Indian Journal of Pure & Applied Physics Vol. 37, April 1999, pp 285-293

Theory of point defect energetics: A review Y.V.G.S.Murti' & Radha D Banhatti"

* Department of Physics, Indian Institute of Technology, Guwahati 781 00 I

**Institut fur Physikalische Chemie, Westfalische Wilhelms Universitat,

Munster, Sclossplatz 4, D-48149 Munster, Germany

Received 3 February 1999

The energetics of point defects provide the controlling factor in determining the atomistic mechanisms in a wide range of solid state processes. We present here a pedagogical overview of the development of the continuum, quasi-Ianice and lattice theories for different classes of point defects and materials. Varied approaches were followed in the past in modelling the relevant perfect crystals for interatomic forces for nonionic solids and model potentials for ionic materials. The earliest continuum approaches are those of Eshelby and Jost for treating point defects as elastic and dielectric singularities. These were followed by semicontinuum Mott-Littleton techniques and the Kanzaki derect force techniques in application to charged and neutral defect species. However the importance of a correct assessment of the dielectric polarization and the anharmonicity of the forces in the evaluation of the enthalpi es and volumes have been well documented. Numerical computations of the enthalpies are seen to be sensitive to the choice of potential parameters and polarization models to varying degrees. While the theoretical picture is relatively clear in the case of the simpler material s with a near-ideal pure disorder, materials with mixed type of point defect disorders call for a more challenging simulation of defect environments which among other things should take into account the strong inhomogeneities of defect fields . The paper gives an overview of the evolution covering the highlights of these developments.

In trod uction

The fundamental physical , chemical and engineering properties of crystalline materials are

invariably influenced by the state of lattice disorder. While the mechanical and metallurgical behaviour is affeectd primarily by dislocations, with an important

role played by point defects, there are a number of processes - both microscopic and macroscopic in

which point defects act as the essential driving force.

The prominent of these properties are broadly classified in Tables I and 2 . Besides these properties,

the mechanical behaviour of materials - notably the inelastic and an elastic response are sensitively

dependent on the nature and content of the point defects .

Table I -- Ionic processes

Process

Mass transport

Charge transport

Property

Diffusion

Electrical conductivity

Symbol

D

cr

Energy transport Thermal conductivity K

Momentum transport Internal friction Q-I

Tab le 2 -- Electronic processes

Process Property Symbol

Optical Transitions Optical absorption or a emission

Dielectric response Dielectric constant £

Magnetic response Static / dynamic X susceptibility

The role of point defects in transport properties

are ~erhaps among the best atomistically modelled so far

l . . To understand this role, it is necessary to

introduce the concepts of the thermodynam ic parameters of point defects . When a point defect

such as a vacancy is added to the lattice, many changes in the thermodynamic extensive variables will occur. Each defect increases the internal energy

of the crystal by a certain amount called the energy

of formation of the defect, i. The unbalance in force

equilibrium will bring about a relocation of the atoms in the neighbourhood and beyond, depending on the extent of unbalance. These atomic relaxations will lead to a change in the volume of the crystai, by an

286 INDIAN J PURE & APPL PHYS, VOL 37, APRIL 1999

amount vrel . In the case of Schottky defects, there

will be an additional change in volume due to the extra volume (n) occupied on the surfaces where the vacancies originate from. These two contributions add to give the volume of formation of a defect, v f = vrel + n. For Frenkel defects, for e.g, the second contribution does not exist.

When the atoms in the proximate region of the defect have relaxed into new equilibrium sites with a different potential curvature, the frequencies of lattice vibrational modes are altered. The phonon entropy of the lattice is changed by /, the entropy of formation of a point defect.

sf = k,L, In(:: J ... (1)

where Wi is the frequency of a normal mode i in the perfect crystal and the primed symbol refers to the frequency of the same mode in the presence of the point defect. kB is the Boltzmann constant.

All these changes taken together can be used to define the Gibb's free energy l for the formation of a point defect as follows:

... (2)

at any temperature T and pressure P.

When n such defects are formed , independently of one another, true for dilute defect densities, the Gibb's free enerp of th~ . crystal as . a whole is increased by ng . In addition, there IS a further contribution coming from the configurational entropy, which depends statistically on the thennodynam ic probabi lity of arranging the n defects on N sites. The importance of l is the fact that it governs the thermal equilibrium concentration of the point defects according to an Arrhenius law. For Schottky defects in a binary crystal, this has the form

n &-/ f) N = expl-2kIJT .. . (3)

Here g{ is the combi ned free energy of

formation for a Schottky pair. Similar the rmodynamic quan ti ties are a lso defined for other

de fect processes stich as migration (gill) or bi nd ing

(l) between defects. Table 3 lists some of these different parameters.

A physical property such as ionic conductivity can be modelled I . to yield a defect dependent electrical conductivity (0) as a function of temperature, impurity density, pressure and other relevant physical parameters. The model may then be subjected to a detailed comparison with experimental data. It is expected that the experimentation is carried out under controlled conditions of sample preparation, treatment and measurement. It may then be possible to obtain good values of all the thermodynamic parameters of the defects. In a similar manner, measured self-diffusion coefficients

3 (D) can also lead to the defect parameters. Normally, the enthalpy of formation , Il, is extracted from the temperature dependence of the property, while the volumes can only be obtained if the experiments are done under different hydrostatic pressures . It thus turns out that the thermodynamic parameters of point defects are obtainable from experiment no doubt but the process invo lves quite a significant amount of care and thoroughness in measurement and analysis. It is in thi s context that the theoretical evaluation] of point defect enthalpies assumes importance.

Table 3 -- Thermodynamic parameters of point dekcts

Process Formation Migration

Energy J unl

Volume f vm v

Entropy / SOl

Enth alpy 1/ hn!

Free energy gf gn!

What are the different theoretica l techniques availab le towards this purpose? Tab le 4 li sts these techniques and physica l models . Even though we are covering the ground with techniqucs appl icab le to all poi nt defect types , the Schottky defcct in ioni c materials is used as an example.

...

MURTI & BANHATTI: POINT DEFECT ENERGETICS 287

Table 4 -- Techniques and mode ls

Technique Authors Year

Continuum lost 1933

Continuum Eshelby 1954

Force balance Mott and Littleton 1938

Energy method Kurosawa 1958

Fourier method Kanzak i 1960

Shell model Faux and Lidiard 197 1

Green function Tewary 1973

Energy minimi zation Lidiard and Norgett 1974

Modified PP1 Murti and Usha 1976

Extended PP1 Radha and Murti 1991

2 Continuum Models

2.1 Dielectric continuum

Jost4 was the first to show that on account of the vacancy bearing an effective charge, the induced polarization reduces the formation energy drastica lly. This reduction is estimated readily by treating the crystal as a dielectric continuum SUbjected to the field of the defect charge (qd) . The static die lectric constant (E, ) of the crystal is the

relevant response parameter. The displacement jj , polarization P are related to the electric field E by

p = c, -\ q" F c,. 4;r IFI3

The po larization energy is equal to

... (4)

... (5)

.. . (6)

... (7)

. . . (8)

where ~ is the potential of the po larized crystal at the defect s ite and the above is s imply the work done by the dipoles in extract ing the iOIl of charge (-qd) from

the lattice site. In the spirit of the continuum concept,

he potential ~ is calcu lated as

. .. (9)

~ is the lower limit of integrat ion, a quantity hard to define for a vacancy. On substitution of qd with the negative of the ion charge (-Ze), the expression for the polarization energy becomes

. . . ( I 0)

Tak ing ~ to be (rJ 2), where ro is the nearest anion-cat ion separation, a working fo rmu la fo r this energy (in eYj, may be written as

Z2 1 UI' = - 1.44- (1- -) .. . (I I)

r" c.,

with ro in units of nm. Estimates of this quantity are given in Table 5 for a few typical crystals. The po lar ization energy is as large as the the defect formation energy ( fo r eg the energy of formati on of

a cation vacancy, u{" ) and thus it is clear that this

quantity plays an important part in any defect format ion process and requires carefu l assessment.

Table 5 _. Polarization energy (eV )

Crys tal Ul Up r Un

NaCI 7.93 4 .15 5.20

AgCI 9.46 4.75 5.4 1

AgBr 9 .33 4 .54 5.04

MgO 42 .2 24 .53 26.4

CaO 36.7 21.75 22 .3

2.2 Elastic continuum

While the fost model is meant for charged point defects, poi nt defects with or without charge may al so be regarded as sin gularities in an el astic continuum ). The equ il ibrium eq uation for the

di spl acement field u(n , of an elastic so lid with

rigid ity modulus /l , Poi sson ratio v , and under the

action of a body fo rce dens ity .1. is given by

288 INDIAN J PURE & APPL PHYS, VOL 37, APRIL 1999

... ( 12)

where the Lame constant A. is defined by

A. = 2J1v 1- 2v

... ( 13)

I f the point defect preserves the spherical symmetry of the continuum, with the realistic constraint of a displacement field decreasing with increasing distance from the defect centre, and with no body forces present, the solution of the above equation takes the form 5

... ( 14)

where c is an unknown quantity termed the elastic strength of the point defect. Recourse to other input information is taken to arrive at a plausible estimate of this quantity . This strength parameter is essentially a measure of the size misfit of the singularity. For example, for an impurity of radius Rj replacing an atom of radius Ra , the displacement u(ro) is taken to be equal to (Rj-R.). It then follows that c is given by

... ( 15)

It is noteworthy that the displacement field is

long-ranged with the elastic misfit displacement liil decreasing with distance r as I /r 2. In section 4, we w ill give a better definition of c . Eshelby's theory also leads to the relaxation volume of format ion of the defect in terms of c:

rei V = CK

where K is the isothermal crystal.

3 Mott-Littleton Model

.. . ( 16)

compressibility of the

The first successful lattice technique is that introduced by Mott and Littleton (ML) in their celebrated paper6. The technique is still the most used and yet to find a better replacement. The authors split the defect crystal into two regions I and 2. The defect and the neighbourhood to be chosen, represent Region I . The remainder of the crystal constitutes Region 2 . While the dipole moments and' displacements of ions in Region 2 are deduced following essentially the lost treatment of a continuum with some modifications, Region I is treated in a completely atomistic model. Thus the

polarization P as given by the equation in the lost model is apportioned into two parts - the dipole

moments p" for the anions and Pc for the cations

in a given cell in accordance with the polarizabilities.

The displacment 11 of any ion in Region 2 is in proportion to the displacement polarizability . Thus for Region 2,

( -- = ::'''!:':!!''' QP Pea {

J cell

... ( 17)

... ( 18)

... ( 19)

... (20)

qc,. are the charges of the cation or anion. U a and Uc are the electronic polarizabilities of the anion and cation respectively. The interatomic potentials are commonly modelled by the Born-Mayer type of function:

Cil

Irijl6

... (21)

The Cjj are the coefficients for dipole - dipole van der Waals potential. The first term is the energy of repulsion due to overlap of ions and the second term is the Coulomb energy.

Region I solutions were obtained by Mott and

Littleton by balancing the forces , Coul ombic 1;. and

non-Coulombic FNC

. .. (22)

The energy of the defect-disto rted crystal is written as an additive sum of the rigid lattice energy (URL ) , the Coloumb energy (Uc) , the short range contribution (USR ) and the two polarization energies

belonging to the Region I (Uj,I)) and to the Region 2

(Uj,2) ).

U - U U U(I) U(2) U - C + SR + I' + I' + RI. ... (23)

The Schottky formation energy follows as

... (24)

MURTI & BANHATTI: POINT DEFECT ENERGETICS 289

where UL is the lattice energy of the crystal per ion pair.

4 Lattice Statics

4.1 Kanzaki model

Kanzaki 7 modelled the vacancy-distorted crystal in terms of virtual forces (Kanzaki forces) that reproduce the distortion and energy changes arising from a vacancy. Fig.1 a shows a perfect undistorted crystal, Fig.l b the distorted crystal containing the vacancy while Fig.l c shows the Kanzaki lattice which has the same distortion as the defect lattice but without the vacancy. The problem reduces to finding the simulating force distribution of Fig.lc . For this purpose, Kanzaki used the harmonic approximation with the energy U of the crystal under the action of a

set of forces Fer) expressed as

u = U" - I IF:x(r')~a (r') , a

.. . (25)

Uo is the energy of the perfect lattice. ~a (r ' ) are the Cartesian components of displacement of an

ion at r'. I and ( represent lattice cell labels. The elements of the force constant matrix A are defined as spatial second derivatives of the energy.

. cfu Aap(ll) = , '" 10

&,/3.x p ... (26)

The element Aa /3(II) represents the a component of the force acting on the atom located at

r' due to unit displacement in the (3 direction of an

atom located at r'" .

Fig .' a Fig . 1 b

Fig .1 c

The equilibrium equations are

.. . (27)

and they lead to

F:x(r') = I IAap (l{)~p (r' ) ... (28) I" P

giving 3N coupled equations - one each for the three Cartesian displacement components of each of the atoms in the crystal.

Kanzaki's technique consisted of Fourier transforming the real space quantities and the equilibrium equations using a normal coordinate expansIOn.

~a (r') = I IQp(ij)exp(iij.r') . .. (29) p t7

where Qp are the Fourier amplitudes of

displacements at any point ij in the reciprocal space . The corresponding equilibrium equations are given by

NIVap (-ij)Qp(ij) = Ga(-ij) . .. (30 ) p

The matrix V is the Fourier tranform of'A :

Vap(ij) = I Aap(l{)exp(-iij .(r'" -r') ... (31) , - ,"

The generalized forces G are similarly the Fourier transforms of the real space forces:

.. . (32)

The three equilibrium equations in the Fourier space are now uncoupled - essentially a feature of the normal coordinates - and are solved for the amplitudes Q/3 at each point q . The q space to be covered is only the fundamental Bri llouin zone. Further, in this zone we need only deal with the irreducible part as per the symmetry of the crystal. Thus in a cubic crystal with 0" symmetry, this part is 1148 of the zone . Having obtained the Q's, the real space displacements are calculated by he reverse transformation, involving summation over the q vectors. The convergence depends on th-e spatial

290 INDIAN J PURE & APPL PHYS, VOL 37, APRIL 1999

extent of the forces F. The decoupling of the equations was possible because of the harmonic approximation. It follows that defects with relatively small distortion forces such as neutral defect species should be particularly amenable to treatment by thi s technique.

4.2 Kanzaki forces

The Kanzaki fo rces can be obtained by writing the energy of the imperfect lattice relative to the perfect lattice in the following manner:

. . . (33)

This leads to the expression for the energy of the defect lattice in term s of the change in the interaction potential energies as

+~ L¢(lr' +~(r/)-(rl +~(rl»1) 2 /I

... (34)

The energies Va , Vb, Vc refer to the energies of the crystal depicted in Fig.1 a,b,c respectively. Differentiation of the above energy difference with respect to the displacement componeilt ~a leads to the formal expression of the Kanzaki forces .

. .. (35)

The theory has been adopted with advantage in the treatment of vacancies in metal s, strain field interaction of vacancy dimers8

, monovalent impurities and their dimers9

, as well as F centres'O Tewary's Greens function technique" is also an equally good method . This approach is equivalent to the Kanzaki method but having other distinct advantages. Applying the Kanzaki theory to vacancy defects, Hardy and Lidiard'2 obtained for the elastic strength of a vacancy, the following explicit expression:

5 Energy Method , ,

Kurosawa ' was the first to ad pt the energy approach in the treatment of the vacancy prob lem. Norgett and Licliard'4 generalized it to give a method that is generally followed in modern work . The basic principles of the method are recalled here for completeness . The total energy Vd of the imperfect crystal is composed of three parts: V ,W) for Region I , V2W,y) - the energy of interaction between Region I and Region 2 and U,(y) - the energy of Regi on 2.

Here f3 represents all the relaxation s and dipo le moments of ions in Region I. Y represents these quantities for Region 2 ions (See Fig.2).

... (37)

r

• Defect

It turns out that the ML approach is consistent with a harmonic treatment of Region 2. Accordingly the third energy term above can be written as

.. . (38)

Here Hij is an appropriate force constant matrix for the Region 2. On using the equilibrium co nditioJl

.. . (39)

we effectively e liminate the third energy term . .. . (36) provided it is agreed that the equilibrium values of

the relaxation variables of Region 2, viz., y, are taken

to be those (Yo,) given by the ML mode l.

..

MURTI & BANHATTI: POINT DEFECT ENERGETICS 291

While the above serves as the guiding equation for evaluating the energy of the imperfect crystal, more has to be developed before it is ready for application . If the pair-wise interatomic potential between two atoms i and j is represented by ~ij, the energy to be computed is

iel,)e l

iel,)c2

_~ ~ ( 7¢ijCI F; -r; I) _ o¢ijCI Ri -r; I)].r 2 ~ a: a: ,."

;el.)e2 .i j

... (41)

Here R; and F; are respectively the unrelaxed

and relaxed position vectors of the ion i. The partial derivatives are all evaluated at the equilibrium positions.

6 Physical Models

6.1 PPI

The energy changes depend on the details of interactions between the atoms, the effect of distorting the electron distributions of the ions, the effects of overlapping of the ions - both on account of long range forces as well as short range forces. The balanced configuration of the imperfect crystal will depend in a detailed way on all these factors . In the point polarizable ion (PPI) model , a model implicit in the ML work6

, each ion is represented as a point charge with a polarizability. The electronic polarizabilities of the ions can be related to the refractive indices n using the Lorentz-Lorenz equation

n2 - 1 1

-2-- = --NCaa + a c )

n + 2 3£" ... (42)

In their widely cited paperl 5 , Tessman, Kahn and Shockley (TKS) use this equation to determine the electronic polarizabil.ities of a number of ions tn cubic crystals in a self-consistent manner. It is to be noted that the validity of the use, of the Lorentz factor

1 (-) is by no means definite . The local field is

3£" evaluated assuming point dipoles located inside the Lorentz cavity and assuming a cubic symmtery. The low frequency dielectric constant (lOs) of the crystal is larger than the optical dielectric constant (Eopt)

because of the additional contribution of the relative displacement of the sublattices. The polarizability (ad ) for this displacement can 'be determined 111 terms of the short range force constant K uSll1g

.. 2 . K = 17[¢s/I (r;,) + - ¢S/I (r;,)]

r;,

... (43)

... (44)

11 is 4 for the rocksalt structure and 16/3 for the CsCI structure. ~SR above includes the van der Waals interaction potential besides the overlap repulsion potential , both for nearest neighbours. The second neighbours do not. contribute to ad. The primes on the short range potential terms indicate spatial differentiation. Using TKS values of polarizabilities and this value of ad, we may calculate lO s from the C lausius-Mosotti equation.

£ -1 1 _s - = --N(aa + a c + lad) t:., + 2 3£"

... (45)

The calculated values of lOs come out to be too large for several crystals for which the Born-Mayer of cohesion works so satisfactorily. For e.g for aC I, the calculated va lue is 12 compared to the measured value of 5.2. This is essentially a failure of the PPJ model to account for the static polari zation accurately. The field of the point dipoles is evaluated using the formula

F = 3(p.r)r-lrI2 p 47r£"l f I5

... (46)

The PPI model overestimates the po larization because of the neglect of the finite sizes of the iOlls. Overlap reduces the magnitude of the dipole

moments. Thus the values of h{ based on PPI are

underestimated . Besides, the formation enthalpy h{ is unstable under expansion of Region I. In order to overcome this problem , one should allow for a

292 INDIAN J PURE & APPL PHYS, VOL 37, APRIL 1999

coupling of the overlap forces and the induced dipole moments. This is achieved in three different ways:

• Deformation Dipole (DO) Model

• Shell Model (SM)

• Modified PPJ Model (MPPI)

Hardy and Karo l6 developed the Deformation Dipole (DO) Model for application in the dynamics of crystal lattices. The model allows for the reduction in polarization by adding a deformation dipole in opposition to the induced dipole. In the shell model (SM) first introduced by Dick and Overhauser l7 each ion is represented by a core and a shell , the two being connected by springs. The shells of neighbouring ions are also connected by another spring. The force constants of these springs and the fractions of charges included in the shell or core of an ion etc., (the shell model parameters) are determined by matching the crystal properties such as the elastic constants, dielectric constant, refractive index and infrared resonance frequency. It was Faux and Lidiard l8 who first used SM in defect modelling. It must be remarked that we have here not addressed the problem of obtaining the numerical values of .he parameters of the potential corresponding to the overlap repulsion and the van der Waals dispersion effects. The overlap potential is given satisfactorily from a density functional approach based on a uniform electron gas model '9. The evaluation of the dispersion potential is less simple.

6.2 MPPI Model

Murti and Usha20 introduced the Modified Polarizable Point Ion (MPPI) model which while retaining the good features of PPI, remedied the deficiency in respect of the polarization . The model takes care of the elastic and dielectric response and is designed for application to situations in which electric fields of a static nature are present - as in the defect environment. New effective polarizabilities are introduced to account for the reduction in the polarization due to overlap. This model had met with success in a wide variety df defect environments . The MPPI model has the merit of being simple, physical, less parametric and computationally efficient. Another variant of PPI was suggested by Boswarva

d S· 21 an Impson.

6.3 EPPI Model

All the above models, PPI, DO, SM, MPPI treat the polarization of the dielectric in the dipole approximation: The electric field of the point defect is implicitly presumed to be spatially homogeneous -an assumption which can not be justified as the defect fields are indeed inhomogeneous. The Extended Polarizable Point Ion model (EPPli2

-24 was

introduced to remove this inadequacy by including the effects of quadrupole moments interacting with the electric field gradients. Thus the dipole moments

p and quadrupole moments Qi of ions i in the Region I, are given by

- "F-'oc Pi =a i i . .. (47)

Q - qVF/t" i -ai i ... (48)

Here ad is the dipole polarizability and a q is the quadrupole polarizability of the ions. The local field

J;'oc and local field gradient V J;"". are respectively

given by

flOC = Fill + F" + F" I I , I

. .. (49)

vt'"' = VF III + VF" + VF" I I I I ... (50)

The superscripts m, d, q refer to monopoles, dipoles and quadrupoles respectively. In the case of materials containing more polarizable ions, the quadrupolar deformations are large and assume significance. The quadrupole deformation does contribute very significantly to the defect formation process in two ways: Firstly there is an additional polarization energy due to the interaction of the induced quadrupoles with the electric field gradient.

3Q

o~ w =-- --" 4::-(jz

... (51)

The dipole moments are also modified because of the contribution of qlladrllpoles to the local field. Application of EPPI provided a conclusive demonstration25 of the reasons for the dom inance of the Frenkel disorder on the cationic sublattice of Agel and AgBr.

...

MURTI & BANHA TIl: POINT' DEFECT ENERGETICS 293

Table 6 - Defect onergies (eV) in AgCI References

PPJ

EPPI

Experiment

HI

5.01

-2.99

-3 .39

~

2.42

1.48

1.4 - 1.5

I

2

3

4

5

The model also led to insight into the cause of 6

the perplexing - anomalies that persisted in the transport bahaviour in these materials26

•27

• Schottky 7

defects and paired vacancies are seen to be two kinds 8

of important minority species which play significant 9

roles in . the transport processes in the above materials24

•2s

• More recentll8 new findings have emerged from the application of EPPI to Schottky defects 'in the oxide system of alkaline earth metals. When we examine the values of defect energies deduced with and without including the quadrupolar · 14

contributions, it turns out that reliable values of the 15

defect energies result only when the induced quadrupoles are included in the treatment, especially

10

II

12

13

16

for crystals with largely polarizable ions. (Table 6). It has been seen that the quadrupole refinement is markedly effective and therefore can hardly be ignored in any defect modelling schemes.

7 MUDRA

A scientific software called MUDRA

17

18

19

20

(Modelling Utility for Defects by Rapid Analysis) 21

based on EPPI is being developed as a user-friendly 22 tool for modelling defect energetics in ionic materials. The present beta version caters only to Schottky defects and rocksalt structure, with potentials of standard fonn. It is expected that 24

MUDRA in the final form will be usable for other structures, many defect types and user-specified

23

25 potentials.

26

Acknowledgements 27

The authors thank the Department of Science and Technology, New.Delhi for providing financial 28

support for a project to develop MUDRA as a scientific software.

Lidiard A B, Handb Phys, 20 (1951) 246.

Franklin A D in "Point Defects in Solids". Eds Crawford 1 H & . Slitkin L M (Plenum Press), 1972.

Laskar A L in Superionic conductors and solid electrolytes. Eds A L Laskar & S Chandra (Academic Press, London), 1989, p. 265.

lost W. J Chem Phys. I (1933) 466.

Eshelby 1 D, J App Phys. 25 (1954) 255.

Mott N F & Littleton .M 1, Trans Farad Soc, 34 (1938) 485.

Kanzaki H. J Phys Chem Solids. 15 (1960) 39.

Bullough R & Hardy 1 R, Phil Mag. 17 (1968) 833 .

Murti Y V G S & Usha V, Crystal Latt Defects. 7 (1977) 31.

Murti Y V G S & Usha V, Physica. 93(1978) 219.

Tewary V K, Advances in PhYSiCS, 22 (1973) 757.

Hardy 1 R & Lidiard A B, Phil Mag .. '15 (1967) 825.

Kurosawa T,JPhysSocJapan, 13 (1958) 153 .

Norgett Ml,AEREReport. R7650(1974).

Tessman 1, Kahn A & Shockley W, Phys Rev. 92(1953) 890.

Hardy 1 R & Karo A M, The Lattice Dynamics and Statics of Alkali Halide Crystals (Plenum Press, NY), 1979.

Dick B G & Overhauser A W, Phys Rev. 112 (1958) 90.

Faux I D & Lidiard A B, Z Naturforsch. 26a (1971) 62.

Allan N L & Mackrodt W C, Phil Mag. 69 (1994) 871.

Murti Y V G S & Usha V, Physica. 83B (1976) 275 .

Boswarva I M & Simpson 1 M, Can J Phys. 51 (1923).

Banhatti R D & Murti Y V G S, Phys Stat Sol (b). 166 (1991) 15.

Banhatti Radha D & Murti Y V G S, Phys Rev, B48 (1993) 6839.

Murti Y V G S & Banhatti Radha D, Proc Int Con[ on Deficts in Insulating Materials. Eds 0 Kanert and 1-M Spaeth, (World Scientific, Singapore), 1993, p. 198.

Banhatti RD. Ph.D Thesis, lIT Madras, (1993).

Banbatti Radha D & Murti Y V G S, Radiation Effects and Defects in Solids. 134 (1995) 157.

Murti Y V G S & Banhatti Radha D, Bull Mat Sci. 20 (1997) 435-440.

Saharia Munima B., Banhatti Radha D, Bharali Usha, Kadolkar C & Murti Y V G S, J Pure and App Phys (this volume).