J. XHANKER et al. : Third-Order Elastic Constants for Mixed Crystals 75

phys. stat. sol. (b) 104, 7 5 (1981)

Subject classification: 12.2; 22.5.4

Department of Physics, Agra College1)

Evaluation of Third-Order Elast,ic Constants and Pressure Derivatives of Second-Order Elastic Constants for AgC1-AgBr Mixed Crystals BY J. SHANKER, G. D. JAIN~), and S. K. SIIARMA

Six third-order elastic constants and the first pressure derivatives of second-order elastic constants for AgC1-AgBr mixed crystals with varying compositions are evaluated. Suitable interionic poten- tials are developed for mixed crystals starting from an analysis of potentials inpure crystals. Long- range electrostatic forces, the overlap repulsive forces extended up to second neighbours, the van der Waals dipole-dipole and dipole-quadrupole interactions, and three-body forces are considered. Calculated values of the pressure derivatives of second-order elastic constants are found to present reasonable agreement with experimental data.

Es werden sechs elastische Konstanten dritter Ordnung und die ersten Ableitungen der elastischen Konstanten zweiter Ordnung nach dem Druck fur AgC1-AgBr-Mischkristalle unterschiedlicher Zusammensetzung berechnet. Geeignete interionare Potentiale werden fur die Mischkristalle au s einer Analyse der Potentiale in reinen Kristallen entwickelt. Langreichweitige elektrostatische Krlifte, bis zu zweiten Nachbarn ausgedehnte abstoBende gberlappungskrlifte, van der Waals- Dipol-Dipol- und Dipol-Qnadrupol-Wechselwirkungen und DreikorperkrLfte werden beruck- sichtigt. Die berechneten Werte der Druckableitungen der elastischen Konstanten zweiter Ord- nung befinden sich in vernunftiger Ubereinstimmung init experimentellen Werten.

1. Introduction Studies on mixed crystals are quite useful as they are of considerable technological importance in the field of storage cells. Recently Cain [l] has made a detailed ex- perimental investigation of the elastic properties of AgC1-AgBr mixed crystals. It has been found [a, 31 that these mixed crystals are formed as a result of replacement of a Br- ion by a C1- ion such that the crystal symmetry is maintained but the inter- ionic forces are changed. Recently Shrivastava [a] has made an attempt to evaluate the third-order elastic constants (TOEC) and pressure derivatives of second-order elastic constants (SOEC) for AgC1-AgBr niixed crystals by considering three-body forces and overlap repulsive interactions between nearest neighbours only. However, his model is too simple to account for real interactions operative in AgC1-AgBr mixed crystals. There are three main weaknesses in the model adopted by Shrivastava. (i) He has neither considered the repulsive interactions between second neighbours nor the cross interactions between C1- and Br- ions which are important particularly for mixed crystals. It is evident from recent calculations based on Lowdin’s LCAO method [5 to 81 that second neighbour interactions are quite important in ionic crystals including silver halides. (ii) The van der Waals dipole-dipole and dipole- quadrupole interactions which are significantly large for silver halides [9, 101 have been completely neglected by Shrivastava. These interactions become even more important when one is concerned with the evaluation of higher-order elastic constants

Agra 282002, India. 2, University Grants Commission Teacher Fellow.

76 J. SHANKER, G. D. JAIN, and S. K. SHARMA

or their pressure derivatives [Ill. (iii) Although the three-body interactions have been taken into account by Shrivastava the three-body potential parameters are actually determined by him using the pressure derivative of bulk modulus as input dat,a. Since the pressure derivative of bulk modulus is directly related to TOEC and pressure derivatives of SOEC [ 121, the agreement obtained by Shrivastava is rather forced and his met,hod does not reveal a critical examination of the int,erionic force model used.

I n the present paper, we consider the overlap repulsive interactions up to second neighbours including t,he cross interactions between C1- and Br- ions, as well as the van der Waals dipole-dipole and dipole-quadrupole interactions. We have also considered Lundqvist’s three-body potential [13] in order to account for the Cauchy breakdown. The three-body potential parameters are determined from overlap inte- grals [7 ] using Cochran’s formula [14, 151. The interionic potentials thus developed are used to calculate TOEC and pressure derivatives of SOEC for AgC1-AgBr mixed crystals.

2. Theory and Method of Calculation Relevant expressions for TOEC and pressure derivatives of SOEC can be obtained [la] from the potential energy which for an ionic crystal can be written as [13]

where ~ ( k ) is the valence of the k-type ion. The terms in the expression represent the Coulomb energy, the short-range energy, and three-body potential energy, respec- tively. The function f represents the three body-charge transfer parameter related to the overlap integrals of the free ion one-electron wave function. Expressions for TOEC and pressure derivatives of SOEC contain short-range interaction terms A,, Bl, C,, A,, B,, and C2 in addition to contributions arising from long-range Coulomb and three-body forces. We are not rewriting here the expressions for TOEC and pressure derivatives of SOEC as they have already been given in [12]. We only discuss how the short-range interaction terms are calculated for mixed crystals. The long- range Coulomb and three-body contributions can be calculated as in the case of pure crystals by taking into account the change in the lattice parameter caused due to the replacement of a Br- ion by C1- ion. The short-range interaction parameters are defined as

where V;(r) and V,(r) correspond t,o short-range nearest neighbour and next nearest neighbour potentials which for pure crystals can be written as follows [15] :

Evaluation of Third-Order Elastic Constants and Pressure Derivatives 77

and

where Bij are Pauling’s coefficients, T, and rz the ionic radii, cij and dij are the van der Waals dipole-dipole and dipole-quadrupole coefficients, respectively. b and e;j are the repulsive strength and hardness parameters, respectively. In (4) the three-body charge transfer parameter f has been expressed using Cochran’s formula [14, 151

f = f,, exp (2). @+-

The potentials expressed by equations (4) and (5) are more general than the potentials used by previous workers [ll, 12, 151 in that we have used different values of repulsive hardness parameters (@++, p--, and Q + - ) for different pair interactions.

For a system of mixed crystals, we can write

[ V;(r)IAgCI-AgBr = A1[v~(~) lA4gC1 + A~[V;(r)]Agl3r 2

[ v;(r)]AgCl-AgBr = Ai[ v&r)]AgCI + &[vk(r) lAgBr + &&[v;(r)]Cl-Br Y

( 7 )

(8 )

where h, and A, are the probabilities of two types of interactions. For pure AgBr, 2, = 0 and A, = 1 and for pure AgC1, A, = 1 and = 0. For a mixed crystal, say 19.5% AgCl and 81.5% AgBr,il, = 0.195 andil, = 0.815. The last term in (8) represents the cross interactions between C1- and Br- ions which can be further expressed as

Table 1 Values of repulsive hardness parameters eij (A), van der Waals coefficients cij (10-60 erg crnB), and dij (10-76 erg oms)

AgCl AgBr

0.344 0.188 0.472

251 255 252 137 117 157

0.366 0.180 0.514

319 249 419 173 113 259

*) Derived from overlap integrals 171. **) Calculated from the variational method [16, 171.

Values of the repulsive strength parameter b and the three-body force parameter f are determined from the equilibrium condition and SOEC following the method described earlier [15]. Values of the repulsive hardness parameters e+-, Q + + , e-- have

78 J. SHANKER, G. D. JAIN, and s. K. SHARMA

been taken to be those directly extracted from the overlap integrals [7]. The van der Waals coefficients cij and d, are taken from recent calculations based on the variatio- nal method [16, 171. Values of cij, dij, and pij for pure AgCl and AgBr crystals are given in Table 1. Values of input data on lattice parameters and SOEC for mixed AgC1-AgBr crystals have been taken from Cain [l]. It has been shown recently [l8] that the additivity rule holds good for the repulsive hardness parameters. In view of this rule we can estimate the repulsive hardness parameter for the interaction between C1- and Br- ions using the formula

( 10) Q.. - 1 (e . . + v - 2 LZ e d 3

where i and j represent C1- and Br- ions, respectively. The van der Waals coefficients cij and di,j for the interactions between C1- and Br- ions are calculated using the com- bination rules [8, 191

(11)

( 12)

c . . - (c..c..)l/z

dij = (d&jj)lI2 . 13 - L k 33 Y

Calculated values of parameters to be used in the evaluation of TOEC and pressure derivatives of SOEC are given in Table 2.

Table 2 Calculated values of potential parameters for AgCI-AgBr mixed crystals

0 -0.045 0.328 -2.37 5.09 -0.314 -32.2 2.51 -0.215 -19.3 19.5 -0.035 0.323 -3.01 4.61 -0.249 -29.0 4.42 -0.433 -36.2 39.1 -0.027 0.315 -3.63 4.14 -0.182 -25.8 5.93 -0.600 -42.4 56.6 -0.025 0.313 -3.99 3.69 -0.119 -23.1 6.86 -0.702 -56.0 78.7 -0.028 0.318 -3.67 3.13 -0.026 -19.9 7.36 -0.754 -63.6 100 -0.037 0.330 -2.93 2.57 -0.066 -17.0 7.09 -0.711 -63.3

Table 3 Calculated values of third-order elastic constants ( lo1' dyn ern-') for mixed AgC1-AgBr crystals

0 -8.38 -2.00 -0.316 0.567 0.264 0.113 19.5 -8.81 -2.50 -0.630 0.596 0.289 0.136 39.1 -9.31 -2.98 -0.815 0.624 0.315 0.161 56.6 -9.87 -3.41 -1.64 0.646 0.331 0.174 78.7 -8.84 -3.50 -1.17 0.668 0.337 0.171 100 -8.86 -3.10 -1.14 0.682 0.325 0.146

3. Results and Discussion Calculated values of TOEC and pressure derivatives of SOEC are listed in Tables 3 and 4, respectively, for pure AgCl and AgBr crystals and for four mixed crystals with varying compositions. We note from Table 3 that three TOEC (Clll, Cllz, and Glee) are negative whereas other three (Clzs, C144, and C,,,) are positive. This situation is similar to that observed for NaCl structure alkali halides [lS]. Calculated values

Tab

le 4

h rL 3

in A

gBr

s? E

men

tal

E s 2 0

Cal

cula

ted

valu

es o

f pr

essu

re d

eriv

ativ

es o

f se

cond

-ord

er e

last

ic c

onst

ants

for

AgC

1-A

gBr

mix

ed c

ryst

als,

exp

erim

enta

l va

lues

are

ta

ken

from

[l]

g

% A

gCl

dCll

idP

dK

/d P

d#

/dP

dC

4JdP

calc

ulat

ed

expe

ri-

calc

ulat

ed

expe

ri-

calc

ulat

ed

expe

ri-

calc

ulat

ed

ref. [4]

expe

ri-

men

tal

men

tal

men

tal

s.

0 5.

60

7.71

5.

26

6.82

2.

72

3.55

-0

.57

-1.1

4 -0

.29

19.5

6.

97

7.60

6.

20

6.76

2.

78

3.45

-0

.26

-1.1

5 -0

.31

39.1

7.

84

7.48

7.

13

6.74

3.

00

3.34

-0

.29

-1.1

6 -0

.37

7.35

6.

30

6.65

3.

21

3.27

-0

.10

-1.1

8 -0

.39

;

100

7.70

7.

15

6.86

6.

57

2.62

3.

28

-0.0

3 -1

.21

-0.5

07

s 56

.6

7.27

78

.7

8.42

7.

25

7.55

6.

63

2.69

3.

21

-0.0

3 -1

.20

-0.4

5

M

-l

W

80 J. SHANKER et al. : Third-Order Elastic Constants for Mixed Crystals

of TOEC show small but regular variation as the composition of the solid solution is changed. Although the experimental TOEC are not available for AgC1-AgBr mixed crystals, they are in fact directly related to the first pressure derivatives of SOEC [lS] which have been measured experimentally [ 11.

A comparison of calculated and experimental values of the pressure derivatives of SOEC is presented in Table 4. We have listed there the pressure derivatives of Cil, K , S, and C,,. These quantities are related to SOEC as follows:

where Cll, C12, and C,, are SOEC. The calculated values of pressure derivatives of ROEC agree reasonably well with the experimental values. The agreement is satis- factory particularly in view of the fact that we have not used any TOEC or pressure derivative of SOEC as input data. On the other hand, the agreement obtained by Shrivastava [4] is forced as he has used dKldP in determining the potential parameters. Moreover, Shrivastava has not compared his calculated values of dC,,ldP with ex- perimental data. Such a comparison has been presented in Table 4. Since dC,,/dP is a small negative quantity as compared to other pressure derivatives, its calculation depends niore sensitively on the potential model used. We note from Table 4 that our calculated values of dC,,ldY are closer to experimental values whereas those of Shrivastava show larger deviations from experimental values.

To summarise, the analysis of interionic potentials presented in this paper explains satisfactorily the elastic properties of mixed AgC1-AgBr crystals and can be used further in the study of mixed crystals.

Roforences [l] L. S. CAIN, J. Phys. Chem. Solids 38, 73 (1977). [2] T. BHIMASANKARAM, Ph. D. Thesis, Osmania University, 1974. [3] L. S. CAIN, J. Phys. Chem. Solids 37, 1178 (1976). [4] U. C. SHRIVASTAVA, Solid State Commun. 31, 667 (1979); Indian J. pure appl. Phys. 18, 227

[5] J. 1,. CALAIS, K. MAKILA, K. MANSIRKA, and J. VALLIN, Physica Scripta 3, 39 (1971). [6] B. CASTMAN, G. PETTERSON, J. VALLIN, and J. L. CALAIS, Ark. Fys. (Sweden) 3, 35 (1971). 171 M. R. HAYNS and J. L. CALAIS, 5. chem. Phys. 67, 147 (1972); J. Phys. C 6, 2625 (1973). [S] J. ANDZELM and L. PIELA, J. Phys. C 10, 2269 (1977); 11, 2695 (1978). [9] 5. E. MAYER, J. chem. Phys. 1, 327 (1933).

-101 P. S. BAKHSHI, S. C. GOYAL, and J. SHANRER, J. inorg. nuclear Chem. 39, 546 (1977). 111 R. C. HOLLINGER and G. R. BARSCH, J. Phys. Chem. Solids 37, 845 (1976).

-121 V. K. GARC, I>. S. PURI, and M. P. VERMA, phys. stat. sol. (b) 80. 63 (1977). 131 5. 0. LUNDQVIST, Ark. Fys. (Sweden) 9, 435 (1955).

~141 W. COCHRAN, CRC Crit. Rev. Solid State Sci. 2, 1 (1971). [15] R. P. SINGH and J. SHANKER, phys. stat. sol. (b) 93, 373 (1979). [IS] 5. SHANKER, G. G. ACRAWAL, and It. P. SINGIH, J. chem. Phys. 6'3, 670 (1978). 1171 V. C. JAIN and J. SHANKER, Pramiina 13, 31 (1979). 181 J. SHANKER, D. P. AGIRAWAL, and R. P. SINGH, Solid State Commun. 31, 765 (1979). ~191 5. 0. HIRSCHFELDER, C. F. CURTISS, and R. B. BIRD, Molecular Theory of Gases and Liquids,

(Received August 13, 1980)

(1980).

Wiley, New York 1954.