearing
an imprint on the structure of grand unification and the nature of the breakdown of
unification [12].
3. Supersymmetric unification
Supersymmetry is the unique symmetry that has non-trivial commutation relations with the
generators of the Lorentz group. Supersymmetries enjoy non-trivial anti-commutation
relations amongst each other. Their action on representations of the supersymmetry algebra
interchange the statistics between the members, Linear representations of the
supersymmetry algebra in relativistic field theory are realized in the Wess-Zumino model
[6|. Imponant representations include chiral multipicts and vector mullipleis, which form
the basis of the extension of the standard model to various supersymmetric versions of the
standard model. Since supersymmetry is not manifest in nature, it must be broken, either
spontaneously or explicitly. It appears that the second option is more favored, certainly
more popular, wherein supersymmetry is broken explicitly but softly. The requirement of
Status of supersymmetric grand unified theories
499
soft supersymmetry breaking is in accordance with the requirement of the well-known
properties of supersymmetric models including the cancellation of quadratic mass
divergences for scalars.
In the context of grand unified model building, the existence of scales and Mq
separated by several orders of magnitude renders the mass of the elementary Higgs of the
standard model unstable and would drive it to the unification scale, without an un-natural
fine tuning of parameters of the Lagrangian. The cancellation of quadratic divergences in
manifestly and softly-broken supersymmetric theories renders supersymmetric versions of
grand unified models attractive candidates for unification. The program of writing down a
supersymmetric version of the standard model, which is then embedded in a grand unified
scheme, (alternatively, a supersymmetric version of a grand unified scheme] may be
realized by replacing every matter and Higgs field, by a chiral superfield whose members
carry the same gauge quantum numbers, and by replacing every gauge field, by a vector
super-multiplet. Supersymmetry also requires that the standard model Higgs doublet is
replaced by two Higgs multiplets. This is turn leads to the introduction of another parameter
lan /3 which is defined as the ratio of the vacuum expectation values of these two Higgs
fields, V 2 /V 1 where ^2 are the vacuum expectation value of the Higgs fields that
provide the mass for the up-type quark and the down-type and charged leptons respectively.
All ihe interactions of the resulting model may then be written down once the
superpotential is specified. Note that gauge invariance and supersymmetry allow the
existence of a large number of couplings in the effective theory that would lead to proton
decay at unacceptably large rates. An ad hoc symmetry called /?-parity is imposed on the
resulting' model which eliminates these undesirable couplings and such a version has
received the greatest attention for supersymmetry search. More recently models have been
and are being considered where /?-parily is partially broken in order to study the
implications to collider searches. However such models arc constrained by bounds on
fiavor changing neutral currents as well as by the standard CKM picture, also as it applies
10 CP violating phases.
In what follows we recall some of the essential successes of the recent investigations
113] in the theory of supersymmetric unification. This was spurred by the confrontation of
the ideas of unification by the precision measurements of the gauge couplings of the
standard nxidel at the LEP [14]. A highly simplified understanding of this feature may be
obtained from a glance at the non-loop evolution equation for the standard model gauge
couplings, more correctly the gauge couplings of the minimal supersymmetric standard
model assuming that the effective supersymmetry scale is that of the weak scale,
NVllh
dtx , Of f
^ 6 , , ft, = 33/5, ^2 = ^3 = “3* where we have assumed three
generations. One may then integrate these equations to obtain : —
^ a, (M2) a, (Me)
^ ^ One may then use the accurately known value of (M 2 ) = 1 / 128, with
72A(6)-6
S()() B Ananthanarayan and P Minkowski
the identity +1/0(2 accounts for the normalization imposed by
unification; and the values of 03 (M2 ) ■* 0.12 to solve for the unification scale Mq and the
unified coupling constant 9 0(t 2,3(M^ ). One then has a prediction for sin^ Bq, at the
weak scale which comes out in the experimentally measured range. Sophisticated
analysis around this highly simplified picture up to two and even three loops taking into
account the Yukawa couplings of the heaviest generation which contribute non-trivially at
the higher orders, threshold effects, etc., vindicate this picture of gauge coupling unification
which today provides one of the strongest pieces of circumstantial evidence for grand
unification [I5|.
Predictions arising from (supersymmetric) unification such as for the mass of the
top-quark have been vindicated experimentally. It turns out that unification based on
50(10) is a scheme with great predictive power not merely in the context of top-quark
mass but also with implications for the rest of the superparticle spectrum. The primary
requirement that is imposed is that the heaviest generation receives its mass from a unique
coupling in the superpotential h 16 . 16.10 where the 16 contains a complete generation and
the complex 10 the two electroweak doublets [16], Wheh the Yukawa couplings of the top
and /7-quarks and the r-lepton are evolved down to the low energy and tan P pinned down
from the accurately known T-mass, one has a unique prediction for the b and top-quark
masses for a given value of h. If h is chosen so as to yield ^^(m/,) in its experimental range,
the top-quark mass is uniquely determined up to these uncertainties. Now tan P s nijiiif,,
and the top-/? hierarchy is elegantly explained in terms of this ratio coming out large
naturally. *
It is truly intriguing that this picture yields a top-quark mass in its experimental
range, with in the range of the LEP measurements despite the complex interplay
between the evolution equations involved, the determination of the unification scale,
running of QCD couplings below the weak scale. Note that this requires that the lop-
Yukawa coupling must also come out of order unity at M2. It is also worth noting that due
to the nature of the evolution equations and competition between the contributions to these
from the gauge and Yukawa couplings, this number ni,(m,) lies near a quasi-fixed point of
its evolution, viz, there is some insensitivity to the initial choice of /i [17]. Moreover, if the
50(10) unification condition is relaxed to an 51/(5) one where only the /7-quark and T-
lepton Yukawa couplings are required to unify at Mq, comes out in the experimental
range while preserving m/,(m/,) in its experimental range for tan p near unity. In this event
also the top-quark Yukawa coupling lies near a quasi-fixed point which is numerically
larger compensating for the smaller value of sin P that enters the expression for its mass :
nif = hf sinP 174 GeV, Another interesting connection arises in this context between the
values of the Yukawa couplings at unification and that of the gauge coupling when onc-
loop finiteness and reduction of couplings is required : such a program also yields top-
quark masses in the experimental range [18j.
Besides the vindication of top-quark discovery predicted by susy guts, another
strong test takes shape in the form of its prediction for the scalar spectrum. In the MSSM
Status of supersymmetric grand unified theories
501
ihc mass of the lightest scalar is hounded at tree level by M/ since all quartic couplings
arise from the D-ierm in the scalar potential. The presence of the heavy top-quark
enhances the tree-level mass, but the upper bound in these models is no larger than
140 GeV.
Other predictions for softly-broken susy models arise when a detailed analysis of the
evolution equations of all the parameters of the model are performed and the ground stale
carefully analyzed. In the predictive scheme with 50(10) unification, the model is further
specified by Mi/j, niQ and A, the common gaugino, scalar and tri-linear soft parameters [5].
[I turns out that in this scheme M^/i is required to come out to be fairly large, at least
~5CK) GeV implying a lower bound on the gluino mass of a little more than a TeV and
providing a natural explanation for the continuing absence of observation of susy particles
Irom scenarios based on radiative electro- weak symmetry breaking [19], [An extensive
study of the NMSSM with 50(10) conditions has also been performed [20]]. Considerably
greater freedom exists when the 50(10) boundary condition is relaxed [21]. In summary
many predictions and consistency of the MSSM and its embedding in a unified framework
have been vindicated; however, it is important to continue theoretical investigations and
cheeks to the consistency of these approaches and extensions to include the lighter
generations [22].
4. Monopoles
Thl^ discussion is somewhat off the main stream of the discussion above. Furthermore, if
one were to discuss spontaneously broken gauge field theory at finite temperatures, when
leiTiperaiurcs reach the scale of symmetry breaking, then phase transitions are expected to
occui which restore broken symmetries. Indeed, at such phase transitions, one expects the
loimaiion of topological defects which may be characterized by certain topological
piopcriies known as homolopy groups of the coset space : C/W, where C is the gauge
gioiip that IS broken to the subgroup H. Examples of topological defects are domain walls,
Mr mgs and monopoles, which may have been produced in the early universe as the universe
urolcd to present temperatures. This is an example of an aspect of gauge field theory that is
uutside the realm of perturbation theory. However, certain interesting preliminary
investigations indicate that monopoles arc inconsistent as asymptotic slates; they arc
Lonfincd even if in the topologically parallelizable sector the gauge theory serving as
non-ahelian basis to the classically acceptable monopole solutions is broken |23]
KHher examples of standard model physics that the outside this realm is that of
liie lormation of fermion condensates that arc required to spontaneously break chiral
Miiimctiy that lead to the generation of massless pions when the quark masses are set
/vio|.
Acknowledgment
thanks G Zoupanos for discussions.
502
B Ananthanarayan and P Minkowski
Rererences
[1] S Glashow NucL Phys. 22 579 (1961); S Weinberg Phys. Rev. Lett. 19 1264 (1967); A Saiam in
Elementary Panicle Theory t6. N Svartholm (Stockholm ■ Almqvisi and Wilscll) p 367 (1969)
(2] For a Comprehensive Discussion, see e.n T P Cheng and L F Li Gaufie Theory oj Elementary Particle
PhysK s (Oxford . Clarendon) (1984)
[ 3] See e y G G Ross Grand Unified Theories (Californio . Menlo Park) The Benjamin/Cummings ( 1 985)
[4] For a Collection of Repons, see Supersymmetry and Superf>ravity, A Reprint Volume of PImics Reports
ed M Jacob (Amsterdam/Singapore Nonh-Holland/World Scientific) (1986)
[5] H-P Nilles Phys. Rep. 110 I (1984) repnnlcd in Ref. [4)
[6] See €.}• M Sohnius Phys rep. 128 39 (1985) reprinted in Ref [4]
[7] See e , 1 * M Green, J Schwarz and E Witten Superstrinf* Theory 1.2 (Cambridge Cambridge University
Press) (1987)
[8] S Ganguly Indian J Phys 72A 527 (Invited Talk at this Conjerence) (1998)
[9] M Narayon {Invited Talk at thn Conference)
flO] H Fnizscii and P Minkowski A/i/j, Plivs 93 193(197.5)11
1 1 1] For a Recent Review, ^ee e ^ H Murnyaina Nucleon Oeeax in GUT and non GUT SUSY Models
hep-ph/9610419
|I2| For a Recent Review, see e f! ? Minkowski Neutrino Mass and Mixinf> (Bern University, preprint,
BUTP-05/22)
1 13 1 For some Recent Reviews see e ^ Li Hall The Heavy Top-Quark and Super.symmetrs hep>ph/96052S8;
F Zwirncr Extensions of the Standard Model hep •ph/9601300, S Pokorski Status of the Minimal
Supersymmetric Standard Model hep-ph/9510224, S Dimopoulos Beyond the Standard Model ICHEP
1904 93-106 ((X'Dlbl H5l 1994)
fl4] U Amalili, W de Boei and H Fiirsienau Phys Lett B260 447 (1991). P Langackcr and M X Luo
Phys Rev D44 817 (1991 ). C Giiintii, C W Kim and U W Lee Phw Lett A6 1745 (1991)
[151 Foi updates, see e W de Boci The C(mstrained MSSM hep-ph/96J1394. Glolnil Fits to the
MSSM and SM to t.lci troweak Precision Data hcp-ph/96 11395
[16] B Ananthanarayan. G Lazandcs and Q Shaft Ph\s Rev D44 1613 (1991). Foi a Recent Update see
U Sand Piecision Top Mass Measurements v\ Yukawa Unification Predictions hep*pli/9601300
[17] Fora Review, see e.)’ B Schrempp and M Wimmci Top Quark and Hifiyis Ro.son ma.s.ses Interplay
between Infrared and Ultraviolet Pliy su s hcp-pli/9606386
[18] J Kubo, M Mondragdn and G Zoupanos Top Quark Mass Predictions from Gauj^e Yakasva Unijication
hep-ph/9S12400
[19] D Ananthanarayan, G Lazarides and Q Shali Phys Utr B300 245 (1^93), B Ananthanarayan, Q
and X-M Wang Phys Rev 1)50 5980 (1994) and references therein
[20] For a Comprehensive .Analysis of the NMSSM with Lar^e tan /) ree B Ananthanarayan and P N Pandiia
Phvs U-tt 8353 70(1995), /9/vi Lett 8371 245(1996)
[22] For other Recent Directions see e T Hlazek et ul A Global X Analysis of Eleciroweak Data in SO(IO)
.SUSY GUI's, hep-pli/9611217, M Catena et al Uottom-up Approac h and SUSY Breakinfi hcp-ph/9610341
[23] M Striebcl Magnetic Monopoles in a Constant Backfiround Gauf'e Field (University of Bern thesis)
(unpublished) (1987)
Indian J. Phys. 72A (6), 503-514 (1998)
UP A
- an i nleniaiionttl journal
Results from LEP 1
S N Ganguli
Tata Institute of Fundamental Research, Homi Bhabho Rood,
Culaba. Muinbai-4(X} 005. India
Abstract : The large electron positron collider, LEP. at CERN is running since 1989 and
Its purpose wiis to study the properties ot Z particle during the first phase called LEP 1 . Some of
the results from LEP I are described in this article
Keywords : LEP
PACS Nos. : n 10 -♦■q. 13 38.Dg. 14 70 Hp
1. Introduction
End of LEP' s era '
A chaplcr ol’ LEP, Large Electron Positron collider at CERN, Geneva, got closed
duiing ihc l u st week of October 1995. This was the first phase of LEP, called LEP 1, and
Us put pose was to study the properties of Z particle and related eleclroweak parameters
with := 45 GeV e" beam colliding with = 45 GeV e+ beam yielding centre of mass energy of
collision of s/1 =1 90 GeV. The Zera has been a great success from physics achievements
pomi of view as well for CERN as a major centre for particle physics. In the second phase
of LEP. called LEP 2. from 1996 to 2(XX) the centre of mass energy is gradually upgraded to
cross the W pair production threshold by incorporating superconducting RF accelerating
cavities in several stages. During 1996 the data has already been taken at LEP 2 with centre
ot mass energies as 161 and 172 GeV.
First beam in the LEP ring was seen on July 14, 1989, with a shon pilot physics run
•Junng mid August 1989 when Z events from e'*'e~ interactions were recorded by the four
I HP detectors ; ALEPH, DELPHI. L3 and OPAL- First physics run took place during
September 20 to October 10, 1989. Each of the four LEP experiments recorded =. 30000 Z
events which led to the determination of mass and total width of Zas : Af^ = 9L161± 0.031
^eV and T = 2.534 ± 0.027 Ge V [ 1 ].
© 1998 lACS
504
S N Ganguli
LEP detectors :
As mentioned earlier there are four detectors at LEP and these detectors have 471 geometry
and the general concept is very similar. For the momentum measurement of charged
particles there is a magnetic field (0.5 to 1.5 T) parallel to the colliding beam direction.
Basic components of these detectors are summarised briefly in the order of increasing
distance from the interaction point : (i) Vertex detector : silicon microvertex detector with a
fine spatial resolution of -10-20 |im. (ii) A multiwire drift chamber to track charged
particles with momentum resolution of rs 5-10%. (iii) An electromagnetic calorimeter to
delect c", e^ and photons. The energy resolution at 45 GeV varies between 1-3%. (iv) A
hadron calorimeter to detect energies deposited by hadrons {n, K, p, ...) through total
absorption. The typical energy resolution for a total energy of 90 GeV is = 10%. (v) Scries
of wire chambers outside tbe hadron calorimeter to detect muons with momentum
resolution for a 45 GeV muon as 2-6%. (vi) For the measurement of luminosity there are
electromagnetic calorimeters placed on either side of the interaction point and very close to
the beam pipe to detect small angle Bhabha scattering (e'‘‘e“-> 6 *^ 0 ").
Standard Model :
The understanding of the mechanism responsible for the electroweak symmetry breaking
leading to massive IV and Z gauge bo.sons is one of the central problem in particle physics.
The simplest mechanism for this is realised in the Standard Model which contains a single
complex Higgs doublet with one physical neutral scalar Higgs particle. The four vector
bosons describing the electroweak interactions are : y, 29, IV*’ and IV". The mixing of yand Z
is described by electroweak mixing angle sin^^. The Standard Model assumes three
fermion farmilies/generation (6 quarks and 6 leptons).
Indian participation in LEP :
The Experimental High Energy Physics group of TIFR joined the LEP-L3 collaboration in
early 1983. The group members took active part in the following activities, (i) Hardware
contribution ; 1100 brass tube proportional chambers for the hadron calorimeter were
fabricated in the laboratory. Precision stainless steel housings for the chambers were
fabricated at BARC central workshop. For the L3 upgrade the group fabricated 7500 wire
fixation blocks and assembled 3000 readout PCB’s for forward/backward muon chambers,
(ii) Group members are taking part in data taking and monitoring of detector, (iii) For the
software development group members contributed towards reconstruction, simulation,
database packages, (iv) For the physics analysis group members are carrying out extraction
of electroweak parameters, QCD, heavy flavour physics and search for Higgs and SUSY
particles.
2. What do we observe in e'^e' interactions ?
Interactions of e'*^e"lead to a pair of fermions in the final state. Lowest order diagrams, sec
Figure 1 (a), are due to y and Z exchanges plus their interference terms. The r-channel
diagrams valid only for e'^e'in the final state are not shown. The contribution of y exchange
Results from LEPt
505
m Vs c: M. is negligible. Examples of diagrams due to virtual and real photons aiv shown
in Figure 1(b) and examples of weak corrections due to fermion loop, box and vertex are
V( M
Figure 1. Lowest order diagrams : (a)
due to y and Z exchanges, (b) virtual
and real photons, and (c) weak
corrections due to loop, box and vertex.
shown in> Figure 1(c). It is important to mention that (i) the weak radiative correction is
proportional to where M,op is the mass of the top quark, (ii) radiative corrections
result in the modification of experimental quantities like total and partial widths of Z,
asymmetries, T^polarization, eicctroweak mixing angle etc. In order to make experimental
measurements independent of theoretical weak radiative corrections we measure all
quantities dressed with electroweak effects.
Z decay modes :
Various decay channels of Z into fermion anti>fermion pairs are summarised in Table 1 .
Table 1. Z decay modes.
ta) Z leptons
Decay channel
Observed particles
Branching fraction
c*e~
= 3.3%
= 3.3%
t*t-
low multiplicity
finai sute
= 3.3%
v.Vj. v,\!.
none
= 20%
(b)Z -» hadrons
uu, dd. ss, cc. bb
2, 3, 4 high multiplicity
jeu of hadrons
= 70%
506
S N GanguU
3. What do we measure in e^e‘ interactions ?
Some of the experimental measurements are summarised below :
(a) a ivv 'G ;
Experimentally one measures cross sections (O) as a function of the collision energy for
the following final states : e'^e" — > hadrons* e*e~ -> e'^e*, e^e“ — > and e'^e
Typical energy scan is carried out between 88-94 GeV around the Z-mass. This is termed
as the Line shape measurement of the Z peak.
The basic physical parameters that describe the cross section are the mass of the Z,
My, its total width Fy and the partial widths /} for decay into fermion pairs. From the
lineshape measurements one measures three quantities : (i) position of the peak which
defines My, (ii) height of the peak which is proportional to F^Tf and (iii) width of the
distribution which gives the total width F
I ' ‘ .
b 11-
_ - - *
B 1 ^ •• •
I 09 -
^ 08 [- , . I ■ . . ■ . , ■
86 88 90 92 ' 94 96
VS (GeV)
Figure 2. Cross section for e^e- -4
hadrons as a function of collision
energy. Hi to the data is shown ns solid
curve Quality of fit can be seen from
the bottom plot
Figure 2 shows the L3 data for the variation of cross section as a function of
collision energy and fits to the data are shown as curves.
(b) Fom'ard-backward asymmetry :
The forward-backward asymmetry A^g of the final slate fermion arises due to vector
j ■ ■ ■ L A — ~ vvhcrc
and axial-vector nature of the Z coupling and it is given by : Afi? - ’
(or (Jfl) is the forward (or backward) cross section when the fermion is in the torw-io'
direction (or backward) direction with respect to the initial e' beam direction.
Results from LEPl
507
measuFement ofApg leads to the determination of the electroweak mixing angle as can be
seen from the following simplified expression evaluated at V 5 = :
3(1 -4sin^ )
Afb =
1 + (l-4sin2 dw)^
(>-4|G/|sin2 0^)
I + (l-4|ef|sin2 ewf
( 1 )
where Qf is the charge of the fermion under consideration.
The following asymmetries are measured at LEP : (i) lepton asymmetries : + e~ — >
e*+ e", + p", + T", (ii) bb asymmetry : e*" + e" -4 b + b, (iii) cc asymmetry : e^*^ + e“
c + 5 and (iv) quark charge asymmetry (< Qpg >).
(c) r Polarization :
T leptons from decay of Z are longitudinally polarized, and the decay of the r via
ihe charged weak current serves as a natural analyser of the r polarization. The momentum
spectrum of the pion from T decays, T —> 7t~Vp gets modified due to its polarization
and consequently the polarization, is measured from the form of the pion energy
spectrum :
J_
N,
dN
dX,
= 1 +P,(2X.-1).
( 2 )
where X* =
’'beam
The polarization measurement at the Z peak determines (i) the relative sign between
the vector {gy^) and axial-vector coupling of Z and (n) the electroweak
mixing angle ;
2g vx / 8 ai
Px - -
^ (s vx / Sar)
2 (1 - 4 sin 2 )
1 + (1 - 4sin2
(3)
(4)
id) Left- right asymmetry :
The left right asymmetry deals with measurement of cross sections with a longitudinal
polarization for the e‘ beam and it is given by : Am = where Oi^n are the cross
s^ections for + e*^ X, where X can be any channel. Am has been measured by the
SLD collaboration at SLC. Am has the advantage of being extremely sensitive to sin*
insensitive to QED radiative corrections and it depends on the Z coupling to initial
^^<^«.e.,toc+e-
508
S N Ganguli
Alh is related to the experimental measurement by the following relation ;
= ^exp / ’ where is the measured longitudinal polarization of the e" beam. The
clectroweak mixing angle is derived from :
2(1 - 4sin2 0eff )
^lr = ;; (5)
1 + (l - 4sin2 dcff )
Another quantity of interest is the fo ward- backward polarized asymmetry which depends on
the Z coupling to the final state and it is given by
pol _ (gp f - O.p f ) - (CTp fl - G-p,b )
f ) + (gp fl + ^
3 2(1 - 4sin2 grff )
= T T' H)
I + (l - 4 sin 2 )
4. Electroweak results from e^e~ interactions
Data ;
Data has been collected over the years 1990 to 1995 as a function ofVi around theZ mass.
During 1990-1991 the energy range covered was | V? - < 3GeV; in 1992 the data
collected at the Zpcak; in 1993 at j V? - Mz \ < I SGeV; in 1994 at the Zpeak and in
1995 at \^^s - M z \ < 1.8GeV. The total number of Z events collected by thc^four LEP
experiments during 1990-1995 is:=16.10^ and the break-up is given in Table 2 [2].
Tabic 2. Number o( Zevenis
l>tecioi
Z — > hadrons
z-^i^r
ALEPH
4.2 X 10^
0.5 X 10^
DELPHI
3 6x 10^
0.4 x 10^
L3
3 4x 10^
03 X 10^
OPAL
3 4x 10^
0.5 X 10*
4. /. Mass, width and number of neutrinos :
The precision measurement of the Zlineshape (gvj Vi ) yielded the mass and width of 2
which are summarised in Table 3. The number of light neutrinos is determined to be three
with a precision of 0.3% [2].
4. 2. Determination of electroweak mixing angle :
The asymmetry measurements lead to the determination of the effective electroweak
mixing angle, sin^ ^fr- Results from different measurements are summarised in Table 4 [2]
It may be mentioned that the LEP average of sin^ 61efr 0,23192 ± 0.00027 is to be
Results from LEPI
509
compared with ihe SLD measurement of 0.23055 ± 0.00041 ; they differ from each other by
2.8 standard deviation.
Table 3 . Macs, width and
Parameters
Measurements
M^GcV
91 186± 0 002
TzCeV
2 495± 0 003
0/ MeV
K3 89± on
ThadMeV
1743 5 ± 2 4
r.nv MeV
499 H ± 19
^/ = ^had/f'/
20 783 ± 0 029
Number of neutrino species
2 992± 0 011
Table 4. Values of sin^
Measurements
sin^ f/pH
Api^ leptons
0 23068 ± 0 0005.S
A j from Pj
0 23240 ± ()0(K)8.S
A^ from Pj
0 23264 ± 0.00096
A/rg /7-quark
0 2323.S ± OOfKMO
A frg L -quark
0 23I5.S ± 0,00111
< Qfb>
0 2322 ± 0fK)l0
Auf (SLD)
0.230.S.S ± 0 00041
4 Measurement of /?/, :
Ki, IS defined as the ratio of the b quark partial width of the Z to its total hadronic
pariial width : Rf, = /^had- An important aspect of this ratio is that most of the
Standard Model corrections common to all quarks drop out in this ratio except the h
quark vertex correction which depend on mass of the top quark. Measurements available as
end 1995 showed a positive deviation of 3.7 standard deviation from the Standard
Model One of the exciting explanations was supersymmetric contribution from light
chargino.
During 1996 all the 5 experiments (ALEPH, DELPHI, L3, OPAL, SLD) made a
dciailed study of the measurement of /?/,. Some of the new points are : (i) usage of several
^idlcrcni tags for b quark, in particular the inclusion of invariant mass tag, (ii) probability of
< quark fragmentation is used from the LEP data itself, (iii) detailed study of various
^Vsicinatics is carried out, (iv) all available data are used and (v) results are obtained by
510
S N Ganguli
carrying out 13 parameter fit to the LEP and SLD data. This leads to the following value
lor/?,, 12.31,
/?/,(LEP) =0.2 179 ±0.0011,
(8)
/?„(SLD) =0.2 152 ±0.0038,
(9)
(LEP + SLD) = 0.2 1 77 ± 0.001 1 .
(10)
The expected value of /?,, from the Standard Model is 0.2158 to be compared with the
LEP+SLD measurement of 0.2177 ± 0.0011; this leads to a deviation of 1.8 standard
deviation.
4.4. Top and Higgs in standard model framework :
The precision achieved in the eleclroweak measurements at LEP and SLD can be used (o
check the validity of the Standard Model. The Standard Model basically needs the
lollowing 4 quantises : Mz. A/,„p, and a,. The other quantities which it needs arc
known. Mz and a, are measured at LEP. and thereby fitting all the electroweak data one can
determine the values of A/, op and A^Higgs- The accuracy of LEP measurements makes them
sensitive to A/, op and A^Higgs weak loop corrections; the dependence on is quadratic
while the leading A^ Higgs dependence is logarithmic.
Results of fits are shown in Table 5 [2]. The second column of the table summarises
fitted values of A/, op and A/niggs using LEP data alone. The third column summan.scs fitted
results using all data which include measurements from LEP, SLD. direct measurements ol
M\[' (80.37 ±0 10 GeV) [4] and A/, op (175.6 ± 5.5 GeV) at pp collider [5], and sin-
measurement from vN interactions |6].
Tables. %lgg^
Parameiers
LEP
All data
op (GeV)
172 7 ±5,4
^Higgs (GeV)
70!j,r
I27_72
It is interesting to note that all the existing data show a low mass for Higgs. Wc
show in Figure 3 the observed values of s X‘ - Xmm ^ function of A/H,gg^ for the
fit with all data. This yields 465 GeV as the one sided 95% confidence level upper limit on
the mass of Higgs. It may be mentioned that direct search of Higgs yielded 66 GeV as the
lower limit on A^Higgs- There are other estimates on the upper limit of A/H,pg< 17].
5. x-Physics from e'^e" interactions
There were problems (a) in the experimental data for r decay branching ratios, in particular
‘l-prong deficit’ was noted in 1984 and (b) the predicted branching ratios (Be =
B{e~v,.v^) or Bu s B{p~v^ w^)) assuming unitarity can be predicted from ma.sse.s
Results from LEPi
511
and lifetimes of the muon and tau. Theory and predictions have differed significantly
since 1986.
Figure 3. The observed values of ~X mm
,is a function of Higgs mass are shown from the Tit
with all available data.
T-^ universality test
Figure 4. Plot of tau lifetime branching ratio of
tau decay via electron mode is shown. The I99.S
world average values (I995 W.A ) of tau lifetime
and branching are in good agreement with the
measured tau mass [II].
During the last few years new measurements of branching ratios |8] and lifetimes [9]
at LEP [8] and mass of the tau lepton at BES [10], Beijing, have significantly improved the
precision. Figure 4 shows the plot of tau lifetime vs B(t -> ); the agreement
between the measured values and prediction is clear [11].
6. Some miscellaneous results from e‘*'e' interactions
6.y. Upsilon production :
r (s bb) production in Z decays requires emission of energetic gluons and hence the
production is highly suppressed. There are two production mechanisms and they are briefly
discussed below ;
(a) Colour Singlet Models ; Here the b quark fragmentation is the dominant mode,
see Figure 5, leading to Z Tbb with branching fraction as Br = 1 .7 x 10-\
(b) Colour Octet Models : This was introduced to explain high production rate of T at
the Tevatron [12]. In this model [13] Upsilons are first produced in colour octet,
sec Figure 5, then they evolve non-perturbatively into colour singlet. The
dominant process is the ‘gluon fragmentation* : Z — > Tqq with a branching ratio as
Br = 4.1 xlO-5.
The OPAL collaboration [14], at LEP, from a sample of 3.7 million Z decays identified
^ighi T candidates from their decays into and pairs. The estimated background in
512
S N Ganguli
ihc signal region is 1 .6 ± 0.3 events. The following branching ratio is obtained for inclusive
y production :
Br(Z-> r + X) = (!.() ±0.4 ±0.1) X l(H (II)
It may be mentioned that none of the 8 candidate events is associated with bE thereby
supporting the colour octet model. The above experimental measurement is to be compared
with theoretical expectation of 5.9 x I0'\
colour-singlet colour-octet
b-quork froqmcntotion
gluon fragmtniolion
Figure 5. Production mechanisms of
upsilons in Z decays from colour
singlet and colour octet models
6.2. Measurement of A polarization :
In the Standard Model, down-type quarks from Z decays arc produced with high
longitudinal polarization :
2(l -4|G^|sinJ0.„)
= - \ ( 12 )
1 + (l - 4|G,|sin2 e,„)
For a strange quark the polarization is = -0.94. Hard gluon emission and hadronization
processes reduce the polarization P^. The quark contents of the A baryon are strange (j).
up (u) and down (d). In the simple quark picture the A is supposed to carry the spin of the
constituent .v quark (light quark pair 'ud* is supposed to be in spin = 0 and isospin » 0 state)
and therefore the As formed from primary s quark will cairy the polarization of s quark.
Results from LEPI
513
The ALEPH collaboration [IS] measured the longitudinal polarization of A to
be : Pa * -0.32 ± 0.04 ± 0.06 which is to be compared with the expected value of
(-0.39 ±0.08).
6.3. Exclusive decays of A/, (* udb ) :
Exclusive decays of A/, have been searched for at hadron colliders and at LEP [lb]. The
DELPHI collaboration [17] from a sample of about 3 million Z decays have identified four
fully reconstructed A/, events ; three in the A*k~ decay channel and one in the A^o,"
channel. The A® beauty baryon mass is measured to be 5668 ± 16 ± 8 MeV.
7. Summary
The Large Electron Positron collider at CERN is an unique machine running for the last
seven years. The first phase of LEP, called LEPI, came to an end during end October 1995.
There arc four mammoth detectors (ALEPH. DELPHI. L3 and OPAL) which arc collecting
data. During the LEPI phase these four detectors together have collected 16 million
Z events. Some of the results from LEPI are : (i) The mass and the width of the Z boson are
measured to a precision of : = 2.10"^ and = 1. 10 "V (ii) The number of light
Ml I 2
neutrino species is measured to be three with a precision of 0.3%. (iii) The electroweak
mixing angle is measured from the asymmetry measurements at LEP and SLD and the
values are ; 0.23192 ± 0.00027 (LEP) and 0.23055 ± 0.00041 (SLD); they differ from each
other by 2.8 standard deviation. <iv) Results presented on the measurement of /?;. in 1995
showed a discrepancy of 3.7 standard deviation from the Standard Model. This discrepancy
IS now reduced to 1.8 standard deviation with new techniques /methods used at LEP and
SLD. (v) The precision of electroweak measurements at LEP and SLD, and with the
measurements of and at the Tevatron collider, the mass of the Higgs is determined
to be \21Vi2^ GcV with the upper limit as 465 GeV at 95% confidence level.
RefierciKci
[1) ALEPH Collaboraiion Phyx Leti. B235 399 (1990); DELPHI CollolHwaiion ; Pins Un B241 42.S
(1990); L3 Collaborauon : Phys Un. B237 1.36 0 990); OPAL Collaboraiion ; Phw Un B240 497
(1990)
J2) LEP Electroweak Working Group. LEPEWWG/97-01 . 7 April 1997
f3] ALEPH Collaboraiion . CERN PPE/97-0t7. CERN PPE/97-0IB; OPAL Collaborauon CERN
PPE/96-167. CERN PPE/97.06; L3 Collaborauon : L3 Noie 2033. L3 Noie 2066; SLD Collaboraiion
P RowKon, talk presented at Moriond 97
1^1 A Gordon Talk presenied ai XXXtInd Rencontres Je Monond. Us Arcs. 16-22 March 1997
13] CDF Collaboration : J Lys. Talk presented at ICHEP96. Warsaw. July 1996. DZERO Collaboraiion .
S Protopopcscu. Talk presented at ICHEP96. Wunutw. July 1996
[61 CDHS Collaboration : Fhys. Rev. Utt. 57 298 (I9B6). Z Phys. C45 361 (1990); CHARM Collaboration ;
Phys. Un. BIT? 446 (1986); Z Phys. C366I I (1987); CCFR Collaboration : Pnneed XV Workshop on
Weak Interocfions and Neutrinos T Fi:ance and G Bonneaud ei al ed.s. Tufts Univernry and LA L. Or.sa\
Vol Ilp607
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1 7 1 A Gunu rhvs. Lett. B385 415(1 996)
[K] DELPHI Collaboration ; PIm. Lett. B3S7 715 (1995); ALEPH Collaboration . Z P/m. CTO 561 (1996),
OPAL Collaboration . Phys. Uii. B369 163 (1996). L3 Collaboration : Phys. Lett B352 487 (1995);
CLEO Collaboration : Phys Rev. DS3 6037 (1996)
19] DELPHI Collaboration ; Phys Lett B365 448 (1996); OPAL Collaboration . P/m. Lett B373 341
(1996); ALEPH Collaboration ; Z Phys.ClQ 549 (1996); L3 Collaboration : Phys. Rep. 236 I (1993),
CERN-PPE/96- 124; SLD Collaboration : Phys. Rev. D52 4828 (1996)
[ 1 0] BES Collaboration . Phys Rev. D53 20 ( 1 996)
1 1 1] H S Chen Proceed. XVII Ini. Symp on Lepton Photon Inierai tion.s. Beijing, August 1995
[12] CDF Collaboration ; FERMILAB-PUB*95/27I -E
f 13) M Cacciari ei id Phys Rev Lett 73 1586 (1994); Phys Ijett B356 553 (1995); P Cho and A Leibovich
CALT 68-1988. CALT-68-2020. E Braatcn et al Phvs. Rev D48 4230 (1993), Phys Rev Uit 71 1673
(1993). V Barger €■/«/ P/m Pev. D41 1541 (1990), K Hagiwara et «/ P/ivv Utt B267 527 (1991); P/m
Uti B316 631 (1993); K J Abraham Z Phys. C44 467 (1989); J H Kuhn and H Schneider Z. Phys Cll
263 (1981); W Y Keung Phys. Rev. D23 2072 (1981)
1 1 4J OPAL Collaboration : CERN-PPE/95- 1 8 1
1 1 5] ALEPH Collaboration ; CERN-PPE/96-04
(16) UAI Collaboration Phys. Utt B273 540(1991). R422 Collaboration . Nuo\ Con 104A 1787 (1991).
DELPHI Collaboration ■ Phys Lett. B311 379 (1993). ALEPH Collaboration Pins IaHI B278 209
(1992), OPAL Collaboration : Phys. Lett. B281 394 (1992)
(17) DELPHI Collaboration : CERN-PPE/96- 16
Indian J. Phys. 72A (6). 515-532 (1998)
UP A
— an international journa l
Physics at LEP 200
A Gurtu
Tata Institute of Fundamental Research, Colaba,
Mumbai-4(X) 005, India
Abstract ; The Large Electron Positron collider LEP at CERN recently achieved centre of
mass energies much above the Z-pole. Recent experimental results from the four LEP
experiments, ALEPH, DELPHI. L3 and OPAL at these hitherto unexplored energy regime in
e'*'e~ interactions are presented.
Keywords : e'*“c” physics. W mass, Higgs search, test of standard model, SUSY searches.
PACS Nos. : l4.70.Fm, 14.80.Bn, l4.80.Ly
1. Introduction
The Large Electron Positron collider (LEP) ran at a centre of mass energy above the Z mass
region for the first time in November 1995 : at V7 values of 130, 136 and 140 GcV. During
1996 the energy was enhanced first to 161 GeV, just above W*'W' production threshold
during June- August 1996 and later to 172 GeV during Ociobcr-November 1996. Each LEP
experiment collected ~5 pb”‘ during November 1995 and ~10 pb”' at each of the two
energies during 1996.
A reminder of the goals of LEP200 :
• Continuing study and precision measurements of Standard Model processes,
• Precision measurement of W Mass and Width,
• Search for SUSY.
• Search for SM and non-minimal Higgs.
• Measurements of Triple Gauge Couplings
• LCX)K FOR THE UNEXPECTED
72A(6).8
'.c wm lACs
5i6
A Gurtu
Figure I depicts the cross sections of typical SM processes as a function of centre ol
niasN energy at LEP [ 1 ].
The lollowing topics will be covered in this talk.
(i) Fermion pair production
(ii) W mass measurements
(iii) ALEPH excess ot 4-)el events
(iv) Search for Higgs and SUSY particles
(v) QCD studies and a,
2. Fermion pair production
This is a continuation of the Z lincshape study begun at LEP 1 00. Apart from testing SM
predictions the main interest is to determine belter the hadronic y/Z interference term
using off’peak points at which the cross section is much more sensitive to it. In a compleiely
model independent (5-niatrjx based) fit the value of highly correlated with/‘‘‘‘ Thus
including off-peak data in such a fit leads to the best model independent values ol
well as Note that the usual Breit-Wigner fits at LEP 100 assume the SM value Icn this
interference term.
Physics at LEP 200
517
Before describing the results 1 would like to point out that inspite of moving away
from the Z peak in centre-of-mass energy, a large proportion of the events at LEP200
energies still “remember” the Z. These are called "return to the Z' events and are due to
initial state radiation (ISR) in which the hard photon takes away just enough energy
to produce a Zas a recoil. Figure 2 shows the L3 distribution of the ’reduced’ or effective’
Figure 2. The reconsirucied effective centre-of-inass energy. >/T'. for the
selection of (a) -> hadrons (y ) events, (b) e*e‘ -> (y) events.
(c) c'^c" -> (y) events and (d) -+ c'^e” (y) events.
centre of mass energy, for the e + c hadrons, c^e-, Except for
the e^e” e'^e' final state, in which the t-channel is dominates, the Z is clearly
seen. An easy way to remove this background is to apply a cut on the value
oi V7,
Each of the LEP experiments collected -2000 events at 161 GeV and -•1000 events
at 130-140 GeV. Of these -40% are true high energy events with V7 > 0.85. Comparison
ot measured cross sections with the SM expectation is shown in Figure 3 for the s-channel
hnal states. All measured data (cross sections and lepton forward/backward asymmetries)
is in good agreement with the SM. A fit to all LEPIOO + LEP13(>-140 daU in the S-matrix
^^onnalism leads to
Mz* 91 193.6 ± 4.0 GeV
;h«d«. 0.21 ±0.20
( 1 )
( 2 )
518
A Gurtu
This value of/"* is -2a away from the SM value of +0.23. Inclusion in the fit of TOPAZ
data from KEK at VI = 58 GeV yields
Mz = 9ll9l.2±3.5 GeV,
= _ 0.07 ±0.16.
(3)
(4)
Figure 3. Leptonic cross section and forward -backward asymmetry measurement.s and
compari.son with .standard model
As pointed out earlier, there is a high correlation between these two parameters :
corr(A/ 2 ./“‘) = -78%.
*
3. Determination of W Mass
Pair production of W bosons at LEP became possible in summer 1996 when the LEP energy
was enhanced to 161.3 GeV, just above WW production threshold. At that time the world
average of Mw was 80.36 ± 0. 13 GeV from pp experiments at CERN and FNAL. Mw is
fundamental eleciroweak parameter and any improvement in its precision helps, firstly, m
testing the internal consistency of the SM and, secondly, in constraining the value of
within the SM framework.
During 1996 LEP operated just above WW threshold during summer, at 161 .3 GcV,
and at 172 GeV during fall.
3. 1 . Identification of WW signal :
IV pairs leading to the hadronic 45.6 %), semi-leptonic {qqlv{yj, 14.6% each) and
leptonic (/v/v(}), 10.6% total) final states were identified. Briefly the following selection
procedures were followed :
qq^<i(y) •
This is a purely hadronic final stale. The signal strength is = 1.6 and 5.5 pb at VI = 161 and
172 GeV respectively. The main background is due to QCD processes, e^c qqW'
whose cross section is = 150 pb.
Physics at LEP 200
519
The first step is to reject radiative “return to the Z* events, which are fairly easy to
reject by imposing an (s'/s) cut.
An example of a e*c“ jets event observed by the ALEPH
collaboration is shown in Figure 4. A typical identification procedure followed for this
final state is
• Selection of high multiplicity events without missing energy,
• Forcing of event to four jets,
• Imposition of energy -momentum conservation to carry out a 4C kinematic fit.
The residual (QCD) background is due to qq qq gluon bremsstrahlung 4 jets in
which
• the bremsstrahlung gluons tend to follow the parent quark direction
• they mainly have smaller energies relative to the four decay quarks coming from
W pair production.
This is removed either by use of multi-dimensional procedures by the ALEPH [2], L3 [3]
and OPAL [4] collaborations or, in the case of DELPHI [5], by the use of a single variable
constructed out of fitted energy and angle variables of the event.
ciqlv( y) final state :
The cross section for this final state at 161 and 172 GeV is = 0.5 and 2.5 pb
respectively and the main backgrounds are due to e'*’e~ -4 and 4-fermion processes,
e^e 'with one lepton undetected.
520
A Gurtu
An example of a e’^e~ -4 q^fiV -4 2 jets -f /i event seen by the OPAL collaboration
is shown in Figure 5.
The selection procedure is
• Identify hadronic event with one high energy, isolated lepton with e, // and r
tagging as at LEP I
• Quster the remaining event into 2 jets
• Determine the missing momentum vector due to the neutrino (p^)
• Apply selection cuts on the kinematics of the reconstructed 4- fermion
system:
- angles between lepton and jets
- magnitude and direction of missing energy
- energies of lepton and jets
- hadronic and leptonic invariant masses , Af ^ ]
/v/vf ^ final state :
The expected signal cross section at 161 and 172 GeV isr=0.4 and 1.9 pb respectively and
the main backgrounds are dilepton events from e^e~ Z(>). Bhabha scattering and events
due to 2 photon interactions.
An example of a e*c" -> c*|l’ event detected by the 13 collaboration is shown in
Figure 6. As is evident from the figure such events are rather easy to detect and select
owing to the presence of two very high energy leptons accompanied by large missing
transverse energy and acoplanarity between the lepton directions.
The selection strategy is then to
• Exclude hadronic events using the multiplicity criterion
• Identify 2 leptons using the e, and T tagging as at LEP I
• Apply selection cuts on
- acoplanarity angle between the 2 leptons
- missing transverse momentum (or energy) in the event
Extraction of WW Cross sections :
For a selected number of events ^ of a particular final state, the cross section is written as
a*
e.A.ji
(5)
where is the number of expected background events, € is the signal selection
efficiency, A the detector acceptance and ^ the integrated luipinosity. The values of € and
A^bfd tiepend on the Monte Carlo programs used for signal and background event generation
leading to systematic errors in addition to the statistical error. Some sources of systematic
OTor are : variation of selection cuts around nominal value, model parameter variation—
Physics at LEP 200
Plate I
Fiuure 5. Example of an OPAL qq^v
Physics at LEP 200
521
signal & backgrounds, model to model variation, IV mass dependence, differences between
data and Monte Carlo and limited Monte Carlo statistics. At present, the overall errors are
dominated by statistics.
Run# 667108 ivMii 606 Vs ■ 172.3 QeV
Figure 6.
The physical cross sections one is interested in arc those corresponding to ete”
i.e., where W pair production takes place, the so-called CC03 processes. On the
other hand the final states that one detects can some times arise from other SM processes.
Most of the events due to these background diagrams are rejected by suitable invariant mass
cuts. The residual background is corrected using Monte Carlo programs which can generate
both signal and background events given the cuts applied. Typically the correction factors
are around 10%.
Summary of selections :
A summary of the selection efficiencies (e) and the numbers of events selected (Nevi) by
each of the 4 LEP experiments is given below (Table 1 ) :
Table 1. Selection efricicney and numbers of e'*'e~ W*'W“ candidate events
at 161 and 172 GeV.
Final
State
161 GeV
172 GeV
e
£
-60
9-15
75-85%
5.5-65
qqlviYt
60-80%
11-16
60 - 90%
40-50
/v/v()^
40-70%
2-6
45 - 80%
5-10
Total
22-36
95-120
522
A Gurtu
Thus all the 4 LEP experiments together detected -100 WW events at 161 OcV,
-400 WW events at 172 GeV, leading to a total 1996 event sample of -500 events.
3.2. W mass using the threshold method :
This method consists of measuring the W pair production cross section, a^nrw* just above
threshold and determining Mw using the dependence of Oww on It can be shown that
the maximum statistical sensitivity occurs at VJ *2x +0.5 GeV, i.e., just above 161
GeV and that is why LEP was run at 161 .3 GeV. The values of -+ cross sections
in various final states at 161.3 GeV are summarised in Table 2.
Table 2. c'^c~ cross sections.
Final Resulu at 161 GeV
Final
CC03 Cross Section (pb)
ALEPH
DELPHI
L3
OPAL
qqeviii
0.62
qqpvil^
qqTv{)i
0.22
qqlv(Y)
l,8S^|{±.06
I.77:SS±.10
/v/v(y)
0.68«;’J±.03
0.3l2jJ±.09
0.39t},’
qqqq^i^
1.80 ±0.50 ±019
1.56:*]’ ±.13
0.98
Total
4.23 ±0.73 ±0.19
3.67 tS ±.19
3.625JJ±.16
In order to obtain a combined LEP average cross section each of the four LEP
experiments provided its value of 0(e'^e“ -> W*'W~) with a symmetrical statistical error
based upon the number of expected events from SM predictions. The common systematic
error was taken as the smallest experimental systematic error of the four. The average LEP
161 GaV
UftSpb.
3.^±l».45pb
commam 1.14
Figure?.
cross section for the process c'^e" W^W at 161.3 GeV was determined to be 3.69 ± 0-^^
pb; the error includes a common systematic of 0.14 pb. This agrees very well with the SM
Physics at LEP 200 ^^3
prccJiclion of 3.80 pb [6]. The measuremcnis and ihc average arc shown in Figure 7. The
S%J based dependence of Oww on and ihe value of the LEP average so derived is
shown in Figure 8. Figure 9 depicts the individual determinations oi M^.
ALRTII
OCLTHI
U
OPAL
LIP
N MJ II iij
"wICtVI
••.i4S;g.v
■•■i»3nciv
NACtojaciv
■Mwn UTG«V
Ki|i>uri‘ H. »nv from avv^ivat 161 GeV Figure 9. LEP 161 GeV VV mass
. 1 .1 \V' mass from reconstruction method :
In principle the procedure is simple after WW identilication has been made.
• Calculate jet-jei, lepton-neutrino invariant masses
fov ijijlv (/ ) channels life is simpler : no combinatorics and small background
under the signal in plot.
• Apply beam energy constraints to improve reconstructed mass resolution.
This results m a 4C fit lor qqqq. a IC fit for qqlviy).
• Application of the beam energy constraint leads to an anti-correlation between
the 2 reconstructed W masses. To take care of this effect one
- either, sets A/wi = leading to a 5C fit for qqqq and a 2C fit lor qqlviy)
- or. studies the fitted Mwi - correlation in MC and applies a correction
• U.^c a Brcii-Wigncr plus parametrized (or actual) background and fit lor and
possibly (additionally)
’ / Sources of systematic enors on M\x :
The systematic errors on Mw using the reconstruction method are given below.
• The use of beam energy constraint to improve mass resolution leads to t\Kv»
sources of systematic error
I- A LEP energy uncertainty. A£lep — MeV. leads to a mass uncertainty ol
similar magnitude.
72A(6)-9
524
A Curru
2. Iniiial Suic Rudiulion decreases (he elTective Va. Thus using (he nominul
value of V!y resulis in an increased Afw Modelling uncerluiniies ol'lSR |ca(j
lo an error on its average value of 2 : 10 MeV leading to a similar uncertainty
in the fitted
9 Modelling QCD background under the signal : the background also peaks just
under the peak. Very detailed studies still being made.
• Detector elTects : miscalibration of energy of leptons and mismatch between
M.C. and data lor energies/ angles of jets.
• Fit type dependence :
- Relativistic vs non-rel BW,
- different parametrisation for backgrounds,
- variations in fitting procedures (4C. 1C vs 5C, 2C)
These elTecis total to :=3(>-5() MeV systematic error on Mw
The main problem at present is one ol low statistics. This makes it dilTiculi to
di.sentanglc statistical from .systematic elfccts.
J 2. T/ieorencal syMenuitics in qqqq :
Owing to the short lilciimc of the W bosons, to .start with the 4 decay quarks arc in close
MCinity in the qqqq Imal slate. Thus “colour reconnection'’, due to the possible gluon
exchange between quarks Irom the decay of the two dilTcrent W^s, leads to a di.sloriion ol
the reconstructed W mas.ses. In principle if one could calculate this dhstortion theoretically
then a suitable correction could be applied. Unfortunately the prc.scntly available models
give divergent resulis on this correction and this uncertainty is translated into a theoretical
systematic error on Mw determined using (he qqqq final slate.
Another similar cnccl of distortion of the reconstructed W mass distribution could be
due to Bosc-Einsiem correlations between identical bosons 7f) produced as decay
products ol the two Ws because the hadromsation regions of the W's overlap. Here again a
good theoretical understanding of this problem is lacking.
The overall theoretical uncertainty in determination due to both these ellecis is
estimated to be -100 MeV in the qqqq final .stale [7|. For a rc.sult combining roughly equal
luiinhcrs ol qqqq .md qqlv[ events the uncertainty will be -50 MeV,
4 f-'niiiie pwspci !\ for W mass ai LLP :
In (he short term, the 172 GeV data will be analysed and results presented by the time ol the
European Winter Conlercnccs in March 1997*. In the long term each experiment at LEP
expects to collect -5(M) pb ' ol data. II the colour reconnection and Bosc-Einsicin clicch
... ol Jul> Ihf .iseiaiic LEH ul .V/^ Iroiii 172 GeV dai;i is HO 62 ± 0,26 GeV Irom ihc
chaniii l HU 46 ± 0 24 GeV Iroir the 1 channel, and averaging wilh Irom 161 CcV. the ovcral
.v/^^ = HU4H ± (J,I4 GeV This agrees well wiili (he laicsi resulis Irom the ffp expennicnis M0.4I
leading 10 a grand woild a\erage ol HU 43 ± U UK CcV.
Physics at LEP 200
525
iire broughl under control theoretically, then using all final stales the linal error on
I'roin LEP is expected to be os low as -35 McV. On the other hand, if these elTeeis remain
un-undcrslood then one may not be able to use the qqq(^ y ) final state and the error may
remain -45 MeV based only upon the ) channels.
4 , The A LEPH 4- jet events
While searching for a possible e + e- hA bbbh (4 jet) signal in 5.7 pb'' data at 130
and 136 GeV, the Alcph collaboration observed an excess of 4-jei events and an
enhancement in the sum of the two di-jcl masses around 105 GeV [81, Their selection
was tailored to minimize SM backgrounds and preserve efficiency for an l\A signal Ibr
Ml, = = 55 GeV :
I . Hadronic final state requiring > 8; ^ 10% Vi
2 Reject radiative return to the Z events
3 Cluster event to 4 jets
4 Require each jet to have M^^ > I GeV as.suming charged particles are ;r-. ncuiials are
massless
5 To reduce QCD background, require all di-jei masses > 25 GeV. the sum ol masses
of the two lightest jets, M\ M 4 , > 10 GeV and the sum of their charged
multiplicities. > 10 .
They determine the signal selection efficiency to be 42‘/( and their backgiound
iL'Icciion iq be better than 99.5%. They then select that pairing of jets which minimises the
dillcrcncc in mass between the two di-jcis. When they plot the sum of the masses ol these
iwo di-jets they observe a clear peak at the expected value of 1 10 GeV in the signal Monte
Cailo sample. The width of the peak is 1.6 GeV This is depicted in Higuie 10. Applying the
Ftfiurc 10. Sum of masses ol the iwo di-jeis
in selceied ALEPH 4-jei events using
Monte Carlo Sec text for details
W 80 too 120 l«0 160
U (Wv/e*)
cuts to the data the plot shown in Figure 1 1 is obtained. A [vak at lv)5 GeV i^
'‘hsorved with 9 events contained within two I GeV bins. The expected background is only
526
A Gunu
I event. If this was indeed the searched for signal, i.e.j e^e“ HA hhhE, then these
events should be rich in b quarks. However, at most one event is lound compatible wiih
having 2 jets due to b quarks using the lifetime lagging algorithms.
o
i
M
I
l*J
3 h
2 h
NX1 Standard Proceisai
O Data
nn n
60
D n
BO
100
120
ALEPH
140 160
EM (C«V/c*)
Figure 11. Sum ol musses of the rwn
di-jeis in selected ALEPH 4 -|lm
events using their data sample See
text for details
Since that publication ALEPH has continued to see this enhanccmeiu, aibcii
with smaller statistical significance [9| at the higher energies at which LLP has lun
ALEPH
PiihIkliMl Mnahrsfa q q : YES
and WWrekctlQO WW; YES
zr: VES
140 110
■«(ao»/0
1995 1 1996 ttoi 3
I
34 obaerved ]
24.S expected
Peak : 18 observed
3.1 expected
A111996.daa
18 observed
17.7 expected
Peak . 8 observed
2.1 expected
iMiaw/t*)
Figure 12. Extended ALEPH analysis on 4-icl events including I99.S+I996 daiu,
Physics at LEP 200
527
161 and 172 GeV. At these energies they have introduced some additional cuts in order to
remove the "background" of WW events. Including all 1995 and 1996 data they now
observe 34 4-jel events whereas they expect to see 24.5. The number observed in the
peak is IK with a background of 3.1. This is shown in Figure 12. A gaussian fit to the data
yields the peak position to be 106. 1 ± 0.8 GeV with a width of 2.1 ± 0.4 GeV.
The other three LEP experiments have .searched for the ALEPH type events
following closely the selection criteria- used by ALEPH. None of them finds any
enhancement either in the number of 4-jct events or in the distribution of sum of masses of
ihc iwo di-jeis | lOJ. As a cross check for possible detector resolution effects etc., ALEPH
provided the four-vectors of their events to the other LEP collaborations who propagated
these through their detectors and confirmed that 65-70% of these events actually ended up
in a similar peak. This is a similar percentage as what ALEPH themselves found for their
own events. Thus, the detector resolutions and other effects cannot account for the fact that
other LEP experiments don’t sec these events. With accumulation of more data at higher
energies one will sec if the effect persists or fades away.
5. Search for HiRg.s and SUSY
When an accelerator progresses into a higher energy regime it is always a time of great
excitement to liH)k for particles which are expected and not yet discovered (SM Higgs) and
lor particles which are theoretically favoured to exi.st particularly if earlier data hints at such
Figure 13. OPAL search for siundard model
Higgs.
a possibility. The latter was the ease for light charginos (SUSY) owing to the existence of
the K,, anomaly.
5 . 1 SM HififiS :
While the LEP runs at 130 and 1.36 GeV were of too low luminosity (-5 plv') to provide
any improvement over the limit set by LLPMH). the OPAL collalMiralion has lieen i|uiek to
use their 161 GeV data to obtain a new lower limit on the mass of the SM Higgs boson of
h.^i CieV III). Figure 1 3 shows the OPAL result’.
.IS III July I'MJ? ihc bcsl lowci liiiiil on IIk‘ .SM Higgs i\ hoiii Al.LI'H 1 121 ' 70,7 la-V :il confidence level
||sln^: daia lU uH LEP energies (ineludmg 172 l leV)
528
A Gurtu
5.2. Search for SUSY :
5.2. 1. Rh anomaly and light charginos :
Summer 1995 saw the height of the Rb anomaly. The experimental value of this
ratio of in Z decay was 0.2205 ± 0.0017 even after fixing Re to its SM
value of 0.1715. Thus it was 3 <t away from the SM expectation of 0.2156. This
disagreement provided fertile ground for theorists to suggest that such a situation
may be naturally explained within the framework of the MSSM (Minimal Super
Symmetric extension of the SM). In the low tan /3 scenario, the Z -> ► 6F with a
light providing the tx^b triangle at the Z® -> bB vertex would do the trick and in
the high tan^ scenario a light A® at the Z® bE vertex would have the same effect.
The former (low tan^ scenario) is preferred as it leaves completely untouched the
SM prediction of . For a light chargino mass. ^ 65 GeV. one could obtain a value
of Rb = 0.219 within the MSSM framework which was 1. 5a away from the measured
value.
Experimental signatures :
The basic assumptions which have gone into the mainstream LEP searches are
(i) R-parily conservation which ensures that the Lightest Supersymmetric
Particles (LSP) will not interact or decay and will escape detection. The
lightest neutralino, is favoured to be the LSP. This assumption leads to a
very powerful experimental signature : that of missing energy (t).
(ii) That the sneutrino, v, is heavy and the charginos are Higgsino-like. This ensures
large production cross sections for e'^c” X\ X\ •
(iii) That the decay followed is xt -» + IV* . where W -*ff' are the usual W
decay modes.
Thus, in addition to the ^ signature due to two undetected X\ fhe three topologies
one looks for are
(i) an acoplanar lepton pair with opposite sign leptons.
(ii) a highly unbalanced hadronic event,
(iii) an isolated high energy lepton accompanied by 2 jets.
The main backgrounds are e+e- -» Zir' Z/r* or WVT of Wevai Zee or ZJY a
two photon interactions. Suitable selections reduce the backgrounds very effectively. The
signal efficiency varies between 5% and 60% depending on the mass of the xf ^ ‘he
mass difference, AM(jj'), between and^J*.
Interpretation within the MSSM framework is done in terms of the five
imxlel parameters tan^, the gaugino mass parameter, M u the higgsino mixing
529
Physics at LEP 200
pftramcter, /i, the sparttcle mass parameter,
sectorM.
'Wo, and the trilinear coupling in ihe Higgs
Modd ^ MSSM ^
L3
Upper Limit on Chaiylno Cidm Section
MSSM Piramtaf Space
Ffpm chMplMand iMuir^ MMch;
Figure 14. Chorgino search by L3 and limits on MSSM parameier space.
^iX) = 20-60 GeV and set a lower limit A/-, Z 84 GeV at 95% C.L. assuming u
sneutrino mass > 200 GeV^
5.2.2. Searches for sleptons, neutralinos, stop, sbottom :
As expected all the LEP experiments have carried out extensive searches lor all
these SUSY particles. As mentioned above the dominant global signature is one ol
niissing energy (^). For slepton search the event signature is a pair of acoplanar oppositely
charged leptons with large i and f. For neutralino search one assumes a pair production
of the lightest, X\^ with the next heavier, with the latter decaying as
/ being a normal fermion. Again the search limits depend upon the mass of the searched
particle and the difference in mass between it and the LSP. To cut a long story short no
SUSY signal has been discovered. Model independent as well as MSSM bu.scd limits
^10 complete the stoiy on the anomaly, much of the problem has lost its urgency as the experinicnial value is
now 0.2179 ± 0.012 which ii leu than 2fffrom the SM expectation.
530
A Gurtu
and plois arc obtained by the LEP collaborations [13] which may be referred by the
interested reader.
OPAL
CroM-fction Umite
MSSMMassUmlts
» m ^ m .BD -y
mttj) (Q#V)
) > ^-5 QeV mo > 1 TeV
fn(jef ) > 62.0 GeV mo minimal
9 95% c.l.
assuming AM > 10 GaV, tan/? = 1.5
Figure 15. OPAL results on Chargino and neulnilino searches
6. QCD studies at LEP
Not to belabour the point, the status of QCD at LEP above Z energies continues to be
satisfactory. Variation writh centre-of-mass energy of two important event shape
quantities, the thrust (< T >) and the average charged multiplicity (< njh >) is shown
in Figure 16. The same figure also shows the fraction of 2, 3, 4 and 5 jets as a
function of at 161 GeV. As one can see QCD models describe the data
very well.
Finally, I say a few woids on the continued evolution of the strong coupling constant,
a,. For example, the value mea.sured by L3 at 161 and 172. GeV is 0. 103 ± 0.005 ± 0.005
and 0. 104 ± 0.006 ± 0,005 respectively 1 14|. The variation of o, with centre-of-rhass energy
Physics at LEP 200
531
or Q is shown in Figure 17. As is evident, the dau is well described by the expected QCD
evolution.
Thrurt, Multiplicity and Jet Rifs
F|Mr?I^QCPgiidlC» ■11^200. See icxt for
f Eiwfi^:;E>«oliitlon
Figure 17. Energy evoloiion of and cowp a rieon with QCD pndiciion.
Rtferwce,
[I] Standard Model Processes : E Accomando et al, CERN Yellow Report ^-01. 19 Feb 1996, Vol 1
P 207 (1996)
ALEPH CoUabomikm : R Barate et at. Pre pri nt CERN-PPE/97-25, 4 March 1997, Submitted to
Un. B.
532
A Gurtu
1^1 L3 Collaboration ; M Acciarri ti al. Priprint CERN-PPE/97-14, 3 February 1997, Subnutied lo
Phyx. Lett. B.
(4] OPAL CoUabomiion . K AckeniafT et al, Phys. Lett. B389 416 ( 1996)
[5] DELPHI Collabonttim : P Abreu etal. Phys. Lett. B397 198 (1997)
(61 GEHrLE/4fan Versiim 2 : D Bardin et ai, CERN Yellow Report 964)1. 19 Feb 1996, Vpl 2 p 26
[7] Determination of the Mass of the W boson ) A Ballestien) et at, CERN Yellow Report 96-01, 19 Feb
1996, VollpUl
[8] ALEPH Collaboration ; D Buskulic et al. Z Phys. C71 179 (1996)
[9] ALEPH Collaboration ; F Rugusa ei al. Presentation at LEPC, November 19. 1996
[ 10] L3 Collaboration : M Pohl et al, Presentation at October 8. 1996;
OPAL Collaboration : N Watson et al. Presentation al LEPC. October 8, 19%
1 1 1 1 OPAL Collaboration : K Ackerstaff et al, preprint CERN-PPE/96-161 , November 1 8. 19%, submitted to
Ph\.x. Lett B
[12] ALEPH Collaboration : R Banite et al, preprint CERN-PPE/97-070, June 17, 1997, submitted to '
Phy.K Lett. B
[13] ALEPH Collaboration ■ R Barate et al. Searches for Scalar Top and Scalar Bottom Quarks at LEP2.
CERN-PPE/97-084, July 17, 1997. submitted to Phys. Lett. 0.; R Barate et al. Search for Sleptons in
e^e~ coUmons at centre-of-mass energies of l6l and 172 CeV, CERN-PPE/97-056, May 27, 1997,
submitted to Phys. Lett B.
DELPHI Collaboration . P Abreu et al. Search for Neutralinos, Scalar Leptons and Scalar Quarks m
e*e~ interactions atyfs - 130 and 136 CeV. Phys. Lett. B387 651 (19%); P Abreu efd/. Search for the
Lightest Cliargmo at V7 = 130 and 136 GeV Phys. Lett. B382 323 (19%)
U Collaboration M Acciam et al. Syarch for R-parity breaking sneutrirw exchange at LEP. CERN-
PPE/97-099. July 28. 1997, submitted to Phys. Lett. 0.; M Acciam et al. Search for Supirsynwietric
Particles at 130 GeV<{s< 140 GeV at LEP. Phys. Lett. B3T7 289 (19%)
OPAL Collaboration . K Ackerstaff et al. Search for Scalar Top and Scalar Bottom Quarks Using {he
OPAL Detector at LEP. CERN-PPE/%- 133, Submitted to Phys. Lett. 0.; K Ackerstaff et al. Search for
Cluirgino and Neutralino Production in e*e~ collisions af = 161 GeV. CERN-PPE/96- 1 35, Submitted
to Phvs Lett B . K Ackerstaff et al. Search for Charged Scalar Leptons Using the OPAL Detector at
= 161 GeV. CERN-PPE/96-182, Submitted to Phys. Lett. B.
( I4j U Collaboration M Acciarri et al. CERN-PPE/97-042, April 25. 1997, Submitted to Phys. Lett. B
Indian J. Phys. 72A (6), 533-545 (1998)
UP A
— an intemadonal journal
New physics at e+e- colliders
Saurabh D Rindani
Theory Group, Physical Research Laboratory, Navrangpura,
Ahinedabad'380 000, Gujarat, India
Abstract : Possibilities of observing new physics, i.c . of observing new particles, or
unexplored properties of known particles, at future electron-positron colliders arc reviewed
Some general properties of linear colliders are reviewed first The main lopics covered under
new physics are measurements of anomalous gauge-boson couplings and of various properties of
the top quark
Keywords : Electron -positron collisions, lineai colliders, clcctrowcak gauge bosons,
top quark
PACS Nos. : 1 3 90 +i, 1 2.60 . 1 4 70 , 1 4 65 Ha
I. Introciuction
In tills talk. 1 will review ihc possibililies of observing “new physics", i.e. of observing new
panicles, or unexplored properties of known panicles, ai future colliders. 1 will dwell
on signaluics of new physics, rather than discuss origins ol new physics in any detail.
Moi cover, due to the limited lime available, I will mainly concentrate on gauge boson and
lop quark properties.
The c^c' colliders presently operational at high energies (at or above theZ mass) are
SI (' (Stanford Linear Collider) at SLAC, Stanford, USA, and LEP (Large Electron Positron
Cullidci l at CERN, Geneva, Swii/erland, with LEP being in the higher energy (LEP2)
piKisc ( I6i GeV and alx)vc) in recent times, planned to reach 190 GeV. The next generation
•'I c\' colliders, which would be of the linear type in the centre-of-mass (cm) energy range
ol 3(K) GeV and above, have been discussed with regard to their feasibility, characteristics,
and physics capabilities for quite some time now (I “3]. Possible locations
^‘ nsidcred are at SLAC (Next Linear Collider, or NLC) DESY (TESLA and the S-Band
i incar Collider, or SBLC), KEK (Japan Linear Collider, or JLC), CERN (CERN Linear
f '^llidci. orCLlC) and Budker Institute, Proivino/Novosihirsk (VLEPP)'. Also considered
"'1^ uilk. ihc lerm NLC will refer lo any one of these, and not nccess.'irily the
Pu.poscd f«i SLAC,
‘ "'■111 Sauiahh^'prl.crnet m
© 1998 lACS
534
Saurabh D Rindani
are options like e~e~,Ye and yy colliders. The photon beams of high energy and intensity are
proposed to be obtained by back-scattering of high energy electrons by low-energy photons
obtained from an intense laser beam [4].
The advantage of e'^e~ colliders over hadronic colliders is mainly in the cleaner
environment. By using leptonic initial states, electroweak interactions are more
conveniently studied because there would be no spectator jets which arise in the case of
hadronic colliders. A fewer number of kinematic cuts to suppress backgrounds are needed
because of the cleaner environment, and thus the effective luminosity is better than at
hadronic colliders. Moreover, theoretical uncertainties due to partonic distribution functions
arc also avoided.
Despite the spectacular success of the standard model (SM), there are still some
outstanding questions, which future experiments can help to answer. One of the questions is
regarding the mechanism of electroweak symmetry breaking. If it is the orthodox Higgs
mechanism, the Higgs particle must be found. In that case experiments can determine its
mass, its CP properties, and its couplings. In particular, the. couplings should be
proportional to the mass of the particles the Higgs couples to. If the symmetry is broken by
some dynamical mechanism without explicit scalars, signatures of this mechanism
should be revealed by experiments. For example, new resonances are predicted in
technicolour models. In any case, the top mass being close to the Fermi scale,
electroweak properties of the top quark may give important clues to the symmetry
»
breaking mechanism.
A related issue is the strength and nature of gauge-boson interactions. If there is no
Higgs with mass below about 1 TeV, gauge-boson interactions would become strong, with
new non-pcrlurbative effects. Even if the interactions arc weak, nonstandard effects like the
presence of heavy particles or compositeness could alter the nature and magnitudes of the
triple and quanic couplings of gauge bosons from those predicted by SM. Presently these
are measured at the pp collider at Tevatron with large errors. It will be the task of future
colliders to improve upon this accuracy.
Extensions of SM which have been widely considered are grand unification,
supersymmetry and technicolour. All these predict new particles, which under certain
circumstances may be in the accessible range of e'*'e" accelerators in the range of
500 GeV - 2 TeV.
2. The Physics posiibHitics
We summarize below a possible physics programme for a future linear e*e' collider. While
it will not be possible in this talk to go into the details of all the topics included in this
summary, the topics of new top-quark physics and electroweak gauge boson couplings will
be dealt with at greater length later on.
New physics at e'^e~ colliders
535
(i) Top properties :
The cross section for e+e' tt increases rapidly just above threshold, and a threshold scan
can be used to measure the top quark mass up to an accuracy of Am, < 500 MeV. The
couplings of the gauge bosons (y, Z, g) to tf, including anomalous magnetic and electric
dipole couplings (together with their weak and colour counterparts) could be measured with
good accuracy in e'^c~-^ tt(g). Similarly, the Yukawa coupling ttH can be measured
directly in e'^e’ ttH. In the decays of t and i produced in e^e" collisions, the chirality of
the fh charged current can be tested.
(m) TestofQCD:
The running of the strong QCD coupling a, {q^) can be measured at higher energies and
compared with theoretical extrapolations from lower energies. The nature and magnitude of
the gluon couplings to ft and to other gluons can be investigated.
(lit) Electroweak gciuffe bosons :
Tuple and quarlic couplings of the electroweak gauge bosons can be studied wilh
great accuracy in a number of production processes, principally, e^e" — » V^VT. Masses
and couplings of a new gauge boson Z' occurring in extensions of SM can be studied in
.// (/ stands for a fermion), with // arising from a real Z', if light, or from a virtual
y.Z.Z' 151.
f/rj Higgs boson :
Higgs particles with masses uplo 2()0 GeV would be accessible for 4s = 500 GeV through
(he reaction c^c ZH, e'^c -> vvH , etc. Once discovered, the mass, CP properties and
eiHiplmgs of the Higgs can be determined [6].
( » ) Supersymmetry :
Supersymmetry, needed to stabilire the light scalar mass in the presence of a hierarchy of
scales as in grand unified theories, predicts a rich spectrum of new particles. The extended
Higgs sector and the supersymmetric partners can be studied for a wide range of masses and
uihor parameters.
fni Additional fermions :
Charged and neutral fermions predicted in extensions of SM could be produced in pairs, or
ill association with ordinary fermions. A range of masses between 4s /2 and 4s can be
IMobed, depending on the production mechanism.
3. Characteristics of the colliders
To avoid prohibitive losses of energy due to synchroton radiation the circular colliding<ring
has to be discarded for e^e” colliders beyond LEP2. The high energy colliders will
to be linear colliders.
536
Saurahh D Rindani
It is expected that the linear e^e" colliders will be realized in two phases. The first
phase will cover the cm energy range from LEP2 energy to 500 GeV. In the second phase,
the energy will be moved up to I to 2 TcV. The luminosity at V? = 500 GeV would be of
the order of 10-^^ cm~- sec”'.
Cross sections would be of the order of a (e^c" -4 » 500 /b at = 500 GeV.
At a luminosity of V = 10^^ cm”^ sec'*, for a running time of 10^ sec (l/3 of a year), the
integrated luminosity would be lndt = 10 fb~\ which is equivalent to 5000 iffT pairs. For
higher energies, the luminosity must be scaled up as the square of the energy to keep up the
same production rates.
A high luminosity is achieved by squeezing and e' into bunches of extremely
small dimensions. As a result, large electromagnetic fields arise, which acting on an
individual c' or e'^ as it traverses a colliding bunch, bends its trajectory. Thus large amonui
of radiation is emitted during the crossing of bunches, and the effect is known as
■ beamstrahlung”. This not only results in loss of cm energy, it also implies that the initial
sharp spectrum is smeared. Moreover, radiated photons produce spurious events, some of
which could also be hadronic. Thus the cleanness of (he e'^e" collider could easily be
destroyed (7).
In narrow-band beam designs, the effects can be reduced to the level of \%. Also,
the hadronic events produced by photons are of the same order as those induced by ordinary
bremstrahlung. The beamstrahlung photons would also produce background C^^e* j^airs,
concentrated in cones of half-angle of about 10^’ around the beam pipe.
Longitudinal polarization of e" is possible at linear colliders. For example, SLC
usTS strained Ga-As cathodes to polarize electrons, which are then accelerated without
loss of polarization. A high degree of polarization can be achieved, exemplified by -80%
polarization at SLC. Polarization of e"^ is not so easy; proposals for it do exist, however
This is in contrast to circular colliders, where transverse polarization is more natural, and
longitudinal polarization is difficult to achieve. The longitudinal polarization of the electron
beams would be useful in discriminating between different types of couplings of quarks
and gauge bosons, as well as in improving the .sensitivity of experiments to certain
anomalous couplings.
4. Anomalous gauge boson couplings
Although the standard electroweak model has been verified in recent years at LEP and SLC
to a high degree of preceision, non- Abelian self-couplings of weak vector gauge bosons
haNC not been tested directly with significant precision. Tevatron results from two-gauge-
bo.son production have not yet reached a precision better than order unity. Ongoing
measurements at LEP2, future measurements at an upgraded Tevatron and at LHC will
improve upon this precision considerably, but cannot match the expected precision ol a
5(H) GeV NLC. much less that of a I TcV or 1 .5 TeV NLC. '
New physics at colliders
537
There exist indirect constraints on anomalous couplings from precision
measurements at the Z resonance, arising from gauge bosons in the loop. But the
calculation of these diagrams suffers from ambiguities. The anomalous couplings
could arise, for example, due to unexpected contribution of new particle propagators
in loops.
4. 1 . Parametriwion of triple gauge boson couplings :
An effective Lagrangian for the WWV (V =Z, y) vertex is written as [ 8 ]
i'wwv/gwwv =ig^{wl^Wi‘V'' -WlV^W»^) + iKvWlW,V>‘''
Mw
+ + iyWlW.Vf'’
( 1 )
Here ^ - d,W ^ , V - dyV ^ and , The
normalization factors are ” “i? '^he couplings include 3 CP-
violating ones ; ^ v. -ind one CP even but C and P violating coupling . In most
studies only the 3 CP even as well as P even couplings are considered.
10 °
10 '"
^Ky
10 ^
10 '^
10 '^
Ttv.
(a^
iLEPIll
180
LHC
|NLC
15001
Figure 1. Companson of limiUi on ononiatous inple gauge-boson couplings at various colliders,
from ref. [10]
In SM at tree level, g,'' =Ky =1. Ay ==^4 = 5 ^' =0- The couplings
should actually be written as form factors with momentum dependent values. However,
for a process like e*e" where the W*, VV“ and the virtual photon and Z always
Have the same momenta, the form factors have fixed values. The couplings for q- = 0.
538
Saurabh D Rindani
where q is the moinentum of the virtual photon, are related to static properties of the
IV as follows :
W electric charge :
W magnetic dipole moment :
W electric quadrupole moment :
=0) = I
Qw = ^
A particular form of effective Lagrangian which is more restrictive than the most
general one possible was considered by Hagiwara et al [9], which is known as the HISZ
scenario, after the initials of the authors. This Lagrangian is the linear effective Lagrangian
in which the coupling of gauge bosons is obtained by gauging an effective Lagrangian for
new physics which is invariant under SU(2)/^ x U(l) x SlJ(3)c. with the further restriction
of equal couplings for SU(2) and U(l) terms.
c^e" at NLC can be used to test the HISZ hypothesis by determining the
Y and Z couplings independently.
4. 2 P resent nieas u rements :
Al Tcvairon, so far a few events have been observed for WW and WZ production and ij'' (10)
events for W /production. The.sc arc consistent with SM. The.se can be u.sed to obtain limits
Figure 2. Feyninann diagrams tor the process W^VT
on corrections to the gauge bo.son couplings. These limits are of the order of unity. For
example, the DO collaboration has obtained the 95% C.L. limits of -1.8 < tsKy < 19
(assuming Ay= 0), and -0.6 < Ay < 0.6 (assuming AiCy = 0) [10]. Here the parameter A used
in the parameiriyaiion of the form factors is assumed to be 1 TeV. After the main injector
upgrade, Tevatron will collect l-lOy/r'. With an integrated luminosity of \0fb the
limits will be compciciive with those from LEP2. Al LEP2, with ji'dt a 500 pfr"', 95% C.L.
limits of the order of 0.1 are expected on the anomalous couplings, considered one at a
lime. Al the present time there are already some results from LEP2 available. Howcvei the
limits arc as yet poor.
New physics at e^e- colliders
539
When LHC goes into action, its higher cm energy will result in considerable
improvement of accuracy. For example, with an integrated luminosity of lOOyZ?-', limits of
the order of 5-10 x 10"*^ are expected to be obtained.
The limits that would be obtained from various colliders, including NLC, are shown
in Figure I. taken from [10].
4, 3. Measurement at NLC :
4.3.1.
The process e'*’e“ — > is the simplest process involving the triple vector couplings. The
amplitude gets contribution from three diagrams shown in Figure 2. Of these the first two
can gel extra contributions from anomalous WV/V couplings, whereas the third one gives
[he same contribution as in SM.
Due to the absence of spectator partons, W pair events can be reconstructed better at
NLC than at hadron colliders. To a good approximation, full energy and momentum
conservation can be applied to the visible final states.
An e’^e"— > VAW event can be characterized by 5 angles : The production angle 0 of
ihc W' with respect to the electron beam, the polar and azimuthal angles and 0* of one
daughter of the W" in the W" decay frame, and corresponding decay angles B * and 0 * of
one ot the W* daughters. (In practice, initial-state photon radiation and final-state photon
and gluon radiation complicate the picture, as does the finite width of the W).
At high energies, e^e‘-> W^W is dominated by the r-channel V', exchange, leading
primarily to very forward W' s. This makes a majority of the events difficult to observe.
Figure 3. Angular disiribulkin of W pairs wiih differeni polari^anon
combinations in . L R and I denote lefi handed, nght-handed,
and longitudinal polarizations. The differential cross sections are given in units
of R at = I TeV. This figure is taken from [3]
However, the amplitudes affected by anomalous couplings are not forward peaked. The
central and backward W*s are measurably altered in number and heheily by these
72A(6).[|
540
Saurabh D Rindani
couplings. W'helicity analysis through the decay angular distributions can be used to probe
them. Figure 3 shows the angular distributions of W pairs of various polarization
combinations.
The most powerful channel is the one in which one IV decays leptonically and the
other hadromcally. The branching ratio for this is about 30%. With this channel, full
momemtUTn reconstruction is possible. Although the branching ratio for a totally hadronic
channel is larger, discrimination power is lost because of the inability to tag fully the charge
of the quarks. The purely leptonic channel has branching ratio of about 0.05, and suffers
from kinematic ambiguities due to two undetected neutrinos.
Initial-state radiation and finite W width leads to some degradation, particularly
when imposing cuts to suppress far-off-shell events and low effective cm energy events.
A comparison of the capabilities of LEP2 and NLC in measuring the anomalous
gauge couplings AKy and Ay in the HISZ scenario is shown in Figure 4. Figure 5 shows
simultaneous limits on y and Z couplings at NLC. These are taken from f 1 1 ].
-0 OCW 0 CvV ' "COJ 0 004 0 004
u,
Figure 4. 9S% C. L. contour in ihc HISZ scenario ¥i%un 5. 95i% C L conioun lor siimiluiiieous Ins
The outer contour in (a) is for v7 = 190 GeV and ui V7 = 500 GeV and 80 /tr'
0.5 //>"’. The inner contour in (a) and the outer
contour in (b) is for V7 = 500 GeV with 80 fb~^
The inner contour in (b) is for V7 = 1 .5 ToV with
190 /fc-'
In general, precision at NLC is 0 i'or^ = 500 GeV, and 0 (few x 10"*)
for V7 = 1.5 TeV. Electron beam polarization helps to disentangle couplings and check
HISZ.
The possibility of studying CP violation in the process c+e" -4 has been
studied by Chang etal [12], Mani et a/ [13] and Spanos and Stirling [14).
New physics at e*e- colliders
541
4.J.2. Other reactions at NLC :
Various other processes have been considered, which have different relative importances at
dillerent values of yfs. Particularly important are the ones with one massive gauge boson
production :
e"^e~ e* V VV*",
(2)
e'^e" e^e‘ Z,
( 3 )
c + c" yZ,
(4)
e'^c" vvy,
(5)
e'*^e“ — > vvZ.
(6)
The last process (6), together with decay of Z in to gq has recently been considered by
Choudhury and Kalinowski [15]. They point out that this process can give bounds
comparable to those expected from e*c’ This process has also been examined
from the point of view of CP-violating couplings. It was shown in [16] that a forward-
backward asymmetry of the 2, which if observed would signal CP violation, singles out the
/*-cvcn, C-odd coupling g
5. Top quark physics
The top quark is so much heavier than the other quarks that much of the intuition of
ordinary hadronic physics is simply invalid when applied to tt systems. The first
nnijor difference is that r decays to an on-shell W boson, and has a lifetime short
compared to typical hadronic scales. The decay width is given approximately by the
CKpression
ru-
16 M
Of ni: ML
1 +
2Ml
1 - 2.9 —
n
*= (1.4 GeV)
175 GeV
]’
( 7 )
Thus the lop decays before non-perturbative strong interaction precesses have time to
act 117]; — ! — « 10'^^ sec, whereas -p « 3,6 x 10"^^ sec,
^QCD '
This implies that the top quark is amenable to perturbation theory. Moreover,
>11 production and decay processes, the top quark retains its spin orientation. The decay
f W/; can then be used as an analyzer of top polarization.
I Gauge couplhifis of the top quark :
Tcm of non-standard couplings to elcciroweak gauge bosons can be addressed at e^e"
colliders by exploiting the large forward-backward and polarization asymmetries in rr
542
Saurabh D Rindani
production and decay. These reflect very different couplings of the left- and right-handed
components. For example if £cm ^
-►«■) = 2^[|/u. 1^(1 + cos +l/u|'(l-cos«)3), (g)
where
2
(l - sin^ «»)(/’ -
ysin^ 6w)
'3
sin^ dw cos^
= 1.4 {otele^ tjn
= 0.2 fore^ej -► (9)
with H = L, R, = left-handed electron beam dominantly produces
forward-moving, left-handed top quarks. In a more realistic case, the angular distribution of
tl pairs in c^e^ « for Vs = 500 GeV is shown in Figure 6, taken from ref. [3].
- 1.0 - 0.5 0 0.5 1.0
cose
Figure 6. The angular disiribuiion of ti pain of various heliciiy combinations
in e ^ e rf at cm energy .VX) GeV. token from (3).
Deviations from the predicted angular distributions can signal anomalous couplings
parametrized by :
i’ = gnv^FaiytLV^ + F2L:^fa>‘''tLV^, + (L ^ «)J- (>0)
V s y, Zand . It may be noted that CP invariance implies Fn = F^r^
and the difference between Fn and Fir \s proportional to the CP- violating electric or weak
dipole moment of the top quark. Expcrimcnially, signals of CP violation would be CP-
violating asymmetries or correlations amongst final state momenta.
Various anomalous quantities which can be investigated are magnetic
electroweak dipole moments, and wrong chirality component in the coupling to W.
New physics at e^€~ colliders
543
5.2. Anomalous magnetic moment :
Since in SM there is only a small number of t^ produced in the backward direction, the
backward direction is sensitive to small anomalous magnetic moment. The angular
dependence can be used to bound the magnetic moment to a few percent [18].
5 ..?. Electric and '*weak" dipole moments :
The measurement of these CP-violating dipole moments necessarily needs decay
disiribulions. A measure of CP violation is NUlIl) - N(t^iff), the difference in the
numbers of like hclicity top and antitops. This number -asymmetry can be converted to
asymmetries in the energies and momenta of decay products [19,20]. CP-odd correlations,
with and without beam polarization can be used to measure or bound the dipole moments
|21-25]. a simple asymmetry in the scmileptonic decay products may be used to probe
ihc imaginary parts of the dipole moments. This is simply the charge asymmetry in the
number or leptons ; [A(T(/^) - Aa{l~ )]/Aa [26]. In this case an angular cut on the
forward and backward directions is needed for a nonzero answer. Another simple
asymmetry is the sum of forward-backward asymmetries of the I* and /“ in semileptnnic
events [A<7f_a(/'*’ ) + A(Tf_fl(/” )]/A(T [26). Limits on dipole moments of the order of a
lew times 10' e cm would be possible with the use of polarized electron beams.
5 4 Chirality of the fh current :
The lepton energy distribution in the semilcplonic decay t bW^ hl^Vi depends
sensitively on the chirality of the current ;
dr xii\-.xi) (otV-A
d.Xi {xi - fj- ){ \ - Xi + p - ) for V + A.
2 £,
where p- <.\/ = < 1, with p = Mwjm,. Deviation from V~A leads to the
siilTcning of the energy spectrum, with a nonzero value at the upper end of the energy
Jisiribution.
5 . 5 . Higg.S‘top Yukawa coupling :
A direct way to obtain the Hti Yukawa coupling is to look at the process e^e' tiH ,
where Higgs is produced by brcmsslrahlung off a / or r in e*e"-4 tt [27]. SM predicts a
iLMsonablc number of events for Higgs mass of about 100 GeV or less.
For Af// > 2m,, the process e^e'-> Zti gels an extra contribution from e'^e'-* ZH,
^ tt. This would produce an enhancement in the cross section around the
mass 1 28]. However, this effect is large for lower top masses, and if the top mass
linger than 175 GeV, as it nows seems to be, the enhancement may not be easy
In observe.
544
Saurabh D Rindani
6. Concluding remarks
An aiiempi has been made to describe the important new physics that can be studied at a
future high energy linear e'^e' collider. While the topics of top quark proi)enies and gauge
boson interactions have been described in some detail, certain other important topics like
supersymmetry, Higgs searches, extra gauge bosons and heavy fermions could not be taken
up because of lack of time. Reviews of these can be found in [2] and references therein.
References
[ 1 1 For a compact update of the operational features of linear colliders, see Physics Monitor, CERN Courier
37 April 1997 p 16
[2] Reviews di<:cussing the physics capabilities include ; PriKcedinfis, e'^e~ Collisions at 500 GeV : The
Phyllis Poretituil (Munich Annecy Hamburg 1991/9.1) cd. P M Zerwas, DESY 92-l23A‘«‘B. 93-I23C,
M E Peskin in Pnu Int. Workshop on Phys. and Expts. with Linear Colliders (Saoriselkil, Finland, 1991)
eds R Orava, P Ecrola and M Nordberg (Singapore : World Scientific) (1992) SLAC-PUB-.S798 (1992);
Proceedings. Phy.\ and Expts with Linear Colliders (Waikoloa, Hawaii 1993) eds. F Harris.
S OKen. S Pakvasa and X Tala (Singapore World Scientific) (1993); P M Zerwas in PrtH:eedinf(s. ECFA
Workshop on Linear Colliders (Munich 1993) ed R Settles. P M Zerwas in "Les Rencontres de
Physique de la Vallee d'Ao^re" (La Thuilc 1994) Editions Frontifcres, cd. M Greco, DESY 94-001,
updated May 1996. H Murayama in Pro< 3rd Int Workshop on Phys and Expts. with c^c" Linear
Colliders (Sept 1995. Morioka, Japan) LBL-3R89I, UCB-PTH-96/21 , NLC ZDR Desinn Croup and
NLC Physiis Workitiff Group (S KuhIman et at) SLAC-R-0485, June 1996, hep-ex/96050l I.
E Accoinando et al. DESY preprint DESY 97-100(1997)
(31 H Murayama and M E Peskin Ann. Rev Part Sti 46 .533 ( 1977) hep-ex/9606003 »
|4j H F Ginzburg. G L Koikin. S L Panfil. V G Serbo and V I Telnov Nucl. Instrum. Meth 219 5 (1984)
[5] Sec. tor example. T Rizzo SLAC PUB-7279 (1996), A Lcike and S Riemann hep-ph/960432 1 and
hcp-ph/96073()6. m Proi Phys nith e*e~ Linear Collider Work.^hop (Annecy-Gran Sosso- Hamburg.
1995) ed P Zerwas. A Djouadi hep-ph/95l23l I. in Proc 3rd. Int. Workshop on Phys and Expts
with e'*e~ Linear Colliders (Monoka. Japan. 1995). J L Hewcil and T G Rizzo Phys. Rep C183 193
(1989)
(61 Sec. tor example, Y Okada m Pro< 3rd Int Workshop on Phys and Expts. with Linear Colliders
(Scpi 1995. Monoka. Japan) KEK-TH-469, A Djouadi, short write-up of lectures given at XXXVI
C’liicow School of Theoretical Physics (Zakopane. Poland. June 1996) PM/%- .34. KA-TP-27.1996
(71 M Drees and R M Godbolc Phys Rev Lett 67 118 (1991). Zeii. Phys. C59 725 (1993). P Chen.
T L Barklow and M E Peskin Phys Rev D49 3207 ( 1994)
(K) K Hagiwora. R D Pecei, D Zeppcnfcld and K Hikasa Nucl Phys B282 253 (1987)
(91 K Hapiwara. S Ishihara, R Szalupski and D Zeppendfeld Phys Lett B283 353 (1992)
f Id) H Aihara et of in Elettroweak Symmetry Breakoifi and New Physics at the TeV Scale eds. T L Barklow.
H E Haber, S Dawson and J L Siegrist (Singapoa* World Scientific) (19%)
1 1 1 1 T Barklow al MADPH 96-975. SLAC-PUB-7366. UB-HET-96-05. UM-HE-96-26. hep'ph/961 1454
(to appear in 1996 Snowmass) [2]
1 12| D Chang. W-Y Keung and I Phillips Phy.s. Rev 48 4045 (1993)
[13] H S Mam. B Mukhopadhyaya and S Raychaudhury MRI. Allahabad preprint MRI'PHY/9/93
1 14) V C Spanos and W J Stirling Ph\s Un. B388 371 ( 1996)
(1^1 I) Oioiidhury and J Kalinowski Nuit Phvs. B491 129 (1997) hep-ph/9608416
New physics at e^r colliders
545
[I6| S D Rindani and J P Singh Physics Letl B419 357 (1998) hcp-ph/9703380
[17] I Bigi and H Krascmann Z Phys. Cl 127 (1981); J Klihn Acto Phys. Ausir Suppl XXIV 203 (1982);
IBigi«/fl/.P/JV.T. Uii 8181157(1986)
( 18J C Schmidt and M E Pcskin Pnn , Workshop on Phys. nnd Expts with e'^e' Linear Colliders (Sauriselkll)
cds. R Orava, P Eerola and M Nordbcrg (Singapore ■ World Scientific) (1992). [2]
1 1 9] G L Kane, G A Ladinsky and C P Yuan Phys. Rev. 1)45 1 24 ( 1 992)
[201 J F Donoghue and G Valencia Phys Rev Utt 58 451 (1987); C A Nelson Phys. Rev. D41 2805 (1990),
C R Schmidt and M E Pcskin Phy.s. Rev. Lett 69 410 (1992); C R Schmidt Phys Utt. 8293 1 1 1 ( 1992)
[211 W Bemrcuthcr and P Overmann Z Phys. C61 599 (1994); W Bcmreuthcr, A Brandenburg and
P Overmann, hep-ph/9602273 and references therein
[221 0 Atwood and A Soni Phys. Rev D45 2405 (1992); hcp'ph/960941 8 and references therein
[231 t) Chang, W-YKcung and 1 Phillips Vue/ Phys 8408 286 (1993); 429 255(1 994) (E)
[24] B Grzadkowski Phyx Utt. 8305 384 (1992), B Grzadkowski and Z Hioki, hep-phy9604301 Nucl.
Phys. 8484 17 (1997), hep-ph/9608306. Phy.s Utt. 8391 172 (1997), hep-ph/96 10306 and references
therein
[25] P Poulose and S D Rindani Phys Utt 8349 379 (1995); F Cuypers and S D Rindani Phys. Utt.
8343 333(1995)
[26] P Pouiosc and S D Rindani Phy.s. Rev D54 4326 ( 1996), Phys. Utt 8383 212 (1996)
[271 A Djouadi, J Kalinowski and P Zerwas Mod Phys. Utt A7 1765 (1992), Z Phys C54 255 (1992)
(2H1 K Hagiwara. H Murayama and 1 Walanabc Nucl Phys 8367 257 (1991 )
IndUm J. Phys. IIK (6), 347-S66 (1998)
UP A
— an miemational journal
Structure functions — selected topics
D K Choudhury
Department of Physics, Cauhati University,
Guwaliati‘781 014, Assam, India
Abalract : We summarise a few topics of DIS like double asymptotic scaling, and spin
and diffractive structure.
Ke y wo rds : Structure function, HERA, Low X
PACS Noa. : 13.88.4e, 13.60.Hb. 12.38.Bx, l2.38.Lg
1. Introdiiclioii
Deep Inelastic lepton scattering experiments have made very important contributions to the
understanding of the structure of matter. The long tradition of experiments of deep inelastic
scattering started with the experiment at the linear accelerator at SLAC in 1968, where
an approximate scaling of the nucleon structure function in a dimensionless variable x
gave first evidence for scattering on charged pointlike constituents of the nucleon. In
the 70’ s and 80* s. beam energies upto several hundred GeV become available and allowed
to measure precisely the logarithmic scaling violation in the structure functions which
become instrumental for testing QCD. In 1992, the ep collider HERA was put in operation,
where centre-of-mass energy of 300 GeV can be reached [1] compared to about 30 GeV
in fixed target experiment. This makes it possible to explore a new domain in x and
Specially low x regime (x S 10^) has received intense theoretical and experimental
attention [2]. Similarly, probiifg the structure of the Pomeron at HERA [3] through
diffractive structure function has opened a new dimension in the physics of deep
inelastic scattering. In the spin physics on the other hand, new information on gi has
been reported [4,5).
The present talk deals with following few selected topics of deep inelastic
scattering :
• Double asymptotic scaling
• Gluon and longitudinal structure functions at low x
72A(6H2 © 1998 taps
548
D K Choudhury
• Diffractive structure functions
• Spin structure functions.
2. Double asymptotic scaling
As early as 1974, it was shown that with reasonable boundary conditions [6], perturbative
QCD predicts a universal growth in the gluon momentum density at large / (f = In ~)
1 ^ ^
and small x faster than any power of In j but slower than any inverse power of .v. More
recently, Ball and Forte [7] brought this perturbative to the phenomenological front. They
have recast the result of reference [6] in two asymptotic variables
HERA [8] provides excellent agreement with both the scaling predictions and confirm the
perturbative results [6]. The asymptotic behaviour of FiiO. p) is then
N^ and fiy being the number of colours and flavours respectively. The unknown function y,
which depends on the details of the starting distribution lends to one for sufficiently small
values of its argument. /V is an o priori undetermined normalisation factor. For nf=4 and
/V, = 3,5-1.36.
In order to test this prediction, data [9] are presented in the variables crand p. taking
the boundary conditions to be jcq = 0.1 and fij = 1 GeV^. and A^q = 185 MeV. The
measured value of F 2 are rescaled by
Rf (a,p) = 8.1 exp ^5 j + •jln(a)+
to remove the part of the leading subasymptotic behaviour which can be calculated in a
model independent way; In {RfF 2 ) is then predicted to rise linearly with a and with a
slope 2 y
Structure functions— selected topics
549
Figure (la) shows such a linear rise. A fii to ihe data gives the value ol
2.22 ± 0.04 ± O.IO for Ihe slope. The result agrees well with the prediction ol the slope
2y = 2.4 lor four flavours. Figure (la) contains data with p^>\.5 only.
p
Figure 1. The rescaled siruciure funclions \og(R'f:F 2 ) and ^ ploned venii<i the vuriable.s
cr and p deHned in the text Only data with p^ > I 5 arc shown m (a).
Scaling in pcan be shown by multiplying F 2 by the factor
( 6 )
This re.scaled structure function should scale in both a and p when both lie in the
asyinpiotic regions. Figure ( I b) shows the scaling in p which sets in for p > 1 .2.
The prediction for RfrF 2 as a function of ponly depends on the gluon density at Ql .
While for a soft starting gluon distribution, scaling for the full asymptotic region is
predicted, a hard gluon input would lead to scaling violation at high p [7]. The data shown
in Figure [I (a, b)l are well described by the asymptotic behaviour derived from soft
boundary conditions.
Mi^re recently, Ball and Forte [10) developed the double scaling formalism with
NLO effects. In this case, (land pare defined as
''^here a,((J-) is 10 be evaluated at the two loop level.
550
D K Choudhury
( OM
- p In In «
with = 102 -
In order to obtain the structure function within the NLO DAS formalism, one defines
besides the usual logarithmic QCD evolution variables t = In and $ = In (^), the
evolution length T of (x^iQ^) from a starting point Qq to
a.(G')
To leading order, Tis simply In ( 7 ^).
For large f and and an F 2 (x, which at Qq is not loo singular in x, the NLO
double asymptotic expression forF 2 (x, Q^) is [ 10 ]
f 2 -yVf(l-/NLo)«p[2yVlr-«r+ l|nT- (II)
The normalisation coefficient Nr is
For fif = 4 and = 3, it gives y = 516, 5 = ^ and Nf = 0.038.
The NLO correction term /nlo *s
/m, - ^lE(«,(ei)-».(0=))-‘5“.ie’>
r206/i; 6^,^
' - [-IT- * I]'
The leading order formula is recovered by setting /nlo = 0 and A =* 0. Defining leading
exponent as ’
Jli = 2y4^ (15)
and the subleading term as
a= - dT+ |lnr- -^In^ (16)
4 4
one rewrites F 2 as
Structure fimctionsselected topics
551
The leading term in the double asymptotic formula for F 2 corresponds to the double leading
log approximation DLL [1 1] of the DGLAP equations [12], It generates the growth of the
structure function with falling Jt proportional to A
(18)
The subleading term also falls with x but slower than the leading term growth.
This formulation has been used recently by HI Collaboration [13].
The value » 0.1 as suggested earlier [7,9]* waa found to be a good choice while
0 Q is set at j2o = 2.5 GeV^. To visualise the double scaling, it was proposed to rescale F 2
with factors Rf and Rp related by ( 6 ) but the explicit term is modified as
where
8.1 exp
/?F(cr,p) =
-2y(7+ + \\n iyo) + In I -
(19)
(20)
( 21 )
( 22 )
Figure (2a) shows RfF 2 versus p to the data with > 3.5 GeV^. The value of A for four
favours is chosen to be A = 263 MeV. Approximate scaling is observed for GeV^
i^nd p ^ 2. At high p, the low data tend to violate the scaling behaviour which is clearly
seen. from the data at 3.5 GeV^.
In Figure ( 2 b), In F 2 shown for p ^ 2 and 5 GeV^ as a function of <T.
The data exhibits the linear growth with a. A linear fit to the data gives a value for the
slope to be 2.50 ± 0.02 ± 0.06 (2.57 ± 0.05 ± 0.06) for < \5 GtV^ (Q^ > 35 GeV^)
and 4 (5) flavours. The results are in agreement with the (JCTD prediction : 2.4 and 2.5
for rif = 4,5 respectively. Compared to the result presented in references [7,9], the
eittraction based on the 2-Ioop formalism [10] is in better agreement with QCD
expectation.
One can therefore conclude that low x, low measurements for ^ 5 QeV^
show scaling in p and a The double asymptotic scaling is a dominant feature of F 2 in
this region.
552
D K Choudhury
In a recent analysis [14], A and a, are determined by fitting the expression (9)
lor Fiix, Q-) to the latest measurement of the proton structure function by the Hi
Figure 2. The rescaled structure functions (a) RfF^ \ eruisp and (b) log( A?): Fj ) (T using
NLO-DAS rormalism Only data with Q->^ GeV - and p > 2 are shown in (b)
experiment [13] at HERA yielding A -248 MeV, a, ) = O.l |3 ± 0.(K)2 {stat) ± 0.007
(.vv.\7) at = 1.12 GeV^. The authors also attempt u QCD inspired parametrization with
leading exponent of (9) :
F.Jx.Q^) (23)
with Hf = 4.
The NLO double asymptotic expression (11) and the modified DLL form (23) are
shown in Figure 3. The modified DLL form (23) is fitted with two parameters = 0.365
± 0.026 {star) ± 0.048 (syjr) GeV^ and A= 243 ± 1 3 ± 23 MeV.
Let us conclude this subsection with a caution. In a recent work, Buchmuller and
D Haidt [15] obtains an equally good fit of the recent data [16] with a simple double
logarithmic form
FiUG-)
, Q‘ . xo
= a -F m n — - In —
GJ ^
(24)
with
a = 0.078,
m = 0.364, A'o = 0.074, Qq = 0.5 GeV^.
(25)
Hence the characteristic feature of double asymptotic scaling, a growth stronger than
any power of In -j, cannot be confirmed from the present HERA data. This more singular
Structure fwtctions— selected topics
553
fM
%
1.5
I
0.5
0
1.5
/
0.5
0
1.5
I
0.5
0
1.5
1
0.5
0
1.5
I
0.5
n
Figure 3, The prolon structure function F^ix.Q^) as measured by the HI experiment at HERA
together with a fit to the NLO double asymptotic expression (II) (full line) for > 5 CeV^ and
with a fit to the modified DLL expression (23) (dashed line) in the full range.
behiiviour should become visible, if al given the range in x is extended at least by one
order of magnitude. In that small x range, the more singular, BFKL [17] power behaviour
nijy afso perhaps be distinguished. This corresponds to an increase in the centre-of-mass
energy squared by one order ol magnitude, which could be reached at future colliders, such
as LEP ® LHC or at a 500 GeV^ Linear Collider® HERA.
O'- 1.5
}
O'- 2.5
V
O'- 3.5
V
....j ....j ....j ...^ ...
O'- 5
V
_J ...J ...-J .._
O'- 6.5
V
O'- B.5
\
O'- 12
V,
O'- 15
o'- 20
i,V
O'- 25
..V
O'- 35
\
O’-
V
o'- 60
V
0*. go
O'- 120
\
...J .._J ....J
\
■ \
\
O'- 150
o'- 200
V
o'- 250
o'- 350
O'- 500
\
O'- 650
\
iMmt umd itM
O’- 000
\
• .IMli Ul^ IIM
O'- 1200
iM^ jiM alia
O'- 2000
: \
H
iiiial jiMd Jiiai iiiJ
O'- 5000
L
L
gynt
...aJ .«J .iiJ tmd ,u,
lo foioio ' lolo lo io ' loioio'fo ' lo foioio ' loioio h '
3. Measuring Gluon and longitudinal structure functions
i I. Approximate relation between gluon and longitudinal structure functions :
In leading order in a,, the longitudinal structure function Fiix, (^) is given by [12)
rr fs f' ^
Fi(y.Q^)
'^hcrc e ^ djnoigj (|,g charge squared of the panons.
( 26 )
554
D K Choudhury
In the low X limit, it yields [18] for four active flavours
- |5.8[^Fi(0.417x.e2) _ ^F,(0.75;t.Q^)]- (27)
Neglecting the quark contribution [ 1 8], one obtains
Fi(0.47it.CM= ^f^JrC(;r.e')- (28)
which directly relates gluon density to longitudinal structure function. NLO correction to
(28) has been reported by Zilstra and Van Neerven [19]. Recently [20] (28) has been used
to test the gluon density with factorisable jc and dependence by predicting the
longitudinal structure function. Such a factorisable gluon has the universal limiting
behaviour at low x
G(x,f) = C(x,ro
(29)
In Figure 4, the predictions of using (29) are compared with those obtained with
collinear [19] and kr factorisation approach [21] at = 20 GeV^. Prediction of (29) are
Figure 4 . Comparison of using (29) with the prediction of collinear 1 19| and *7 faciorisaiion
(2 1 1 approaches.
found to be higher than those of [19] and [21]. The difference increases as .r decreases
However, as the cross-over of gluon distribution (29) with LO-ORV [22] occurs in the
range Jt - 10"' - 10"^ for - 20 GeV^ the prediction may not be reliable tor j: ^ 10""
Structure functions— selected topics
555
3. 2. Measuring the Gluon density directly from structure function :
Instead of a direct relation between FjOr, and the gluon distribution CU, Q^) will be
more interesting from experimental point of view. Prytz [23] has initiated such a
programme of study.
Using Taylor expansion approximation of GLAP equation [12], one obtains [23]
dF2{x)
d\nQ^
20 .
(30)
The method has later been extended [24] to include the NLO corrections as well :
dFAx,Q^)
dinQ^
G(2x)^^[i + ^3.58
9 An [3 An
QJ).
(31)
where A/(.r, Q^) is given explicitly [24], The result for four flavour in the MS scheme
explicitly yields (25)
C(v 0=) - dF.(xll.Q^)ld\i^Q^
~ (40/27 + 7.96a, /4n:)(a, /4;r)
(20/9)(a, /4Jr)Af(jc/2.e2)
40/27 + 7.96a, jAn '
An alternative method of extracting gluon density was proposed by Ellis Kunszt and
Levin (EKL) [26].
In the EKL method [26], the gluon momentum density and Fi are assumed to behave
as \ , which leads to the following form for the scaling violation of Fj :
=P'^''{a)o)T(x,Q'-) + pF<=(Wo)g{x,Q^) (33)
d\T\Q- "
with
I F2(X,Q^)
i'i) ’
(34)
where {e^ ) is the average of the squares of the quark charges ( for four flavours).
The non-singlet contributions are neglected. The evolution kernels and P^^ are
expanded upto third order in (NNL)
P^^{(Oq ) « a,P^^ + a-P," +
P^^(tt)o ) » <XsPq^ (36)
The LO and NLO results arc obtained by keeping in (35) and (36) the terms upto O(o;)
‘ind 0(aj ) respectively. The coefricicnls P" andP,^^ depend on the parameter qjo
^2A(6)-I3 .
556
D K Choudhuty
und are tabulated in [26] for a range of values. The actual value of cub must be extracted
from data.
In contrast to the Prytz method [23,24], quark contribution is included i^
the EKL method (33). The expression for the gluon momentum density for four
flavours is
C(x, )
18/5 UFi(x,Q^)
(37)
In the EKL method, in contrast to the Prytz method, the gluon density at jr is
calculated using the structure function and its logarithmic slope at the same value
of -V.
Figure 5 compares the results of Prytz [23] and EKL [26] methods, with that
of the LO global GLAP fit [25] at = 20 GeV^. The results are consistent among
each other.
ZEUS 1993
Ftgure 5. Gluon momenium density as a function of jr at = 20 GeV^
determined from the ZEUS data using the method of Prytz [23] and EKL [26].
Solid line is the LO GLAP fit.
Figure 6 shows the gluon momentum density obtained in NLO. Good agreement
between the results of the three methods is observed. The shaded band in Figure 6 indicates
the uncertain ity of the gluon density from the global GLAP fit as estimated by adding ih®
statistical and systematic errors in quadrature.
Structure functions—selected topics
557
A relation dlicmative to (30) have also been suggested recently [27] which reads
dF2(.x.Q^)
d\nQ^
^|c(4./3)
(38)
The difference arises due to the choice of the expansion point of G( Q^) occurred in
the GLAP equation [12].
ZEUS 1993
Figure 6. The gluon momentum densiy as a function of x al = 20 GeV^
determined from the ZEUS data using the methods of Prytz [24] and EKL [26]
in the next to leading order. The solid line shows the result of the NLO GLAP
global fit. It also shows the gluon distribution (hatched region) determined by
the NMC experiment.
Gay Ducati and Goncalves [28] later obtained the expansion of the gluon distribution
G( -j^) at an arbitrary point z = a Retainig terms only upto the first derivative In the
expansion, they get in the limit jr 0.
^ (1 9g idF2{x,Q^)
'[l-oU" JJ 5a, 2 d\nQ^
This reduces to Prytz relatioa.(301for ® while for a * 0, it yields
dFi^.Qt) _ 50,
d\nQ^ “ 9* 2"'l2 J
which is the corrected version of (38).
(39)
( 40 )
558
D K Choudhury
Recently Kotikov and Parenlc [28] present a set of formulae to extract Q^) from
F 2 ix. 0^) and
dFj (x, )
t/IngJ
directly. Assuming behaviour of the parton densities at low
X, they obtain the following formula for Nf= d {a = a,(Q'^)l :
1 5 45 j
which will be useful in extracting longitudinal structure function directly from the structure
function and its derivative, instead of gluon distribution as (28).
4. Spin structure functions
Hadronic tensor defined in deep inelastic lepton nucleon scattering has two spin
structure functions g| and ft [29] :
Ap.q) = ^d*xe«i^{p,s\[j^ (A:),y^(x)]|p, j)
P^Pv
= -guvf^\ +
p.q p.g ^
S^g\ +
p.qs^ - s.qpP
P‘Q
'82
(42)
where j" is the spin of the nucleon and other symbols have usual meaning. For
longitudinally polarised beam and target, one measures the longitudinally polarised
asymmetries
and
-
w\\
^1
-f
2xMf,
[~)
82
gl'(x.Q^)
(43)
f.U.GM
“ ±«r
(44)
For transversely polarised nucleon with polarisation perpendicular to the beam direction
the corresponding polarised asymmetry is
2xM
= ±-
-(^i +52)*
(45)
Conventionally, g| is called longitudinal spin structure function while g 2 is called the
transverse structure function. g| has the interpretation of incoherent supi of parton
probabilities
8\(x,Q^) = + A^(x, C^)]. (46)
where = <? T - ^ i . On the other hand, g 2 has no such simple partonic interpretation. It
differs from zero because of the masses and the transverse momenta of the quarks. It has a
Structure functions— selected topics
559
unique leading order sensitivity to twist-3 operators, Le. quark gluon correlation effects in
QCD. Thus g 2 will be a unique probe of higher twist effects [30].
In general, ^2 can be written as the sum of a contribution , directly calculable
Irom [31] and a purely twisi-3 term g 2 [30]
=«2^(jr,C^) + «2(x,C^) (47)
with -gi(x,Q^)+ (48)
JjT t
£q. (48) is called Wandzura-Wilczek relation.
A sum rule for g 2
\ g2(x.Q^)(lx = 0 (49)
JO
was derived by Burkhart and Cottingham [32] using Regge Theory. It has been regarded as
a consequence of conservation of angular momentum [33]. At present, the validity of the
derivation of the sum rule is in question [30,34] and it is clearly important to test it
experimentally. SMC [35] have reported measurement of spin structure function g 2 as well
as ihe asymmetry Aj defined as
^2 = ‘*■^ 2 )- (50)
Results of has also been summarised by SMC [35] as shown in Table 1 .
Table 1. Results on the spin a.symnietry A 2 and the structure functions K 2 ^d ^ reponed
in [3S].
X interval
(^)
((?MGeV2))
^2
n
ww
0.006- 0.01. S
0.010
1 4
0.002 ±0.083
1.2±6I
0.73 ±0.10
0.0I!5-0.0!S0
0026
2.7
0.041 ±0.066
70±I2
0.47 ±0.09
0.050 - 0.1 SO
0.080
.5.8
0.017 ±0.091
0.2 ±2.9
0.I5±0.02
0.1 .50 - 0.600
0,226
11.8
0.149 ±0.1.56
0.5 ±0.8
-0.10 ±0.02
E143 Collaboration [36] has measured structure functions g 2 andg^ the
range 0.03 < jc < 0.8 and 1 .3 < < 10 (GeV/c)^ Figure 7. In the same figure the twist-2
calculation is shown using g|Cr, Q^) evaluated from a Tit to world data [37] of
asymmetry and assuming negligible higher twist contributions. Also shown are bag
model predictions [38,39]. At high jc, the results for indicates a negative trend
i^onsisteni with the expectation for g^^ . By extracting the quantity gjU.Q^) *
fii (x, ) -g^*^ (x, ), one looks for possible quark mass and higher twist effects.
This can be seen from the difference between the data and the solid line in Figure 7. Within
560
D K Choudhury
the experimental uncertainity, the data are consistent with g 2 being zero, but also g 2 being
of the same order of magnitude as
0.05 U.l 0 5 10
X
Figure?. Mea.sureineni ot (a) ^"‘l ' X: EI4.^ cxpcriiiK’iu 1.U>1
El 43 Collaboration [37] has also evaluated the integrals ;
Jo 03
f g'lix)dx =
Jo 0.3 ^
-0.013 ± 0.028’
-0.033 ± 0,082
(31)
(32)
These results are consistent with zero and conforms to the expectation of sum rule (49).
More recently [40], results are reported from the HERMES experiment at HERA, on
a measurement of the neutron spin structure function (J‘) using 27.3 GeV
longitudinally polarised positrons incident on a polarised ^He target. The data cover the
kinematic range 0.023 < jc < 0.6 and I (GeVA < G’ < 15 (GcV/c)-. Evaluating at a lixcil
of 2.5 (GeV/c’)^, experiment reports
f"'" gUx)djc= -0.034 ± 0.013 ± 0.005 (JV5/).
J0.023
Assuming Regge behaviour at low .r, the first moment comes out to be
r;' = = -0.037 ± ().oi.^(«of)
± 0.005 (m/) ± 0.006 (exinipol)
(.S3)
(.S4)
Structure functions — selected topics
561
5. DilTractlve structure function
The obseivalion ol' "dinVaclivc " deep inelastic scattering (DDIS) events with a large
rapidity gap [41 ) has opened up a new field on structure functions in the last few years.
While in the non-diMractive deep inelastic scattering (DIS) where a virtual photon
probes a parton (Figure S), in dilTractivc deep inelastic scattering (DDIS), a virtual photon
f
Ki|;uri' H. L suai deep inclasin. <^ + /» — K' + X
Figure 9. DiMraciive deep iiielaMie seailciini: e + /^ — » e + + A
piobcs a colour neutral object emitted from the target proton (Figure 9). This colour neutral
ohjcct IS called "Pomcron".
In usual DIS, the standard kinematic variables are
Ip.q p.k
W-={p + q)-, g- = -(/-• (55l
"Ikic P.k.k'miiii = k-k' arc the lour momcnia of the proioii. mcidcni lepton, linal
Irpum anj the virtual photon.
562
D K Oumlluiry
In DDIS, ihc proion rcmnan! emerges wiih momcnlum P\ As u result, one
iiiiioduccs the additional kinematic variables |34]
C/P 2q.{P-P')
l=(P-P’)' (56)
Furthermore
Q- Mx- - t
P =
(57)
“ Q- + W2 - Ml
Q'^ M\ - t
\Uia-c Ml = -Q^ + (P-P')-. For
f5H)
-H
(59)
p = . ■' .
M-v + 0- V;.
(60)
and
besides
In this liinii |34|,
p
Xp = Fraction ol the Proton’s four momcnlum transferred to the Pomcron
and
P = Fraction of the Pdaieroii's four momentum earned by the quark
entering the hard priKcss.
The DDIS cross section lor the process c p ^ e + X +p is given by
ilCTiP, Q- ,.\ /. )
(IP (IQ- (h (l.x P
2k(x~
PQ*
I + (I ->)- ]f 02, JtpC
(61)
vvheie /■i^'^'(P,Q-. \p) is called the diHraciivc structure lunction, integrated over the
variable r. The uninlcgraied version is denoted by ,Xp).
The main experimental lealurcs of ■
iui \p dependence
This IS shown in Figure 10 which is m the range 2 x I0^< A7 >< 2x I0‘^.
fit to the ZEUS data (16) yields
j
u iih
a= 1.4b ± 0.04 ± ().()«.
(63)
Structure fum tians—selei ted topics
563
Corresponding analysis of H I cxpcrimcnl [ 16| fixes ri ai
fl= 1.19 ±0.06 ±0.07.
(64)
• ZEUS 1993 ~ this analysis
oZEUS 1993 - previous onolysis
■ HI
(/)) P dependence :
li is shown in Figure 1 1. The largest range of P is covered in the cxperiineni is M =
U.OO.V Figure II shows that ff” rises as /} decreases, which is expected Iroin QCD
evolution of parton densities ol proton.
The uniniegraled diffractive structure function F'"*' and the Poineion siruciuic
lunciion are related via the Factorisation ansau [42 1
f (.x> . jS, r. e- y . nr up. q - )
(65)
72A(o).i4
564
D K ChoiuUmry
uIkmc is ihc Ilux factor describing the llux of Pomcrons in the proton, which
L\in be extracted from hadron-hadron scattering assuming universality ol the proton llux.
ZEUS 1995
8
FiKurc 1 1. The diflraclive struciure function / as a function of fi at =
t)(M)3 ji = 14 and 31 GeV- The full line, dashed line, dashed dotted and
dotted lines arc model predictions discussed in ( 16]
The Pomeron structure function has the parton decompi)sition
where f iP.Q- )) probability of finding a parton (antiparton) ol
ll.iNiiLii If NMih momentum Iraclion /3 inside the Pomeron. For “hard Pomeron and soli
Pomeron" they have simple forms [431
iP,Q^‘ ) - Pi \ - p) • Pomeron
~ ( 1 - /J ) : Soft Pomeron
(67)
Q- dependence of Ff is expected to be weak and is neglected.
I’aianiciri/ailoiis liir Pomeron llux laclors arc also reported in the literature. The
Injielmen-Schlein lorni of the llux factor |42| is parametriitcd by a lit to UA4 data |4.11
//.(.V,
/) = [0.3«f'*' + 0.424e'' ]■
( 68 )
On the other hand in the Donnachie-Landshoff model 144]. the flux lactor is
^Pi)
/ /• t A . f ) - “ r ^ I ^ ^ ■ I ■' /*
4/r -
( 69 )
where P^^ = I.S Go V ' and F,(/) is ela.stic form factor of the proton.
Structu re functions—selected topics
565
The Poincron irajcciory ap{t) occurring in (69) obeys a linear relation :
apit) = ap{0) + apt. (70)
Dcl'iiiing ap as a pit) averaged over /. the exponent a defined in (62) obeys the relation
— a+\
«/» = — (71)
I'rom the IV dependence of the diffractive cross section
ap = 1.23 ± 0.02 (stat) ± O.O^isyst) (72)
which is in between the soil Pomeron dip - 1.05 occurred in hadron>hadron collisions [441
jnd the hard or BFKL Pomeron (17) with ap - 1.5.
6. Conclusion
With the start of the HERA experiments, a novel era in the investigation of the proton
siniciurc has began. The Double Asymptotic Scaling, methods of measurements of gluon
and longitudinal structure functions, diffractive structure functions (Jiscussed in this talk are
ihc topics which evolved mostly during the present-HERA years. Although the study of
spin structure functions dates back to the sixties, experimental information on gi has
become available only during last few years. Coming years with HERA, LHC and LEP ®
LHC will undoubtedly throw new light in the .structure of the nucleon.
7. Acknowledgments
1 graiclully acknowledge financial support from the Department of Science and
Tcihnology, Government of India. I also thank Abhiject Das for helping me preparing the
m.inuscript.
Rtk’ rentes
111 NPavel. DESY9VI47
121 WJ Stirling. hcp-ph/960«4 II
i M ''fi: lor example. ZEUS Collaboralion, DESY 96-018
Ml .SMC Collaboration. D Adams et a! Ph\\\. Leu. D336 I2S (1994)
Ml E 1 4.1 Collaboralion. K Abe ei al Phys. Rev. Leu. 76 .587 ( 19%)
IM A l)c Kujula. S L Glashow. H D Poliizer, S B Tnemon. F Wiiczek and A Zee Phys. Rev. DIO 1649
(1974)
n\ K I) Ball and S Forte Phy.y Leu. B335 77 (1994); Phys. Utt. B336 77 (1994)
I « I ZEU S Collaboralion. Phys. Leu. B316 412(1 993), H I Collaboration. Nucl. Phy.s. Wffl 5 1 .5 ( 1 993)
l^^l H I Collaboration. S Aid ei at Phys. Lett. B354 494 (199.5)
I '•'! K D Ball and S Forte Prm’, XXXV Cravow ScIum/I ofTheorelicai Physics (Zakopane, June 1995); CERN
TH/95-32.1
I’M L V Gribov. E M-Levin and M G Ryskin Phys. Rep. 100 I (1983)
S66 DKChoudhury
[12] G Altaitlli and G Parixi Nucl. Fhys. B126 298 (1977): V N Gribov and L N Lipatov .SVn J. Nucl. phy^
15 438 (1972): L N Lipatov Sov. I Nud Phyy 20 94 (1975)
[13] HI Collaboration. S Aid et al DESY 96-039 ( 1996)
[14] A De Roeck, M Klein ond T Naumann DESY 96-063 ( 1 996)
[15] W BuchmUlIerond D Hoidt DESY 96-061 (1996)
[ 16] ZEUS Collaboration, M Derrick et al DESY 96-018: H I Collaboration. S Aid et al DESY 96-039 ( 1996)
[17] E A Kureav. L N Lipatov and V S Fadin Sov. Pliys JETPAS 199 (1977). Ya Ya Balitsky and
L N Lipatov Sov. J. Nucl. Phys 28 822 (1978)
[18] AM Cooper Sarkar er al. Z Phyx C39 281 ( 1988)
[19] E B Zilstra and W L Van Neerven Nucl. Phys. B383 552 ( 1992)
[20] R Deka and D K Choudhury Z Phys. C75 679 ( 1997)
[21] J BlUmlein Nucl Phys. B. Proc. Suppl. 39 BC 22 ( 1995)
[22] M Gliick. E Reya and A Vogt Z Phys. CS3 127 ( 1997)
[23] K Pryiz Phyit. Lett B31I 286 (1993)
[24] K Prytz Phys. Utt. B332 393 ( 1 994)
[25] ZEUS Collaboration. M Derrick et al. Phys. Utt B345 576 (1995)
[26] R K Ellis. Z Kunszt and E M Levin Nucl. Phys. B420 5 1 7 ( 1 994)
[27] K Bora and D K Choudhury Phys. Utt. B3S4 15 1 ( 1995)
[28] A V Kotikov and G Parente US-FT/19-96. hep-ph/ 9605207 (1996)
[29] Sec for example, F E Close in An Introduction to Quarks and Parions' (New York Academic)
(1979)
[30] R L Jaffc Comments Nucl. Part. Phys. 19 239 (1992)
[31] S Wandzura and F Wiiczek Phys. Utt. B72 1 95 ( 1 977)
[32] H Burkhon and W N Coningham Ann. Phys. 56 453 (1970)
[33] R P Feynman in Photon-Hadron Interactions' (New York Benjamin) (1972)
[34] L Mankiewicz and A Schdfer Phys Utt. B265 167 (1991)
[35] SMC Collaboration, D Adams et al. Phys Utt B336 1 25 ( 1 994)
[36] El 43 Collaboration. K Abe et al. Phys. Rev. Utt 76 587 ( 1996)
[37] EI43 Collaboration. K Abe et al. Phys. Utt. B364 61 ( 1994)
[38] X Song and J S McCanhy Phys Rev. D49 3169 (1994).
[39] M Siratmonn Z Phvs. C60 763 ( 1 993)
[40] HERMES Collaboration. K Achcrsiaff et al. Phys Utt B404 383 (1997)
[41] ZEUS Collaboration. M Demck ei at. Phys. Ull. B3IS 481 (199.1); B332 228 (1994). B338 483 1 1994);
H I Colloboraiion, T Ahined ei al. Nucl. Phys B429 477 ( 1 994)
[42] C Intelmm and P Schlein Phys. Uii. B152 2.16 (198.1)
[43] UA4Collabontion. M Bozzor/a/. PA.v.t. Lftf BI36 217(1984)
[44] ADonnachieandPVUindshoffNitt/. Pb.vi B3a3 634 ( 1988); P/nt. Li-it B285 172(1992)
Indian J. Phys. 72A (6), 567-578 (1998)
UP A
— an intemationaJ joumai
Nuclear structure functions
D Indumathi*
Centre for Theoretical Studies. Indian Institute of Science.
Bangalore-S60012. India
Abstract : We present a general review of currently popular models of bound nucleon
structure functions. The dependence predicted by various models is highlighted; in principle,
this can be used to experimentally distinguish between various models.
Keyword : Nuclear suucture functions. dependence
PACSNoa. : l3.60.Hb. 24 85.+p, 25 30.Mr
1. Introduction
Structure functions of bound and free nucleons are not equal ; this is called the EMC
effect [1]. Although this discovery was made nearly fifteen years ago, the origin of the
EMC effect is still an open problem [2]. In deep inelastic scattering of leptons off a
nucleus of mass A, the average nuclear structure function, Fj {x, ), was thought to be
an incoherent sum ;
(X, ) * 7 [Zf 2 Z)Fl (jc. )]•
where the kinematic variables, x = Q^/{2p.q), - (^ = represent the Bjorken scaling
variable and the momentum transfer from the lepton to the hadron of momentum p. Here
represents the proton (neutron) structure function respectively.
This assumption was made, because corrections due to nuclear binding (for a typical
potential well depth of around 40 MeV) were expected to be about 1-4%. For nuclei with
equal number of protons and neutrons, i.e., Z= A - Z = A/2,
F^(x,Q^)= j(f^(.x,Q^) + F^(x,Q^)} ( 1 )
’Present Addibu : Mehtt Research Instiniie. Allahabad-221 506. India
© 1998 lACS
568
D Indumathi
which is to be compared with the average free nucleon structure function,
F? (X. CM » I (Fj'’ {X, Q-t ) + Fj" (AT, )|.
( 2 )
Hence, at first glance, it appears as if the ratio of nuclear and free nucleon structure
functions,
(3)
for all jc, Nuclear targets were therefore used to improve the statistics in the experiment,
since the total cross section is proportional to ^4. It was expected that there would be
deviations from this value, at very small and very large x values, due to nuclear shadowing
and Fermi motion respectively. However, when the first data was taken by the EMC
in 1982 [1], it was seen that was, in general, not equal to 1 (see Figure 1). An attempt
to explain this phenomenon led to the development of various models of nuclear structure
Figure J. The ratios of the bound and free nucleon structure function as First
determined by the EMC Collaboration [I] The solid curve shows the
theoretical expectation at that time.
functions. All of them have various predictions for R^, the latest data for which come from
the NMC and E665 collaborations [3,4] for the nuclei, He, Li, C, Ca, Sn, etc. (See Figures
4, 6 and 7 for the data for some of these nuclei). It is seen that R^ is typically smaller than
one for small x,x< 0.05, and for very large > 0.3, and larger than one for intermediate
values of x. The small- and intermediate-jc regions are usually called the shadowing and the
antishadowing regimes.
Data also exists for the Drell Yan ratio in p A collisions from the E772 collaboration
[5]. This indicates shadowing of the sea quarks, but no antishadowing. Information on the
nuclear gluon distribution is available from J/ Vf production in both /iA and pA collisions,
however, the results are fairly controversial and we shall not discuss them further here.
Nuclear structure functions
569
There are many models that describe the modification of the parton distributionB
inside a bound nucleon. Each model is based on different phenomena and applies in
different kinematic ranges. Due to lack of time, we will discuss here only some models
(typically representative of a class of similar models). A list of models and their region of
applicability is neatly represented in the schematic shown in Figure 2, taken from Ref. [2];
Figure 2. Regions of applicability of various models of bound nucleon
structure functions, taken from the review (2].
many more models are discussed in this review. Finally, we would like to emphasise that
all parts of the data cannot be explained by any one phenomenon. We believe that
modification of parton densities in bound nuclei is due to multiple effects occurring in the
nucleus. Hence, current models are mostly hybrid in nature. We shall concentrate on the
small and intermediate x regions in our discussions, ignoring Fermi motion effects at very
large jc values. We begin by discussing the rescaling model, which was chronologically one
ot the earliest models to explain the “tradiiionaf’ (large-x depletion) EMC effect.
2. Rescaling models
These use nuclear binding to explain the modification of nuclear parton densities. The
rescaling can be either in x [6] or [7.8). Their characteristic feature is an increase in the
confinement size in a bound nucleon, Hence, in a bound nucleon is effectively
increased by an amount,
or, equivalently, x increases by a factor 1 + £ /Ms* where £ is the average one-nucleon
separation energy, "and Mfi the mass of the nucleon. This results in a decrease of ^ at
570
D Indumathi
large x but cannot explain the small x shadowing. There is no "explanation" for the
change of scale; the model only provides a framework for discussing it. Furthermore,
it is not clear whether the sea densities are depleted as well or just the valence
densities.
These were initially discussed prior to the availability of any data [9]. However, the
models have undergone many modifications in detail. The underlying idea is that a
parton with momentum fraction jc of a parent hadron with momentum p, is localised
to within ^ from the uncertainty principle. On the other hand, the average
intemucleon separation (in the Breit frame) is - IR^Mn/p^ where is the nucleon
radius. When Az •~Azs* partons of different nucleons start to overlap spatially. This
happens when
JKX;, - \H1RhMn) ~0.1;
(5)
the effect saturates when
(6)
where R^t the nuclear radius, is not to be confused with the ratio of structure functions
or densities, R^. The idea is that overlapping partons can interact and fuse, and so
(a) reduce the parton density at small x < 0.1, and (b) correspondingly increase it
at intermediate x. Hence the ratio of bound to free 'nucleon structure functions is
parametrised as
■ 1 ; jc^ < jr < I ;
Here K is an unknown, free parameter and {x^/x -I) is the number of overlapped
nucleons. This was more of a geometric counting approach, and did not discuss the
origin of shadowing, i.e., the mechanism of fusion. That is, the K factor was fitted to
data. Soon a QCD'based purely perturbative calculation appeared [10,11]. The usual
DGLAP evolution equations [12] for free nucleon densities are linear in the densities.
The GLR-based MUller-Qiu equations are non-linear. The nonlinear terms arise when
the overlap of partons (or an increase in density) allows two gluons or a quark-anliquark
pair to fuse to one gluon, in a process w* ich is like the inverse of the usual parton
^'splitting" diagrams. The resulting evolution equations for quarks and gluons appear
as follows :
Nuclear structure functions
571
<?lne2 2n
27o?
— ^ 0(jcn - x)(.xg)^ + HDT;
16002 RJ
^In02 2n{''xi®^+ Pf,
2n
81a:
( 8 )
The first term on the RHS corresponds to the usual DGLAP term and HDT refers to higher
dimensional gluon terms [11]. Note that the extra terms due to fusion come with a relative
negative sign, and so deplete the densities at a given x (the equations are valid for small
X < Xn)- The effect of quark-gluon fusion is rather small, i.e, quark shadowing is indirectly
CompansuMu with |BMS Osi* far X« (*) iMl EMC dan
for Sn |29|.Calciifai«dnMlli m far Xc. ^4*2.24.
Figure 3. The ratio. R^, according to the model [13] in comparison with data for
Ca. Xe and Sn.
driven by the more dominant gluon-gluon fusion. Finally, these extra terms arc associated
with a l/Q^ factor so that the depletion at small x must decrease or even disappear with
increasing The observed quantity is the ratio of the bound to free nucleon structure
functions The free nucleon structure functions do not have any modification of l/(?
nature, but only the usual log behaviour. (The probability of parton fusion is considered
to be much smaller within a single nucleon). The bound nucleon structure functions,
according to the above model, have a leading logC^ behaviour with a depletion term at
small x which has a \/Q^ behaviour. Hence, the small-x shadowing, although predicted to
decrease with increasing will vanish at a rate in between that of a log^ and a l/Q}
behaviour, according to this model. This model is in fact one of the most popular models to
explain small-jr shadowing behaviour in nuclei. The predictions of the hybrid model of
Kumano and Miyama [13], which combines the ideas of rescaling and parton fusion, is
shown in figure B.-in comparison with available date for various nuclei. The fits are good;
72A(6).|5
572
D Indumathi
however, the model predictions are extremely sensitive to the initial from which the
densities are evolved.
4, Vector meson dominance models
This class of models also attempts mainly to explain the small x shadowing. Here the
basic idea [14] is that the interacting (virtual) photon fluctuates into a quark-antiquark
pair, or, equivalently, a meson, which then interacts with the target proton or nucleus.
Hence /^(A) scattering can be viewed as hadron-hadron scattering, with the photon
propagator being expressed as
/propagator 5 7 ; V=p, (9)
The vector meson-nucleus cross section is obtained by Glauber multiple scattering; every
scattering turns out to have an amplitude opposite in phase to the previous one, and of
decreasing magnitude :
A/^Ao-A,+A2+--^[l-(flOL
where a < I, thus leading to shadowing. Hence, at low x, the extra contribution to the
nuclear structure function is [14]
S^FUx,Q^)
1 y ^VA
^ " T + fv
9 »)
The model is again valid only at low x and cannot explain the conventional EMC effect. It
not only, predicts a significant decrease of shadowing with but also predicts that
shadowing decreases linearly as \/Q^, disappearing totally by about -10 GeV^. This
may not be borne out by Drell Yan data [5]. The model predictions at low x for various
nuclei are compared with data in Figure 4.
5. Nudemr effects and the parton model
This class of models [IS] continues to use the linear DGLAP perturbative evolution
equations with no fusion terms. Shadowing is then obtained by appealing to nuclear
binding. Since' bound nucleons lose typically an amount (■ the binding energy per
nucleon ^ 15 MeV) due to binding, bound nucleons have a larger spatial extent than free
nucleons [7]. If is the relative increase in radius of a bound nucleon compared to a free
one, due to the uncertainty principle, the momentum di^bution (x distribution) of bound
nucleons is different from free ones. Howevor, at the starting low scale from wbcm the
parton densities are evolved, the number density of partons as well as the total momentum
carried by each type, remains conserved. These three constraints are sufficient to fix the
bound nucleon densities in terms of the free parton distributions [15] aiid which is afise
Ca
Nuchar stntciunfiuictums
573
ttfm 4. The depeedeiK* of the mio, « diffeitm viloes of a. according lo the
apdd (14) ta compifiioii wilh NMC dm for Cl [31.
p,«new to die model. IlK «suU of this modification is a “pinching” of .he . distribution,
u shown in Figure 5.
Tim btodtog eoergy. b, conesponds to loss of energy of the bound nucleon, it is
auumed that due eMrgy teas is taken from toe “mesonic” component or the sea qutuks of
toe nocleou to this model. Hie bound nucleon sea density is Urns reduced from the free
nucleon to
when (S„), it toe momentum ftnction carried by the sea in a free nucleon a. the input
Kile. Since the nMOQS ere soft, tluB is s simll’is effect.
574
D indumathi
Hence, swelling prescribes the bound-nucleon densities at the input scale, The
sea densities are additionally depleted due to binding effects. These distributions are then
evolved to any scale, using the DGLAP equations.
Figure 5. The effect of nucleon swelling on tiK calcium input dismbutions
[15] ‘ the ratios of the modified to unmodified densities are shown for the
valence (u^., dy), .sea (5) and gluon (g) densities with respect to the GRV [16]
distributions for the free nucleon, and 64 = 0 . 1 .
»
Thefc is a further depletion of the sea densities which occurs at the lime of
scattering, due to nucleon nucleon interaction, arising from parton-nucleon overlap.
As discussed in the parton fusion models, whenever the struck parton has a small
enough momentum x < its wave function can overlap neighbouring nucleons. The
subsequent interaction due to the overlap was seen to deplete the small x distributions
by an amount K (see eq. (7)), where K was not calculable. Here, K is computed by
analogy with binding. Let the energy loss due to overlap of sea quarks with one other
nucleon be
U,(Q^ ) = PMn JJ* {X, ) -msiSA ))j-
and assume that the strength of this interaction is the same as that due to binding,
viz.,
. uaQ^) Ui^^) ( 10 )
Mn{Sa(Q^))2
U{^) being the binding energy between each pair of nucleons. Here, the possible ^
dependence of /Jis ignored. The only r6le of here is to provide the impulse which allows
the parton-nucleon overlap to occur. Then the extent of depletion of the sea at the scale Q
Nuclear structure functions
575
due to this overlap (called second binding effect) is given by eq. (7) with K a 2/3. The
model predictions for the jt, 0^, and A dependences of the ratios for He/D, C/D and Ca/D'
are shown in Figure 6.
Figure 6. The structure function ratios as functions of x for (a) He/D. (b) Li/D,
(c) C/D and (d) Ca/D according to the model [15], in comparison with data [3].
The dashed, full, broken and long-dashed curves correspond to = 0.5. 1, 5 and
1 5 GeV^ respectively.
Since the model has just the usual log dependence, the ratio has very little
dependence on for a fairly large range. Hence, this model predicts a similar
behaviour for both bound and free nucleon distributions. Earlier data typically was
consistent with little or no 2^ dependence. Recent data on Sn/C from the EMC
collaboration [18] seems to show a significant 2^ dependence. This is the only data for
which detailed 2^ dependences are available, with very high statistics, and consequently
small errorbars. This model is so far compatible with the data [17] as shown in Figure 7.
However, continued evidence for a significant 2^ dependence, especially at low x, will
mdicatc that the 2^ dependence of free and bound nucleon structure functions is not the
same.
We add, in brief, that the model can be straightforwardly extended to the spin
dependent case. Results [15.19] confirm that the ratio of the spin dependent bound and free
"iiruciure functions is similar to the unpolarised ratio, /?*. This has positive implications [19]
376
D Indumathi
for the extraction of spin dependent structure functions from lepton-nucleus polarised deep
inelastic scattering experiments.
Figure 7. The model prediction [17) for the dependence of the structure function ratio for
Sn/C, in comparison with data from the NMC [18). with statistical and systematic errors added
I . quadrature. Average (central bin) values of x are shown.
6. Siuninary and comments
Wc see tliat most models can fit the bound nucleon structure function, (or, equivalently,
the ratio, R*), as a function of x over most of the x range over which data is available.
However, these models generally differ with respect to the dependence, especially at
tmall X. This may be used to discriminate between them when more data becomes available
at small x. over a substantial range. This will esublish if higher twist terms are
Nuclear structure functions
577
significant, and enable the estimation of the bound nucleon gluon density, g^Oc, from
3F{/d\n about which very little is currently known.
F 2 and Drell Yan data are complementary in nature. Hence, we cannot cross check
the two sets of measurements against each other or establish the validity of any given
model. Semi'inclusive tc K,... hadron production in deep inelastic lepton-nucleus scattering
experiments can yield information on the valence combination. (uy+ dy), inside a nucleus,
at all X, by measurements of suitable combinations of cross sections PO]. Such
measurements can, in principle, discriminate between swelling and rescaling models.
Recently, uncertainty in AB collisions has been recognised to be due to nuclear
absorption effects [21]. It may be possible to separate these from conventional (initial state)
nuclear effects, provided the latter are well understood.
Many technical advances have recently occurred in the field of nuclear structure
functions. This gives hope that "parametrisations" of bound nucleon parton distributions
will soon be available, comparable in accuracy with free nucleon ones (like GRV [16],
MRS [22], CTEQ [23], etc). This is important in the light pf recent interest in the
knowledge of bound nucleon parton densities, not as a tool in understanding
nuclear/binding forces, but in order to be able to make suitable corrections to heavy ion
collision cross sections, in the ongoing search for Quark Gluon Plasma [24].
Acknowledgment
1 thank the organisers for giving me the opportunity to present this talk.
Rerercnces,
[ I ] J J Aubert el al The EMC Phys. Uu B123 275 ( 1982)
[2] For a recent review, sec M arneodo Phys Rep 240 301 (1994); see also L L Frankfurt and M 1 Strikman
Phys. Rep 100 235 for a more theoretical review ( 1988)
[^] P Amaudruz et al The NMC Nud Phys B441 3 (1995); M Arneodo el al The NMC Nud Phys. B441 12
(1995)
[4] M R Adams ei al The E665 Collaboration, Z Phys. C67 403 (1995)
[5] D M Aide etal The E772 Collaboration, Phys. Rev. Lett. 64 2479 (1990)
161 M Ericson and A W Thomas Phys. Utt. B128 1 12 (1983)>C H Llewellyn Smith Phys. Lett. B128 107
(1983); S A Akulinichev, S A Kulagin and G M Vagradov Phys. Lett. B158 485 (1985)
[7] R L Jaffc Phys. Rev. Utf. 50 228 (1983)
[8] R L Jaffe. F E Close, R G Roberts and G G Ross Phys. Lett. B134 449 (1984); Phys Rev D31 1004
(1985)
[9] N N Nikolaev and V I Zakharov Phys Lett B55 397 (1975); V I Zakharov and N N Nikolaev 5ov. J.
Nud. Phys. 21 227(1975)
[10] L V Gribov, E M Uvin and M G Ryskin Phys. Rep. 100 1 (1983)
Ml] A J Mueller and J Qiu Nud Phys. B268 427 (1986); J. (Jiu Nud. Phys. B291 746 (1987); K J Eskola
A/uc/./>6yj. 8400240(1993)
M^l V N Gribov and L N Lipatov Sov. J. Nud. Phys. 15 438 (1972); ibid, 675; Yu. L Dokshitzer, Sov. Phys.
JETP 46 641 -(1977); G Altarelli and G Paris! Nud. Phys. B126 298 (1977)
578
D Indumathi
[13] M Miyama. S Kumono Phys. Rev. C50 1247 ( 1994); Phys. Rtv. C48 2016 (1993) (the mutts pmenled in
Figure 3 were taken from this paper); S Kumono, M Miyama Phys. Lett. B378 267 (1996); R Kobayaiht
S Kumano, and M Miyama Phys. Lett B354 465 (1995)
[14] W Melnitchouk and A W Thomas Phy.s. Lett. B346 165 (1995): Phys. Rev, C52 3373 (1996) (the myiu
presented in Figure 4 were taken from this paper); G Filler. W Ratzki, and W Weiae Z Phye, A3S2427
(1995)
[15] W Zhu and J G Shen Phys. Uti B219 107 (1989); W Zhu and L Qian Phys. Rmv. C45 1397 (1992).
D Indumathi, W Zhu Z fUr Physik C74 1 19 (1997) (the results presented in Figures 5 and 6 were taken
from this paper)
[16] M GiUck, E Reya. and A Vogt Z Phys. C48 471 (1990) ; Z Phys. CSI 433 (1995)
[17] D Indumathi On the dependence of Nuclear Structure Functions. (Doftmuod Uidveiilty prepilat)
DO-TH-96-17. (1996), Z Phys. C769I (1997)
[18] NMC preliminary data on Sn/C; A MUcklich. PhD. Thesis, Ruprecht-Karia-Univeititic, Hekblberg.
1995; Michal Szleper, private communication
[19] D Indumathi Phy.s Utt. B374 193 (1996)
[20] D Indumathi and M V N Murthy Nuclear Effects on Parton Densities in Deep Inelastic Lepton Hadron
Scattering. Extended Abstract, DAE Symposium on Nuclear Physics, (Aligarh. India) (1919)
[21] D Kharzeev, H Satz Phys. Lett. B366 316 (1996); D Kharzeev. C Loiuenco. M Naidi. H Salz,Z
€74 307(1997)
[22] A D Martin, R G Roberts and W J Stirling Phys. Rev. D506743 (1994)
[23] J Botu et at The CTEQ collabontion, Phys. Utt. B304 159 (1993)
[24] T Alber el al NA49 collaboration, Phys. Rev. Utt. 75 3814 (1995); E. Scomparin et a/ NAM
Collaboration. Nucl. Phys. B610 331c (1996)
Indian J. Phys. 72A (6), 579-600 (1998)
UP A
— an intertmti onal journal
Heavy flavor weak decays
R C Vemia*
Department of Physics, Punjabi University,
Pauaia-147 002. India
Abstract : Weak decoys of heavy flavor hadrons ploy a special role in our understanding
of physics of the Standard Model and beyond. The measured quantities, however, result from a
complicated interplay of weak and strong interactions. Weak leptonic and semileptonic decays
are reasonably well understood, whereas weak hadronic decays present challenges to theory. In
this talk, we review the present status of exclusive weak decays of charm and bottom hadrons.
Keywords : Quark Model, Heavy flavor, weak decays*
PACS No. : 13.25.-k, l4.40.Cs, 1 1 30. Hv
1. Introduction
Soon after the discovery of Jj V^(cc) meson in 1974, weakly decaying pseudoscalar charm
mesons (D®, and ) were produced. Data on these hadrons have been collected at e'^e”
colliders and at fixed target experiments [1]. Study of B-physics began in 1977 with the
discovery of slate. However, further progress in measurements in naked bottom
sector could occur only in the last decade with the development of high resolution silicon
vertex detector and high energy colliders [2,3]. Three bottom pseudoscalar mesons
and /JP) have been studied whereas the fourth meson is also expected to be produced.
In the baryon sector, a few weakly decaying charm baryons and and one
bottom baryon A/, have been observed experimentally [Ij. A number of charm and bottom
baryons are expected to be seen in future experiments.
The weak currents in the Standard Model generate leptonic, semileptonic and hadronic
^laays of the heavy flavor hadrons. An intense activity on theoretical [2-7] and
^experimental [8-1 1] studies of these hadrons has been going on in this area. Experimental
J^iudies have mainly focused on precision measurements of branching ratios tor their
'^'^ak decays. Regarding their lifetime patterns, inclusive decays, exclusive leptonic and
e mail rcv@pbi.eniet.in
‘^ 2 A ( 6).16
1998 lACS
580
R C Verma
semileptonic decays, a complete picture is beginning to emerge [4], though a few
discrepancies yet remain to be explained. However, a theoretical description of exclusive
hadronic decays based on the Standard Model is not yet fully possible [3,5] as these involve
low energy strong interactions. Weak decays of heavy quark hadrons provide an ideal
opportunity to probe strong interactions, to determine the Standard Model parameters and to
search for physics lying beyond the Model.
In this review, present status of exclusive weak decays of heavy falvor hadrons is given.
We first discuss their lifetime pattern, leptonic and semileptonic decays. Then weak
hadronic decays of charm and bottom mesons are presented. Particularly, emphasis is given
on the factorization hypothesis and relating the hadronic modes with the semileptonic
decays. Finally, baryon decays are briefly introduced. In preparing this short talk, it has
been difficult to make a complete presentation of all the aspects of weak decays. For further
information, reference is made to some review articles [2-6].
2. ' Lifetime pattern of heavy flavor hadrons
At quark level various diagrams can contribute to the weak decays (Figure I ). These
are generally classified as (a) Spectator quark, (b) W-exchangc, (c) W-annihilation and
a a
Figure 1. Various quark level weak procc.sses : (a) .Spectator quark diagram,
(b) W-exchange diagram, (c) W-annihilation diagram, (d) Penguin diagram
Heavy flavor weak decays
581
(d) Penguin diagrams. W-exchange and .W-annihilation diagrams are suppressed due to the
helicity and color considerations. Penguin diagrams, contributing to Cabibbo suppressed
modes, are also expected to be small in strength. Thus the dominant quark level processes
seem to be those in which light quark/s behave like spectator. This simple picture then
immediately yields decay width for a hadron containing a b quark,
r =
Gjrml
192 *’
(I)
where Fpg is a phase-space factor. There is also a term with which is very small and
has been neglected [2]. Thus all the bottom hadrons are predicted to have equal lifetimes.
For charm hadrons also, the spectator quark model leads to equal lifetimes. Though order of
estimate of life-times is alright, the individual values |l] do show deviations from a
common lifetime ;
T(D+)-2.5T(D«)»2.5T(D;) = 5.0r(A;. )«3.0r(£;)= I0r(£“). (2)
These differences seem to arise from many considerations [6], like
(a) interference among the spectator diagrams (color enhanced and color suppressed)
which enhances D* life-time;
(b) nonspectator diagrams, life W-exchange diagram, which yield the following life-
time pattern for the charm baryons :
t(£?)<T(A:)<T(£;), (3)
Applying these considerations to the bottom hadrons, following observed pattern can be
obtained [12] :
r(/l*)<r(fl'>)-T(2>f)-T{fl*). (4)
However, an exact agreement with experiment for B meson and A/, lifetime ratio is difficult
to obtain. Recently, this ratio is described [13] by a simple ansatz that replaces the quark
mass with the decaying hadron mass in the nig factor in front of the hadronic width.
However, there is yet no theoretical explanation for the ansatz.
3. Weak leptonk und semilcptonlc decays
In the Standard model, leptonic and semileptonic decays naturally involve factorizations of
their amplitudes in terms of a well understood leptonic part and a more complicated
hadronic current for the quark transition. Lorentz invariance is then used to express the
matrix element in terms of a few formfactors which contain the nonperturbaiive strong
interaction effects [4]. Explicit quark models [14-20] have been constructed to construct the
hadron states which are then used to calculate the formfactors. In the last few years, a new
iheoretical approach known as the Heavy Quark Effective Theory (HQET) has emerged for
analyzing heavy flavor hadrons. In the limit of heavy quarks, new symmetries [21] appear
582
R C Verma
which simplify the calculaions of the formfactors. Nonperturbative approaches like lattice
simulations [22] and QCD sumrules [23] have also been used to calculate the formfactors.
Weak quark current generating the charm hadron decays is
Jf’-' ( 5 )
where q'q denotes the V-A current (I - ys)? and represents the corresponding
Cabibbo-Kobayashi-Maskawa (CKM) matrix element. Selection rules for these decays are:
Ag = - 1 , AC = - 1 , AS = -1 for Cabibbo enhanced c -» j + / + v/ process,
AC = - I , AC = -1 , AS = 0 for Cabibbo suppressed c J + / + i// process.
Similarly, the weak quark current
= VAcb) + V^(ub), ( 6 )
gives the following selection rules for bottom hadron decays ;
AC = I . A6 = 1 , AC = I for CKM enhanced ^ c + / + v/ process.
AC = I , A6 = 1 , AC = 0 for CKM suppressed h u + / + u/ process.
3. 1, Leptonic decays : P[j^ = 0 ” ) -> / + .
These decays are the simplest to consider theoretically, and are usually helicity suppressed
particularly when lighter leptons are emitted [24,25]. Decay amplitude for a typical <lecay
D tvf involves the decay constant /q defined as
<0\a^\D{p)> =ifDP^ ( 7 )
which measures the amplitude for the quarks to have zero separation. This leads to the
following decay width formula :
r{D(qc) ^ I flmomf I -
For D* decay, all the theoretical values [4] for ranging from 170 MeV to
240 MeV, are well below the experimental limit [26] :
/^< 310 MeV. (9)
For D* Ivf decay. Particle Data Group [1,27] gives the following values :
fos = 232 ± 45 ± 20 ± 48 MeV, 344 ± 37 ± 52 ± 42 MeV.
430![^± 40MeV. C®)
using the Mark and CLEO data. Potential models [4] give f p, between 190 MeV and
290 MeV. Lattice calculations [22,28] yield ; 220 ± 35 MeV, which matches with
Heavy flavor weak decays
583
QCD sumrules estimates [23]. More recently. E653 collaboration [29] has obtained
f Ds * 194 ± 35 ± 20 ± 14 MeV and CLEO result has been updated to [30] ‘ f ds •
284±30±30±10MeV.
For B-mesons, leptonic decays are strongly suppressed by the small value of
Lattice simulations give fg = 180 ± 40 MeV whereas the scaling law derived
inHQET [21],
fp = ^ (1 + 0 ( 0 )+ •■■)] ( 11 )
predicts a rather lower estimate /u ® 120 MeV [28] which is expected to increase due to
the radiative corrections. Potential model values [4] range from 125 MeV to 230 MeV.
QCD sumrules estimate : /^ = 180 ± 50 MeV is in good agreement with those from the
lattice calculations. Thus, theory predicts [3]
-4 » 4.0 X 10-^
tor the most accessible of the leptonic B decays because the large T mass reduces the
helicity suppression. Experimentally, the following upper limit is available [1] :
< 1.8 X 10-^
Measurement of fg decay constant at future b<factories would have a significant impact
on the phenomenology of heavy flavor decays. A precise knowledge of fg would allow
an accurate extraction of the CKM matrix element I. Moreover, it enters into
many other B-decay measurements, notably B-B mixing and CP violation in B-decays
[3,31,32]. The standard model allows B, B ^ 1*1' leptonic decays via box or
loop diagrams. Theoretical values [3,33] for such modes, are well below the present
experimental limits [1].
3.2. Semileptonic decays : P -> M[J^ =0" or 1“ ) + / + .•
With the enormous data samples now available for charm and bottom mesons,
iheir semileptonic decays, particularly emitting a pseudoscalar meson or a vector
meson, are well measured. These decays occur via spectator quark diagram and involve
no final state interactions. So these decays are the primary source of the CKM
elements and various formfactors. Decay amplitude for P{^'Q)“^ M{q'q)lvi is
given by
<M|y^(9fi)|i»>(v,r'‘O-rs)0- (‘ 2 )
Using Lorentz invariance, the hadronic matrix elements arc described by a few formfactors
'vhich are also needed in the analysis of the weak hadronic decays.
584
R C Verma
3.2. J. Semileptonic decays of charm mesons :
D-¥ Plvi Decays ;
When the final state meson is pseudoscalar, parity implies that only the vector
component of the weak current contributes to the decay, whose matrix clement is
given by [6,14),
(p(.')IMd(p))
(p*p')u -
- 9*1
P\(q^) +
9pPo (9^.
(13)
where Fi(0) = Fo(0) and ={p~ p')^ The formfactors for Cabibbo enhanced transition
represent the amplitude that the final stale (qs) pair forms a K meson. Energy of K meson
in the rest frame of D meson is linearly related to q^.
Ek =
mn + ml -
2m j
(14)
At q^ = q^^ = (nip - , the K meson is at rest in the rest frame ol D meson.
Then the overlap of the initial and final state is maximum and so the formfactor is ul
its maximum value. At q^ = 0, is maximum and so the formfactor is at its
minimum value. This q^ dependence is usually expressed through the pole domirfance
formula [14],
^(9^)
F(0)
I-9V"'?’
(15)
which is studied by measuring the differential decay rale [4). Present data [4,34 1 on
differential decay rale for D-> Riij yields, for IV,, I = 0.974 and the pole mass m* =
2.0010.11 ±0.16 GeV,
= 0.75 1 0.03. (16)
Quark model values lie between 0.7 to 0.8 [14-20], lattice calculations give 0.6 to 0.9 [22]
and QCD sumrules approach gives 0.6 [23] for this formfactor.
Decay width ratio of Cabibbo suppressed decay D— >7r/iij and the D^Klvf
serves to deliver j . Mark 111 and CLEO data [34] yield the following respective
values :
= 1.0:J»±0.l, 1,29 ± 0.21 ± 0.1.
These results are consistent with theoretical predictions which range front 0.7 to
1.4 [4).
Heavy flavor weak decays
585
D — > V[j ^ = 1 ■ )lvi decays:
When ihe final slate meson is a vector meson» there are four independent form
factors [14] :
{v(p\e)\j,\D(p))= £,.pae-^P'’p'<’V{g^)
-‘^"<v^-;p^q^{A:i(q^)-Ao(q^))\, (18)
where is the polarization vector of the vector meson, and = p')^ is the momentum
iransfer. Total decay width r[D -^KUvi)\s dominated by A, formfaclor. Ratios of other
rornifaclors V and Ai with A| are determined from the angular distribution [2-4]. Present
data [34] yield: .
A,^^* (0) = 0.56 ± 0.04, (0) = 0.40 ± 0.08,
(0) = 1.1 ± 0.2. (19)
Theoretically quark models [14-20] give large values Ai(0) = 0.80 to 0.88 and A2(0) = 0.6
to 1 2, whereas the predictions for ^(0) range from 0.8 to 1.3 in good agreement with
experiment. Lattice calculations [22] and QCD sumrules estimates [23] are in better
agreemept with experiment [4].
For Cabibbo suppressed mode, experimental value [1]
B[D* B{D* = 0.044:®^ ± 0.014, (20)
IS consistent with theoretical expectations [4,18] within the errors.
Scmilepionic decays of strange-charm meson -> 0/^/^]' + / + v/) have also
been measured [I]. These decays appear to follow the pattern of D decays in terms of the
torm factor ratios [4].
2 2. Semileptonic decays of B mesons :
For B-decays, following data i^ available for CKM enhanced mode [1] :
ij(B°-»D-rv)=1.9±0.5%,
B(fl“ -> D*-rv) = 4.56±0.27%.
B(fl- -»5®ri/) = l.6±0.7%.
-»D*®rv) = 5.3±0.8%.
586
R C Verma
Using \VJ = 0,038 ± 0.004, present data yield (34,35)
A|(0) = 0.65 ± 0.09, V(0)//\,(0) = 1 .30 ± 0.36 ± 0. 14,
y42(0)M ,(0) = 0.64 ± 0.26 ± 0. 1 2, (2 1)
which are consistent with quark models estimates (4).
In nonperlurbativc problems, exploitation of all the available symmetries is very
important. For the heavy flavor physics, the use of spin-flavor symmetries, that arc present
when masses of the heavy quarks are » Aq, leads to considerable simplifications [21] In
going to the limit all the formfactors are expres.sed in terms of one universal
function called Isgur-Wise function
= H‘1') =A2(q^)
■M,
2VMb/W„
C(w).
( 22 )
where ( 0 = \)g These reimions are valid up to pcrlurbative and power corrections
[4.28|. Theoretical difficulty in making predictions for the form factors lies in
calculating these corrections with sufficient precision. At present, in the presence ol
the.se corrections, 1.30 and 0.79 are obtained (4,35| for the ratios VIA\ and
respectively. ,
Charmless scniileplonic branching Iraction is expected to be around \*'/t of tinii ol
the .scniileplonic decays emitting charm meson ba.scd on the pre.sent estimate |V'„/,/f J =
0.08 ± 0.02 ( 1 1. Heavy quark symmetry is less predictive for heavy light decays than it is
for heavy — > heavy ones. Experimentally two branching ratios have been measured ieccml_\
by CLEO collaboration 136] :
= (1.8 ± 0 4 ± 0,3 ± 0.2) X
BiB^^ p /M-) = (2.5 ± 0.4:};" ± 0.5) x 10
which arc consistent with theoretical expectations.
In addition to single meson emission in the final slate, scmilcplonic decays
also permit the production of two or more me.sons. Quite often these mesons are
produced through decay of a meson icsonancc produced in the weak decays |1J. for
D-mesons, known resonant exclusive modes come close to saturating the inclusive
scmilcplonic rates. In B decays, there is some room for nonresonanl niulli-hadron
final Slate. Semilcptonic decays of charm and bottom baryons have al.so been observed
However, experimental results currently have limited statistical significance. Much
larger data on these decays arc expected In the future, allowing tests of various theoretical
models [37].
Heavy flavor weak decays
5VI
4. Weak hadronic decays
Weak hadi'onic decays of heavy flavor hadrons are considerably complicated to treat
iheoretically. At current level of understanding these require model assumptions. Even if
ihe short distance effects due to hard gluon exchange can be resummed and the effective
Hamiltonian has been constructed at next to leading order, evaluation of its matrix elemetits
IS not straightforward. Various theoretical and phenomenological approaches have been
employed to study weak hadronic decays. Broadly, these are :
(i) F lavor symmetry frameworks :
In the flavor symmetry frameworks, initial and final state mesons and weak Hamiltonian
belong to their irreducible representations. Using Wigner-Eckart theorem, decay amplitudes
arc expressed in terms of few reduced amplitudes. Thus useful sumrules among different
decay amplitudes are obtained [38] using isospin and SU(3) flavor symmetries. However,
SU(3) violation has been shown by the charm meson decay data [39].
(mJ Quark line diagram approach :
Quaik diagrams appearing in the weak decays are classified according to the topology of
weak interaction with all the strong interaction effects included. With each quark line
diagram, a corresponding parameter is attached and appropriate C.G. coefficients are
iiiiioduced depending upon the initial and final state particles [40]. Using experimental
values, relative strengths of various quark diagrams are then obtained.
(ml Reldtivistic and nonrelativistic quark models :
Lxplicii quark model calculations have been done to determine the strength of various
quark level processes. These models usually employ factorization [41] which can be used to
I elate hadronic decays to the semileptonic decays [42].
(ivj Nonperturbative methods :
QCD sumrules [23] approach has provided the general trends but agreement with present
data IS poor at a quantitative level. Lattice QCD calculations [22], though promising, are
sidl m progress. Further these methods have their own uncertainties.
At present extensive data [ 1 ,43] exist for weak hadronic decays of charm and bottom
mesons; though in the baryon sector, only a few decay modes of and have been
studied experimentally [ 1 ,44]. The heavy flavor hadrons have many channels available for
then decay involving two or more hadrons in their final states. However, for charm hadron
decays, two-body decays dominate the data as multibody decays sho.w resonant structure.
to the considerably larger phase space that is available in bottom hadron decays and to
die much higher number of open channels such a feature cannot extend to the bottom
hadrons. Nevertheless these are expected to make up significant fraction of their hadronic
decays.
12A(6)-17
588
R C Verma
Most of the observed two-body decays of heavy flavor mesons involve pseudoscalar
(P) and vector (V) mesons (s-wave mesons) in their final state : P PPIPVIVV. in
addition, some of the decays of charm mesons emitting axial (A), Scalar (S) and tensor (T)
mesons (p-wave mesons), like P -> P + AISIT have also been measured [I]. Bottom
mesons, being massive, can also decay to vector meson and another p-wave meson, or two
p-wave mesons. In addition to these modes, weak decays accompanying photon (like B
AT* + )^ are also observed.
4. 1 . Weak hadronic decays of charm mesons :
The general weak ® current weak Hamiltonian for hadronic weak decays in terms of the
quark fields is given by
(23a)
for Cabibbo enhanced mode,
,^aC=-l.^=0 ^ ^ ^ y^ y.^
for Cabibbo suppressed mode, and
(23cl
for doubly Cabibbo suppressed mode. Since only quark fields appear in the weak
Hamiltonian,' the weak hadronic decays are seriously affected by the strong interactions
One usually identifies the two scales [6] in these decays : short distance scale at which
W-exchange takes place and long distance scale where final state hadrons are formed
As the hard gluon effects at short distances are calculable using the perturbative QCD,
long distance effects, being nonperturbative, are the source of major problems in
understanding the weak hadronic decays. The hard gluon exchanges renormalize the
weak vertex and introduce new color structure [6]. Effective weak .Hamiltonian thus
acquires effective neutral current term. For instance, weak Hamiltonian for Cabibbo
enhanced mode becomes
„ ^^.V^V',[ct(ud){sc) + C2{sd)(uc)\, (2'”
where the QCD coefficients C| = ^(c+ + c. ), cj =
-| /2h
with £/ = = 8 and ^ = 1 1 - 4 ^ ^ , A// being the number of effective
cxjml)
flavors, //the mass scale [6].
Heavy flavor weak decays
389
4.1.1. D -» PP/PV/W decays :
Decay width for a two-body decay of D meson is given by
r(D -» M, -(■ Mi ) = C^(CKM factors)^
X ^ (mass factors) I ((M, Mj )l0, |D)|^ (25)
where / denotes the angular momentum between the final state mesons M| , M 2 , and i
denotes the helicity of these mesons. The operators O, correspond to the quark processes
responsible for the decays. In the evaluation of matrix element of the weak Hamiltonian,
one usually applies the factorization hypothesis [6,14] which expresses hadronic decay
amplitude as the product of matrix elements of weak currents between meson states.
(P.Pi ID) - (/>, \J, |0>(Pi |D), (26a)
(PV\HJD) - [{PMJ0)(1'|7**'1D) + (V|yj0)(P|ytx(D)]. (26b)
(V,Vi|H„|D)~(v,|y,|0)(l/l|7*»‘|D}. (26c)
Matrix elements of weak current between meson and vacuum state are given by eq. (7) and
(V(p.e)lyjo) (27)
Meson to meson matrix elements appearing here have already been given in eqs. (13) and
(IK). Thus factorization scheme allows us to predict decay amplitudes of hadronic modes-in
icrrn.s of the semileptonic formfactors and meson decay constants.
For the sake of illustration, we consider Cabibbo enhanced decays D PP.
.Separating the factorizahle and nonfaciorizable parts, the matrix element of the weak
Hamiltonian, given in eq. (24), between initial and final states can be written [6,45] as
(/>,/>i|H„|D) = ^V„,V;(fl,(P,|(Hd)|0)(/>2|(Jc)|D)
+ 02(^2 |(Jd)|0)(P,|(uc)|D)
+ (c2{P,/’2|H»|D) + c,(P,/>2|W1|D))_^,__^' (28)
where fl ,,2 = c ,.2 +
In addition, nonfaciorizable effects may also arise through the color singlet currents [46].
Matrix elements of the first and the second terms in eq. (28) can be calculated using the
luciorization scheipe. So long as one restricts to the color singlet intermediate states.
590
R C Verrna
remaining terms are usually ignored and one treats a\ and 02 as input parameters in place
of using Nf. = 3 in reality. The charm hadron decays arc classified in three classes,
namely
(i) Class I transitions that depend on ay (color favored),
(ii) Class 11 transitions that depend solely on ^2 (color suppressed),
(iii) Class 111 transitions that involve interference between terms with a\ and
It has been believed [6.14] that the charm meson decays favor » limit, />.,
Gy ~ 1.26, ai = -0.51. indicating destructive interference in D* decays.
4.1.2. Long distance strong interaction effects :
The simple picture of spectator quark model works well in giving reasonable estimates for
the exclusive semilcptonic decays. However, success in predicting individual hadronic
decays is rather limited. For example, spectator quark model yields the following ratios :
r(D«
for Cabibbo enhanced mode and
r{D^
f = 0, 9 (2 . 5 ± 0 , 4 Expt. ) (29b)
r(D«
for Cabibbo suppressed mode.
Similar problems exist for D—^ K*7tf K*p decay widths. Besides these, other
measured decays, involving single isospin final state, aLso show discrepancy with theory.
For instance, the observed K^^rj andD® decay widths are considerably
larger than those predicted in the spectator quark model. Also measured branching ratios
for r\lr \' are found to be higher than those predicted by the
spectator quark diagrams. F\ir rj/ q' + 7t*, though factorization can account for
substantial part of the measured branching ratios, it fails to relate them to corresponding
semileptonic decays > t?/ + consistently [47].
In addition to the spectator quark diagram, factorizable W-exchangc or W-
annihilaiion diagrams may contribute to the weak nonicptonic decays of D mesons.
However, for D — > PP decays, such contributions arc helicity suppressed. For D meson
decays, these are further color- suppres.sed as these involve (}CD coefficient C 2 , whereas for
PP decays these vanish due to the conserved vector (CVC) nature of the isovector
current (mJ) [47J.
It is now established that the factorization scheme does not work well for the charm
meson decays. The discrepancies between theory and experiment arc attributed to various
long distance effects which are briefly discussed in the following.
Heavy flavor weak decays
591
(i) Final state interaction effects :
Elastic final state interactions (FSI) introduce phase shifts in the decay amplitudes [48],
which can be analyzed in the isospin framework. For instance, the isospin amplitudes 1/2
and 3/2 appearing in Kn decays may develop different phases leading to
A{D'^ -^K-n*)= ].
(30a)
(30b)
A(D* = V3-4j/ 2 e'*’" .
(30c)
Similar treatment can be performed for Kp, K*p modes. These decays are
seriously affected as their final states lie close to meson resonances. Experimental data on
these modes yield [48,49] :
|'^i/ 2 |/|^ 3 / 2 | = 3.99 ±0.25 and ^ 3/2 - ^ 1/2 = 86 ± 8 ° for /f;r mode.
|^i/ 2 |/|>^ 3 / 2 | = 5.14±0.54 and 63/2 - 5 i /2 = 101 ±14^^ for /f*;: mode,
hi/ 2 |/|^i/ 2 l = 3.51 ±0.75 and ^ 3/2 - 5|/2 = 0±40® for/fpmode,
hi/ 2 |/h 3 / 2 | = 5 .l 3 ± 1 . 97 and 53 , 2 - 5,/2 = 42±48‘^ forF*pmode.
loi Cabibbo enhanced mode, and
|'^o|/M 2 | = 3.51 ±0.75 and 5o ~ (52 = 0 ± 40° for ;r 7 r mode,
|4o|/|/4,| = 3.51±0.75and5o -5, = 0 ±40° for mode,
tor Cabibbo suppressed mode.
In addition to the elastic scattering, inelastic FSI can couple different decay
ihannels. For example, £)-♦ /C*;rand D -> ^p decays are found to be affected by such
inelastic FSI [48].
ill} Smearing effects :
further, in certain decays a wide resonance is emitted, like D-^ Kp. The large width
uf ihe meson modifies the kinematical phase space available to the decay. These effects can
be studied using a running mass (m) of the resonance, and then averaging is done by
introducing an appropriate measure r(m^) like Brcii-Wigncr formula. For instance,
pp decay width is calculated as [50] :
r(D->Pp)» f r{m^)r(D^ Pp{m^))dm^,
Jim,
( 31 )
592
R C Verrtia
Such effects can be as large as 25%. For example,
r{D° ^K-p-^)/r(D0 ^K-p*) =o.n. 02)
Smearing effects have been studied [51] forD -> VValso.
(Hi) Non facto rizable contributions :
Indeed factorization, combined with the assumption that FSI are dominated by nearby
resonances, has been in use for the description of charm meson decays. Recently, this issue
has been reopened. In the factorization scheme, one works in the large limit, and ignores
the nonfactorizable terms, which behave like l//V^.. However, this approach has failed when
extended to B meson decays [52]. So D-meson decays are being reanalyzed keeping the
canonicar value N^ = 3, real number of colors. Efforts have been made to investigate the
nonfactorizable contributions. It is well known that nonfactorizable terms cannot be
determined unambiguously without making some assumptions [45] as these involve
nonperlurbaiive effects arising due to the soft-gluon exchange. QCD sumrulcs approach has
been used to estimate them 153], but so far these have not given reliable results. In the
absence ot exact dynamical calculations, search for a systematics in the required
iionlaclorizable contributions has been made using isospin [54] and SU(3)-navor-
symmetrics [46].
4.1.3. D P(O’) + p - wave meson (0'^, !*, 2^} decays
Weak hadronic decays involving mesons of intrinsic orbital momentum / > 0 in final state
arc expected to be kinematically suppressed. Some measurements are available on^hese
decays. Contrary to the naive expectations, their branching are found to be rather large [1].
Estimate for formfaclors appearing in the matrix elements <p- wave meson \ J \D> are
available only in the nonrclativistic ISGW quark model [17,18]. In general, theoretical
values are lower than the experimental ones [55].
4 2. Weak hadronic decays ofB-mesons :
Weak Hamiltonian involving the dominant b c transition [2,3] is given by
= ^\V,,Vl,(cb){du) + V,i,Vl,(cbHsu)
+ 1/,* v;j midc) + 1',* V'; m (?c)]. (33)
A similar expression can be obtained for decays involving b u transition by replacing
cb with ub . Following ^ = 1 decays modes arc allowed :
(i) CKM enhanced modes :
4C= 1,AS = 0, and4C = 0,dS = - I; (^*^
(III CKM suppressed modes :
4C= l,4S = - l,andzlC = 0. 4S = 0:
(34b)
Heavy flavor weak decays 593
(iii) CKM doubly suppressed modes :
^Ca- 1, ASa-l,anddCa-l. d5 = 0. (34c)
These provide a large number of decay products to B-hadrons. Including hard gluon
exchanges, the effective Hamiltonian can be written as
Hat = V'*j|a|[(rftt)(cfc) + (?c)(cfc)]
+aj[(c«)(d/») + (cc)(Jfc)]}. (35)
In the large limit, one would expect :
fl, «C, a 1.1, fl2 “^2 = -0.24.
4.2. 1. Determination of a i and ^ ;
Like charm meson decays, depending upon the quark content of mesons involved, B-meson
decays can also be classified in the three categories. Several groups have developed models
of hadronic B-decays based on the factorization hypothesis [2,3]. Recently, it has also been
argued that the factorization hypothesis is expected to hold better in the heavy quark limit
[56], for some decay channels, as the ultrarelativistic final state mesons don't have time to
exchange gluons. Present data seem to go well with theoretical expectations for most of the
B-meson decays [3]. For instance,
= (36.)
B(B^
^ = 3.4 (4.5 ± 1 .2 Expt.). (36b)
B(B^
By comparing B~ and B^ decays, i I, \ 02 I and the relative sign of ^ 2/^1 can be
determined. Thus B^ D^n- / D^p' / / D*^p- yield :
Ifl, 1= 1.03 ±0.04 ±0.16, (37a)
decays yield :
I fl2 1 = 0.23 ±0.01 ±0.01, (37b)
and data on 5“ clearly yield [3,52] :
0.26 ±0,05 ±0.09. (37c)
Note that though magnitude of the ratio is in agreement with theoretical expectation, its sign
>s opposite indicating constructive interference in B' decays. Other uncertainties of decay
constants, FSI and formfactors may change its value but not its sign [3]. This situation is in
contrast to that in the charm meson decays, where the ratio 02/01 = - 0.40 implies
destructive interference in decays. Interestingly, the constructive interference enhances
the hadronic decay width of meson and reduce its semileptonic branching ratio [57]
bringing it closer to the experimental value.
594
R C Verma
4.2.2. Final state interaction :
Factorization breaks down in the charm sector due to the presence of final state interactions.
The strength of such long distance effects in B-decays can also be determined by
performing the isospin analysis of related channels, such as B D;r decays. At present
level of experimental precision, there is no evidence for nonzero isospin phase shifts in
B-decays, as the data gives [3] cos (5|/2 - S 3 / 2 ) > 0.82 for B Dn.
4.2.3. Tests of factorization :
Since a common matrix element {M j J j B) appears in both semileptonic and factorized
hadronic decays, the factorization hypothesis can be tested by comparing these two
processes. Eliminating the common matrix terms in these decays, the following relation can
be derived 1 2,3,57] ;
r(/?o
= 1.22 ±0.1 5 (theory), 1.1410.21 (Expt.). (38a)
Here, we require that the lepton-neutrino system has the same kinematic properties as docs
the pion in hadronic decay. Similar relations can be obtained for B^ — ► D*pand B^ ->
D'ii\ decay. s, ,
(B"
dq
</ - = ni '
= 3.26 ± 0.42 (theory), 2.80 ± 0.69 (Expt.),
r(B« -->D‘-^ar)
dq-
= 3.0 ± 0.5 (theory), 3.6 ± 0.9 (Expt.).
Theory agrees well with experiment within errors.
(38b)
(38c)
4.2.4. Application of factorization :
Having factorization tested, one may exploit this to extract information about poorly
measured semileptonic decays. For example, integrating over ^^-dependence and using
experimental value B{B~ D**^Tr) = 0. 1 5 ± 0.05, one obtains [3] :
B{B -4 D**/v) = 0.48 ± 0. 16% ( 1 .00 ± 0.30 ± 0.07 Expt.).
(39)
Heavy flavor weak decays
595
Another application of relating hadronic mode with semileptonic decay is to determine .
For instance, D*^D; ) = 0.93 ± 0.25% gives [3]
fo, =271±77MeV. (40)
using B(D^ -> 071*) = 3.7%. Similarly, one can obtain
/^. =248±69MeV. (41)
4.2.5. Results from heavy quark effective theory :
Spin symmetry, appearing in the limit of heavy quark mass, combined with factorization
relates different decays [3]. For instance,
B{B^-^D*7t-)
; = 1.03(1.11 ±0.22 ±0.08 Expt.), (42)
-» D*p- )
= 0,89 ( 1.06 ± 0.27 ± 0.08 Expi.). (4.1)
B{B"^D'*p-)
llsiiig a comhinaiions ol' HQET, factorization and data on semileptonic decay B — > D*/iv,
Manuel etui (581 have obtained the following predictions for
-» D^p-)
B{B'> -» D*7t-)
= 3.05,2.52, 2.61
(44)
loi ihrcc dilTereni parunietcrizations of the Isgur-Wise function. Experimental value for
this I alio IS
-♦ D^-)
= 2.7 ±0.6, (Expl.)
Similarly prediclions have al.so been made lor B — > DD. j D D. j D
fleCcivs (31.
4 2 f) Rare B-(leea\ ^ .■
C'liarnilcsN decays involving b u transition, like B KTdKplKK arc important to find
1 jv to probe penguin contributions and to study CP-violation [3,591. Weak radiative
B-ineson decays present a verysensitive probe ot new physics, like Supersymmetry particle
coninbutions. Precise measurements of exclusive radiative decays, like B K*^ would
ilnovv light on elements [2,3]. B-mesons have enough energy to create p-wavc mesons
'il^o Branching ratios ol such decays have been estimated using the ISGW model (60].
13 mesons provide an unique opportunity to study baryon-aniibaryon deeays ol a meson.
Hinve\er. only a few upper limits are available experimentally [1,61 1. There is now a
considerable experimental evidence lor B - B oscillations, which can be used to determine
l ,./and elements [2.3|.
^2XifO-lS
596
R C Verma
4. 3. Weak hadronic decays of baryons :
For heavy flavor baryon decays, data has only recently started coming in. Two-body decays
of the baryons are of the following types :
l/2-^)/Z)(3/2^) P(0-)A'( I -).
Experimentally, branching ratios of almost all the Cabibbo enhanced -> )
+ ^(0“) decays have now been measured [1,44]. A recent CLEO measurement [62] of decay
asymmetries of A* -> give the following sets of PV and PC amplitudes (in
units ofCfKwV'c.x lO-^GeV^):
a(A* -> =
or
-4.3tS.
B(A* An*) =
+ 12.7!^?
or
+ 8.9!^ i;
= +1-3:!??
or
+5.n?.
B(a* -^1* n°)
= -17.3!?^
or
-4.l!5o-
Recently, CLEO-II experiment [63] has measured B{E^ = 1.2 ± 0.5 ± 0.39{.
This small data has already shown discrepancies with conventional expectations In the
beginning, it was thought that like charm meson decays, charm baryon decays may he
dominated by the spectator quark priKCss. This scheme allows only the emission of if Iff
and mesons. However, observation of certain decays* like
A* —> K * / Ln, Lq gives clear indication of the nonspectator contributions. In fact. W-
exchange quark diagram, suppres.sed in the meson decays due to the helicity arguments, can
play a significant role due to the appearance of spin 0 two-quark configuration in the baryon
structure. Due to the lack of a straightforward method to evaluate these terms, tlavoi
symmetry [64] and model calculations [65] have been performed. So far no theoretical
model could explain the experimental values
Study of bottom baryon decays is just beginning to start its gear. So lar. only
exclusive weak hadronic decay A/, J f y/ + A has been measured. Recent CDf
Collaboration experiment |66| gives BiA/, —^Jlyz + A) = (3.7 ± 1.7 ± 0.4) x 10^ which
is consistent with theoretical expectation 167].
5. Conclusions
In the last several years, tremendous progress has been achieved in understanding the heavy
flavor weak decays We make the following observations :
(I) Leptonic decays arc the simplest to be treated theoretically, but base very .small
branching ratios. Since a direct determination of meson decay constants is highU
desirable, particularly for B-B mixing, it is impormnt to improve their
measurements as larger data samples arc accumulated.
Heavy flavor weak decays
597
(2) Semileptonic decays are next in order of simplicity from theory side. Here all the
strong interaction effects are expressed in terms of a few formfactors, which arc
reasonably obtained in theoretical calculations, based on quark models,* HQET,
lattice simulation and QCD sum-rule approaches. However, higher precision
measurements are needed to find
(3) Weak hadronic decays experience large interference due to the strong interactions
and pose serious problems for theory, particularly for the charm hadrons. Though
qualitative explanation can be obtained for these decays, discrepancies between
theory and experiment indicate the need of additional physics. For instance, final
state interaction effects play significant role at least in the charm meson decays.
Smearing effects due to the large width help to improve the agreement when a wide
resonance, like p, is emitted in a decay.
(4) Results from CLEO II have significantly modified our understanding of weak
hadronic B-decays. Data on their branching are now of sufficient quality to
perform nontrivial tests of factorization hypothesis. It seems to be consistent at the
present level of experiment. Large sample of B-decay data will be obtained in next
few years which will present more accurate tests for the factorization scheme.
(5) The ratio a 2 la\ is demanded to be positive for bottom meson decays in contrast to
what is found in the charm meson decays. This has opened the issue of
nonfaciorizable terms for the weak hadronic decays. It is now clear that significant
nonfactorizable contributions are there in the weak hadronic decays of charm
mesons. For bottom sector, a quantitative estimate .of their size require precise
measurements of their decays. Study of rare decays, like radiative decays and
charmless B-decays, has a good potential to throw new lights on our understanding
of the penguin terms and CP violation.
{(^) Weak hadronic decays of charm baryon have recently come under active
experimental investigation, though search for bottom baryon decays is merely
begun. These decays are difficult to treat theoretically. Observed data for decays
clearly demand significant W-exchangc contributions. More data on baryon
decays, which will be accumulated in the near future, is expected to confront
iheory with new challenges.
Acknowledgments
• 'luncial assistance from the Department of Science & Technology, New Delhi (India) is
'hiinktully acknowledged.
Kcitrtnctt
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598
R C Verma
|31 T E Browder and K Honscheid ‘B Mesons'. UH 51 1-816-95, appeared in Prog. Nucl Part. Phys 35
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|K| S N Ganguli 'Physics for LEP / ' and A Guitu ‘Physics for LEP 2'. presented at this symposium
1 9] V Jam Retent Re.sults from CLEO'. presented at this symposium
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1 1 1 1 P (Quintas ei al 'The Standard Model and Beyond', Fermi lab-FN -640 (1996). S D Rindani 'New PHysk v
at e'*'i~ colliders', presented at this symposium
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1671 H Y Cheng 'Nonlepionic Weak Decays of Bottom Baryons ’. lP-ASTP-06-% (19%)
Indian J. Phys. 72A (6), 601-619 (199S)
UP A
— an intenulioiuil journal
Quark gluon plasma— current status of properties
and signals
CP Singh*
Depanmenl of Physics, Banaras Hindu University,
Varanasi-221 005, India
Abstract : The current status of properties and signals of quark gluon pla.sina (QGP) is
reviewed. We find that the simple equations of state (EOS) used for the description of QGP os
well 08 hadron gas (HG) shold be properly modified to account for the interactions present. We
briefly review the promising signatures existng in the literature. We find that the recent
suppression observed by NA50 experiments gives a clear hint for the deconfinemeni
phase transition. We discuss how the variation of 0/{p * at) with energy density can yield an
unambiguous signal for QGP formation. Finally we discuss the recent work connected with the
baryon-density inhomogeneity created in the early universe due to a quark-hadron phase
transition.
Keywords : (^ark gluon plasma, heavy ion collisions, equation of state, hodron gas.
early universe
PACSNos. : 25.75.-q,2l 65+f. l2.39.Fc
1. Introduction
QCD predicts the existence of a phase transition from an ordinary hadronic matter to a
plasma of quarks and gluons (QGP) whenever the energy density £ exceeds its value
existing inside a proton (£p = 0,5 GeV/fm^). A large energy density can be achieved in two
ways : cither by compression which means putting more and more particles in a given
volume, or by heating the vacuum which means increasing the particle kinetic energies. In
the new phase hadrons dissolve into weakly interacting quarks and gluons and an ideal
colour-conducting plasma of quarks and gluons is formed. In a QGP medium, the long
range colour force is Debye screened due to collective effects in the same way as noticed in
the case of an -electromagnetic plasma. In QCD, the potential consists of two parts ; one
i^oulombic part and the other is a linearly rising connnemcnl potential. The long range
behaviour of this potential is screened in a dense medium as (jf — ► (1 - Thus at
'ery high density when r $ r©' colour screening will dissolve a hadron into its coloured
xiail cpsingh 9 bananK.emei.in
© 1998 lACS
602
C P Singh
quark constituents so that a colour-conducting phase appears from a colour-insulator phase
through a deconfining phase transition.
The discovery and proper understanding of QGP is of paramount significance for
QCD since it foretells the long-range behaviour where the theory is still poorly understood.
The two different phases correspond to different states of vacuum in QCD ; the perturbative
vacuum in which quarks and gluons propagate almost freely and the physical, non-
perturbative vacuum which is relevant for the low-energy hadrons. Phase transition in QCD
at high temperature is useful in the cosmological studies, e.g. possible formation of a
baryon number inhomogeneity which persists to the epoch Ter 100 MeV could affect
primordial nucleosynthesis yields of light elements. Such inhomogeneity can also result m
the formation of a stable or metastable strange quark nugget and it can also explain the dark
matter. It has also been suggested that primordial black holes and primordial magnetic-
fields can originate in a first order QCD phase transition. The QGP phase of matter can also
provide a key information about the inner core of a neutron star having a very large nuclear
density.
Ultra-relativistic heavy-ion collisions provide us the opportunity to search for the
QGP formation in the laboratory. A large number of particles produced in a finite volume
of the collision give rise to a large value of energy density. Such a large energy density can
be achieved in two ways (which is shown schematically in Figure la) : either by heating the
nuclear matter so that the kinetic energy of the particles bficomes higher or by compressing
the matter so that the baryon density becomes extremely large. The evolution of produced
matter is governed by relativistic hydrodynamics which is shown in Figure lb. In bothMie
cases the hadrons come closer to each other and the distance between the quarks decreases
resulting in a very weak force between them. The phase diagram of hadron gas and QGP
has been shown in Figure 2. One expects that the hadrons exist in the low density, low
temperature region while the high density, high temperature region is populated by quarks
and gluons. The precise determination of the critical line is done by using the Maxwell’s
construction of the first-order equilibrium phase transition.
The simplest (JCD motivated model which indicates the formation of CJGP is the bag
model of hadrons. In the bag model, a hadron consists of a set of quarks moving freely
inside a bag of finite dimension and quarks acquire infinite mass outside the bag. Free
quarks and gluons can only propagate where the complex structure of QCD vacuum has
been destroyed. The value of the vacuum pressure B represents the energy required as a
result of such kind of vacuum re-alignment. In other words, the-phenomenological quantity
summarises the interaction effects which arc responsible for a change in the vacuum
structure between the low temperature and the high temperature phases. Minimising the
energy of a spherical bag, one gets the equilibrium energy density inside a proton Eq = ®
which is also the latent heat density required for a deconfining phase transition. If the
pressure of the quark matter inside the bag is increased, there will be a point when the
outward pressure arising due to the kinetic energy of the quarks becomes larger than the
inward vacuum pressure B. In such situations, the bag cannot confine the quarks and it will
Quark gluon plasma— current status of properties and signals 603
result into a new deconfined phase. The pressure of the quark matter increases when the
temperature of the matter is large enough and/or the baryon density is quite large,
CENTRAL NUCLEAR COLUSiONS
Btlore colHtlon
CipKt maximuin baryon donsity (pj to bo acMavad In
atoppad nuclei at e. 10 QaV/N for uranltini.
Figure 1(a). Stheinaiic diagram
ol ultiu-rclativiMic nucleu<«-nucleus
collision.
Cato 2: Nudaar “TRANSPAMENCr'
Caniral roglon
<maaon-rtch, p, a* 0)
Eipact minimum baryon danalty In control roglon oftar
nudol paia throiigb aacb other at E » 30 QaV/N
(aquivalMt to E, » 2 TaV/N).
Figure J(b). Space nrrte evolution diagram in u nucleus-nucleus collision.
72A(6)-19
604
C P Singh
The QCD Lagrangian is written as
where the indices a, j. k are the colour-indices (a » I - ■ - 8; ^ s 1 ■ • ■ 3) and the covariant
derivative is
D';^ =S,,d>‘ +ig{T,),tG^ ( 2 )
Similarly the gluon field tensor is
Fr = d>‘G:-a^G!!-gj^^.Gi;c:: O)
Here are the gluon fields, T„ are the SU(3) colour generators, is the strong coupling,
is the quark mass matrix and are the structure constants of SU(3)c- An important
symmetry of VgcD eq. ( I ) is the chiral symmetry. For massless fermions i*QCD is invariant
under global flavour rotations U/t and Ul for right and left handed spinors. For Nj flavours,
c.iily iiiiivi.i:>c
baryon denaiiy p •*'^0 • * ■
Figure 2. Phase diagram of the strongly inicracling matter showing the
hadionic phase ai low temperature and baryon density, the transition region
(mixed phase), and the QGP pha.se
the matrices are Nf x /V^and they form a group U{Nf) x U{Nj^ which has a proper chiral
subgroup SU(A^/1 x S\J{Nf). For = 2, the group is SU(2) x SU(2) and two Noether
currents for a combined llavour and transformations arc V" = /2)V^ and
- \pY^ Xs ( /2) where r^^’s are the Pauli isospin matrices. The vector current is
conserved corresponding to isospin-invariance. However, partially conserved axial current
satisfies PCAC relation where is the pion field. It means that the
vacuum is not invariant under isoaxial rotations and hence the vacuum expectation value of
operators ^ 0.
2. Lattice calculations
Phase transitions can be examined through the behaviour of an order parameter. A
discontinuous change m the order parameter at the critical point characterizes a first order
Quark gluon plasma— current status of properties and signals
605
phase transition. For a second order phase transition, the order parameter changes
continuously. In lattice gauge theory, one evaluates [5] the partition function.
I/T
Z(r, V) s I d\ifd\pdGft exp | drjd^xi^ (4)
on a discretized space-time as a lattice of points with and a^ as lattice spacings in
space and time directions, respectively so that V = iN^aaf and T = {N Thus
infrared and ultraviolet divergences are properly handled in the discretized lattice
formulation with yt and \p as site variables. The link variable between two sites
Ufi (x) = exp (^)] represents rotations in colour space. Finally, we find
that the structure becomes equivalent to spin system and can be evaluated in an analogous
way.
In order to determine the order of deconfinement phase transition in lattice
calculations, one evaluates the order parameter as the expectation value of Wilson
loop variable < L > = exp [-F/T\ where F is the energy of a quark. Since F — > in the
confinement regime, we get < L > = 0 but in the dcconfined region, < L > = constant (> 0).
Similarly order parameter for chiral phase transition is (ytyt) and is a constant greater than
zero for the constituent quarks but is zero for current quarks. In Figures 3(a<b), we show the
Figure 3(a). Deconfincment measure <L> and chiral .syrnineiry measure
< ipyf >, on Q 8-^ X 4 lattice.
results of lattice gauge calculations and these results suggest that both these phase
transitions occur almost at the 8ame temperature. They are also first order phase transition
because they involve a sudden change in energy density £. Similarly, a quantity (£- 3P)/T^
which yields a measure for an ideal gas behaviour is not zero even at Tr: 27^ and thus it
involves considerable plasma interactions. When there are quarks in the theory, there is a
606 CP Singh
big difference in physics for NfS 0. 2/3 flavours respectively. Critical temperature depends
on the number of flavours (T, « ). Similarly for yV/= 2 massless quarks, we get a
continuous transition. For 3 massless quarks, the phase transition is of first order but
for Nf^ 3 with two massless u, d quarks and > 0, wc again get a continuous transition.
However, lattice simulations for n® ^ 0 involve some technical troubles and the calculations
are not unambiguous ones.
Figure 3(b). The energy density e and the pa*ssure }P, normalised lo ihe ideal
Boltzmann ga.s limit, according to lanice calculations
»
One can construct first order phase transition by using Gibbs criteria tor equilibrium
phase transition Phg = ^qgP’ ^hg = ^ogp* /4^G = A^ogp critical point. In other words,
one uses the condition
^HG (^C’ ^c) = ^QGP Me)
However, one can obtain pressure in hadron gas (HG) or in QGP phase provided one has
the proper knowledge of realistic equations of slate (EOS) in both phases separately. If one
simply write P„ = Pq -P for the case ^ = 0 in eq. (5) where B summarizes the interaction
effect in QGP phase, one gets 7^ = 0.72 or 7^=1 140 MeV tor = 200 MeV.
3. EOS for QGP
Recently there were some suggestions for modifying the EOS for (JGP by giving a p and 7-
dcpcndence to the bag constant B. Let the QGP hadronize at fixed 7 and p to a hadron gas
(HG). If we calculate entropy per baryon (S/ B) ratio, we find that
This, however, violates the second thermodynamical principle. In order to cure this
problem, either one should change 7 and /i during mixed phase which is often reterred as
subsequent dilution and reheating, or one can fix 7 and p during phase boundary an
Quark gluon plasma-^current status of properties and signals
607
assume isentropic equilibrium phase transition so that {S/B)qqp = t^/^)HC- price one
has to pay is to assign a T and jU dependence to the hag constant. Thus
■^OGP “ -^OGP ShG
«0OP - "qcp "ho
(7)
u r oc ^ . dB{fi,T)
where the correction factors 5^p = — — and n^p = — —
partition* function for (JGP with massless u, d quarks and gluons :
TlnZgop rinZ^ rinZj*
V V V
(8)
where
^ In 2***“' - i i ^ 5fil T)
V '"^0 - 9o" ' ^ 9 ^ l62jrJ ’
(9)
1 Inz^"* = -a,(/i.rvj^*r« + +
n* 1
81 jr’
(10)
and
, ... ntrf. f0.8(/i2/9) + 15.62272'!'
«.(/i.r)= 29 ['"[ 42 J
-1
(11)
For T -> large and ^ 0 case, we gel expression for B(jU, 7) as
xjeosh (iilT) - l)
9 9;r^ 81;r^
( 12 )
Similarly in the large density limit {Le. p large, T -¥ 0), we gel
17 I 1
fl(/i.r) = 5, + g«T« + ^M^r^-gTV
T*n^
124 [?J [“4 " fl’J I 2 ■
(13)
'►here « = (/i ^ - m ^ We have shown the variations of B(#t T) with fi and
^ in Figures (4-5) using eqs. (13) and (12). respectively. “We find that fl(/i. D decreases and
608
C P Singh
|T-50Mrt/l
Figure 4. Vanation of bag pressure Biji, T) from eq (13) with baryon chemical potential (/i)
at a temperature T = 50 MeV Curve A represents the free QCP EOS with = 235 McV
Curve B represents the interacting QCP EOS with QCD scale parameter A & 100 MeV* and
bI/^ = 235 McV and curve C for A = 150 McV and Bq^ = 235 McV. In curve D we have
used bI^^ = 170 MeV and A = 100 MeV.
goes to zero also. Similar behaviour has also been obtained in other models as well. From
eq. ( 1 2), it is obvious that at = 0 we get
fi(/i = 0. 7) = Bo
4
where Tq
9Bo
ll;ra,(0. T)
■ Similarly at 7= 0, we get from eq. (13) :
(14)
B(/i,7= 0) =Bo
(15)
where
SlTT^
a, (A/, 0)
Bo . These kinds of equations have also been obtained in other
models like sum rules etc.
In Figure 6 we have shown the critical phase boundary obtained when B(/i^ 7^) = 0.
This phase boundary signifies the transition from hadronic matter to a plasma of completely
free quarks and gluons. For comparison, we have also shown the phase boundary obtained
Quark gluon plasma — current status of properties and signals
609
I >1 » 50 MeV
Fl^rt S. Variation of bag pressure as obtained from eq. (12) with temperature (D at constant baryon
chemical potential = 50 MeV. The notations are the same os in givemin Figure 4.
F^nn 6. OitictI curve obtained from dm equolion ^ ^ ^
two extroine i^giona of luge low Uc respectively. The conesponding critical
curve obtained for the Gibbs condition of the pressure equality is shown by a dashed line (curve B).
610
CP Singh
using Gibbs criteria in isentropic phase transition. In Figure 7, we have demonstrated the
variation of (€ - 3P)/T* with temperature T and wq find the curve agrees with that obtained
in lattice gauge calculations.
4. EOS for hadron gas
Wo can make attempt to modify the equation of state (EOS) for a hot and dense hadron
gas. At a large T and K a large number of resonances are also present in the hadron gas
giving rise to a large interaction. Attractive interactions can be accounted well by
considering a large number of resonances in the HG. The main problem is how to
account for the repulsive interactions in the proper EOS for HG. It has earlier been
demonstrated that we must consider repulsive interactions in the HG at large ^ and T,
otherwise we do not get a unique phase transition point. Repulsions have been considered
cither by considering mean field type approaches [14] or by using excluded volume
approach [9-13] in which we give a finite hard-core volume to each baryon. In the mean
field approach, the repulsive interaction results from (u( 78 3 Me V) meson exchange potential
V{r) = - ^exp(-mair) between two nucleons and hence it generates a potential energy
in a hadron gas with a net baryon density /i/^. In the case of early
universe = 0, so Wing) = 0 although HG contains a very large number of nucleons, anti-
nucleons and pions. In such situations, excluded volume approach is more successful and
we write the expression for free volume V' - V - Z, /V, where is the total number of
Quark gluon plasma — current status of properties and signals
611
baryons ol i-ih kind and V, is the proper volume of one such baryon. Thus ihe excluded
volume L, N, V, is subtracted from total volume. Thus Cleymans and Suhoncn [lOJ got for
the net baryon density n, = /(I -f Z, /]“ V, ), where n^ is the net baryon density of i-
ih pointlike baryon species. Kuono and Takagi suggested that the repulsion exists either
between a pair of baryons or between a pair of antibaryons and thus we get ;
/!« =
(16)
In the Hagedron model, we gel excluded volume proportional to pointlike energy
density so that
n
0
R
I + fO /4B'
£
£0
I + fO /4B
(17)
Al large 7', l4B )) I, so that e = 4B. However, all these approaches lack
thermodynamical consistency, Rischke et al [W] proposed the following modification to
the grand partition function
Z^-iT.p, V-VbN) = V - NV ii)0(V - NV g) (18)
N=0
and then define the pressure partition function as
Z= j^^Ve-i^Z^'^T.^i.V-VgN) (19)
II we put p=p-TVfj^ = p- V , we shall gel :
/. = jdxe ^'Z^-{T,p,x) ( 20 )
so ihut wc get a transcendental equation as
I + Va««(r,);)
( 21 )
obviously these equations are difficult to solve.
Recently, we formulated one unique way to incorporate excluded volume correction
in a thermodynamically consistent way by directly doing the volume integral :
. 7 _ S,A,
dkk*
exp
( 22 )
where A, is thd fugagity of i-lh species, /, is the remaining part of the integral. Thus we can
write the following equation of n , ;
72A(6)-20
612
C P Singh
/I, =(l (23)
where R = ^^jVj is the fraction of the occupied volume. After simple manipulation,
we get
/f=(l-/f)5^/.A,V/. (24)
We can solve the differential eq. (24) by using the method of parametric line and get
R =
(25)
where and G(r) = /(fl 2 +/.iV'3r) . Thus the values of some
of the parameters arc fixed arbitrarily. However, we can use the assumption that the
number density of /-ih species depends on its fugacity A, only and then dRj BX, =
\ Bn,
r IT,
V, . Thus we get a simple differential equation as
^ +H,[(l//,V,AJ) + {I/A,)]=j
/V,X,
(26)
Its solution can be written as
>1, = /(A,exp(-l//,V',A,) -O,)]'
(27)
f
where Q, = exp (-1/ /, V, A, ) JA/
Jo
R = ^ X/(l + X)-
X = ^fl/(A,exp(-l//.V,A,)-G,)-
In Figures (8-9), we have shown the variations of baryonic pressure with temperature
T and chemical potential ^g, respectively for a multicomponent HG. We find that the
prediction of our recent model lies close to the model of Rischke er al [1 1) and ditlers
considerably from the inconsistent Kuono-Takagi model at large T and/or fjg. In Figure 10
we have demonstrated the prediction of our model for the variation of the entropy
per baryon S/B ratio with baryon chemical potential ^g and we find maximum value ol
S/B in the present model. This means that the present experimental value of 5/^ around
sixty can only be obtained in our present model which is thermodynamically consistent
one.
Quark gluon plasma — current status of properties and signals
613
FiRurc 8. Baryonic pressuiv » v buiyun chemical poicniial ai / = 200 MeV m a
multicomponeni HC with M A. A X- X*- A* . Sbaryonic coinponenK and K. K* meson'*
Cor strangeness conservation Curve A gives the prediction ol the inconsistent uppiuach
of Kuono and TaWagi B yields the results of our present calculation and C that ol
Rischeke et ol D lepresenis the curve obtained in the Uddin-Singh model
Figure 9, Vonaiion of baryonic pressure with temperature at = 0 MeV The notations
for A. B. C. D are given in Figure 8.
614
C P Sififih
z
o
OOC ZOCCC ^OCOO 600 :g 800 00 1000 oc
3ASY0N chemical POTENTIAL
Figure 10. Vanaimn of the entropy per baryon ratio S/B with.baryon chemical
poieniial ///^ at T - 200 MeV in the incunsi<stenl model of Kuono and Takagi (curve
A), oui piesent theimodynamieally consistent model (curve B) and the model of
Kischke i‘i (//[II] (cuive C)
5. QGP signals
Tlic siafiiLtnl incllioci for icslirig (he QGP signals is lo compare’lhe predictions of nucleus-
nuL'icLis collision models incorporaling the presence of a QGP with the prediction of motlels
li.ised cnlirely on the dynamics ol colour-singlci hadrons. Unfortunately we do not have a
pioper iheoiciical understanding ol high energy nuclear collisions [I). The quantitative
Lindcrslaiulmg ol the background processes in the hadron gas is essentially a prerequisite
The models of ihe HG vary from the ihermal models where HG is considered as
cqiiilibraied statistical system lo transport or cascade models which do not involve
equdihnum concepts but are based on the superposition of hadron-hadron collisions. No
one has yet devised a clean and unambiguous signal of QGP formation. Moreover, it is
ama/mg lo see that the existing data are explained both in Ihermal HG picture as well as in
cascatic model approach.
Strangeness abundance has been suggested lo be one such signal. The idea is simple.
In a bai yon-dense hadron gas, we have
When 1 . 1^1 > 200 MeV and m, = 150 MeV, >n-. Similar results one obtains for qq
sNiiimeinc (/ (' = 0) system as well when T » Moreover, it is easier to produce i-
Quark gluon plasma^urrent status of properties and signals
615
quark in QGP than AT-meson in HG because and degeneracy t'aclor for s quark is
larger than that of K meson. Larger n? /n ,7 ratio in QGP means a larger ratio after
hadronization of the QGP. Recent CERN experiments using sulphur-nucleus reactions have
reported a strangeness abundance 3-5 times larger than observed in pp reactions. However,
this rise in the strange particle production can be explained either by using thermal models
or by hadron cascade models. Moreover, the rise in production can also be explained
by considering a medium modiHcation of hadronic parameters e.g. masses m .
Asher Shor [15] inferred from large s and J quark densities that there would be an
abundant production of 0 mesons. Since 0 production from ordinary hadronic prcK'esses,
^ PP — >0PP suppressed due to Okubo-Zweig-Iizuka (OZI) rule, so
0 mesons from a hadron gas without QGP formation would be far too smaller in magnitude.
The OZI rule forbids processes if they involve disconnected ‘"hairpin" type quark-line
diagrams. However, OZI rule is not exact and we invoke unitarity diagrams to explain such
breaking. In the dual topological unitarization (DTU) scheme, the twists in the quark lines
involved in the s-channel unitarity diagrams give rise to a cancellation mechanism and thus
the suppression of amplitude is explained. However, the twists in the t-channci quark lines
do not give rise to such cancellation mechanism. In other words, (j) production in the
fragmentation region (large ^-regime) is more suppressed than in the central region (small
p region). Thus we suggested that the variation of the ratio 0 / co with the baryon chemical
potential can serve as a signature because this ratio rapidly decreases for a HG without any
(JGP formation whereas in the presence of a (}GP matter, it is almost a constant [16].
However, we cannot infer about p and T in a nucleus-nucleus collision unless we use an
EOS for the matter. Recently we have suggested [17] that we can study the variation of
0 /(p + (y) either with baryon density ng or energy density e and this will give a potential
probe for a QGP formation. In our calculation, we have used hadronic EOS propt)sed earlier
and get .the energy density £ * 7IS + p n, - P. We take K, K. t], p, o), rj', K\ 0 , p. n. A, L, E,
4 1* and A* (1405 MeV) in our calculation for the quantities of HG.
For the calculation of the ratio 0 /(p + (U) from a QGP, we consider production of
quarks and anti-quarks in the plasma during the equilibration of gluons in the mid-rapidity
region and the probability for the emission of a particle with q quarks per unit of phase
space volume is
P ■ exp(-£q /T*)*
where fg is the probability for creation of quarks by gluon fragmentation, ni^ is the mass of
the quark q, Xq is the chemical fugacity, gg is the degeneracy factor, Yq is the relative
equilibration factor and Eg is the energy of the quark. Thus after some assumptions, we get
We have used the same £7 window for the particles. In Figures. ( 1 1-12), we have shown
the variation of the ratio with respect to e and rig* respectively. We find for QGP fonnation,
616
C rSinffh
0/(p + 10 ) reiiluins consianl around 0.4. However, lor a HG wiihoui QGP, ihis ralio
increases wiih e and reaches a constani around 0.25. The variation of the ratio with also
Figure II. Variation of the ratio ^/{p^^aP) with the baryon number density at u fixed energy
density t = 1 GeV/rni;\ The solid line represents the QGP coninbuiion. the dash-dotted line
indicates thermal hadron gas calculation and dashed line represents the contribution Iruin the
superposed hadron-hadron model
0.4,
0.3
0.1
0 . 0 ' ^ '
0.0 0.1
A
€
A —
B —
C--
■1G*V/fin3
OGP
— Tbtnnol Hodron goa
SuptrpoMd hodron-hodrop
•eottvrmg
-I 1 1 ■ I ■ '
0.2 0.3 0.4 0.9
Fixurc 12. Vanalion of Ihe nuio ^/{p^ * (iP) with Ihc energy density e el t constant ag '
0.2.^ riir\ solid line represents Ihe QGP contribution ond dash-doned line lepresenls the ihemul
hadron gas calculation. Eaperimenial data ore taken from |26|.
Quark gluon piaxnia — current status of properties and signals
617
shows u similar behaviour. Moreover, wc find that variation of 0/(p + cu) with Hg can also
distinguish, between a thermal picture and cascade picture of HG. Similar conclusions can
be derived for other strange particles 118].
One important signature 1 19| for the QGP formation has been suggested as photons
and dileptons production. Since these particles arc subjected to electromagnetic interactions
only, their mean free paths arc larger and they arc unaffected by the hadronizalion of the
system. Since they reveal the thermal status of the fireball, these particles are known as
ihcrmomcters. For dileptons from QGP, we have mainly the process qq whereas
Irom a HG rtK Pp-^ 0 as well as Orel I Yan processes pp p'^pr
X can contribute. Our main motivation is to identify some particle ratios or particle spectra
which are much different from the HG background. Similarly for the photon production,
qq qx qy^ qx q y contribute in the QGP whereas nn py, np ;ry;
p -> TTTry, (i) — > nyeu. contribute in the HG case. The situation as standing at present tells
(hat HG contributions almo.st match with the QGP contributions.
One signiUcani signature lor QGP foimalion was the suppression in 7 /*P production
as suggested by Malsui and Satz. The idea is that J/H* is mainly produced in the pre-
cquilihrium colli.sion stale. Since it has a large mass, its production from a thermally
equilibrated system is not significant unless wc have a very large temperature. J/H* then
passes through a deconfming medium in the case of a QGP formation. So the resonance
melts into cZ quarks and they arc .separated by a large distance depending on the size of the
Figure 13. The 7/ survival probabilily after ub.sorpiion through nuclear
matter. a.s a* function of where A and 0 are the mass numbers of
ihe colliding objects The full line (doited line) is the survival probability for
ihe proton- nucleus (nucleus-nucleus) systems, calculated with a cross section
- 6.2 mb.
deconfining medium. Thus the possibility for a recombination into J/^ '\s very small. In
i^i)mpari.son, for a HG without QGP formation only some of J/^ will be lost due to
■^-'scattering. So a suppression in production will signal a QGP formation. NA38
experiments with O-U at 200 GeV/A clearly show such suppression for 7/ ^ peak in the
618
C P Sinfih
masN Ji>iribuiion ol'dilcplons around 3.1 GcV. Moreover, 7/ suppression occurs more at
lower Pi as expected lor QGP formation. However, conventional explanations with nuclear
absorption oi rescatterm*: can also explain 7/V' suppression.
Recently. NA5() collaboration has reported |20J a strong suppression ol J/H'
production in Pb-Pb central collisions at 158 GeV/A. The suppression is much stronger
than the expected one from conventional explanations |2I | which explain the previous data
foi O-G or S-U central collisions, as well as lor the hadron-nucleus collisions. Thus NA5()
data as shown in Figure 13, show a clear deviation from the previous situation 122J. It is
believed ihal.this new observation has given us the first clear hint for the dcconfincmeni
occurring in the ultra-relativistic heavy ion collisions.
6. Farly universe scenario
In the early universe, the coloured quarks and gluons were deconlined and the matter
existed 111 the loim ol QC*F- As the univcr.se expanded, the temperature dropped through the
eiitieal temperatuie /' lor the phase transition where the QCjP could exist in thermal,
mechanical and chemical equilibrium with a den.se. hot HG. This could induce a large
isothermal baryon number fluctuation which would change the standard sccnaiio loi
primordial nucleosynthesis. The ratio of baiyon number densities in the two phases is
represented by a baryon contrast ratio K = evaluated at T = 7', and the baryoii
chemical potential Pn = l()"'‘’7', W R » I, we get a thermodynamically lavourable condition
!oi the baryon number to reside piedommainly in the QGP phase. Recentiv. scNeial
attempts hace been made to determine the values ol l< using \arious types ol equatiTiii ol
stale (LOS) lor the Q(jP as well as HG pha.scs. These studies amply make it clear that ilie
quark-hadion pha.se transition induces a large fluctuation in the baryon density in the eail\
urn \ el se 1 23-25 1.
Acknowledgment
The author is gratelul to the Department of Science and Technology (DST). Nev^. Delhi loi
.1 lesearch giant.
Ml C I^SiinJi Kir 236 147 1 1
|:i t PSiiiiilW/a / Moil rii\s A77I4S(|W2)
PI B Miillci Kip /'"'V /'/n\ 5H6I I ilWs)
|4| S A Boiioineiio .iiuK) Kiin.iiii) /^//^ \ A’i/» 22H 17^
Dl H \lL‘>ci-Oi1m.iiiiis At'i Mini Khw 6H47i(hMf>)
lf>| .A Lcniiido\. K Kcelikh H Sai/. l£ Siihoncn ,iiul (i Wchei /V/w Hev D504()s7 i |W4)
| 7 | B k I’ali.i.iiuU' f'Smyli /’/nH Wti 1)54 4SS | i |‘Nh,
I S I B K Paira .iiul C P Singh / /Vm ^ C74 im 1 1 ^)S‘7 >
P'l K Haijciloin / rh^^ C'42 26.^ ( IVXli
Quark gluon plasma — current status of properties and signals
619
( lOj J Clemans and E Suhoncn Z. Pirn C37 51 ( 1987)
[ M ] D H Rischke. M I Gorcnstcin. H SUk’kei and W Cireiiier / Ph\ \ C51 485 ( 199 1 )
(12) S Uddm and C P Singh Z Phys C63 1 47 ( 1 994)
( 1 3J CP Singh. B K Patra and K K Singh Phys Leu H387 680 ( 1996)
1 14] J I Kapusta. A P Vischerand R Vcnugopalan Phw 'Aev. C51 901 1 1995)
[15) A Shoi- Phys. Rev Uli 54 II 22 ( 1 985 )
1 1 6] CP Singh Phys. Rev. Lett 56 1 750 ( 1 986), Phys Lett 11188 369 (1987)
1 1 71 CP Singh, V K Tiwari and K K Singh Phvs Lett B393 188(1 997 )
1 18] V K Tiwan, S K Singh. S Uddin and C P Si.ngh Phy.t. Rev CS^ 2388 ( 1996)
[19] Jan-c Alam, Sihaji Raha and Bikosh Sinha Phys Rep 273 243 and rclciviK‘;s ihcicin i I99(i)
[20| NA50 collaboration, M Govin el ul in Proceedin^\ of Quark Mutter VO cdik-d In P Oi.iuni Muiumgfi
et at Nurt Phys A (to be published)
[21] J P Blaizot and J Y Ollilraull Phvs Rev Utt 77 1703 (1996)
[22] N Arrneslo, M A Braun. E G Femio and C Paiaics Phw Re\ l.eu 77 373(i ( 1996)
[23] C P Singh. B K Paha and S Uddin Phw Rev D49 4023 (1994)
[24] B K Patra. K K Singh. S Uddin and C P Singh Phvs Hei D53 993 (1996)
[25] B K Patra and C P Singh Nud Phys. A6I4 337 ( 1997)
[26] J Stachel and G R Young Ami Hei Nia I Pm / Si lem e 42 537 ( 1 992)
72A(6)-2I
Indian J. Phys. 72A (6), 621-634 (1998)
UP A
— an inicrnaliona l journal
Blackhole evaporation — stress tensor approach
K D Krori
MuiliciiijiiLul Physic* Forum. Colton College,
Guwahuli-7KI (K)l, India
Abstract The stress lenspr approach to black hole evaporation has been reviewed i
this talk
Keywords Black hole evupoialiun. stress tensoi
l*A( S Nos. (W 20 Cv. 9*1 10 .S( 97 60 Lf
I. liUnxIuction
\s Inst pointed out by Hawking (1,21, the gravitational field of a collapsing object will
iiuIljcc the qiuunum ( reutkm of fiarticles so that the object radiates with a thermal spectrum
*ii *1 icmpcruturc inversely proportional to the mass of the object.
Lill i icr, ealculaiions ol this effect examined the behaviour of the quantum
iichls m)l\ near infinity. Consequently it was nol clear precisely where the radiation
IS being ercaicd, and what is happening near the horuon of the "black hole". Davies,
lulling and Unruh (.1| pointed out for the first time that a knowledge of the
encigy-momentum tensor ol the quantum licld in the vicinity of the object would
lielp in clarifying the details of the creation process. Unfortunately, this quantity is
iilwuw fomiaily diverffent, and the meaningful physical component must be extracted
a regulurisation procedure. Such procedures always contain ambiguities which
xuisi be resolved by the application of additional criteria, such as physical
i^MMMiablencss.
Besides the problems of regulurisation, mathematical complexities have prevented
'^l^'Uiiled discussion of quantum field theory near the surface of a blackhole. However, it is
pusMblc lo circumvent the latter problem by studying a simple two-dimensional model of
hijckhole. This model has the advantaf(e oj ix)ssessinf( a conformally flat metric so that
© 1998 IACS
622
KDKrdri
the mode functions for the quantum field can be explicitly evaluated everywhere, while
retaining the essential features of the Hawking evaporation process. The highly plausible
character of the "renormalised" energy-momentum tensor for this simple model encourages
the hope that the qualitative features of the full four-dimensional collapse are contained in
this treatment.
2. Stress tensor
The metric for any two-dimensional space-time is conformally flat and may be
written as
ds^ = C(u, v)du dv, (1)
where u, v are null coordinates. The massless scalar field, 0, for this metric obeys the
simple equation
du dv^
= 0 .
( 2 )
The solutions of this equation are
0=f{u) + g(v), (3)
where f(u) and g(v) are, in general, arbitrary functions, restricted only by the spatial
boundary conditions.
It is intended to calculate the expectation value of the operator
in some quantum state. In expanding the operator 0 in normal modes, we assume that
there exist null coordinates u , v such that the ingoing and outgoing parts of a normal
mode are respectively
(5)
The stale which wc have examined is the one annihilated by the operators with modes
a)> 0 in the field expansion.
If the geometry is initially static or has an asymptotically flat region at intinity,
this state is made unique by the requirement that the modes reduce to ordinary plane
waves in that region. This slate is then that in which no particles are present initially
(before the collapse begins as in the problem of Davies, Fulling and Unruh [3] discussed
in § 3), and is conventionally called the "vacuum" or "in-vacuum" state,
On rcgularisation (on physical grounds), the expectation value of 7^^
(also designated by is
R
( 6 )
Blackhole evaporation — stress tensor approach
623
where R is the curvature scalar and 9^^ as evaluated in the special u , v coordinates
has the components
0iri7 = -(12;r)-'C2(C’^).i7ir,
= -(12;r)-'C^(C'i).irr, (7)
0|7r = 0ri7 = 0.
The regularisation scheme adopted for derivation of (6) stimulated some controversy,
because it involves discarding certain ambiguous terms which inevitably arise as an artefact
of the regularisation process (these terms are ambiguous because they depend on the
direction of point-splitting). Because of this controversy, Davies [4] adopted an alternative
procedure to confinn the result (6). Remarkably, however, it is possible to determine <T^^>
uniquely without regularising infinite quantities at all, provided that we assume that the
stress-tensor possesses a non-zero trace. Here, it is important to mention that in two
dimensions, quite general arguments imply that conservation, zero trace and particle
production are incompatible.
Let us consider the metric in u , v coordinates in the conformally flat form
ds- = C(m, v)du dv. (8)
The only non-vanishing ChnslolTcl symbols arc then
• rL =(• ri, =c-'chc. (9)
The siics.s-tcnsor is defined to be covarianlly con.served,
fA'v = 0, (10)
which in terms of C becomes
=0 ( 11 )
logether w'ith a similar expression for T-- with m and i ’ interchanged.
The trace in ( 1 1 ) is assumed to be non-zero, even though the stress tensor operator
lor massless .scalar fields is known to be traceless. The appearance of a trace in the vacuum
expectation value of a (formally divergent) traceless operator is known as a conformal
anomaly, because it breaks the conformal invariance. Conformal anomalies are to be
expected on general grounds in quantum field theory [5J. Here we only need assume that
IS a non-vanishing local quantity. It is a scalar quantity with the dimensions (Length)'-
(in units fi ~ c - 1 ) so it must consist of teims which are quadratic in derivatives of C.
As there is no conformal anomaly in flat space-time, must vanish for certain choices
the conformal factor C. This requirement suttiecs to determine the trace to within an
624
K D Krori
overall numerical factor. First, it is noted that, if C is a function of a or v alone, space-
time is flat, because a simple rescaling of one null coordinate reduces the right-hand side
of eq. (8) to dudv. Hence can only contain a linear combination of the factors
r}-Cr?j:C and Because the theory does not contain a characteristic length, a
simple scaling argument shows that ^ must be a homogeneous functional in C of
degree- 1 . Consequently
Ti", =aC-^a;9;!C+bdsCdzC.
( 12 )
Next we note that the choice C = corresponds to the Milne Universe, which is just
Minkowski space in disguise, so we require the right-hand side of (12) to vanish in this
case. This fixes a = -/?. so
=
(13)
where R is the scalar curvature
Equation (II) may now he written in the form
( 14 )
which may be immediately integrated to give
r,-! = +/(?),
(1^)
where f \s an arbitrary function of i7. To determine f{u), it is noted first that, as T-~ is
local, f can depend on the geometry only through C and its derivatives at the point (u, u) of
interest. Now if ^ 0, C will be a function of both u and v, so /(«) is generally
independent of C because it is a function of u alone. At most /can be a constant. Davies
|4| has omitted this con.stant.
To fix up the value of a, Davies has appealed to a special case, the case of a moving
mirror, emitting radiation and obtained the value (with/= 0)
a=-(24;r)'' (16)
Hence the complete strc.ss tensor Is
T^,. = e„. + (4uy' Rg^,. ( 17 )
where 0^,, are as given in (7). T-- follows from T-- by interchange of u and v and the
values of the conformal anomalies.
r;; = ( 247 t)-'R
No rcgularisation has been used to obtain these results.
( 18 )
Blackhole evaporation— stress tensor approach
625
3. GnivltaCioiial coHapse of a spherical shell
Now (6) [or (17)] is applied to the collapse of a spherical shell. The two-dimensional metric
is obtained by eliminating the angular coordinates from that of a four-dimensional shell of
matter which collapses at high velocity. Inside the shell, the space-time is flat, whereas
outside the shell the metric takes the Schwarzschild form
(19)
There exist three useful sets of null co-ordinates for this problem. In the first, given
outside the shell by
ii« r- r
v*t^r*
r"*r-¥2Mln{r/2M- I)
the external metric takes the simple form
ds"^ S5
dudv
( 20 )
( 21 )
where r is an implicit function of u, v by eqs. (20).
The second set. (/, V, is defined so that the interior metric takes the simple form
dS^^dUdV. ( 22 )
The relation between the u, v and the U, V co-ordinates has been obtained by Unruh [6] by
demanding continuity of the metric across the boundary.
Finally, we have the co-ordinates «, tJ which arc to appear in mode solutions (5)
and in the determination [eq. (7)] of the energy-momentum tensor. Following Unruh [6],
relations are obtained which lead to an expression for the external metric in u,
co-ordinated and to values for For retarded times u before the onset of the collapse,
one obtains
,4,2 , ^1 _ lE'^dSdv (23)
that is, the conformal factor, C(u. u), to be used in eq. (7) is
Cm I -IE (24)
in the external region of space-time. The values of in this region expressed in
u.v coordinates are
7..
626
a: D Krori
( 25 )
For retarded limes, m, long after the collapse, the external conformal factor in u, v
co-ordinaies takes the form
C(i7.U) = (j-^)[^+0(l)], (26)
where 0(1) are terms of order unity in u and /I is a parameter such that u = A is the
equation for the future horizon. Evaluating outside the shell, transforming to u, i;
co-ordinates, and neglecting terms which die off for large values of u, one obtains
T.,„ = (24;r)-'
2r*
_W 1
32M2
= + (27)
with and 7, remaining as in (25).
Comparing (27) with (25) one finds that the effect of collapse is to add a constant
term to which appears at large ras a flux of energy defined by Unruh [6] of magnitude
1768;rAf^] 'Mhis is just the energy flux one would expect on the basis of Hawking’s
»
arguments 1 1 ,2] as applied to this model.
From (27) and (25), one finds that the flux of energy is given by two components.
Near the infinity it is dominated by an outward null flux of energy (given by 7„u)* , Near the
horizon, however, it is a flux of negative energy going into the horizon of the blackhole
(represented by for r near 2M)*
4. Hiscock’s model of evaporating blackholes : calculation of stress tensor
components
Hiscock [7] modelled the Hawking process of evaporation of a spherically symmetric
blackhole with a Vaidya metric [8] which represents imploding null fluid. The metric of the
model space-time is
ds^ = -^1 -
where )Vf(i>) = 0, u<0,
M(u)^0, Uq > u>0, (29)
Af(i>) = 0, v> Vq.
+ 2<fuir + sin^ (28)
Oil ihe other hand, near r = IM, /"yys -(768;rAf^r’
BlacWlole evaporation— stress tensor approach
bn
The mode\ EBH (evaporating Wackholc) space-time is initiaiiy flat, empty Minkowsi space
for all t) < 0. Then, at v = 0, a collapsing ball of mass M(i)) = m is formed. Negative-
energy-density null fluid then falls into the hole at a greater or lesser rate, depending on the
choice of M(u). such that the mass of the biackhole is reduced to zero at The final
stale is again flat, empty Minkowski space for all v>Vq.
Hiscock took two examples for (29) in one of which
0, u < 0, Phase I
M{v) = w^l - t>o > > 0» Phase II (30)
0, v> Vq, Phase HI
so that with a 0 = constant, 0 = constant slice through the model FBH space-time to get a
two-dimensional metric, we are left with
dv^ ^Idvdr..
(31)
With the following substitutions
j f - -In (uo _i))
»? = C + 2z', z’ = |(z^ - 2/tt’ + 2z) dz, (32)
H^mfVo
(3 1 ) reduces* to the form
- 2/rz + ^y^dn. (33)
This is the metric for phase II (m)> u > 0). The metric for phase I is
ds^ = - dudv, (u < 0) (34)
and, in phase in. the final Minkowski space-time is
J.v2 = -d(/Ju, (u>tb). (35)
The two-dimensional stress tensor for a quantised massless scalar may now be computed by
relating these three sets of null co-ordinates cqs. (33-35) to the canonical set (m.u) in
which the vacuum state is defined. The results m, for phase / (u < 0)
r,„=o.
for phase II (Vo> v> 0) :
4mu2
i73
12m 2 1)2
(37)
72A(6)-22
628
KDKrori
and phase III (u > :
T'uu -
mVo ( 3 m ^ - 2Vou - 6mvo )
67tu^(U - Uo)'
= Tuy = 0 .
( 38 )
( 39 )
(40)
(41)
The stress tensor in phase II is observed to be finite everywhere except z = + oo
and/or m = 0 at the curvature singularity. The (T). 0 co-ordinate system behaves poorly
as z Z+, rj -^oo (the event horizon), but examination of the stress tensor components in
a Kruskal-type co-ordinate system regular on the event horizon shows that they are
finite there.
The stress-energy in phase III consists solely of a stream of outgoing radiation
whose energy density diverges as £/ -4 i.e., as one approaches the Cauchy horizon.
[“The Cauchy horizon, simply, is the place where the Cauchy problem breaks down;
usually it occurs accompanied by a naked singularity (a pathological causal structure)”—
Kaminaga [9]]. Note also that the integrated energy density diverges as 1/ Do The energy
density is always positive for 1/ < Ub- Since the stress tensor for phase II is Finite all along
the event horizon, it is natural to associate this diverging energy flux with the naked
singularity.
Hiscock’s model has been extended to an evaporating charged blackhole by
Kaminaga [9].
5. Balbinot’s formula for Tyv
Balbinot [10-13] extended the work of Davies et al and Hiscock to a physically general
line-element describing a spherically symmetric evaporating blackhole of the form
ds^ = +2e'>'dvdr + r^(d9^ +sin2 8d^^), (42)
where ^rand m are functions of D and r. In four dimensions m is the total gravitational mass
of the system as viewed from infinity. Taking a B = const., 0 = const, slice, (42) reduces to
<£,2 + 2«*'</udr
If V^= 0 and m =Af = const., (43) describes a two-dimensional Schwarzschild space-time in
advanced time, Eddington-Finkelstein coordinates.
Blackhole evaporation— stress tensor approach
629
A new set of null coordinates (L/, VO defined by
dU = g\
dr.
V^v
are introduced, where g is an integrating factor which satisfies
In terms of U, V, (43) becomes
ds^ = -
8
2m;
r >
dUdV.
(44)
(45)
(46)
which is manifestly conformally flat.
Now. in a two-dimensional space-time having a line-element
ds^ = - C{u,v)dudv (47)
the expectation value T^^,o( a massless scalar field in the vacuum state |0) defined by the
normal modes, exp (-im) and exp (-icov) is given by (6) and (7).
In general, the |0) does not represent the correct vacuum state for an
evaporating blackhole so that one cannot simply use (6) as it stands. A prescription for
how to define the correct vacuum stale, call it |^), for an evaporating blackhole in a
non-staiionary space-time having a line-element (46) is not yet known but, for his
purpose, Balbinol considered it sufficient to use some general properties of for
ihe spacetime (46).
In fact, requiring that v^and m are well-behaved at past null infinity (i,e., the space-
time under consideration is past asymptatically flat) the scalar field modes for the |^)
vacuum will have the form exp (-/cuV') on /" This gives the relation
(48)
which is valid everywhere in the space-time. By (48), the ingoing normal modes for
|0) and |§) vacua coincide, so (T\/y)o = (^w)^ C**' dimensions there is no
scattering of massless particles by the geometry). The outgoing modes do not contribute
to Tyy and, by construction, both vacua reduce to the usual Minkowski vacuum on /'.
For the state |i5} one must further require that the invariants (e.g., ) of
be well-behaved on the event horizon of the blackhole. This condition
requires (Ti/y)^ and (Ti/u)^ to vanish there; away from the horizon their form will
depend on the exact definition of the fields outgoing normal modes for the state |§) and,
of course, on g. Balbinol was only interested in finding the flux of negative energy
630
KDKrori
going down the hole. Thus, it is sufficient to look for the VV component of (Tfiv)^ and
this does not depend either on the choice of the outgoing modes of the field or on g; it
is fixed by the metric (43) and by the boundary condition (48).
From (46),
I
1
C2 =
2
(49)
Then
H}
1 _1 1
w - jC - -
c-iie
dv
(50)
where
(51)
remembering that
A = ± _ 2m'
dV 3v 2 { r ,
1-
1 9r
(52)
Hence, the expression for (Tw)^ given by (6) and (7) is
(rw)^ = -(12;r)-'[lc2_ (53)
and does not depend on g. '
Following Bardeen [14], Balbinot chose to be roughly constant and m( v, r) • m( v)
near r = 2m so that, from (53), he got
{Tyy)^ =(24;r)-'
This reduces to the third equation of (25) obtained by Davies et al [3] for a collapsing
• I
shell if m = ^ = 0. Furthermore, as previously stated one expects (Tyy)^ to vanish on
the event horizon and to give a non-vanishing, positive, outgoing flux across time-like
surface r ^ 2m (the apparent horizon of the dynamical model, the event horizon being
located somewhere inside it [15]).
One can associate with the flux (54) a blackbody temperature T which should be
considered as the effective temperature of the hole, since by the energy conservation (which
satisfies) one expects this temperature to reflect the radiation content emitted at infinity-
From (54) we have
r= (i2|rvv|»-')f
m
7T
3m^
+ »— r-
(54)
(55)
BlackhoU evaporation — stress tensor approach
631
If, however, we have a metric in an arbitrary form
ds^ * -A(v,r)dv^’¥2B{v,r)dvdr (56)
then we have 7w in the form
. ^ \\dB \SA^
where ^ = -5 -t“ +
Bldv 2 dr j
(57)
(58)
and F{v) is a function of v to be determined by a boundary condition T^,y on past null
infinity.
6. Some applicatioiia of Balbinot's formula
(a) Evaporating blackhoies in the presence of inflation :
Mallet [16] has taken the following metric for a model for the dynamical evolution
an evaporating blackhole in an inflationary universe :
+ 2 dvdr + ^ 5in2 J (59)
where M{v) is some decreasing mass function and x effective cosmological constant
associated with the de Sitter inflationary phase of the universe.
The twO'dimensional space-time associated with eq. (59) is obtained by taking
Ob const, and const, with the result
Jj2 - + 2dvdr.
Applying (56) - (58) to (60) leads at once to (with v) = 0)
T
vv
= (24»)-'
M(v)
rl
M(v) 3 MHv)
r' -2 r*
(W))
From (61), the following picture emerges. Near the event horizon ot the blackhole. there is
a negative-energy flux into the hole due to the first term in (61) and this is interpreted by an
observer outside ihf event horizon as an evaporation of the hole. On the other hand, since
632
KDKrori
> 0, ihe second term indicates that the net effect of inflationary environment is the
introduction of a positive energy<t1ux of radiation into the hole causing a slight decrease in
the evaporation process.
(b) Evaporaiing blackhotes with acceleration :
Recently, Krori et al [17] have studied the effect of acceleration on an evaporating
hlackholc by the stress tensor approach. The two-dimensional metric obtained by taking
6= const, and 0 = const, is [18]
ds- = - Hdv- + Idvdr, (62)
where H=\- + 6Antiv)p + ArG, p - r^Gip), (63)
A = acceleration parameter,
(J{p) =!-/>-- lAmiv)p' = sm- (64)
G,p= -2p - (iAm{v)p- . (65)
Tabk I. An esiiinait; of 7',,.
Tabk I. An esiiinait; of 7',,.
(ul Coniribution of
(b) Coniribuiion of
Sum ot (u)
(he rirsueiiii 1. )
(be second term | )
and (h)
0 T = l,g
of (66)
of (66)
»
0 8.15.1
0 8.151 m
m -
III -
0 } 46 III
0 11x10-'
+
m2
6 84x10-'
6 95x10-'
* 1
III -
T
III -
0 20661 m
0 20661 m
m2
1
III “
nfl 22m
14.9x10-'
+ ; —
7.999x10-'
m 2
6 901x10-'
^
w-
m 2
0..1.14I2 m
0 .1.1412 ill
m -
m2
n 1 73 m
. 12 8x10-'
5,66x10-'
m2
7 14x10-'
m-
m 2
Blackhole evaparatiori’^stress tensor approach
633
Now, applying (56H58) to (62) leads at once to (with F(v) = 0)
-(I 2 ;r)-
+ SL. -
3
* 2 r’
4
f(C.p )2
- —C
16
2 r ^
-^rG(/,)C.p+ -^r2C2(p)+-^C(p)^l-
+ j(6mp + rG,p-Ar^G(p))x.^^ +/4^G(p)j| (66)
Near the Schwarzschild surface, i.e., r s the first term within circular
bruL'kets in ( 66 ) represents a negative energy flux into the hole. On the other hand,
the second term within curly brackets is the contribution due to acceleration parameter
A and varies with 9. As Table 1 will show, the net energy flux, Zw • negative. Hence,
a net positive out-flow (/.e., radiation) will occur in accordance with energy
conservation.
For numerical estimates, wc shall take Am ~ - 7 == and use some relevant data
V54
from Farhoosh and Zimmerman [18]. We shall consider three specific directions,
9= 0, ;r/2 and 7t.
The table (Table 1 ) shows that for practical purposes, (0), Tyy {n/2) and Ty^ (n)
(for 9=0, nh and K respectively) are equal for small m (Davies et al [3]). However,
strictly speaking, \Tyy{7t) I appears to be maximum. The table also reveals an interesting
feature. The contribution from the second term of ( 66 ) so tampers that from the first term
that Tyy has practically the same value for 0=0, 7c/2 and ;r(for small m.
Rvrernices
1 1 ] S W Hawking Nature (Umdtm) MS 30 (1974)
(2) S W Hawking Com. Math. Phys. 43 199 (1975)
[3] P C W Davies. S A Fulling and W G Unruh Phys. Rev. DU 2720 (1976)
[41 P C W Davies Prw. Roy. Sw. Umd. A354 529 ( 1977)
(.^1 S Deser. M J Duff and C J Isham Nuct. Phys. Bill 45 (1976)
[6] W C Unruh Phys. Rev D14 870 (1976)
[7] W A Hiscock Phys. Rev. D23 28I3 (1981)
[8] P C Vaidya Proc. Indian Acad. Set A33 264 (1951)
634 KD Krori
[9] Y Kaminiga Class. Quani. Grav. 7 1 135 (1990)
[10] R Balbinoc PHys. Uti. 13CB 337 (1984)
[11] R Balbinoc Class. Quant. Grav. 1 573 ( 1984)
[12] R Balbinoi II Nuov. dm. 86B 31 (1985)
[13] R Balbinoc Rhys. Rev. D53 161 1 (1986)
[14] J M Boitleen Phys. Rev. Uit. 46 382 (1981)
[15] D N Page Phys. Rev. DU 1 98 (1976)
[16] R L Malice Phys. Rev. DU 2201 (1986)
[17] K D Krori. K Pathak and A Purkayastha Das (Deb) Submitted for publication (1996)
[18] H Forhoosh and R L Zimmerman Phys. Rev. Dll 317 (1980)
Indian J. Phys. 72A (6). 635-640 (1998)
UP A
— an international journal
Light-front QCD : present status
A Harindranath*
Theory Group, Saha Institute of Nuclear Physics,
l/AF Bidhannagar, Calcutta-TCX) 064, India
Abstract : We review the present status of light-front Hamiltonian approach to solve
Quantum Chromodynamics (QCD). After providing a brief motivation for the use of light-front
dynamics, we di.scus.s a recently propo.sed similarity renormalization group approach to QCD
We surnmanze recent advances made in the study of confinement in this approach The features
of chiral symmetry breaking on the light-front are highlighted A new approach to the study of
deep inelastic structure functions combining coordinate space and momemtum space techniques
IS bncfly outlined Lastly we mention some of the open problems in the field.
Keywords : Light-front QCD, high energy scaiienng
PACSNos. : ll.lO.Ef, 11.10 Gh, 12,38 Bx
1. Why light-front ?
I.ighl-front dynamics (1 ] was introduced by Dirac in 1949. He found that one may set up a
dynamical theory in which the dynamical variables refer to physical conditions on a light-
Iront X* = +x'^ = 0. is the light-front time and x' is the light-front longitudinal space
variable. Transverse variable jc^ = (.r*,jr^). For an on-mass shell particle, longitudinal
(^1 )2
momentum >0 and energy Jt" = — -p . From this dispersion relation,
wc observe that large energy divergences occur from large k^ and small k* and since they
appear not addilively in the expression for energy, one can expect nonlocal counlerterms
which results in a complex renormalization problem. Thus one may legitimately ask : why
bother ?
To answer this question, we have to take a look at the symmetries of light-front. First
consider the boosts. Under a longitudinal boost, Thus longitudinal boost is
simply a scaling operation which leaves jc* = 0 invariant. In canonical field theory,
generators of longitudinal boost and scale transformations obey identical commutation
relations. Since longitudinal boost invariance is an exact Lorentz symmetry, it cannot be
e moil : hari(ij>tnp.s^ha.eniet.in
72A(6)-23
© 1998 lACS
636
A Harindranath
violated by masses which is in sharp contrast to usual scale invariance. On the other hand,
transverse boosts are exactly Galilean boosts familiar in non-relativistic dynamics which
also leave x^ = 0 invariant. The fact that boost symmetry on the light-front is kinematical
has interesting consequences, for example, in the computation of the elastic form factor of
composite systems [2].
Since only carry inverse mass dimension jr and have to treated differently in
the scaling analysis. It immediately follows that power counting is different on the light-
front [3].
Next consider rotations. Rotations in the transverse plane are kinematical (light-front
helicity is kinematical) whereas transverse rotations change = 0 and hence are dynamical
and as complicated as Hamiltonian.
An attractive feature of the light-front is the apparent triviality of the vacuum. For a
massive on-shell particle, ^ 0. On the other hand vacuum processes receive contributions
only from ^ = 0. If = 0 is removed (say, by imposing a cutoff k* 2: e) then Fock space
vacuum is an eigen state of the full Hamiltonian. Thus, to build a hadron we need not worry
about the ground state of the theory. Thus the constituent picture of hadrons which
underlies ever popular quark models of hadrons may find justification in quantum field
theory.
( j^l )2 ^.^2
From the dispersion relation it " = -.k* near e which corresponds to
long longitudinal distances along the light cone appears as ultraviolet (large) divergences in
energy. This offers a possibility to address long distance effects (nonperturbative issues)
through renormalization.
After this brief introduction to the features of light-front dynamics, we take a look at
the canonical Hamiltonian of light-front (JCD.
2. Light-front QCD
2.1. Canonical structure :
Choosing the gauge A* = 0, the canonical Hamiltonian of light-front QCD can be
constructed from either the Lagrangian density or from light-front power counting.
+ {.idi, ) - imf (idi,
^ 2§^r‘'5) +
Lighi-firont QCD : presens status
637
At the tree level itself, canonical Hamiltonian exhibits processes which are sensitive to k*
near zero for gluons and processes sensitive to k* near zero for quarks.
2.2. Similarity renormalization approach :
To investigate the low energy structure, namely the bound state problem, one may visualize
solving the eigen value equation
P-\H')
M2
|*P).
( 2 )
with the state vector IV') expanded in terms of the muiti-parton wave functions.
Unfortunately this is a never ending series in field theory and direct diagonal izaiion is too
difficult to tackle. It is clear that one needs to make approximations. Any cutoff
Hamiltonian necessarily violates the sacred (Lorentz and Gauge) symmetries of the theory
and we have to figure out how to restore them. The important question is how to get finite
answers that are sensible.
Similarity Renormalization group approach [3] to tackle this problem was introduced
by Glazek and Wilson. Given the bare cutoff canonical Hamiltonian, to solve the bound
state problem, a two-step process is devised. First, effects at relativistic momenta are
computed using perturbation theory and possible structures of the counterterms are
identified. Second, the effective Hamiltonian at an appropriate low energy scale is
diagonalized to yield low energy observables. The effective Hamiltonian at the low energy
scale is constructed from the bare cutoff Hamiltonian using a similarity transformation
which is designed so that no vanishing energy denominators appear in every order of
perturbation theory and the effective Hamiltonian does not cause transition between low
energy and high energy states.
At the second step, by lowering the energy scale, particle degrees of freedom are
eliminated in favor of effective interactions that do not change particle number. If we
choose the energy scale to be just of the order of hadronic mass scale, the character of the
bound state problem changes from a field theoretic computation with arbitrary number
of constituents to a computation dominated by potentials. At that level, the coupling
does not run, we can choose it to be weak, and model the bound state calculation after
that of QED. By increasing the scale, we bring back relativistic processes and hope to get
closer to QCD.
2.3. Alternatives :
Alternative methods with the same goal in mind have been devised in the past. The Discrete
Light-Cone Quantization (DLCQ) program [4] of Brodsky, Pauli and collaborators attempts
a direct discretization in momentum space k^). The transverse lattice Hamiltonian
approach [S] of Bardeen, Pearson and Rabinovici treat x* and jt continuous while treating
the transverse space as discrete.
638
A Harindranath
2.4. Confinement:
A second order analysis of processes sensitive to small tC gluon in the similarity
renormalization (SR) scheme has lead to the emergence of logarithmic confinement [6].
Conventional perturbation theory leads to a complete cancellation of small divergences
in the single quark self energy and one gluon exchange processes. But SR perturbation
theory analysis leads to a partial cancellation. In an analysis with the small longitudinal
momentum cutoff {k* > e) both contributions contain log f plus finite terms. For color
singlet states log e terms cancels between the two type of processes. The remaining finite
terms behave like log I jr I for large jr and log I oH- 1 for large x-^. Utilizing this confinement
mechanism first principle calculations have been performed recently for the spectroscopy of
heavy quark systems [7].
2.5. Chiral symmetry' breaking :
For the cut off theory (k* = 0 mode removed) vacuum is trivial. This means mechanisms for
the effects associated with spontaneous symmetry breaking are very different in this theory.
Further, chiral symmetry is exact for free quarks of any mass which means that mechanisms
for the effects associated with explicit breaking are also different.
The second statement above may appear rather strange for .someone unfamiliar with
the features of the light-front. On the light-front it turns out that chirality is simply helicity.
The basic reason behind this remarkable property is the fact that on the light-front the four
component fermion field can be decomposed as -f i/r. The component is
dynamical and yr is constrained. In = 0 gauge the constraint relation is '
\f/~ (.v“ --j J dy' e{x~ -y )[a-^ gA-^ ) +
xy/''(y-,.r-L ). (3)
The fermion mass enters the Hamiltonian only through yr . Introducing the two
component field rf
¥*= I • ( 4 )
the free fermion Hamiltonian density is given by
)- + m-
We note that the fermion mass enters the free Hamiltonian as m^ and gamma matrices do
not appear in this case. There is an explicit chiral symmetry breaking term in the interaction
part of the Hamiltonian which is linear in the quark mass and is given by
( 6 )
Light-front QCD : present status
639
Since in the chiral limit we need to avoid degenerate pion and rho, it is clear that we
need noncanonical terms in our Hamiltonian that explicitly violate the chiral symmetry
and survive the chiral limit. At present investigations are under way to study this
problem.
3. High energy scattering
It is well known that the various structure functions one encounters in deep inelastic
scattering are Fourier transforms of equal jr^ correlation functions and in the gauge = 0»
they are amenable to very clear physical interpretation which leads to the celebtrated parton
picture. Also, light-front power counting which is based on light-front symmetries treat x~
and differently which is natural for deep inelastic processes. Recently we have attempted
to combine coordinate space techniques (Bjorken-Johnson-Low (BJL) expansion plus light-
front current algebra) with momentum space techniques (Fock expansion plus ultra-violet
fenormalization) to address problems at the interface of soft and hard physics. The former
leads to hilocal form factors and the later utilizes multi-parton wave functions. The aim is to
unify the description of both perturbative and nonperturbative physics using the same
language, that of multi-parton wave functions.
As an example, consider the twist two part of the structure function F 2 . Utilizing
BJL expansion and light-front current algebra one arrives at
(7)
where ^ = ^P^y~ . The bilocal form factor
V|((5)= ■^^(/’l[?(v)r*V'(0)- r(0)r*v^()')]|7’)- (8)
Considering a meson like state for the target, we expand the state |P) in terms of the quark-
antiquark amplitude ^ 2 ^ cjliark-antiquark-gluon amplitude ‘P 3 etc. A straight forward
evaluation leads to
ei
Utilizing the fact that the slate |P) obeys the eigen value equation, the high energy limit of
the structure function, can be computed perturbativcly from the knowledge of the high
momentum behaviour of multi-parton wavefunctions. In this approach we have investigated
18] various issues, namely, suppression of coherent effects at high energy, cancellation of
collinear singularities, emergence of factorization, etc. We have also clarified the parton
interpretation of the bad (-L) component of the bilocal vector current [9] and shown the
important of quark mass in the computation of the transverse polarized structure function in
perturbative QCD [.10].
640
A Harindranath
4. Open problems
Instead of a summary we list some of the immediate open problems in the field. In order to
probe the fate of logarithmic confinement one has to study higher orders in SR scheme.
Since the logarithmic confinement in second order is not rotationally invariant, one has to
see whether and how rotational symmetry is restored by higher order corrections to the
effective Hamiltonian. The study of chiral symmetry breaking on the light-front is in its
infancy. One has to study the origin and role of non-canonical operators and their
renormalization. The phenomenological consequence of such operators are also worth
investigating. Regarding the program for high energy scattering the crucial question is ; Can
one consistently calculate ? To answer this question, of course, we need to compute higher
orders in the BJL expansion. This is especially important for the study of higher twist
observables.
References
[1] For a pedagogical introduction see A Harindranath An Introduction to Light-Front Dynamics for
Pedestrians, id Light-Front Quantization and Non-Perturbative QCD eds. James P Vary and Frank Wolz
(distributed by HTAP. ISU. Ames, I A, USA) (1997)
[2] See for example, S J Brodsky and G P Lepage Exclusive Proces,%es in Quantum Chromodynamics,
in Perturbative Quantum Chromodynamics, eds. A H Mueller (Singapore : World Scientific) (1989)
[31 K G Wilson. T S Walhout. A Harindranath. W M 2^ang, R / Perry and St D Glazek Phys. Rev. D49
6720(1994)
[4] S J Brod.sky, H C Pauir and S S Pinsky Quantum Chromodynamics and other Field Theories on the L^ht
Cone, SLAC-PUB-7484, hep-ph/9705477
[5] W A Bardeen and R B Pearson Phys. Rev. D14 S47 (1976); W A Bardeen, R B Pearson and E Rabinovici
Phys. Rev. D21 1037 (1980)
[6] R J Perry in Hadron Physics 94 : Topics on the Structure and Interactions of Hadronic Systems
eds V E Herscovitz et al (Singapore : World Scientific) (1994)
[7] Wei-Min Zhang Phys. Rev. D56 1 528 (1997); M M Brisudova R J Perry and K G Wilson Phys. Rev, Lett.
78 1227(1997)
[8] For an overview of this approach, see A Harindranath and Rajen Kundu preprint (19%) hep-ph/9606433;
A Harindranath, Rajen Kundu and Wei Min Zhang hep-ph/9806220
[9] A Harindranath and Wei Min Zhang Phys. Lett. B390 359 (1997)
[lOJ A Harindranath and Wei Min Zhang hep-ph/9706419: Phys. Lett. B 408 347 (1997)
/ndteR I Phys. 7U (6), 641-661 (1998)
UP A
— an iinemniio nal journal
Methods of thermal field theory
SMallik
Sahi InsdtuiB of Nuclear Physics, l/AF, Bidhannagar.
Calcuna-700 064, India
Ataatmct ! We introduce the basic ideas of thermal field theory and review its path
integral formulation. We then discuss the problems of QCD theory at high and at low
temperaiiiius. At high temperature the naive peituibation expansion breaks down and is cured by
lesumnMtloo. We illustrate this improved pertuibation expansion with the theory and then
sketch itt application to find the gluon damping rate in QCD theory. At low temperature the
hadronic phase is described systematically by the chiral perturbation theory. The results obtained
from this theory for the quark and the gluon condensates me discussed.
Kapwnrds ; Thermal Held theory, QCD theory, chiral perturbation expansion
PAGSNoa. . : ll.lOWx, 12.38.Lg. 12.39.Fb
1. iDtroducdoD
ThennaJ field theory has grown into a vast subject. There has been a number of theoretical
developments, like the resummation at high temperature, chiral perturbation theory at low
tempenture, non-equilibrium formalism, ere. It has also been applied to topics ranging from
cosmology to heavy ion collisions in the laboratory. For most of the applications, however,
it is difficult to formulate the problem in a way which is realistic and at the same time
amenable to an easy theoretical study.
In tl^ review we shall not deal with any of the applications in particular; instead, we
shall discuss some of the theoretical developments in the QCD theory at high and at low
temperatures. That is, we discuss the methods available to find the properties of the QCD
medium in thermal equilibrium and of panicles propagating through it in the hadronic and
in the quark-gluon phase. Several well-written and much more complete reviews exist in
this area [1-3].
When the temperature is low, the system consists predominantly of pions. Chiral
perturbation theory is eminently suitable to evaluate all the physical properties of the
<S) 1998 lACS
642
SMallik
system. As the temperature is increased, the interaction among the pions become strong and
heavier hadronic degrees of freedom are excited. At some point a phase transition
presumably takes place giving rise to the quark-gluon plasma. There is no analytic method
based directly on QCD theory to discuss this transition region. When the temperature is
high enough, the quark-gluon interaction becomes weak so that the usual perturbation
expansion is expected to be valid. However, this naive expectation is not realised ; whqi the
external momenta are small compared to the temperature, loop contributions are of the
same order as the tree level contributions. A resummation is thus needed to restore the
validity of the perturbation expansion.
In Section 2 we discuss the basic ideas of thennal field theory and bring out its
similarity with the conventional (zero temperature) field theory. In Section 3 the path
integral formalism is obtained which gives rise to the real and the imaginary time versions.
An application of the real time formalism to the thermal state in the early universe is also
included here. In Section 4 we describe the resummation procedure at high temperature. In
Section' 5 we briefly introduce the elements of the chiral perturbation theory and discuss the
results of the quark and the gluon condensates it predicts at low temperature. We conclude
in Section 6.
2. Basic ideas
Conventional (zero temperature) quantum field theory describes the interaction of a few
fundamental particles in the vacuum. Thermal field theory extends it to describe (he
interactions in a statistical system in thennal equilibrium. (We do not discuss non-
equilibrium conditions in this review, except for a fecial type of non-equilibrium
appropriate to the early universe.)
Despite apparent differences, the conventional and the thermal field theories can be
developed in close parallels. This is because the Boltzmann weight factor, becomes
the time evolution operator in quantum mechanics on identifying the inverse temperature P,
with the imaginary time, -fir. It is the choice of the time path which distinguishes the
different formulations of the thermal theory as well as the conventional one.
The basic quantities are the thermal averages of operators. Thus for an operator A
we have
< A > =Tr(e-^A)/Z, (2.1)
where the Tr (ace) is to be evaluated over a complete set of states of the system and Z is the
partition function, Z = Tr For an operator in the Heisenberg representation, we have,
suppressing the space dependence,
= A(r + r').
e/^«A(0«-^" = A(r-i^).
so that
( 2 . 2 )
Methods afthermal field theory
643
Consider now the thermal average of the product of two operators A and B,
(A(f)B(f')) = z-' Tr(«-/w>i(Ofl(f'))
= (fl(/'-ij8M(0). (2.3)
where we have used the cyclicity of the trace in the second line and eq. (2.2) in the third
line. This equation, expressing thermal equilibrium condition, is called the Kubo-Martin-
Schwinger (KMS) condition.
A correlation function like (2.3) is not defined everywhere in the complex time
plane. To sec this, evaluate the trace over a complete set of eigenstates of the Hamiltonian,
^l^fi ) = l^n ) then insert the same complete set between the operators to extract
their time dependence,
= Z-' (m|A(r)B(l')|m)
m
01,11
The finiteness of the individual terms for define the strip of analyticity,
-/3 1 Im (/-/') < 0. (2.4)
Now consider the time ordered thermal propagator for a real scalar field of mass m,
• {T,<^M^(x')) = ec(t-t'){4>(x)<ti{x')) + e,{<l>(x'mx)) (2.5)
or, Df(x-x') = ec(l-t')D*(x-x') + e,(.t'-t)D^(.x-x'). (2.6)
Here Bg generalises the usual ^function to an oriented contour. The KMS condition (2.3)
applied to the thermal propagator becomes
*-*')■ (2.7)
For imaginary times the same condition shows that the euclidean propagator can be
continued outside the interval (0, ^ as a periodic function of euclidean time.
Note that the thermal propagator satisfies the same differential equation as the
one at T* 0,
(p-m^)D^(x-x') = S*(x-x'). ( 2 . 8 )
It is only in the boundary condition that the thermal propagator differs from the 7 s 0
propagator. The fact that the thermal propagator satisfies the same differential equation
as at 7 as 0 implies that the short distance singularities of the propagator is the same as at
7 s 0. Thus the same renormalization counterterms, needed to remove the ultraviolet
divergence of the theory at zero temperature, will also suffice to make the thermal field
theory divergence fme.
72A(6)-24
644
SMallik
3. Path integral formulation
Let I 0(x), t) be the basis ket in the Heisenberg picture, being the eigenstate of the field
operator 0(x, t) with eigenvalue 0 (r).
0(jr,f)|^(*).f) = ^(x)l^(x).»).
The basis kets evolve in time as
|0(x)./) =e'»|^(x)).
The Feynman functional formula giving the transition amplitude in going from 0|(r) at time
fi to (hQc) at time t 2 is
( <l>2 (*). h I 01 (*).'! ) = ( 02 (Jf) k-'""’ -'■ > I 01 (*) )
(5,,
where L is the Lagrangian. To arrive at the partition function, we let i(t 2 -t\) = P and
set ti = -T, where T is arbitrary at the moment [4]. The trace operation requires J [d<l>] to go
over all periodic paths, 0 (-7, x) = 0 (-7 ~ x) and to integrate the resulting expression
over the end values of the field. We then get formally the functional representation for
the partition function as
Tre-fif* = j</^{0k'^''|0)
= n\ (3 2 )
Aperiodic
Note that we have not yet specified the path of integration over time connecting the end
points -7 and -7 - ip. In principle it can be any path as long as it is within the analyticity
strip and slopping downward. The so-called imaginary and the real time formalisms result
from two convenient choices of the time path.
Imaginary time formalism :
It results from choosing the contour along the imaginary axis in the complex time plane
from 0 to -1)3 (see Figure la). Setting / = - it, (3.2) becomes
Z = W J ■ (3.3)
Being periodic in T, 0 (x, T) admits a fourier expansion in T,
0(X. T) = ^ X ^
P
= — V f t* **i".f A (t)
P
(3.4)
Methods of thermal field theory
64S
f:i
(a) (b)
Figure 1. The lime contours (a) and (b) for the imaginary and the real time fonmlitm
respectively.
Now the free action in (3.3) may be worked out to get the propagator [4]. Alternatively
recall that the propagator is a periodic function in T of period so that it has the
Fourier series,
C(jr-*',T-T')= (3.5)
Noting that
S(T-T')=
eq. (2.8) gives
G(k,u)J
1
0 )^ +m2
(3.6)
It is clear that the Feynman rules in the imaginary time formalism are the same as in
the conventional field theory with the replacements.
f d^k ^
J 2;r^ ^ " (2n)^ ’
*0
(f-> -ix)
Real time formalism :
If one is interested in Green’s functions with real time arguments, the imaginary time
formalism is not convenient, as it has to go through a non-irivial process of analytic
646
SMallik.
continuation. It is then useful to have the time integration over a path including the real
axis [5]. The propagator with such a time path can be obtained by solving (2.8) subject to
the KMS boundary condition (2.7), we found earlier in the operator formalism. (It can also
be obtained in the path integral formalism.)
Introduce the spatial Fourier transform,
Dp(x-x') =
where the parameter r runs on the time CSniour. Now Dp satisfies,
(j)] =k'^ + m'^.
The most general solution is obtained by adding homogeneous solutions to any
particular solution, which we take to be the (zero temperature) solution with Feynman
boundary condition,
bp(.T,T',k)= - +
The KMS condition (2.7) now gives
getting finally
+ {e,(T'-r) + (3.7)
A contour which gives rise to a symmetrical propagator is shown in Figure lb.
It starts at -T and runs along the real axis to +7 (segment Cj), drops vertically from +T to
+r - iP/2 (segment C 3 ), returns parallel to the real axis to -T - ifi/l (segment C 2 ) and
finally again drops vertically to -7 - ip (segment C4). It can be shown that as 7 the
generating functional factorises into a contribution from C] and C 2 and a contribution from
C 3 and C 4 . Thus for the computation of the real time Green’s function, the functional
integral involving C 3 and Q behaves like a multiplicative constant and may be dropped.
In momentum space the elements of the 2 ® 2 matrix propagator can be easily
obtained from (3.7)
=D^‘(k)i2 = D(k) + 2m(a^)S(k^ -m^), (38)
=D^(Jt)2, = 2m{C0ii)eP<“/^S(k^ -m^),
where n (a)^) is the Bose distribution function, n(to* ) = - I)"' and D(k) is the zero
temperature Feynman propagator, D(k) = i/{k^ +i£).
Methods of thermal field theory
647
Although the propagator has now a matrix structure, the topological and the
combinatorial structures are the same as in the zero temperature theory. The matrix
structure, arising out of the two segments C| and Ci in the lime path, may be interpreted as
due to a doubling of the degrees of freedom : the field of type 1 living on the segment Cj
and the field of type 2 on the segment C 2 . the 'thermal ghost’ field. There is no direct
coupling between the two types of fields and the Feynman rules for the two kinds of
vertices differ by a minus sign.
Of course, we are interested only in Green’s functions of type 1 fields, but the
perturbation expansion brings in type 2 vertices along with the type 1 vertices. If we use
only the type I vertices, pathological terms appear. But the contributions of type 2 vertices
Figure 2. The double loop diagrams in the two point function for (a) the
physical and (b) the ghost vertex
just cancels these terms. As an example, consider the two 2>1oop diagrams of Figure 2. The
contributions of these two diagrams separately are.
2 2 f ^4/ f
where the propagators are multiplied with the matrix T, which is diagonal with elements
1 and -1. It takes into account the sign change at the type 2 (ghost) vertex. Each of the
above expressions has a pathological term - (5(4^ - ))^ . However the two terms can
be added to give
It IS helpful to use the so-called mass derivative formula [6]
DpT=(Dptr^'
U) write (3.9) as
which is a well-defined expression.
(3.10)
(3.11)
648
SMallik
Non-equilibrium in early universe :
Though a discussion of the thermal r m-equilibrium situation is outside the scope of this
review, we nevertheless wish to point out an application of the real time formalism to a kind
of non -equilibrium, which presumably took place in the early universe.
In the expanding universe there is no strict definition of thermal equilibrium.
However, operationally, an equilibrium condition ia reached around a time fQ, say, when the
collision rate of the particles far exceeds the expansion rate of the universe. Then the
density matrix is given by
In the Heisenberg representation the density matrix is constant (even if the Hamiltonian has
explicit time dependence, as is the case here). Thus the thermal average of an operator 0
continues to be given at later times by the expression
(0(0) =Trp(fo)0(l) (3.12)
even if the system ceases to be in thermal equilibrium.
Let us describe the matter in the early universe by a single real scalar field. The
action for the matter field in an external gravitational field is given by
S= (3.13)
where the mass includes the thermal contribution. In the standard cosmology the metric
is taken to be homogeneous, isotropic and spatially flat,
ds^ = dt^ - a{t)^ dx^.
In the path integral formulation, the factor is represented as a functional
integral involving the Euclidean action associated with H (to). The time evolution of the
field 0 (x, r), on the other hand, is analysed in terms of a minkowskian path integral
involving the action (3.13). To evaluate a quantity like (3.12), the two types of functional
integrals need be glued together. One thus gets the time path of Figure 3, as proposed
originally by Semenoff and Weiss [7].
The propagator now becomes a 3 9 3 mauix satisfying
= (3.14)
where Ki = -^rr + 3—-^ + ,
dt'^ a at I
and tu 2 (/) = -2 2 + A/ 2 - o)(tQ ).The boundary conditions to be imposed ait
obtained by matching the components of at the meeting points of the three segments.
Methods of thermal field theory
649
The resulting propagator will have additional short distance singularities
compared to the zero temperature propagator, due to the fact that the density matrix
is specified sharply at t » (q. They are similar to those at zero temperature in the
case of a background geometry for which the derivative of the scale factor a(t)
changes abruptly at t » to. This additional singularity makes the field theory non-
renormalizable.
This problem may be avoided [8], if we ihermalise ihc system at a lime prior lo
Iq in a fictitious, static background and ^hen follow the dynamical evolution of the
Greens function as the fictitious geometry smoothly goes over to the geometry of
interest. Below we first assume such a deformed geometry and then assess its ellect on
the propagator.
The plane wave decomposition of the propagator may be written as
r d^k , -
( 3 . 15 )
where
l2»I(f)]■’'^ a = 1.2
[[2«Io]■’'^ 0 = 3
We first write the Minkowski space mode function.
( 3 . 16 )
650
S Mallik
with ihe boundary conditions /^(/q) = l./*(^o) = They are then extended to
functions defined on the complex contour by
/*(0
fHt). a=l2
e±o>of ^ a = 3
It is now simple to write the propagator on the complex contour as
G(k. T,r') = {eA.r-t') + B}f*{T)f-(T')
+ {6,(r'-t) + B}f-{T)f*{T') (3,17)
with - I )■' . Note the remarkable similarity of this propagator with the flat-space
propagator in (3.7). The density function refers only to the time (q. It can now be easily cast
in the form of a 3 3 matrix.
Coming back to the question of using the deformed scale factor, we note that it
cannot affect the mode functions significantly as long as
- a
atf-fo, (3.18)
cr a
which for M - gives
(3.19)
a
This condition should be compared with the condition for maintaining the iherrMal
equilibrium' in the expanding universe. The collision rate - while the expansion rate
-'ala. Thus thermal equilibrium requires,
-. (3.20)
a
Thus once condition (3.20) is satisfied, (3.19) is automatically satisfied.
The reader may recall the “infrared problem", encountered in setting up quantum
field theory on curved space-lime. If the momentum kfa and Mjit) are so small that
the curvature term dominates in cq. (3.16) for the mode function, we cannot define
them to belong to positive and negative frequencies. The resulting field theory appears
ambiguous.
The present formulaiibn of the quantum field theory in the cosmological context
avoids this ambiguity. It is, of course, essential that the evolution passes through a phase
where the condition (3.18) holds, i.e., the effective mass is large enough compared to the
expansion rale. It ensures the existence of positive and negative fcqucncy mode functions
around the time Iq. Later on. the scale factor and the mass may well develop in such a way
that the expansion rate exceeds the mass. But once the system is in a thermal stale at /o- H''
evolution can be traced on the basis of the thermal propagator, irrespective of wheihei oi
not the condition (3. 18) continues to hold.
Methods of thermal field theory
651
4. Resummation at high temperature
At high enough temperature the QCD medium dissociates into quarks and gluons with
simultaneous weakening of the strong interaction, so that the ordinary perturbation
expansion is expected to hold. However, this expectation is naive : Loop corrections tend to
be as large as the tree level contributions at high temperature. Indeed, it is this breakdown
of perturbation expansion which constitutes the earliest example of application of thermal
field theory to particle physics, viz, the restoration of symmetry at finite temperature
[9-111. The gauge symmetry breaking at zero temperature by the Higgs potential is restored
at high temperature when the tree level (tachyonic) mass of the Higgs field is compensated
by its loop correction.
This situation in thermal field theory calls for a resummation and a consequent
reformulation of the perturbative expansion. Although loop corrections were included
earlier in the propagators by several authors [12], the systematic approach to the problem is
due to Braaien and Pisarski [13].
Consider a field theory at high temperature, when the bare masses of the particles are
negligible compared to the temperature. Then one has natural momentum scales 7, gT and
so on, where g is the (small) coupling constant. Restricting to one loop diagrams, we have
two .scales to consider ; the hard, of order 7 and the soft, of order gT, A momentum is called
hard, if the magnitude of any of its components is of order 7; it is called soft, if the
magnitudes of all its components are of order gT. The resummation is needed only for
amplitudes with soft external momenta. One finds that the contribution of one-loop
diagrams to such amplitudes, which are of the order of the tree graphs, arise from hard
inicrnal momenta. The dominance of these so-called hard thermal loops (HTL) is
understood by recalling that the temperature in the density distribution provides the cut-off
tor the otherwise divergent momentum integrals. In addition, the Landau singularities also
contribute to the enhancements.
Although the non-abelian gauge theories present all the aspects of the resummation
programme, the basic ideas can be illustrated by a consideration of the scalar field theory
1 14], to which we now turn.
Sailar field theory :
Consider the field theory of a single massless scalar field,
(4.1)
Let us examine the one-loop contributions to different n-point functions. The self energy to
one loop is given by the tadpole diagram. Figure 4a. Subtracting off the zero temperature
pan, it is given by
Wl"“’ ■ ^ -
(4.2)
72A(6)-25
652
SMallik
Summing over the series of one particle reducible tadpole diagrams, we gel the elTeciivc
propagator to one loop,
*D(k) =
I
(4.3)
Thus if the momentum kfj is hard, the effective propagator is, to a gcKxl approximation, the
same as the bare one. On the other hand, if k^ is soft, the loop correction is as large as the
bare inverse propagator.
(^t)
(1-)
Figure 4. The two poini funciion wiih (a) che bare propagator and (b) the
efiective propagaioi
(.0 ( h )
(<■)
i :i
idi
Fifsurc 5. The lour poini in (al tree level and Ih). ic) and id) ai one loop
Next consider the diagrams for the 4-pomi lunclion in Figure 3. The hare xerlex is
e". The eonlribulion of the other three diagrams are similar. To illustrate the nature ol the
contribution expected, consider Figure 3b Let p = p\ + Pz = P\ + P^ Then the diagram
conn ibutes as
, f r/V I ,
J 2£,2f, ' ■
!>„ - f, - /)„ + £| + £; ;
I r ^
+
p^, + p^ + E^-E2 )
(44)
where £, = I £ I, £> = I /? - A 1 and and //i are the corresponding Bose densities. The
retarded amplitude is given by replacing pQ by />(, + i£. Consider now p^^ > 0. At T = 0. the
absorptive part (given by I m the factor (I + /i| + 112 ^^ corresponds to both the inteinal lines
ha\ mg positive energies. But lor 7> 0, there arise additional contributions corresponding lo
one of the lines having positive and the other negative energy. It represents Landau
damping, where one particle is absorbed from the heat bath and the other emitted into 11
The Landau terms have discoiuinuilies below the light cone, while the non-Landau leiiiis
give rise to ihe usual discontinuity above the physical threshold.
Methods of thermal field theory
653
We now evaluate the contribution of (4.4) foj|- soft external momentum The
r = 0 contribution, after renormalization, is proportional to In where p is the
renormalisation scale. It is In compared to the bare vertex.
The other terms in (4.4) involve the density function n{k). It is easy to estimate these
terms lor soft loop momentum k, for which n{k) ^T/k. With the internal and external
momenta both soft, the only scale is gT, so that the integral is g'*n{k) It is thus
suppressed by a power of ^ compared to the bare vertex.
The remaining domain of integration is over the hard internal momenta. When p is
soft and k hard,
^1 = |*|. E 2 = \p-k\ - |/t| - |p|cos0,
Po ± (£| +£ 2 ) - ± 2|Ar|, pq ± (E^ - E 2 ) - Pq ± 1p|cos0,
|p|cos0
/i| - /12 - I. fh -/I2 — ni(l + //| ).
Then the Landau terms contribute as while the non-Landau terms given -g'* In (T/p),
where p is the soft momentum. So the contribution of the HTL is also --g'^ In g compared to
the bare vertex.
It is important here to notice that though the Landau terms have energy
denominators larger by a factor of \/g with respect to the those in non-Landau terms, the
lormcr could not dominate because of (/ij - /I 2 ) - p/T, The siioation will be different for the
QCD theory, where the fermion density function is ~ I for hard internal momenta.
Similar analysis shows that the corrections to all higher point vertex functions are
small compared to the their tree level contributions. Thus in the scalar theory the only HTL
IS in the two point function.
Having obtained the effective propagator, we can now construct the effective
perturbation expansion to replace the naive one. Rewrite the Lagrangian (4. 1 ) as
L=Lo + 5L,
where 4 =
SL =
= (4.5)
The effective expansion is obtained by constructing the usual perturbation expansion with
which is the same as the earlier bare one except for the thermal mass term in the
propagator. The counlerterm 5L is a reminder to avoid double counting, i.c., to exclude the
coiuribulion of the hard internal momenta in loops appearing in the propagator with soft
momenta.
654
S Mallik
As an example of the effective perturbation expansion, we calculate the leading
correction to the cITeclivc self-energy to one loop. It is just the same tadpole diagram with
the bare propagator replaced by the effective propagator (Figure 4b),
d'k I
(2ff)’ 2E
2/1 (£■)..
£ = +'"j (4.6)
where again the zero temperature contribution has been subtracted off. For small values of
fti/T, it can be evaluated as
+T = ^1 - + •••j. (4.7)
Since the counterterm subtracts off the hard thermal loop (s <51). we are left with solt
internal momenta k - fiT in cq. (4.6), lor which n(k) - Tfk as we noticed already. Over such
momenta it is of the order of
lilLL
k k
2
which IS the origin of the second term in (4.7). (In the integration region over the hard
internal momenta, (4 6) has also a correction along with HTL. But it is with respect to
the latter )
Hot QCD theory :
The existence of hard thermal loops in QCD theory may be investigated essentially in the
same way as we did lor the .scalar field theory Here the source of complication lies in the
lad that, unlike the ease for the scalar theory where HTL exists only in the two-point
junction, all N-point lunctions ol gluons and all (N-2)-point function of gluons and a quark
pair have hard thermal loops.
Let us illustrate the rcsummation programme for QCD by discussing the gluon
damping fate in a schematic way |I5|. Dropping the colour and the space-time indices,
the bare gluon propagator and the bare three- and the four-gluon vertices arc
wiiiten respectively as Mp), and where the momentum dependence of the
vertices arc also omitted. The corresponding clfcctive quantities arc written as ‘4
and c" r
T,, = r„ + 5r„, /i= 3.4 (4.K)
p- +on
where Sf] and dr„ arc the contributions of the hard thermal loops in the gluon self-energy
and the vertices respectively.
Methods of thermal field theory
655
The cffcclivc expansion Tor ihc gluon scif-cnergy lo one loop can be wriiicn
schemaiically as
•/7(p) +
+ contributions ol the counicrtcrms, (4.9)
where the first two terms correspond to the diagrams of Figure 6. The /.ero of ihe inverse
gluon propagator is given by
( 0 ~ -p2 -Sn-*n{E,p) = 0 , Sn{E, O) S (4.10)
so that the gluon damping rale at zero momentum is given to lowest order in .eby
y(0) = . ' Im'nif. 0). H.ll)
2m,
We now c.stimaie the order of magnitude of the contributions to */7 coming from the soft
and the hard internal momentum regions. In the following wc denote a loop corection as
0(/;f''). if it IS of order with respect to the corresponding tree amplitude.
(!•)
Flgutt 6. Effcciive expansion of the gluon self-energy lo one kwp
656
S Mallik
Lei us lirsi show that the inlegralion over hard momentum in (4.9) does nol give
any coniribuiion lo /(O). li gives (9(1) contribution but the resummation programme
IS just designed lo cancel it with counlerterms. But this region also give terms 0(/»),
Ot.e-) which were neglected in arriving at the HTL contribution. However they
cannot contribute to the discontinuity in (4.11) for kinemalical reasons. When the
internal lines are put on mass shell, both lines will be hard. So the discontinuities
are either of the Landau damping type in the unphysical region or far above the
threshold.
We are then left with soft loop momenta. With internal and external momenta both
soft, the erteclive propagators and the vertices are of the same order as the bare ones. ALso
the only mass scale in the integral \sgT. Thus if the outside factors of ^ and the density
functions arc removed, the remaining integral Hence for soft momenta, the integral
must be -ght{E) {gT)- -g^P. Thus with -gT, we get from eq. (4.1 1), y(0) -g^T.
Nontrivial loop calculations involving effective propagators are required to find the
constani of proportionality f 151.
5. Chiral perturbation theory at low temperature
Chiral perturbation theory (j^PT) [16], so successful in analysing the low energy structure of
(he QCD theory, can naturally be extended to finiic temperature to describe the
thermodynamic and other thermal properties of the theory.
Let us first briefly recall the basis of ;jfPT. Consider only the «.and the d quarks. Tfl a
good approximation, they may be taken to be massless. Then the QCD Lagrangian
becomes,
-^OCD = - iA^(jr))^(A), (5.1)
where q{x) has two components in flavour space and A^{x) is the colour 5(y(3)-malrix
valued gauge potential. It is invariant under independent isospin transformations of the left-
handed and the right-handed quark fields,
qR geSU^L ®SU(2 )k. (5.2)
It is generally believed that this chiral symmetry is spontaneously broken down to
5(7(2) V/ by the ground state- of the theory having a quark condensate. It gives rise
to 3 massless, pseudoscalar Goldstone bosons, lo be identified with the pion triplet.
This symmetry is again broken explicitly by the quark mass term, which gives pion
Its mass.
It can be shown that the above transformation law for the quarks induce the
transformation law.
U{x) -> V^U{x)Vl'.
(5..1)
Methods of thermal field theory
657
t>n the matrix U{x) of the pion fields 0, U{x) = where F can be identified with the
value of the pion decay constant in the chiral limit. Thus although U{x) transforms linearly,
0(a) transforms non-linearly.
It is now easy to construct the effective Lagrangian invariant under the
lianstormalion law (5.3). There cannot be any term without derivatives. The pieces in the
Lagrangian can be ordered according to the number of derivatives,
= + (5.4)
The lowest order term is the one with two derivatives,
(5.5)
Chiral symmetry is broken by terms in which contain the (diagonal) mass marlix of u
and d quarks. To lowest order
= ^F^Bir\m(U + U^)], ( 5 . 6 )
where B is given by Ml = (m„ + nij)B. By ;|fPT one refers to the effective field theory
constructed with this effective Lagrangian, which combines the expansion in powers of
momenta with expansion in powers of and wij.
As already pointed out in Sec. 3, the partition function can be converted to a path
integral formula, leaving the time contour free to choose. To compute the static
thermodynamic properties it is convenient to use the imaginary lime formalism. With the
effective chiral Lagrangian (5.4-6), we get
Z= (5.7)
the path integration extending over all pion field configurations which are periodic in the
euclidean time direction, U{x, x 4 + p) = U{x, X 4 ).
The partition function has been evaluated to 3 loops in ;|;PT by Gerber and Leutwyler
117]. They carry out the entire calculation in configuration space, where the pion propagator
can be written as
C()= ^Mx,x^+nP).
rr = -<»o
where ^(jr) is the euclidean propagator at zero temperature.
For massless quarks the pressure has been calculated to give
P = (1 + ^ 1" i^p/T) + 0(r‘)) (5.8)
where - 275 MeV. The leading (one-loop) contribution is the familiar Bose gas term.
The two-loop contribution is zero, as the nn scattering amplitude vanishes in the forward
658
SMallik
direction, on account of the Adler zero. We refer to the original papers [17] for the details
of the calculation. Below we only discuss die results for the quark and the gluon*
condensates.
To get the quark condensate we have to perturb the chiral Hamiltonian by the
quark mass term
Z = (5.9)
where m is the quark mass. We get
{qq) = Z-'1x€-0"qq = (5-10)
where the volume V goes to infinity at the end. Since In Z = - PV(£q -P), where ^ and P
are the vacuum energy density and the pressure respectively, we get
Thus it is necessary to work out the pressure up to the term linear in the quark mass. One
obtains finally [17]
{qq) = (0|«|0)^1 |jr2- y /F) + •••j,
x=T'^l%F^, (5.12)
where = 470 MeV.
The temperature dependence of the gluon condensate can also be determined [18].
The trace anomaly reads as
0i‘ = Bllg'iG’' C>‘'" ■ -G^,
^ 2gi *
(5.13)
2
where pig) denotes the beta-function of the (JCD theory, P(,g) = — - — j-(ll n^),
(4?r) 3
being the number of quark flavours. To normalise 0^ such that it is zero in vacuum,
we write
-G^ + (0|C2|0)
giving (G^ ) = (0|G^ |o) - ).
(5.14)
The thermal average of 0^ can be related to pressure
e-3P =
The series (5.12) and (5.15) for the two condensates in powers of {TIF) must be treated as
asymptotic [19]. Any non-Goldstone particle of mass M gives a contribution of
which does not show up at any finite order. Though negligible below T - 140 MeV, such
contributions grow rapidly with further increase of temperature.
Corrections from non-zero quark masses and contributions from more massive states
have been included in the formula for the quark condensate. Even then its validity is
expected up to T- 150 MeV, as beyond this temperature the interaction of massive states
with the pions and among themselves become significant. HoweVer since the condensate
falls off rapidly at the upper end of this range, it is meaningful to make an estimate for the
critical temperature from the corrected formula, which gives = 190 MeV.
Although the gluon condensate also melts with growing temperature, the melting
takes place much more slowly than in the case of the quark condensate. The difference may
be traced to the fact that while qq transforms in a nontrivial manner under chiral
transformations, and are chiral singlets. The gluon condensate is a parameter
associated with non-perturbalive scale breaking effects and does not represent an order
parameter.
6. Condusion
We began by reviewing the path integral formulation of the perturbation theory at finite
temperature. The discussion concerned only the thermal equilibrium, except for the special
non-equilibrium situation which is relevant in the context of the early universe. Then we
explained, in the simpler context of theory, the necessity of resummation of the
perturbation series to restore its validity at high temperature. As an example in QCD theory,
we sketched the calculation of*the gluon damping rate.
We must add that the subject of resummation at high temperature is far from being
closed. As already pointed out by Braaten and Pisarski [13]. there appear collinear
divergences when the external particles are on the mass shell or massless. This fact and the
absence of magnetic mass generation in perturbation theory give rise to a number of
problems, which are actively persued at present.
At low temperature in the hadronic phase, the (JCD theory as such is very
complicated and is replaced by its symmetries as embodied in )ljPT. Here we discussed
72A(6)-26
660
SMallik
mainly Che results it gives for the quark and the gluon condensates. Among Che numerous
applications of ;)^PT at finite temperature, we mention the calculation of the effective
masses of hadrons [20].
Not covered in this review is the method of QCD sum rules at finite temperature
[21]. Is has the potential to provide substantial information on the thermal properties of
QCD theory. Unfortunately all the works done so far with these sum rules are incomplete
and hence unreliable in that not all the operators of leading dimension, which appears
in the operator product expansion of the two point functions [22], are included in the
sum rules.
Clearly none of the methods are adequate to analyse the intermediate region of
temperature, where the QCD medium is supposed to undergo a phase transition. The
appropriate method here is the numerial analysis on the lattice, which again is not discussed
in this review.
Acknowledgments
I wish to thank the organisers of the XII DAE Symposium on high energy physics for the
invitation to present this review. I also thank Mr. K Mukherjee for preparing the latex file
for the diagrams.
References
[ I ] N P Uindsman and Ch G van Weert Phys. Rep. 145 141 (1987)
[2] M Le Bellac Thermal Field Theory (Cambridge : Cambridge Univ. Press) ( 1997)
[3] P Aurenche in Proc 4th Workshop on Hif^h Energy Physics Phenomenology eds. A Dutta, P Ghose and
A Raychaudhury (Calcutta ; Allied) (1997)
[4] C W Bemaid Phys. Rev. D9 3312 (1974)
[5] A J Niemi and G W Semenoff A/m. Phys. 152 lOS (1984); Nucl. Phys. B230 181 (1984)
[6] Y Fujimoto, H Matsumoto, H Umezawa and I Ojima Phys. Rev. D30 14(X) (1984)
[7] G SemenofT and N Weiss Phys. Rev. D31 689 (1984); ibid. D31 699 (1984)
[8] H Leutwyler and S Mallik Ann. Phys. 205 1 (1990); See also N Banerjee and S Mallik Arm. Phys. 205
29(1990)
[9] D A Kirznitz and A D Linde Phys. Lett. 42B 471 (1972)
[ lOJ S Weinberg Phys Rev. D9 3357 (1974) ~
[11] L Dolan and R Jackiew Phys. Rev D9 3320 (1974)
[12] HA Weldon Phys. Rev. D26 1 394, 2789 (19|2); V V Klimov 5^. J. Nucl. Phys. 93 939 ( 1981)
[13] E Braaten and R D Pisarski Nucl. Phys. B337 569 (1990)
[14] R D Pisarski Nucl. Phys. A525 175c (1991)
[15] E Braaten and R D Pisarski Phys. Rev. D42 2156 (1990)
1 16] S Weinberg Physica A96 327 (1979); J Gasser and H Leutwyler Nucl. Phys. 11307 763 (1988); Ann. Phys.
158 142 (1984). For a lucid summary of different aspects of xPT see H Leutwyler Lectures at the
Workshop on Hadron (Cramado, RS, Brazil) (1994)
Methods qf thermal field theory
661
[17] P Cert)er and H Leutwyler Nucl. Phys. B321 3(7 (1989); See also J Gosser and H Leuiwyler Phys. Len
BU4 83(I987)
|I8) H Leutwyler PAys. Lew. B2M 106 (1992)
(191 H Leutwyler Lecmrt given al ihe Workshop on Effective Field Theories (Dobogokoe. Hungary)
(1991)
[20] ASchenkPfcyi Rev IM7SI38(l993);CSongPA>’j. Rfv.D49 1556(1993), D48 (375 (1993)
[21] A I Bochkarev and MEShaposhnikovA/ud PAyt. B26( 220 (1986)
[22] S MallikPAyr. Len. B416 373 (1997)
Indian J. Phys. 7ZA (6), 663-677 (1998)
UP A
an inteniational journal
Quantum integrable systems : basic concepts and
brief overview
Anjan Kundu
Saha Institute of Nuclear PHysics, Theory Group*
l/AFBidhanNagar. Calcutta-700 064, India
Abstract : An overview of the quantum integrable systems (QIS) is presented. Basic
concepts of the theory are highlighted stressing on the unifying algebraic properties, which
not only helps to generate systematically the representative Lax operators of different
models, but also solves the related eigenvalue problem in an almost model independent
way. Difference between the approaches in the integrable ultralocal and nonultralocal
quantum models are explained atid the interrelation between the QIS and other subjects are
focused on
Keywords : Quantum integrable systems. 2d statistical models. Algebraic approach
PACS Nos. : 03.65.Fd, 1 1 .55 Ds, 1 1 lO.Lm
1. Introduction
The theory and applications of nonlinear integrable systems is a vast subject with wide
range of applications in diverse fields including biology, oceanography, atmospheric
science, optics, plasma etc. The quantum aspect of the subject is a relatively new
development. However the theory of quantum integrable systems (QIS) today has grown up
into an enormously rich area with fascinating relations with variety of seemingly unrelated
disciplines. The QIS in one hand is intimately connected with abstract mathematical objects
like noncorommutative Hopf algebra, braided algebra, universal /^-matrix etc. and cfn the
other hand is related to the concrete physical models in low dimensions including quantum
spin chains, Hubberd model. Calogcro-Sutherland model as well as QFT models like sine-
Gordon (SG), nonlinear Schr^idinger equation (NLS) etc. The deep linkage with the siat-
mech problems, conformal field theory (CFT), knots and tT:iids etc. is also a subject of
immense importance.
In giving the account of this whole picture within this short span of time, I am really
faced with the problem of Tristam Shendi [1], who in the attempt of writing his
© 1998 lACS
664
Anjan Kundu
autobiography needed two years for describing the rich experience of the first two days of
his life and thus left us imagining when he would acomplish his mission. Therefore I will
limit myself only to certain aspects of this important field and will be happy if it can arouse
some of your interests in this fascinating subject.
We have to start possibly from an August day in 1834, when a British engineer
historian. John Scott Russell had a chance encounter with a strange stable wave in the
Union canal of Edinburgh [2]. Such paradoxically stable solutions will be observed again
after many many years in the famous computer experiment of Fermi, Ulam and Pasta [3].
However only in the mid-sixties such fascinating phenomena will be understood fully as the
solutions of nonlinear integrable systems and named as Solitons [4],
Formulation of the integrable theory of quantum systems started only in late
seventies [7], though today many research groups all over the globe are engaged in active
research in this field.
Mathematical basis of classical integrable systems was laid down mainly through the
works of Sofia Kawalewskaya. Fuchs, Painlev6, Liuoville and others [6]. There arc many
definitions of integrability; we however adopt the notion of integrability in the Liuoville
sense, where integrability means the existence of action-angle variables. That is, if in a
Hamiltonian system H\p{x, r), qix, f)] given by the nonlinear equation
5H 6H
^ Sq' ^ ’
(LI)
it is possible to find a canonical transformation ip{x, t), q(x, t)) (a(A), b{K 0). such
that the new Hamilonian becomes dependent only on the action variables, i.e. H = H[a{k)],
then the system may be called completely integrable. In this case the dynamical equations ;
Slj Sij
- -=7“ = 0, 6 = = CO, can be trivially solved and moreover we gel as the
Ob da
generator of the conserved quantities. The number of such independent set of conserved
quantities in integrable systems coincides with the degree of freedom of the system and in
field models it becomes infinite. One of these conserved quantities may be considered as
the Hamiltonian. The inverse scattering method (ISM) [5] is an effective method for solving
nonlinear equations. The important feature of ISM is that, instead of attacking the nonlinear
equation (1.1) directly, it constructs the corresponding linear scattering problem
r,(jc, A) = L(q{x, r), p(x, f), ^)T{x. A), (1.2)
where the Lax operator Uq, p, A) depending on the fields q, p and the spectral parameter A
contains all information about the original nonlinear system and may serve therefore as the
representative of a concrete model. The field q in ISM acts as the scattering potential. The
aim of ISM is to find presizely the canonical mapping from the action-angle variables to the
original field and using it to construct the exact solutions for the original nonlinear
equation. Soliton is a special solution, which corresponds to the reflectionless (6(A) s 0)
potential.
Qiumium Iniegreble sysienu ; basic concepts etc
665
1 Examples Oltaitcirtbleflyiteiiii
U, » » ^ .Mmptes of ,te u. ^
“* " “> ™™™l» inv™. obiM » 0. l.„^
/. Trigonomefric Class :
I . Sine-Goidon (SO) model (Equation and Lax operator)
«(x.f)„-M(x.f)„ = ^sin(r)H(x.f)),
tsc
= f ‘P-
\ms\n {k + J
msin {X-r}u)\
■n«). -ip }" = "■
2. Liouville model (LM) (Equation and Lax operator)
«(x.f)„ - u(x.t)^ = ieJ-i-u.o, = i
P.
&
-P
3. Anisotropic XXZ spin chain (Hamiltonian and Lax operator)
N
* = X + cosrj(T>’^, ),
ft ^
r sin ( A + fj<j ’ ). 2i sin a(j; ^
^.U) =
(^2isinqtr*. sin(A-Tj(T’)
(2.1)
( 2 . 2 )
(2.3)
II. Rational Class :
1 . Nonlinear Schrddinger equation (NLS) (Hamiltonian and Lax operator)
«y(x.r), + vf(x.r)„ + fj(vr*(x,r)v'(x,0)v'(Jf.O = 0,
jfNLs(^) =
'a.
I \
qiyr
(2.4)
2. Toda chain (TC) (Hamiltonian and Lax operator)
I '
Let us note the following important points on the structure of the above Lax operators.
(i) The Lax operator description generalises also to the quantum case [7,8]. Its elements
depend, apart from the spectral parameter A, also on the field operators u, pot
etc and therefore the quantum L(A)-operators are unusual matrices with
noncommuting matrix elements. This intriguing feature leads to nontrivial
underiyingialgebnic stnictures inQlS.
666
Anjan Kundu
(ii) The off-diagonal elements (as % in (2,4) and (T, <7^ in (2,3)} involve creation and
amihilation operators while the diagonal terms are the number like operators. It is
obvious that under matrix multiplication also this property is maintained, which has
important iiiiplications, as we will see below.
(iii) The first three models, though diverse looking, belong to the same trigonometric
class. Similarly the rest of the models represents the rational class. The fact signals
about a fascinating universal behaviour in integrable systems based on its rich
algebraic structure.
3. Notion of quantum integnibility
Note that the Lax operators are defined locally at a point x, or if we discretise the space, at
every lattice point i. However, since the integrability is related to the conserved quantities,
which are indeed global objects, we also have to define some global entries out of the local
description of the Lax operators. Such an object can be formed by matrix multiplying Lax
operators at all points as
T(A)=nt,(A)=
i»i
A(K)
C(A) D(A)J
(3.1)
Here the global operators B{k), C{X) are related to the angle like variables, while A(A), D(X)
are like action variables and t(A) » trT{X) s A(A) D{X) generates the conserved operators :
In r(A) ~ Ij Cj . For ensuring intcgrahilif f one must show for the conserved quantities
that ] = 0,[C„ ] = 0, which is achieved by a key requirement on the Lax
operators (for a large class of models) given by the matrix relation known as the Quantum
Yang-Baxter equation (QYBE)
(A, n)Lu (A)L2, {fi) = Li, (/i)L„ (A)J?,2 (A. //). (3.2)
with the appearance of a 4 x 4-matrix /^(A, y) with c-number functions of spectral
parameters, satisfying in turn the YBE
R\1 (A. /i)/fn (A. y )«23 (ft. r) * R23 (/*• r)Rii (A. r)Ri2 (A. m )- (3.3)
Due to some deep algebraic property related to the Hopf algebra the same QYBE also
holds globally :
(A. m)T, (A)r2 (/i) - Tj (m)T, (X)R,j (A. n). (3.4)
with the notations Ti^T ®/, T 2 ^I 9T. Taking the trace of relation (3,4), (since under the
trace i? -matrices can rotate cyclically and thus cancel out) one gets \'t(X),x(ji)]^0t
establishing the commutativity of C„ for different n*s and hence proving the quantum
integrability.
The QYBE (3.4) represents in the matrix form a set of commutation relations
between action and angle which can be obtained by inserting in (3.4)
Quantum integrable systems : basic concepts etc
667
matrix (3.1) for T and the solution for quantum /?(A» /i)-matrix, which may be
given by
(fa)
R(X)
1 /.
/l 1
fa)j
(3.5)
The solutions are usually of only two different types (we shall not speak here of more
general elliptic solutions), trigonometric with
/ =
sin (A + rj)
sin A
and the rational with
sinq
sin A
/ =
k + T)
(3.6)
(3.7)
4. Exact solution of eigenvalue problem through algebraic Bethe ansatz
Such generalised commutation relations dictated by the QYBE are of the form
(4.1)
D{X)B{n) (4.2)
together with the trivial commutations for [A(A), A{p)] = [B(A), = [Z>(A), =
[At A), Dill)] etc
It is now important to note that the off diagonal element B{k) acts like an creation
operator (induced by the local creation operators of L(A) as argued above). Therefore if one
can solve the quantum eigenvalue problem
//|m)=£„|/n) (4.3)
or more generally
t(A)|in> -A,(A)|m> (4.4)
the eigenvalue problem for all C,'s can be obtained simultaneously by simply
expanding A(A) as
C,|m) = A;(0)A;'(0)h). Ci\m) = (a;( 0)A-' (0)) jm) (4.5)
etc. The m-particle state lni> n»y be considered to be created by fl(A,) acting m times
on the pieiidovacuiMK)> :
|m>-B(A,)J(A2)-fl(A,)|0>.
nAidyxt
(4.6)
668
Anjan Kundu
Therefore for solving (4.4) through the Bethe ansatz, we have to drag t(A) » A(A) -i- D(A)
through the string of ) ’s without spoiling their structures (and thereby preserving the
eigenvector) and hit finally the pseudovacuum giving A(^) 1 0 > = oik) 1 0 > and D{k) I 0 >
= P(k) I 0 >. Notice that for this purpose (4. 1.4.2) coming from the QYBE are the right
kind of relations, (the other type of unwanted terms are usually present in the LHS in
lattice models ((as "‘in (4. 1,4.2)), which however may be removed by the Bethe equations
for determining the parameters kj, induced by the periodic boundary condition. In case of
field models such terms are absent and kj become arbitrary.) As a result we finally solve the
eigenvalue problem to yield
( 4 . 7 )
y*I j»\
5. Universality in integnible systems
The structure of the eigenvalue AJ,k) reveals the curious fact that apart from the a(a). jS(or)
factors it depends basically on the nature of the function /(A -A|), which are known
trigonometric or rational functions given by (3.6) or (3.7) and thus is the same for all
models belonging to the same class. Model dependence is reflected only in the form of a(A)
and /)[A) factors. Therefore the models like SG, Liouville and XXZ chain belonging to the
trigonometric class share similar type of eigenvalue relations (with specific forms for a(A)
and IXk)). This deep rooted universality feature in integrable systems carries important
consequences.
5.7. Generation of models :
One may start with the trigonometric solution (3.6) for the /^-matrix and consider u
generalised model with Lax operator
L,(X) =
sin (A + T]s^),
^sin T}S'^ ,
sin r]S~
sin (A -ri5^)j
(5.1)
with the abstract operators 5* belonging to the quantum algebra (QA) Ug(su{2 )) :
[S\S^] = ±5^ [5^5-] = [253]
(5.2)
where [x] =
qx ^q-x sin(ocx)
q-q-
sina
, ^ = Following the above Bethe ansatz procedure
the eigenvalue would naturally be like (4.7) and different realisation of the quantum algebra
(5.2) would derive easily the eigenvalues for concrete models belonging to this class. At the
same time the Lax operators of these models can also be generated from (5.1) in a
systematic way.
Quantum integrable systems : basic concepts etc
669
For example,
S* = (5.3)
constructs from (5'.1) the Lax operator of the spin^ XXZ-chain and describes the Bethe-
ansatz solution for the suitable choice of a(A) and /9(A). Similarly,
= = = (5.4)
2
yields (lattice) sine-Gordon model. At A 0
one gets the SG field model with the Lax operator obtained as L„ s /-t- A£(x) + G(A).
All the conserved quantities of the model including the Hamiltonian can in principle
be derived using the Lax operator. In fact a more general form of the ancestor Lax operator
than that of (5.1) exists corresponding to the same trigonometric /^-matrix, the explicit form
of which can be found in ref. [16]. Concrete realisations of such ancestor models generates
various quantum integrable models (in addition to those already mentioned) like quantum
Derivative NLS, Ablowitz-Ladik model, relativistic Toda chain etc. The Bethe ansatz
solutions for these models also can be obtained (with specific case-dependent difficulties)
following the scheme for their ancestor model, which as mentioned above is almost model
independent and same for all models of the same class.
At q 1 limit, and given by the elements (3.7). The ancestor model also
reduces to' the corresponding rational form
with «(«„) =
1 , , ( 1
1 + — m^A^ cos 27J ii„+ —
2 12
Tjs*,
Tjs-
X-TJS^
(5.5)
The underlying QA (5.2) becomes the standard jm( 2) algebra
ts^, [ 5 '*" , J" ] = 2j^ . (5.6)
Such rational ancestor model (or with more general form [16]) in its turn reduces also
to quantum integrable models like spin-'j XXX chain, NLS model, Toda chain etc.
For example, spin- ^ representation 5 " = gives the Lax operator of XXX chain
from (5.5), while the mapping from spin to bosonic operators given by Holstein -Primakov
transformation
leads to the quantum integrable Lattice NLS model. Similarly, the Toda chain can be
derived from the ancestor model of [16]. The Bethe ansatz solutions for these desendant
models mimics also the scheme for their ancestor model with rational K-manix.
670
Anjan Kundu
Thus for both the trigonometric and rational classes one can construct the Lax
operators and solve the eigenvalue problem exactly through Bethe ansatz in a systematic
way. This unifies diverse models of the same class as decendants from the same ancestor
model and at the same time realisations like (5.4) gives a criterion for defining integrable
nonlinearity as different nonlinear realisations of the underlying QA. This fact of the close
relationship between seemingly diverse models also explains in a way the strange
statements often met in other contexts like ‘(Quantum Liuoville model is equivalent to spin
(--^) anisotropic chain’ [31] or 'High energy scattering of hadrons in (}CD is described by
the Heisenberg model with noncompact group’ [30].
5.2. Algebraic structure of integrable systems :
The underlying QA, as mentioned before, exhibits Hopf algebra property. The most
prominent characteristic of it is the coproduct structure given by
A(j3) = j 3 -5± (5g)
This means that if 5* = 5^ ® / and 5^ = / ® 5* satisfy the QA separately, then their tensor
product A(5^) given by (5.8) also satisfies the same algebra. This Hopf algebraic property of
the QA induces the crucial transition from the local QYBE (3.2) to its tensor product given
by the global equation (3.4), which in turn guarantees the quantum integrability of the
system as shown above.
The QIS described above are known as the ultralocal models. They are the stand^d
and the most studied ones. The ultralocality refers to their common property that the Lax
operators of all such models at different lattice points i ^ j commute : [L|, Lij]- Note that
this is consistent with the property : = 0 for the generators of the quantum algebra
described above. This ultralocality is actively used for transition from the local to the global
QYBE, i.e. in establishing their quantum integrability.
Note that the standard matrix multiplication rule
= (AC^BD) (5.9)
which holds due to the commutativity of B 2 = / ® B and C| % C 9 /. remains also valid for
the ultralocal Lax operators with the choice
4=I,„(A), B = L..,(//), C = L.(4 D = L,(/i). (5.10)
Therefore starting from the local QYBE (3.2) at 1 + 1 point, multiplying with the same
relation at i and subsequently using (5.9) with (5.10) one globalises the QYBE
and repeating the step for N times obtains finally the global QYBE (3.4). This in turn
leads to the commuting traces t(A) « TrT^A) giving commuting conserved quantities
C„. n * 1 ■ 2 ■ • • N.
Quantum integrable systems : basic concepts etc 67 1
6. Nonultralocal modeb and braided exteosion of QYBE
However, there exists another class of models, known as nonultralocal models (NM) with
the property [Li, , L 2 j ] ^ 0, for which the trivial multiplication property (5.9) of quantum
algebra fails and it needs generalisation to the braided algebra [9], where the
noncommutativity of Bi, C\ could be taken into account. Consequently the QYBE should
be generalised for such models. Though many celebrated models, e.g. quantum KdV model.
Supersymmetric models, nonlinear a models, WZWN etc. belong to this class, apart from
few [10,1 1] not enough studies have been devoted to this problem. The generalised QYBE
for nonultralocal systems with the inclusion of braiding matrices Z (nearest neighbour
braiding) and Z (nonnearest neighbour braiding) may be given by
Rxi (u - v)Z 2 i‘ («. v)Lij(u)Z 2 l (m. v)L 2 ; (v)
= Zfj' (v, u)L2j(v)Zi2 (v, u)Lij (u)Ri2 (u - v). (6.1)
In addition, this must be complemented by the braiding relations
^2j*l (“) = ^21 (“• v)Lxj(u)Z21 («, V)
xL2j^i(v)Zj-I(u,v) (6.2)
at nearest neighbour points and
^ 2 k MZ 21 (u, v)Lij (li) = Zj/ (u, v)Lij (u)Z 2 i (U, v)
xL 2 *(v)Z 2 -/(ii,v) (6.3)
with it > y + 1 answering for the nonnearest neighbours. Note that along with the usual
quantum Ri 2 (u - v)-matrix like (3.5) additional Z, 2 . Z ,2 matrices appear, which can be
(in-)depcndent of the spectral parameters and satisfy a system of Yang-Baxter type relations
[11]. Due to appearance of Z matrices however one faces initial difficulty in trace
factorisation unlike the ultralocal models. Nevertheless, in most cases one can bypass this
problem by introducing a R(u) matrix and defining. t(u) = tr(R(u)7lu)) as commuting
matrices [ 20 , 11 ] for esUblishing the quantum inlegrability for nonultralocal models.
Though 18 wellfiramcd theory for such systems is yet to be achieved one can derive the basic
equations for a series of nonultralocal models in a rather systematic way from the general
relations (6. 1-6.3) by paricular explicit choices of Z. Z and /^-matrices [11,19]. The models
which can be covered through this scheme are
(i) Nonatelian Toda chain [12]
Z* LZ*/ + tA(e 22
(ii) Curmt algebra in WZWN model [29]
Z » 1 and Z 12 « R;i2 • ^ trigonometric
R(k};maxm.
672
Anjan Kundu
(iii) Coulomb gas picture of CFT [ 14J
Z= 1 andZij
(i v) Nonultralocal quantum mapping [ 1 3]
Z = 1 and Z, 2 (« 2 ) = 1 + ^ ® «oW •
Ii2
(v) Integrable model on moduli space [15]
Z = Z,2 =/?;.
(vi) Supersymmetric models
Z = Z = ^ where ® «aa ® and ^ ■ (-1)^ with
supersymmetric grading a.
(vii) Anyonic type SUSY model
Z=Z=J^ nafigap ■ with iafi “
(viii) Quantum mKdV model [18]
Z = l.Z |2 = Z 21 = ^ 2 and the trigoiKmietric /?(u) matrix
(ix) Kundu-Eckhaus equation [\1]
Classically integrable NLS equation with 5th power nonlinearity
<>i + 2iB(iif^yf),tffm 0, (6.4)
as a quantum model involves anyonic type fields : n > m :
IWm Wl] = 1 'Hie choice Z ^ \2 ~ diag (e^, 1, 1, and and te rational' i? matrix
constructs the braided QYBE. The trace factorisation problem has not been solved.
Other models of nonultralocal class are the wellknown CalogenhSutherland (CS)
and Haldae-Shastry (HS) medels with interesting long-range interactions. Spin extension of
the CS model may be given by the Hamiltonian [34,32]
N
+ l J^(a^-aPj^mxi-xt) (6.5)
j=\ \ij<nN
with Ixj, pt] = where the potential V{xj - x*) ■ r — ^ ^ for nonperiodic and
— 73 periodic model. Pj^ is the permutation operator responsible for
exchanging the spin states of the ;-th and the A-th particles. In the absence of the operator
(6.5) turns into the original CS model without spin.
Quantum integrable systems : basic concepts etc
673
The spin CS model exhibits many fascinating features, namely its conserved
quantities including the Hamiltonian exhibit Yangian symmetry, the eigenvalue problem
can be solved exactly using Dunkl operators, the ground state is a solution of the
Knizhnik-Zamolodchikov equation, the system can be viewed as the free anyonic
gas related to the notion of fractional statistics etc. [32]. Though the satisfactory
formulation of quantum integrability of the model by braided QYBE has not yet been
achieved, this was done through an alternative procedure using operators L and M and
showing [f/„. L] = [UM] [33].
Remarkably, at a -> «> the Hamiltonian of the CS model (6.5) for the periodic case
reduces to the HS model [35]
Hhs
^ P ii
( 6 . 6 )
This discretized long-range interacting spin chain like model seems to be less well
understood and its Lax operator description difficult to find.
7. Interrelation of QIS with other Adds
As I have mentioned in the introduction the quantum integrable system is intimately
connected with various other branches of physics and mathematics. Therefore the
knowledge and techniques of QIS is often hdpful in understanding and solving other
problems. Here we briefly touch upon' some of these relations just to demonstrate the wide
range of applicability of the theory of integrable systems.
7 . /. Relation with statistical systems :
The ( 1 •»’ 1 ) dimensional quantum systems are linked with 2 dimensional classical statistical
systems and the notion of integrability is equivalent in both these cases. For integrable
statistical systems the QYBE (3.2) and the YBE (3.3) becomes the same and leads similarly
to the commuting transfer matrix r(A) for different A.
Let us examine a classical statistical model known as vertex model by considering
2-dimen$ional array of x Af lattice points connected by the bonds assigned with -i-ve (-ve)
signs or equivalently, with right, up (left, down) arrows in a random way. The partition
function Zof this system may be given starting from local properties, Le. by finding the
probability of occurrence of a particular configuration at a fixed lattice point i. For
two allowed signs on each bond, 4 x 4 » 16 possible arrangements arises at each lattice
point. Setting the corresponding Boltzmann weights (Oj » e~^^ as the matrix elements of
a 4 X 4-matrix, we ptU the -matrix with crucial dependence on spectral parameter A
The configuration probability for a string of N-lattice sites in a row may be given by
the transfer matrix t{a,P)^tr(n^R^*^y For calculating the partition function
involving M such strings, one has to repeat the procedure M times to give Z »
tr(n^T)^tKT^)^
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Anjan Kundu
The YBE (3.3) restricts the solution of the /^-matrix to integrable models. However
the R matrix with 16 different Boltzmann weights, representing in general a 16- vertex
model is difficult to solve. Therefore we impose some extra symmetry and conditions on
the ^-matrix by requiring the charge conserving symmetry ^ 0, only when k I
along with a charge or arrow reversing symmetry (see Figure 1).
RZZ
= MA).
Fi^re 1. Boltzmann weights of the 6-vortex model constituting the elements of the R -matrix.
Using an overall normalisation it leads to a 6-vertex model for which the /^-matrix is
given exactly by (3.5), which is turn represents the Lax operator of the XXZ spin- y chain
as constructed above. Thus we see immediately the similarity between statistical and the
quantum systems in their construction of ^-matrices, transfer matrix, integrability equations
etc. This deep analogy goes also through all the steps in solving the eigenvalue problem by
Bethe ansatz in both the systems, e.g. vertex models in integrable statistical systems and the
spin chains in QIS [21 ,22]. *
7.2. Interrelation between QIS and CFT :
There exists deep interrelation between these two two-dimensional systems, first revealed
perhaps by Zamolodchikov [24] by showing that, if CFT is perturbed through relevant
perturbation and the system goes away from criticality it might generate hierarchies of
integrable systems. For example, c = y CFT perturbed by the field a= 2) as W = //*,2 +
hal(7(x)d^x, represents in fact the ising model at T = with nonvanishing magnetic
LM k. Similarly the WZWN model perturbed by the operator generates integrable
restricted sine-Gordon (RSG) model. Under such perturbations the trace of the tress tensor,
unlike pure CFT, becomes nonvanishing and generates in principle infinite series of
integrals of motion associated with the integrable systems.
In recent years, this relationship has also been explored by streching it in a
sense from the opposite direction. The aim was to describe CFT through massless
5-matrix [23] starting from the theory of integrable systems. This alternative approach
based on the quantum KdV model attempts to capture the integrable structure of CFT.
Note that the conformal symmetry of CFT is generated by its energy-momentum
tensor T{u) = - with satisfying the Virasoro algebra. The operators
675
Quantum integrable systems : basic concepts etc
hk~\ = ^ Jo duT 2 jciu\ with r2*(i4), depending on various powers and derivatives of
Tiu) represents an infinite set of commuting integrals of motion. The idea is to solve their
simultaneous diagonalisation problem, much in common to the QISM for the integrable
theory. Remarkably, this is equivalent to solving the quantum KdV problem, since at the
classical limit the field T(u) = -^U(u) with U{u + iTt) = U{u) reduces the commutators of
T(u) to {f/(u), t/(v)} = 2{U(u) - U(v)) 5'(u - v) + 8"iu - v), which is the well known
Poisson structure of the KdV.
Another practical application of this relationship is to extract the important
information about the underlying CFT in the scaling limit of the integrable lattice models.
Interestingly, from the finite size correction of the Bethe ansatz solutions, one can
determine [28] the CFT characteristics like the central charge and the conformal
dimensions. For example one may analyse the finite size effect of the Bethe ansatz solutions
of the six-vertex model (with a seam given by ic). Considering the coupling parameter
q = e* '■+' , one obtains from the Bethe solution at the large N limit the expression
for the ground state energy and
for the excited states. Here A. A are conformal weights of unitary minimal models and
c = \ - . V = 2, 3, ... is the central charge of the corresponding conformal field
theory.
7.3. Link polynomial using integrable systems :
A link polynomial is an invariants corresponding to a particular knot or link and is
extremely useful for classifying them. Jones polynomial is such an example. There
are various ways to construct such polynomials. Interestingly, the Integrable systems
provide a systematic highly efficient way of producing such polynomials, which can
distinguish between different knots, where even Jones polynomial fails. The main i(!|pa
is to start with a trigonometric R(A)-matrix solution of the YBE, which in general is a
X matrix depending on the higher representation of the SU(2) algebra. Then the
task is to find the corresponding braid group representation by taking the A -> «» limit.
Defining now the Markov trace in a particular way one can construct a series of link
polynomials for different cases of N = 2, 3, .... Higher the N richer is the contents of the
polynomial. For example, using V = 2 one gets the same polynomial for the Birman's
72A(6)-28
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Anjan Kundu
two closed braids while TV s 3 by the above method generates two distinct polynomials
for these braids [25].
8. Conclusion without conclusion
Basic notions of the quantum integrable systems are explained focusing on various aspects
and achievements of this theory. The deep interrelations of this subject with many other
fields of physical and mathematical sciences are mentioned. However it is difficuh yet to
draw any conclusion at this stage, since we expect to hear many more surprises in this
evergrowing field. The recent Seiberg- Witten theory might be one of them. The influences
of this theory in explaining high Tc-superconductivity [26], reaction-diffusion processes
[27] etc. are being felt. We expect also to have breakthrough of quantum integrability in
genuine higher dimensions. Therefore let us leave this conclusion without concluding and
keep this task for the future.
References
[ I ] Barton Russel Mysticism Jl Logic Ch. S
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Press) p 978 (1965)
[4] N Zabusky and M D Kniskal Phys. Rev. Utt. 15 240 (1965)
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M J Ablowitz, D J Kaup, A C Newell and H Segur Phys Rev. Lett. 31 125 ( 1973) ^
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[8] H B Thacker Uct. Notes in Phys. Vol. 145 I (Berlin : Spnngcr) (1981); Rev. Mod. Phys. 53 253 (1981),
J H Lowcnstein in Les Houches Lect. Notes ed. J B Zuber el al p 565 (1984), P Kulish and E K Sklyanin
Lect Notes in Phys. cd. J Hietannta et al (Berlin : Springer) Vol. 151 p 61 (1982)
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[ lU] L Freidel and J M Maillet Phys. Utt. 262B 278 (1991); Phys. Utt. 263B 403 ( 1991 )
[11] L Hlavaty and Anjan Kundu Int. J Mod. Phys. 11 2143 (19%)
[12] V E Korepin / Sov. Math. 23 2429 ( 1983)
[13] F W Nijhoff, H W Capcl and V G Papageorgiou Phys. Rev. A46 2155 (1992)
[14] O Babelon and L Bonora Phys. Utt. 253B 365 (1991); O Babelon Comm. Math. Phys 139 619 (1991):
L Bonora and V Bonservizi Nucl. Phys. B390 205 (1993) ~
[15] A Yu Alexeev Integrability in the Hamiltonian Chem-Simons Theory, preprint hcp-lh/93 1 1074 (1993)
[16] Anjan Kundu and B Basumallick Mod. Phys. Utt. A7 61 (1992)
[17] Anjan Kundu J. Math. Phys. 25 3433 (1984); F Calogero Inverse Prob.3 229 (1987); L Y Shen in
Symmetries and Singularity Structures cd, M Lakshmanan (New York ; Springer-Verlag) p 27 (1990)
[ 1 8] Anjan Kundu Mod. Phys. Utt. AlO 2955 (1995)
[ 1 9] Anjan Kundu in Prob. of QFT (D V Shirkov et al JINR publ.. Dubna) p 140 (19%)
[20] E Sklyanin J PIm A21 2375 (1988)
[21] R Baxter Exactly Solved Models in Siattstical Mechanics (New York : Academic ) ( 1 98 1 )
Quantum integrable systems : basic concepts etc
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[22] L A Takhtiuan and L D Faddeev Russian Math. Surveys 34 1 1 ( 1 979)
[23] V V Bazhanov, S L Lukyanov and A B Zamolodchikov Comm. Math. Phys. 177 381 (1996); V A Fatecv
and S Lukyanov Int. J. Mod. Phys. A7 8S3 1325 (1992)
[24] A B Zamolodchikov Pisma ZETF 46 1 29 ( 1 987)
[25] M Wadoti, T Deguchi and Y Akutsu Phys. Rep. 180 247 (1989)
[26] F Haldane J. Phys. C14 2535 (1981); P W Anderson The Theory of Superconductivity in the High T^
Cuprates (Princeton : Princeton Univ. Press) (1997); J Carmclo and A Ovchinnikov J. Phys C3 757
(1991)
[27] F C Alcaraz, M Droz, M Henkel and V Rittenberg Ann. Phys. 230 667 (1994)
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[FS 19] 619 (1987)
[29] L D Faddeev Comm. Math. Phys. 132 131 (1990); A Alekseev, L D Faddeev, M Semenov-Tian-Shansky
and A Volkov The Unraveling of the Quantum Group Structure in the WZWN Theory, preprint CERN-
TH-5981/91 (1991)
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Indian J. Phys, 72A (6). 679-687 (1998)
UP A
— an intemationai journ al
Perspectives in high energy physics
G Rajasekaran
Institute of Mathematical Sciences.
Madras-600 113, India
Abstract : A broad survey of High Energy Physics (HEP) both within as well as beyond
the Standard Model is presented emphasizing the unsolved problems. Inspite of the spectacular
success of the Standard Model, there is a serious crisis facing the field. The importance of
research on new methods of acceleration that can resolve this crisis by taking us to superhigh
energies is stressed. We bnefly review the status of HEP in India and offer suggestions for the
future.
Keywords : Standard model, string theory, future of HEP
PACS Nos. : 1 1 15.^. l2.60.Cn, l4.60.Pq
1. History
The major events which culminated in the construction of the Standard Model of High
Energy Physics are presented in Table 1 in chronological order. Using nonabelian gauge
theory with Higgs mechanism, the electroweak (EW) theory was already constructed in
1967, although it attracted the attention of most theorists only after another four years,
when it was shown to be renormalizable. The discovery of asymptotic freedom of non
abelian gauge theory and the birth of (}CD in 1973 were the final inputs that led to the full
standard model.
On the experimental side, the discovery of scaling in deep inelastic scattering (DIS)
which led to the asymptotic free CJCD and the discovery of the neutral current which
helped to confirm the electroweak theory can be regarded as crucial experiments. To
this list, one may add the polarized electron-deuteron experiment which showed that
SU(2) X U{]) is the correct gauge group for electroweak theory, the discovery of
gluonic Jets in electron-positron annihilation confirming QCD and the discovery of W and
Z in 1983 that established the electroweak theory. The experimental discoveries of
charm, T. beauty and top were fundamental for the concrete 3-generation standard
model.
(S)19981ACS
680
G Rajasekaran
However, note the blank after 1973 on the theoretical side. Theoretical physicists
have been working even after 1973 and experiments also are being done. But the tragic fact
is that none of the bright ideas proposed by theorists in the past 25 years has received any
exp)erimental support. On the other side, none of the experiments done since 1975 has made
an independent discovery. They have only been confirming the theoretical structure
completed in 1973. ft is clear that if such a situation persists for long, it may become
difficult to continue to be optimistic about the future of high energy physics. We shall take
up this point in Section 3.
Tabic 1. History of the .standard model
Theory
Experiment
1954
Nonabelian'
gauge Helds
I960
1964
Higgs mechanism
I960
1967
EW Theory
1968
Scaling in DIS
1970
1971
Renormalizability
of EW Theory
1970
1973
A.symptotic freedom
-♦QCD
1973
Neutral current
1974
Charm
1975
r-lepton
1977
Beauty
1978
"td expt
1979
gluonicjets
1980
1983
W.Z
1980
1990
1994
top
1990
2. Perspectives and highlights of the symposium
The standard model based on the gauge group SU(3) x SIAZ) x f/(l) describes all of
preseqtly known High Energy Physics. How well the standard model fits the data, was
reviewed in the talks of Gautam Bhattacharyya, Somnath Ganguli and Atul Gurtu. This is
the peak where we have reached. From here we can survey the view either below us (Le.
within the standard model) or above us {Le. beyond the standard model). Possible topics in
either view are the following :
Within the standard model :
QCD and hadronic physics
Higgs and symmetry breaking
Perspectives in high energy physics
681
Neutrinos
Generation problem
CP, axion etc.
Beyond the standard model :
Preons
Grand Unification
Supersymmetry and Supergravity
Higher Dimensional Unification
Superstrings
Let me first dispose of the view below the standard model.
QCD and hadronic physics :
Here the questions are the following :
(i) Can we establish QCD to be the correct theory of strong interaction ?
(ii) Can colour confinement be proved ?
(iii) Can hadron spectrum be calculated ?
(iv) Can hadron scatter! ng^the calculated ?
(v) Do glue balls exist ?
(vi) , Does quark’gluon plasma exist ?
Ten years ago I talked on ‘Terspectives in HEP” (Ref. : Proceedings of VIII High
Energy Physics Symposium, Calcutta, 1986, p. 399). The above list of topics and questions
is in fact taken from that talk. Have the questions raised at that time, been answered ? In the
following, I shall enclose the quotations from the 1986 talk as ” ”
“Unfortunately at the present moment, the answer to all these questions is negative.
Answer to the first question will depend on the answers to the next three questions. Lattice-
gauge-theorists are working hard on these problems. Here a word of caution may be
appropriate, concerning the numerical calculation of hadronic properties such as their
masses and couplings. It must be remembered that these properties of hadrons have been
calculated earlier more than once in the history of high energy physics - first within the’
analytic S matrix and bootstrap approach and later in quark potential models. Each time
success was claimed. The real test of any numerical calculation in hadronic physics must be
the prediction of a new number or a new phenomenon in the area of strong interaction,
which is then confronted with experiment. Until that is achieved, success cannot be
claimed. After all, what is the sense of using expensive computer time to calculate the
masses of the hadrons, when these can be obtained with much greater accuracy, by looking
up the excellent Particle Data Tables T Although the main point of these critical statements
still stands, one has to admit that important new developments have occurred. Asit De gave
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G Rajasekaran
a very lucid review of these and claimed that lattice QCD results are just starting to enter
Particle Data Tables. This is good news !
“In the absence of a clean check of QCD in the realm of the dirty hadrons, the
existence of glue balls or the transition of hadronic matter into quarkgluon plasma would be
a direct and strikirtg confirmation of QCD. But distinguishing glue balls from flavour-
singlet quark balls has not proved a clean job. Let us hope that the imminent heavy-ion
collisions will produce the eagerly awaited quark-gluon plasma and that the plasma will
announce its arrival with a clean signal". Heavy-ion collisions have occurred, but people are
still searching for clean signals of QGP ! C. P. Singh reviewed the current status of this
field.
What about continuum QCD ? Light-front QCD appears to be a promising approach
and progress in it was reported by Harindranath. A scholarly review on thermal field theory
was given by Samir Mallik, who pointed out that the infra-red problem for finite
temperature QED has been solved by Indumathi. The status of perturbative QCD and the
structure function of the proton as revealed by HERA was reviewed by Dilip Choudhury
and Rahul Basu,
Higgs and symmetry breaking :
"Is Higgs the correct mechanism of electroweak symmetry breaking ? There are claims
from the axiomatic side that A/ theory may be an inconsistent theory. Should Higgs
mechanism be replaced by some other nonperturbative dynamical symmetry breaking ?
Inspite of much effort, we have not progressed much towards an understanding of
dynamical symmetry breaking. Experiments being planned in the TeV region may reveal
either the presence of Higgs bosons or a new type of strong interactions in the electroweak
sector. In either case, we will have an exciting time". S. R. Choudhury showed how the
triviality of theory combined with consistency can be used to yield bounds on Higgs
mass and D. P. Roy described the ongoing searches for the Higgs boson.
Neutrinos, generations, CP, axion etc :
"Are the neutrinos massless ? If not, what are their masses and mixing angles ? The recent
elegant explanation of the solar neutrino puzzle by resonant neutrino oscillations (the
Mikheyev-Smimov-Wolfenstein effect) must be noted. This explanation needs confirmation
by independent experiments such as that proposed by Raghavan and Pakvasa (1987). Here
one perhaps has a powerful tool for pinning down neutrino masses and mixing angles". The
atmospheric neutrino puzzle has now joined the solar neutrino puzzle and both indicate
neutrino oscillations. Neutrino physics has grown into an important field. Data from the
new generation of neutrino detectors (Super-Kamioka, SNO and Borexino) are eagerly
awaited. Also, long-base-line terrestrial neutrino experiments are being planned.
"How may generations of quarks and leptons exist and what fixes this number ?
Of the various options within the standard model for explaining CP violation, which
is the correct one ?
Perspectives in high energy physics
683
Is Peccei-Quinn symmetry and axion the correct cure for the catastrophe of strong
CP violation in QCD ? If so, where is the axion ?”
“On all these questions, enormous amount of theoretical work has been done, but no
memorable results have come out. So most theorists have gone out of the standard model to
make a living. This is not surprising, for this is what theorists have been always doing. We
did not solve all the problems of atomic physics before moving on to nuclear physics, nor
did we understand nuclear physics fully before inventing a new field called particle physics
and moving into it. After reaching a peak we do not set up our permanent quarters there :
we climb to the next peak. So, we move on to ... beyond the standard model.” I then went
on to describe Preons, SUSY and SUGRA, Higher Dimensions and finally Strings, which
contained the following remark.
“Further, search for consistent theories of even more complicated objects than
strings, for instance, membranes, lumps etc must continue. Any reported “No go”
theorem in this context need not be regarded as a permanent barrier. Remember, without
the invention of SUSY and acceptance of higher dimensions, even string theories
would suffer a “No go” theorem. There will be discovered other things which will make
the theories of membranes, lumps and even objects extending to higher dimensions
consistent”.
This is what has happened now. We are witnessing a Second Revolution in String
Theory which has converted String Theory itself into* a Theory of p-branes (objects
extending to p dimensions).
Following arc a few highlights of this symposium that dealt with “Beyond the
Standard Model”.
Supersymmetry :
Probir Roy, D. P. Roy and Ananthanarayanan presented comprehensive reviews of
supersymmetric theories. We still await their experimental discovery.
String theory :
Sunil Mukhi gave a stimulating talk on the recent developments. Using the web of duality
they are catching a rich harvest of interconnections between various string theories and they
are already getting a glimpse of a so-called M-theory which may be the fundamental source
of all string theories, membrane theories etc.
If string theory is the correct theory of Quantum Gravity it should help us to
understand black holes better and the recent developments have achieved this. It is the
understanding of the solitons and D-branes of string theory that has contributed to this
development and Dabholkar dcali with this topic.
After listening to any talk on this Second Revolution in string theory, I feel so
envious of my younger colleagues who are making such a fantastic progress in this difficult
and highly competitive subject. (I wish I were 20 years younger !)
^>-29
m
G Rajasekaran
Two application of string theory :
(a) Proton stability :
The problem of catastrophically fast proton decay (Tp - 10"^ sec) in supersymmetric
theories, which is due to the existence of colour triplet scalars in these theories, is not yet
solved. Conservation of /^-parity is a possible solution and a few other solutions are
technically possible, but not compelling. No deeper theoretical reason for proton stability
has been found. Jogesh Pati argued that the real solution may require superstrings.
Hopefully, this would provide the deeper reason.
(b) CP violation:
In an interesting talk, David Bailin sought CP violation in the orbifold compact! fication of
10-dimensional heterotic strings. It may be possible to incorporate CP as a geometrical
transformation in a higher-dimensional theory and hence its violation may have a
geometrical origin.
Dualized standard model :
In a beautiful work, Tanmay Vachaspati has shown how the standard model could be
dualized. He starts with SU (5) and breaks it down to a version of SUO) x SU{2) x
The most remarkable aspect of his work is that no fermions are put by hand. The solitonic
monopoles that arise in the theory have precisely the same magnetic charge as the electric
charges on the quarks and leptons of the standard model. So, if we make the jjroper
idenlificarion, the quarks and leptons can be generated as solitons ! This is certainly a bolt
from the blue and deserves further study.
Topological quantum field theory :
Romesh Kaul described how the QFT framework (which we use to describe HEP) can be
used to reveal the topological properties of 3 and 4 manifolds. Thus QFT has enough power
to move the frontiers of Modern Mathematics too! In particular, duality in cohomological
field theory leads to an almost trivial calculation of the famous Donaldson invariants in 4-
D, which are in turn related to instantons. Since 4 is the number of physical dimensions of
space-time in which we live and since Donaldson invariants are related to the infinite
number of differential structures that have been proved to exist only in 4 dimensions, all
this mathematics may have profound consequences for physics!
3. Does HEP have a future ?
We now return to the blanks in discovery mentioned in Sec. 1. The blanks have
remained inspite of the tremendous activity in HEP in the past two decades. The biggest
loophole in standard model is the omission of gravitation, the most important force of
nature. Hence, it is now recognized that Quantum Gravity (QG) is the next frontier of
HEP, and that the true fundamental scale of physics is the Planck energy 10^^ Gev. which
is the scale of QG.
685
Perspectives in high energy physics
Wc are now probing the TeV (Itf GcV) region. One can see the vastness of the
domain one has to cover before QG is incorported into physics. In their attempts to
probe this domain of 10^ - 10*’ GcV, theoretical physicists have invented many ideas
such as supersymmetry, supergravity, hidden dimensions etc and based on these ideas,
they have constructed many beautiful theories, the best among them being the supersiring
theory (or, Af-theory, its recent iifcamation), which may turn out to be the correct theory
ofQG.
But, Physics is not theory alone. Even beautiful theories have to be confronted with
experiments and either confirmed or thrown out. Here wc encounter a serious crisis facing
HEP. In the next 10-15 years, new accelerator facilities with higher energies such as the
Large Hadron Collider (-10^ GcV) or the Linear Electron Collider will be built and so the
prospects for HEP in the immediate future appear to be bright. Beyond that period, the
accelerator route seems to be closed because known acceleration methods cannot take us
beyond about 10* GcV.
It is here that one turns to hints of new physics from Cosmology, Astrophysics &
Nonaccelerator Experiments. Very important hints about neuU'inos, dark matter etc have
come from Astrophysics and Cosmology. Nonaccelerator experiments on proton decay,
neutrino masses, double beta decay and 5>th force are important since they provide us with
indirect windows on superhigh energy scales.
In spite of the importance of astroparticle physics and nonaccelerator experiments,
these must be regarded as only our first and preliminary attack on the unknown frontier.
These are only hints ! Physicists cannot remain satisfied with hints and indirect attacks on
the superhigh energy frontier. So, what do we do ?
As already mentioned, the outlook is bleak, because known aceleration methods
cannot take us far.
To sum up the situation : There are many interesting fundamental theories taking us
to the Planck scale and even beyond, but unless the experimental barrier is crossed, these
will remain only as Metaphysical Theories.
It follows that either, new ideas of acceleration have to be discovered or, there will
be an end to HEP by about 2010 A.D.
It is obvious what route physicists must follow. We have to discover new ideas on
acceleration. By an optimistic extrapolation of the growth of accelerator technology in the
past 60 years, one can show that even the Planckian energy of 10'^ GeV can be reached in
the year 2086 (see my Calcutta talk). But, this is possible only if newer methods and newer
technologies are continuously invented.
Some of the ideas being pursued are laser beat-wave method, plasma wake field
accelerator, laser-driven grating linac, inverse free electron laser, inverse Cerenkov
acceleration etc. What wc need are a hundred crazy ideas. May be, one of them will work.
Lawrence's discovery of the cyclotron principle is not the end of the road.
oeo
U Hajasekartm
4. State of HEP Id India and suggestions for die future
Theory :
There is extensive activity in HEP theory in the country, spread over TIER, PRL. IMSc,
SINP, lOP, MRI, IISc, Delhi University, Punjab University, BHU, NEHU, Guwahati
University, Hyderabad University, Cochin University, Viswabharati, Calcutta University,
Jadavpur University, Rajasthan University and a few other Centres. Research is done in
almost all the areas in the field, as any survey will indicate.
Theoretical HEP continues to attract the best students and as a consequence its future
in the country appears bright. However, it must be mentioned that this important national
resource is being underutilized. Well-trained HEP theorists are ideally suited to teach any of
the basic components of physics such as Quantum Mechanics, Relativity, Quantum Field
Theory, Gravitation and Cosmology, Many Body Theory or Statistical Mechanics and of
course Mathematics, since all these ingredients go to make up the present-day HEP Theory.
Right now, most of these bright young theoretical physicists are seeking placement in the
Research Institutions. Ways must be found so that a larger fraction of them can be absorbed
in the Universities. Even if just one of them joins each of the 200 Universities in the
country, there will be a qualitative improvement in physics teaching throughout the country.
This will not happen unless the young theoreticians gain a broad perspective in the topics
mentioned above and train themselves for leaching-cum-research careers. Simultaneously,
the electronic communication facilities linking the Universities among themselves and with
the Research Institutions must improve. This will solve the frustrating isolation problem
which all the University Departments face. ^
Experiment :
Many Indian groups from National Laboratorie.s as well as Universities (TIFR, VECC, lOP,
Delhi, Punjab, Jammu and Rajasthan Universities) have been participating in 3 major
international collaboration experiments :
• L3 experiment on collisions at LEP (CERN)
• D0 experiment on pp collisions at the Tevatron (Fermilab)
• WA93 & 98 experiments on heavy-ion collisions at CERN.
Highlights of the Indian contribution in these experiments were presented in this
symposium.
As a result of the above experience, the Indian groups are well poised to take
advantage of the next generation of colliders such as LEP2 and the LHC. Already the
Indian groups have joined the international collaboration in charge of the CMS which will
be one of the two detectors at LHC. It is also appropriate to mention here that Indian
engineers and physicists will be contributing towards the construction of LHC itself.
Thus, the only experimental program that is pursued in the country is the
participation of Indian groups in international accelerator based experiments. This is
Perspectives in high energy physics
687
inevitable at the present stage, because of the nature of present-day HEP experiments that
involve accelerators, detectors, experimental groups and financial resources that are all
gigantic in magnitude.
While our participation in international collaborations must continue with full
vigour, at the same time, for a balanced growth of experimental HEP, we must have in-
house activities also. Construction of an accelerator in India, in a suitable energy range
which may be initially 10-20 GeV and its utilization for research as well as student-training
will provide this missing link.
In view of the importance of underground laboratories in v physics, monopole search
p decay etc, the closure of the deep mines at KGF is a serious loss. This must be at least
partially made up by the identification of some suitable mine and we must develop it as an
underground laboratory for nonaccelerator particle physics.
Finally, it is becoming increasingly clear that known methods of acceleration cannot
take us beyond tens of TeV. Hence in order to ensure the continuing vigour of HEP in the
21st century, it is absolutely essential to discover new principles of acceleration. Here lies
an opportunity that oar country should not miss ! I have been repeatedly emphasizing for
the past ten years that we must form a small group of young people whose mission shall be
to discover new methods of acceleration.
To sum up, a 4-way program for the future of experimental HEP in this country is
suggested :
1 . A vigorous participation of Indian groups in international experiments, accelerator-
based as well as non-accelerator-based.
2. Construction of an accelerator in this country.
3. Identification and developmeat of a suitable underground laboratory for
nonaccelerator particle physics.
4. A programme for the search of new methods of acceleration that can take HEP
beyond the TeV energies.
Acknowledgment and Apology
I thank Dilip Choudhury for the invitation to give this talk and excellent hospitality at
Guwahatj. I apologize to those whose contributions could not be highlighted in my lalk.
Indian J. Phys. m (6), 689-700 (1998)
UP A
- an intemaiional journal
Experimental summary — Xn DAE HEP symposium,
Guwahati, 1997
Sunanda Baneijee
Tala Instihite of Fundamental Research,
Mutnbai-dOOOOS. India
Abatract : The experimental talks presented m the XII DAE HEP Symposium, Guwahati
1997 are summarised here.
Keywordi : Standard model, Higgs, top quarks
PACSNo. : l2.10.Dm, l4.80.Bn. 14.6S.Ha
Introductloo
There* have been 16 invited talks and 20 contributed papers in the field of experimental
physics in this conference. These talks can be broadly divided into six physics categories—
new particle searches, heavy flavour physics, electroweak physics, strong interaction
physics and QCO, heavy ion interactions, future experiments and techniques. The break-up
of the talks are summarised in Table 1 . We have learnt new results from several current
experiments from CESR. LEP/SLC, Tevatron Collider as well as fixed target facilities,
heavy ion programme at the CERN SPS and some non-accelerator experiments. There have
been some talks on future experimental activities with the Tevatron Collider (TEV 33) and
the Large Hadron Collider (LHC) at CERN. Notoble omissions are v experiments, ep
scattering at HERA and future experiments in the b-factories.
Tabit 1. of tflki on experimenial high energy phyxics in the conference
new
Puticle
Heavy
Flavour
Electroweak
QCD
Heavy
Ion
Future
Experiment
Long Invited
2
2
1
0
1
I
Short Invited
1
2
1
1
1
3
Coniribeied
6
1
5
3
2
3
0 1998 lACS
690
Sunanda Banerjee
The summary talk is organised as follows. The next section will deal with the
discoveries in the recent past. This will be followed by precision measurements. Then the
null results from a variety of searches will be described. There will be a brief mention of a
few detailed measurements. Finally a couple of ratlier interesting but inconclusive
observations will be discussed.
Discovery
This is the first DAE symposium where the discovery of the sixth quark flavour, top,
has been reported [1]. The first hint of direct observation of top was reported by the
CDF collaboration during the summer of 1994. During March 1995, both the
Tevatron collider experiments, CDF and D0, reported a -5a excess of the top signal.
Since then, the statistics of the signal has almost doubled (> 100 pb~' of integrated
luminosity per experiment) and systematic errors are better understood by both the
experiments.
Top is produced in pair in the pp collisions and top dominantly decays to a b-quark
and W'^. Discovery channels for top are the ones where one or both the W’s decay
leptonically. ConsequenHy one has the two following scenarios ;
Signature 2 leptons + ^ 2 jets + missing Ej 1 lepton + ^ 4 jets + missing Ej
Fraction -5% -2x15%
Signal/BG 3 : 1 (e± ; 1 ; 1 (e+e- -1:4
This indicates that additional handles are required to improve the signal to background ratio
for one-lepton final state. This has been achieved using two distinctive features of top
decays, namely,
1. top is heavy and its decay is symmetric. So cuts in global event shape variables
would distinguish top decays from background.
2. top always decays to a b-quark. b-jets can be identified through displaced vertex
and/or accompanied soft non-isolated lepton.
These additional cuts bring the signal to background ratio in the range 1 .5 : 1 to 4 : 1 .
The number of top candidates as seen by the two experiments in the various
final states are summarised in Table 2. From the observed events, expected background
events and the integrated luminosities, the cross section of top-pair production has
been determined by CDF and Dtf^ to be pb and 5,2 ± 1.8 pb respectively.
They agree with the three possible estimates using next-to-leading-log QCD
calcjulations. CDF has started to look for top in other channels where both W’s decay
hadronically.
Events belonging to lepton + four jet category have been used in estimating the
top quark mass. Kinematic fits have been performed to events belonging to this category,
constraining the lepton-neutrino and 2-jel mass (from W-decay) to W-mass and the
Experimental summary— XII DAE HEP symposium etc
691
two combinations of W and b-jet masses to be the same (t and t having the same mass).
This gives rise to several combinatorics. A multi-variable discriminator is used to choose
Table 2. Number of top candidates found by the two experiments CDF and Dd together with
estimated background and signal events.
Experiment
Channel
# Observed
Estimated BG
Expected Signal
CDF
6
1.21 ±0.36
>1.6
eV
6
0.76 ±0.21
-2.4
(SVX)
/ + 4Jet
16
2.80 ±0.58
m
e+c-
1
0.66 ±0.17
-0.9
^+/i-
1
0.55 ±0.28
-0.5
eV
3
0.36 ±0.09
-1.7
(Event Shape)
/ + 4Jet
21
9.23 ±2.83
-12.8
(li Ug)
/ + 4jct
11
2..58±0.07
-9.0
signal from background. The data are then fitted to the estimated background and signal
using binned Poisson statistics and discrete top mass. Top mass is determinad by
maximising the log likelihood function. The results are summarised in Table 3. One
expects the top mass to be determined to an accuracy of ± 2 GeV with the high luminosity
run alTEV33 [2].
Tabic 3. Top mass determined from direct reconstruction.
Experiment
Top Mass
(GeV)
AM (Stat)
(GeV)
AM (Syst)
(GeV)
CDF
176.8
±44
±4.8
D0
169
±8
±8
Combined
175
±6
Precision measurements
Precise measurements on the properties of the vector bosons Z and W have been
reported from LEP [3,4] and Tevatron [5]. LEP reported analysis of all their data till
1995 including the high energy run at 130-140 GeV whereas Tevatron reported analysis
of the combined Run I data. Certain heavy flavour properties have been precisely
measured at the Tevatron fixed target experiments [6], CLEO [7] and experiments at
LEP and Tevatron collider [8].
Z Boson properties :
LEP [3] made a very precise measurement of the beam energy using several magnetic and
resonant depolarisation measurements. This has been supplemented by very precise
determination of cross section and forward-backward asymmetry in the final states
e^e- f f by the. four experiments ALEPH. DELPHI, L3 and OPAL. The measurements
72A(6>.30
692
Sunanda Banerjee
have made use of 14.4 million events in the hadronic final state and 1.6 million events
in the leptonic final state. The systematic errors have been controlled to a very small
level (for example the systematic error for hadronic cross section measurements is less
man 0.2%). Using improved Born approximation for the Z-exchange contribution
and assuming the photon exchange and Z-y interference from the Standard Model, one gets :
Paraineter
Average Value
»i,(C5eV)
91.1863 ± 0.0020
rt(Gcv)
2.4946 ± 0.0027
Ohad
41.508 ± 0.056
Rz
20.778 ± 0.029
ApB
0.0174 1 0.0010
Mass of the Z-boson has been determined with an accuracy of 2 parts in 10^. However, if
one relaxes the assumption on the Z-y interference term by introducing a scale factor, one
finds a large correlation of mz with the scale factor for hadronic final state, J . This leads
to a larger error on mz. If one now uses cross section measurements where the contribution
of the interference term is relatively larger (at centre of mass energies away from mz), one
gets significant improvement in the jneasurements. This has been achieved by using the
cross section measurements at 130-140 CieV (LEP 1.5).
Measurement mi(GcV) jjjj
LEPl-hLEPl.5 9LI936 ± 0.0040 -0.21±0.20
LEP I + LEP 1 .5 + TOPAZ 91 .1912 ± 0.0035 -0.07 ± 0. 16
Lepton universality has been tested in the charged as well as the neutral current
sector to a high degree of precision from the measurements of asymmetries in production
(for all 3 leptons) and decays (for t's).
The ratio of b and c quark partial widths of the Z to its total hadronic partial
width and the corresponding forward backward asymmetries have been measured.
These measurements created a lot of interest to theorists in 1995 because they deviated
from the Standard Model predictions by nearly 30* s. These measurements have been
done by tagging hadronic Z decays using heavy flavour characteristics (large life
time, heavy mass of the b-quark, fast D*’s produced by c-quark). The 19% analysis reveals
Parameter Average Value
0.2179 ± 0.0012
0.1715 t 0.0056
A®**
0.0979 ± 0.0023
^fb
0.0733 ± 0.0049
Experimemal summary— XII DAE HEP symposium etc 693
The disagreement with the Standard Model has been greatly reduced. Several things have
contributed to this shift in the measurements. The main differences are due to (a) use of
only multi-tag measurements, (b) attempt of using mostly the high energy measurements,
from LEP, (c) increased statistics, (d) improved tagging techniques.
The electroweak mixing angle has been determined from a variety of measurements
as summarised in Table 4. As one sees, all the measurements are consistent with each other
supporting the validity of the Standard Model.
Table 4. Detenninaiion of sin ^ 6^ from different measurements.
Avenge by group
of observations
Cumulative
average
A®-'
0.23085 ± 0.00056
0.23240 1 0.00085
0.23264 1 0.00096
0.23157 1 0.00042
0.23157 1 0.00042
3.9/2
^Q.b
"'PB
0.23246 1 0.00041
''fb
02315510.00112
0.23236 1 0.00038
0.23200 1 0.00028
6.3/4
(Qfb)
0.2320 1 0.0010
0.2320 1 0.0010
0.23200 1 0.00027
6.3/5
4lr (SLD)
0.2306110.00047
0.2306110.00047
0.23165 1 0.00024
12.8/6
W Boson properties :
Measurement of the mass of the W-bo$on has been reported from the Tevatron collider [5]
and LEP [4]. The pp colliders identify W’s through the leptonic decays. The energy and
directions of the lepton and the missing v are. measured and the transverse mass of the Iv
system is determined. The transverse mass is calibrated against the mass of Z decaying to
lepton pair and is then fitted to obtain m^. D0 measures W-mass :
mw * (80.38 ± 0,07 ± 0.08 ±0.17) Ge\
where the first two errors are due to W and Z statistics and the third error is due to
systematics. The systematic error includes errors due to transverse momentum of W and
luminosity. Both these errors will significantly reduce when the analysis is finalised and
one expects the systematic error to become » 0. 1 GeV.
Combining the measurements from UA2, CDF and D0, one gels the current best
estimate of W-mass from hadron colliders :
mw« (80.356 ±0. 125) (jeV
, The centre of mass energy of the e'*’c" system has gone above W-pair threshold in
1996. Preliminary results exist on W-mass from all LEP experiments from the threshold
scan measurement of the cross section o ( c''’e” — > W^W" ). Measuremeiit of W-mass from
direct reconstruction is expected soon. LEP measures
mw » (80.4 ± 0.2 (stat) ± 0. 1 (LEP)) GeV
m
Sunanda Banerjee
WidiW advent of high luminosity run at LEP and the Tevatron Collider, the following
evolution of error on W-mass is expected [2] :
Tevatron nin 1
Tevatron nin 11
LEP2
TEV33
$ too MeV
-40 MeV
-33 MeV
-20 MeV
Width of the W boson has been measured at the pp collider by two independent
methods. In the indirect measurement, one uses the expression
j. ^ r(W -> Iv)
* ” Oz ■ B (Z -) H) ■ R
where is obtained from theory (3.33 ± 0.03), B ( Z -► / / ) is measured at LEP
[(3.367 ± 0.006)%], r(W /v) is taken from the Standard Model [(225.2 ±1.5) MeV]
and the ratio R defined as
^ ^ <TwB(W /V)
{T 2 B (Z — > //)
is measured at the Tevatron collider. This gives
rw = (2.062 ± 0.059) GeV
CDF has looked into the tail of the transverse mass distribution for W (mr > 1 10
GeV). Using the measured pr of the W’s, CDF obtained *
Tw = (2. 1 1 ± 0.28 (sut) ± 0. 16 (sys)) GeV
The non-Standard Model contribution of W width is restricted to < 109 MeV at 95%
confidence level. In future, the error on fw is expected to go down to 45 MeV after Run II
of Tevatron and to 20 MeV at TEV33.
Preliminary measurements exist on triple gauge boson couplings. Using CP
conserving Lagrangian, one obtains two coupling constants each for WWy and WWZ
vertices. The parameters, denoted by K and A, are 1 and 0 respectively in the Standard
Model calculations. One uses radiative W production in pp collisions and all W events in
e'^e'collisons to measure the anomalous coupling of W’s. Event rates as well as kinematic
distributions have been used. The current and future limits of the coupling constants are
summarised in Table 5.
Table 5. Limits on anoinalou.s coupling of W-boson.
9.3% CL Limit
Current
Run 11
TEV 33
LEP 2
LHC
|Ar,l
14
038
0.21
0.24
0.06
0.4
0 12
0.06
0.24
0.01
Experimental summary— XII DAE HEP symposium etc
693
Physics with c Quarks :
The standard Model expects CP violation in D decays at the level of -10"^ in singly
Cabibbo suppressed decays and to be non-existent in Cabibbo favoured and doubly
suppressed decays. Experiments performed in the Tevatron fixed target facilities [6] have
now reached a sensitivity level of -lO"*. The two experiments E687 and E791 have
measured the asymmetry function Acp ;
r(D* D - r(D- -> f-)
^ “ r(D+ -+ f*) + r(D- -» f-)
All measured Acp’s are compatible to 0 and the 90% CL upper limits are summarised
in Table 6.
Tabic 6. Limits on CP violating asymmetries in D decays.
Decay mode
90 % CLfromE687
90% CL from E791
Kkff
-14% < A^p <8.1%
-6.2% < Acp < 3.4%
-7.5% < Acp <21%
-8.7% < Acp <3.1%
K*K
-33% < Acp <9.4%
-9.2% < Acp < 7.2%
RRK
-8.6% < Acp < 5,2%
One expects, within the framework of the Standard Model, the mixing between
D® “ to be small.
r (D® 5° f )
''mi. = r(D® -» f )
is expected to be in the range The experiments have now reached the
accuracy of E791 [6] has measured from semileptonic D** decays
D®jr+ -4 {Krl*vl)n*) where the rates of same-sign 7t/l (mixing) verwi opposite
sign n/l (no mixing) is compared. This measurement yields r„, < 0.5% to 90% CL. E791
has made similar measurements from hadronic D decays where the corresponding limit is
rmx < 0.4%.
Table 7. Limitt on flavour changing neutral cuneol from D® decays.
Decay Mode
90% CL Upper Bound obtained by
CLEO
E6S3
E791
2-3x10"^
6.6 X ir*
2x KT*
l.8x lOr*
3.1x10-5
The experiment E791 [6] also teponed on non-observation of flavour changing
neutral current in D decays. The experiments have reached accuracy in the range of
696
Sunanda Banerjee
KH whereas the Standard Model expectations are < 1 (H. The results ate summarised in
Table 7.
CLEO [7] has used data corresponding to integrated luminosity of 3.6 fb~' to
measure various rare decay modes of D,. The measurements agree well with the theoretical
expectation based on broken SU(3) model using factorisation hypothesis.
Physics with b Quarks :
CLEO [7] reported on their recent measurements of electromagnetic penguins in both
inclusive and exclusive final states. CLEO measures the branching ratio for (6 to be
(2.32 ± 0.57 ± 0.35) x 10^ whereas the Standard Model expectation is (2.8 ± 0 . 8 ) x 10~^. In
the exclusive final states one measures
nntl State
Branching Ratio
(4.4±1.0±0.6)x ir^
(3.8!f;? ±0.5)x 10-5
B-»Jf*y
(4.2 ± 0.8 ± 0.6) X ir®
CLEO [7] also reported on the prospect of observation of the gluor ic penguins (b -4
s + g). CLEO studied final states with very heavy mesons where the meson is too heavy to
come from b c decays. They choose 17 ' decaying to 77 ;r 7 rand study the rate of production
»
as a function of the momentum fraction x. By comparing the rates on and off resonance,
they obtained branching ratio of B to rj\ The low x region would have admixture of gluonic
penguin with B decays to D*Tf. The large jr region (jc > 0.4) will contain pure b s g
decays. The data are consistent with Monte Carlo prediction and the data at large x region is
at the moment limited by statistics.
The Tevatron experiments CDF and D0 have reported [ 8 ] observation of clear B
signals in J/ channel. So there is the potentiality of CP violation studies in the
b-system in the high luminosity runs of the Tevatron. CDF and D^ give limits on pure
leptonic decay mode of B's at the level of 10~^ whereas Standard Model expectation
is<l(H.
Several LEP experiments [ 8 ] have reported on the observation of time
dependent B^-B^ oscillation. If the flavour eigenstates are not mass eigenstates and
if the life times of the mass eigenstates are comparable, the flavour eigenstates are
expected to oscillate with probability given by the difference in mass Am and average
life time r. To study the oscillation, one needs to ug the b-flavour at production as well
as at the decay, b-flavour at production is tagged through jet charge or lepton charge and
at decay is tagged through lepton charge or D* charge. The decay length of the B-meson
is determined with the precise vertex detectors and the boost is measured from the
Experimental summary— XU DAE HEP symposium etc
607
energy measurements. After corrections due to background, mistags, combinatorics,
one obtains
Amd = (0.468 ± 0.019) ps-*
Am, > 9.2 ps"* at 95% CL
Null results
There have been several searches for new phenomena in LEP [4,9] and Tevatron [11].
The searches have yielded null results so far giving rise to limits to various
processes.
Higgs searches :
LEP experiments [3] have reported several electroweak measurements which agree with the
Standard Model. So if one tries to fit all the measurements obtained from LEP, SLC,
Tevatron and low energy experiments to the Standard Model, one obtains a fit
corresponding to rather low mass of the Higgs boson. This can be translated to a 95% CL
upper bound on Higgs boson mass of 650 GeV. However, some of the measurements are
inconsistent giving rise to large one scale up the errors to make the measurements
consistent, the limit loosens to 920 GeV.
LEP experiments [9] also looked for direct observation of Higgs bosons -through the
Bjorken process (associated production with Z). The associated Z will decay to a pair of
fermions (search concentrates on leptonic decay mode of Z) and a Higgs above 10 GeV will
decay dominantly to a pair of 6-quarks. All LEP experiments reported on non-observation
of signals for Higgs boson. An attempt has been made of combining the four LEP
experiments [9]. One requires harder cuts thus lowering the efficiency of individual
experiments. Taking tuned efficiency for 60 GeV Higgs and reducing the efficiency by
25%, one obtains 95% lower bound on Higgs mass :
m^ > 65.6 GeV
At LEP2, one needs to produce the associated Z on shell to get appreciable cross
section. This limits the reach of search to Vr - 100 GeV. With the new run at -JJ =
161 GeV [4], Higgs of mass up to 60 GeV can be probed. OPAL has combined the
results from LEPl and LEP2 searches and improves the limit on lower bound from
59.6 GeV to 65.0 GeV.
LEP2 can look for Higgs up to a mass of 90 GeV. Beyond that mass, Higgs can be
found in the high luminosity run of the Tevatron (21 or at the LHC [10]. In hadron collider,
Higgs search is difficult in the mass range 90-130 GeV. For low Higg s mass, one needs to
look for Higgs in the yy decay mode. The signal to ^Background ratio, the discovery
potenUal. of 10 can be reached with -100 fb-' of integrated luminosity. The background
rate would be smaller if the 2ys are looked with associated 2 2 jets. There the signal to
background is 1 : 1, and the discovery potential would be similar for 100 fb"' integrated
luminosity. .
Sunanda Banerjee
SUSY searches :
Signals for Supersymmetry has been looked into in e‘^e~ and hadron colliders with the
assumption of R-parity conservation. This hypothesis gives rise to the scenario that the
super particles are produced in pair and that the lightest super particle (LSP) is stable. The
LSP interacts weakly with matter and will thus avoid detection. So experimentally one
would look for events with large missing energy. In hadron colliddr, missing energy is not
measurable and hence one looks for large missing transverse energy. For gluino searches,
one utilises the fact that gluinos are Majorana type [10,1 1] and it will give rise to excess of
like sign dileptons.
In the SUGRA motivated SUSY models, the parameter space of SUSY is givea by
the scalar mass mo. gaugino mass M, higgsino mixing //, ratio of the vacuum expectation
value tan p, and a soft trilinear coupling term A. The experiments develop search strategies
which make use of generic topologies of SUSY signal and the parameter space provide a
guideline for the expected signal level.
LEP experiments utilised their high energy runs [4] to look for chargino, neuu^lino,
slepton and squarks. No signal has been observed above the level of expected background
and this provides 95% CL upper bound on cross section of SUSY signal. One can scan
parameter space and exclude the parameter space where the expected cross section is above
the excluded cross section. This essentially rules out charginos and selectrons all the way to
the kinematic limit.
Tevatron experiments [11] looked for squarks and gluinos through jets + missing Ej
signature and also through same sign dileptons. Both CDF and D0 provide limits in squark-
gluino mass plane. Gluinos of mass less than 180 GeV have been ruled out by the
experiments.
Several search scenarios of SUSY signal in the future hadron collider have been
reported [10,12] in this conference.
Test of QCD
There have been several results testing the theory of strong interaction from the pp
collider [13] as well as from LEP [14]. Only a small selection of these results is
included here.
Direct photon production :
CDF [13] has measured direct photon production in ^ collisions in the fiducial range I t| I
< 0.9. The purity of the signal has been estimated to be within the range 25% to 80% for
photon pr of 20-60 GcV/c using Monte Carlo. One finds ■ 20% excess in the data for pr in
the range of 20-30 GeV/c. The direct photons will be background to Higgs searches at LHC
in the H yydecay mode and have to be monitored carefully.
Experimental summary — Xi! DAE HEP symposium etc
m
Strong coupling constant from LEP :
LEP [ 1 4] has used the high energy hadronic data to measure the strong coupling constant
a R . The global event shape variables, thrust, heavy jet mass, jet broadenings, arc compared
to analytical calculation to second order with complete resummed leading and next-to-
leading log terms. The study has been extended to low Vs values by utilising events with
direct photons. The hadronic events with high energetic isolated photons are due to an early
radiation from initial or final state. The hadronic shower formation factorises out. So using
such events and looking into the hadronic subsystem, one can measure a, over a large
centre of mass energies (from 30 to 172 GeV) from a single experiment. The measurements
favour running of ct* ala QCD. The L3 measurements are consistent with o^(mz) of 0.1 19 ±
0.002 where the error refers to experimental'error only. The theoretical uncertainty is at the
level of ± 0.006.
New physics
There have been a couple of observations which cannot be explained by standard processes.
However, the signals are indicative only and not supported by statistics or other
experiments.
ALEPH four jet events :
ALEPH [4] has reported excess of events in the four jet final slate from their analysis of
high energy data. From the 130-140 GeV data, ALEPH looked for events witli two massive
particles of approximately same mass each decaying to a pair of jets. Using energy
rescaling, to improve the mass resolution, they saw a peak in the distribution of £M for the
four jets at 105 GeV. The peak corresponds to 9 entries where only I is expected from
background. The integrated rates are 16 and 9 in data and Standard Model Monte Carlo.
Combining with the high energy data at 16^1 and 172 GeV, the number of entries in the peak
region has increased to 18 whereas background expectation is 3.1. Total number of events
in the 4-jei category is now 34 with 24.5 events expected from background.
This excess was not reported by any of the other three LEP experiments. The
position of the peak is at 106.1 ± 0.8 GeV with a Gaussian width of 2.1 ± 0.4 GeV. The
width is compatible with detector resolution. All properties of these events are similar to
normal hadronic events. For example, there is no large bb excess in these events. There is a
working group set up by LEP to look into these four jet events. The preliminary work has
shown that the other 3 LEP experiments have similar acceptance and resolution for such
events. So they would have observed similar excess. One should wait for the final analysis
of this working group to get a clearer picture of the situation.
CDF special event :
CDF [111 has reported one special event with two energetic isolated electrons, two
energetic isolated photons and lots of missing transverse energy. The transverse energies
of the two electrons are 59 and 36 GeV respectively and those for the photons are 38 and
72A(6)-3J
700
Sunanda Bmerjee
30 GeV. Missing Et is S3 GeV. Such an event cannot be explained by the Standard Model.
One, however, expects such events in a SUSY scenario from slepton pair production. There,
one also expects to see several events with energetic photons and missing Ej. Such events
have not been reponed by CDF or as yet. One needs more data to sort this out.
Outlook
We got a glimpse of several excellent results from the work of many physicists in a variety
of experiments. The current trend of results consolidates the standing of the Standard
Model. We hope some of the hints on new physics can lead to physics beyond the Standard
Model. May be in two years from now, one can hear more of such results In the future DAE
symposium. 6
References
[ I ] M Narayan (Talk given at this confeience)
[2] . U Heintz (Talk given at this conference)
[3] S N Ganguli Indian J. Phys. 72A 527 (1998)
[4] A Guftu Indian J. Phys. 72A 339 (1998)
[5] B Chaudhury (Talk given at this conference)
[6] S Mi.shra (Talk given at this conference)
[7] V Jain (Talk given at this conference)
(81 T Aziz Indian J. Phys 72A 689 (1998)
[9] A Sopezak Indian J. Phys. 72A 469 (1998)
(101 DP Roy Indian J. Phys. Ilk 587 (1998)
(111 N K Mondal Indian J. Phys. 72A 505 ( 1 998)
(12] K Maumdar (Talk given at this conference)
( 1 31 MR Krishnaswamy Indian J. Phys. Ilk 67 1 ( 1 998)
(141 S Sarkar (Talk given at this conference)
A Abbas
Prativa Das
S Abbas
R Datta
Rathin Adhikari
A De
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Sudipta De
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TDe
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Alri Deshamukhy;
T Aziz
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( ii >
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FORTHCQMISG PtMiLlCA'nONS (B)
DECEMBER 1998. Vol. 72. Nii. 6
Rapid Communication
Spccii al sol'tcning due lo winds in accretion disks
Sandip K Chakrabarti
Review
Tropospheric VHP propagation studies over Indian East Coast
SwAiiCHounMLiRY,DDurrAMAjuMOER AND AmitaPal
General Physics
Ground stale of confinement potential in two dimensions
Na/aka'i Ui.i.ah
Optics & Spectroscopy
Surlacc enhanced Raman spectroscopic study of poly(o-melhoxy
aniline) organised in Langmuir-Blodgcit film
JoYDi 1 1» Chowdhiikv. Pkahiu Pal and T N Misra
Phoioqucnching efiect in rigid (cresyl violet) and non-rigid
(disodium fluorescein) dye molecules
Gi-orol C CHhNNArrucHF.RRY, G Ajith Kumar, P R Biju,
C VFNur.oPAL AND N V Unnikrishnan
Ijghl scattering as an alternate probe of fractal structure of the
Agl colloidal aggregates
Trman Nicula
Statistical Physics, Biophysics & Complex systems
Dependence of non-classical behavior of OPl Hamiltonian on the
strength of coherence of initial light
M A Jafarizadkh, A Adibi and A Rostami
Amplitude-squared squeezing and photon statistics in second and
third harvnonic generations
Jawahar Lai. and R M P Jaiswal
I Omi'ti tm nexi pufie }
Notes
Dual riuorescence of indole-2-carboxylic acid and indole-S-
carboxylic acid
Prakrit] Ranjan Bangal and Sankar Chakra vorti
FT-Raman and FT-IR spectra of a fluoroquinolone complex
J Marshell
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[5] U Fano and ARP Rao Atomic Collisions and Spectra (New York : Academic) Vol 1, Ch 2. Sec 4,
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[7] T Atsurm, T Isihara, M Koyama and M Matsuzawa Phys. Rev. A42 639 1 (1990)
[II] T Le-Brun, M Lavolled and P Morin X-ray and Inner Shell Processes (AlP Conf. Proc. 215)
cds T A Carison, M 0 Krau.se and S Manson (New York : AIP) p 846 (1990)
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Pages
Heavy ion physics and quark gluon plasma
Quark gluon plasma — current status ot properties' and signals
C P SiNCJH
Formal field theory
Blackholc evaporation - stress tensor approach
K D Krori
Light-front QCD : present status
A Harindranath
Methods of thermal field theory
S Mall IK
QuaiUiim inicgrable systems . basic concepts and brief overview
Anjan Kundli
Summary talks
IVrspcciives in high energy physics
Ci Rajaslkaran
t xpciimenial summary — XII DAE HEP symposium, Guwahati, 1997
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