K. KUMAR and S. C. GOYAL: Lattice Theory of Fourth-Order Elastic Constants 97

phys. stat. sol. (b) 139, 97 (1987)

Subject classification: 62.20; S9.11

Department of Physics, Agra College1)

Lattice Theory of Fourth-Order Elastic Constants of Primitive Lattices

Application

BY K. KUMAR and S. C. GOYAL

The lattice theory of fourth-order elastic constants for primitive lattices recently derived by Goyal and Kumar is applied to obtain the relations for the FOE constants of CsCl structure type solids within the framework of the Lundqvist many-body potential. The derived relations of FOEC are used to compute the numerical values of the F O E constants and the second-order pressure derivatives of the SOE constants of cesium halide solids.

Die kiirzlich von Goyal und Kumar abgeleitete Gittertheorie der elastischen Konstanten vierter Ordnung (FOEC) fur primitive Gitter wird benutzt, um die Beziehungen fur die BOEC von Festkorpern mit CsC1-Struktur im Rahmen des Lundqvist-Vielkorperpotentials zu erhalten. Die abgeleiteten Beziehungen der FOEC werden benutzt, urn die numerischen Werte der FOEC und die Druckableitungen zweiter Ordnung der elastischen Konstanten zweiter Ordnung von Casiumhalogenidfestkorpern zu berechnen.

1. Introduction

The study of the higher-order elastic constants of solids plays an important role in analysing the various anharmonic properties of the solids. In this direction Garg et al. [l] and Mori and Hiki [2] have formulated theoretical expressions using the homo- geneous deformation theory to study the fourth-order elastic (FOE) constants of cubic structure solids.

Recently authors [3] have also reported a theoretical approach using a macroscopic theory of elasticity [4] and the lattice theory [5] of elasticity to study the FOE con- stants of the cubic structure solids. The present paper deals with the application of the lattice theory [3] to obtain the analytical expressions of t,he FOE constants of CsCl cubic structure solids. The derived relations of FOE constants are also used to evaluate the values of second-order pressure derivatives of the SOE constants of cesium halides. Section 2 deals with the basic steps of the derivations. The numerical application and conclusion are given in Section 3.

2. Theory

Following the notations used by Srinivasan [ 7 ] , the general expression of the FOE constants for the primitive lattice recently derived by the authors (31 is expressed as

h h 0

2 c j p , km, rs, Id = 2 [ C j k , [rs], [Id], p m C p k , [rs], [ I d ] , jm - c j p , [rsl, [ I d ] , krn - h ,-. 0 h

- C [ s d , p m ] d j r s k l - C [ d s , pmlaj l8kr + C[dE, Icm18jdpr - C [ s d , jm$prsk l -

- C[dE, jm]8pldkr + C [ s d , k?n$jr8plI - ,. ,.

- [ c j p , km, sd + C p m , t j , sd + C p j , km, sd - c’j, pm, Ed] &I - l) Agra 282002, India.

7 physica (b) 138/1

98 K. KUMAR and S. C. GOYAL

Lattice Theory of Fourth-Order Elastic Constants of Primitive Lattices 99

In order t o derive the relations for eleven FOE constants of the structure under study using (2) t o (5) we have to select an appropriate crystal potential. The Lund- qvist [8] potential for the crystal is expressed as

where ~ ( k ) is the valence of the k-type ion. The function f is related to the overlap integrals of the free-ion one-electron wave function and is assumed significant only for the nearest neighbours.

This potential (6) has been widely used to account the contribution of many-body forces in various fields in elasticity 19 to 111, in dielectric behaviour [12, 131, in lattice dynamics studies [14 to 161, and others. This potential (6) contains the three contribu- tion viz. Coulombian, repulsive, and three-body. When it is applied in (3) to (5) of C-coefficients we get the following relations for the various contributions.

(a) Second-order coupling coefficients:

Lattice Theory of Fourth-Order Elastic Constants of Primitive Lat'tices 101

* e2E(k) ~ ( k ' ) 'CZZ, [a], 12 = -1.7200 ,

v a a

* TC22, [21], 12 = 0 Y

h e2&(k) ~ ( k ' ) 'Czi, [211,22 = 0.3602 P

vaa

- 2.6633~(k)

( c ) Fourth-order coupling coefficients :

'41, [ill, [111,11 = 21.7979 h e2E(k) d k ' )

, vua

e2 12c, 1 2 4 %, [lll, [Ill, 11 = a Tfill, Ell], [ill, 11 =

= - e2 k.O892e(k) (a3 g) $- 29.6661e(k) (a2 s) a 2t - 69.2881&(k) 4a4

,. e2e (k ) ~ ( k ' ) cc11, [ I l l , [221,11 = - 1.0467 ,

vua

R&l, [111, (221, 11 =

TC1l, [lll, [221, 11 = h

= 4a4 (2.0892&(k) (a3 $) + 4.7028&(k) (az 2) - 34.8889e(k)

102 K. KUMAR and S. C. GOYAL

%, [ZZ], [ZZ], 11 =

TC1l, [22], [ZZ], 11 = A

= & (2.0892&(k) (a3 $) - 3.6182e(k) (az 2) + 21.6942&(k) (a z)},

= 4a4 e2 1.0892&(k) (a3 2) - ll.9394&(k) (aZ $) + 27.5874&(k) (a :)},

A

TC1l, [22],[23], 11 =

_ _ - t4 {1.0446&(k) (a3 g) + 1.8692&(k)

h

l-Cl1, [121, [12], 11 =

= $ k.O446e(k) (a3 g) + 5.4876e(k) (a2 3) - 15.8202~(k) a - , ( 31

A

TCl2, [lZ], [ I n ] , 12 =

Lattice Theory of Fourth-Order Elastic Constants of Primitive Lattices 103

R622,[12], [12], 33 =

A

TC12, [23], [321,12 =

4a4

A

‘C11, [23], [32], 22 = (9)

104 K. XUMAR and S. C. GOYAL

The lattice sums used in the above relations are taken from Garg and Verma [17]. Now the above-derived various order coupling coefficients (7) to (9) are used in ( 2 ) in order to obtain the final relations of the FOE constants of CsCl structure solids. These relations are as follows :

15 ar3 36.19028(8 + 16f(r)) + R, + B, + 2.0892~ (a3 9) +

-2.65058(~ + 16f(r)) + R, + 2.08928 (a3 2) -

e2 4a4

C,,,, = - [0.5134~(8 + 16f(r)) + R, 4- 2.08928

- 9.04598 - ( :::) + 31.09518 (a z)] -2.73208(~ + 16f(r)) + R, + 2.0892~ (a3 2) -

-2.74978(~ + 16f(r)) + R, + 1.04468 4a4

e2 c1166 = [ -1.23988(8 + 16f(r)) f Ri + 4& f 1.04468 (a3 $) -

-2-7497&(& +- 16f(r)) + R, + 1.04468

1.1741e(s + 16f(r)) -+ R, + 2Bl + 1.04468 (a3 $) -

Lattice Theory of Fourth-Order Elastic Constants of Priniitive Lattices 105

-2.7320&(& + 16f(r)) + Rl + 0.5223s

1.9956s(s 4- 16f(r)) + R, + - 9 B, - 16.64198 ( a2 - "2') + 2 ar2

+ 9.60858 (a :)I, e2 C,,,, = 4a4 [ -2.4105~(s + 16f(r)) + R, + B, - 5.5473s

+ 3.2027s (a g)], where

D, - 6C, + 15A, - 15B, 54

R,=-- ~ - ..

The repulsive parameters A,, B,, C,, and D, for CsCl structure are defined by

The derived relations of FOE constants hold the identity

c1123 + 2c145f3 = c1144 + 2c1244 * (12)

3. Numerical Calculations and Discussion

The values of model parameters A,, B,, f (r) , (a(aj/ar)), (a2(a2f/ar2)), and (a3(T3f/8r3)) in (10) are calculated with the help of equations of SOE constants, dkldp, a2C,,/i3p2 and the equilibrium condition of CsCl structure type solids. The parameters C, and D, are evaluated by using their definitions (11) and the Born and Mayer short-range interaction potential. The input data me the same as given in [21]. The results are given in Tables 1 and 2 .

It is clear from Table 1 that the values of FOE constants for all the three cesium halides, i.e. CsC1, CsBr, CsI are much larger in magnitude as compared t o the values of TOE or SOE constants of these halides showing that the expansion of potential function is converging slowly and the higher-order anisotropy is present there. It is also interesting to note that the values of all SOE constants of these halides are positive while TOE constants are negative [18], but again the values for all FOE constants of these solids are positive. Such a trend of variation in sign of these con- stants is not present in case of solids crystallizing in NaCl structure solids [19]. More- over, the calculated values of longitudinal constants C,,,, in case of all three cesium halides are larger than the calculated values of the other ten transverse constants.

106 K. KUMAR and S. C. GOYAL

Table 1

Calculated values of FOE constants (in 1011 dyn/cm2)

CsCl CsBr CSI

Cl l l l 160.0672 ‘11,z 56.5093 ‘1122 62.5484 ‘l,,, 52.2316 Cn44 56.1783 c1166 56.7649 c1244 54.7552 ‘1266 63.4692 Cl45, 56.7990 ‘444, 61.4050 ‘4466 55.8451

134.4529 46.0288 49.1952 39.2973 49.8036 50.3022 47.1025 54.1287 52.4129 53.5894 52.2189

11 1.0086 39.0899 41.2560 33.0548 42.5622 42.9864 40.1622 45.8013 44.9613 45.4754 44.8718

Table 2 Second pressure derivatives of SOE constants (in 10-11 cm2/dyn)

calcu- experi- calcu- experi- calcu- experi- lated mental lated mental lated mental

[18,201 [18,201 P8, 201

CsCl -6.8 - 6.8 If: 1.2 -1.0849 -1.8 If: 1.1 -1.7936 -2.5 0.7 CsBr -7.5 - 7.5 * 1.5 -2.2521 -2.4 + 1.7 -2.2242 -2.2 f 1.0 CSI -9.2 - 9.2 f 1.6 -2.8996 -3.0 & 1.7 -2.8214 -2.5 * 1.0

Due to the lack of experimental data on the FOE constants in case of these halides, we have checked the derived relations by calculating the second pressure derivatives of the SOE constants as the experimental data of these constants for the crystmale under study are available.

It is interesting to note from Table 2 that the calculated values of azC~J8p2 arc in good agreement with the corresponding experimental values and are much betAer than the previously calculated values [6, 21, 221. A comparison of the derived expres- sions of FOE constants with the expressions derived by using a homogeneous defoima- tion theory [ l ] suggests that the expressions of Cllll, Cll,,, ClIz2, and C,,,, derived by Garg et al. [ l ] are erroneous. The present study provides the strengthening to the derived lattice expressions of FOE constants [3] as it can be used to study both FOE constants and the higher-order pressure derivatives of SOE constants of cubic structure solids. Thus the present theory is more general than the earlier reported study of pressure derivatives of SOE constants [6, 211, therefore, it may be useful in the further study of elastic behaviour of cubic structure solids.

Acknowledgement

One of the authors (K.K) is grateful to Council of Scientific & IndustrialResearch, New Delhi for providing the financial assistance to the present work.

Lattice Theory of Fourth-Order Elastic Constants of Primitive Lattices 107

References

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[6] U. C. SEARMA and M. P. VERMA, phys. stat. sol. (b) 102, 487 (1980). [7] R. SRINIVASAN, Phys. Rev. 144, 620 (1966). [8] S. 0. LUNDQVIST, Ark. Fys. (Sweden) 9, 435 (1955). [9] D. S. PURI and M. P. VERMA, Phys. Rev. B 15, 2337 (1977).

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[lo] X. C. GOYAL and S. P. TRIPATHI, J. Phys. Chem. Solids 38, 351 (1977). [ l l ] S. C. GOYAL and R. PRAKASH, Solid State Commun. 24, 121 (1977). [I21 S. C. GOYAL, S. P. TRIPATHI, and R. PRAKASH, phys. stat. sol. (b) 85, 477 (1978). 1131 M. P. VERMA and L. D. AGARWAL, Phys. Rev. B 10, 1958 (1974). [14] M. P. VERMA and R. K. SINGH, phys. stat. sol. 33,769 (1969). [15] H. H. LAL and M. P. VERMA, 5. Phys. C 5, 543 (1972). [l6J R. P. GOYAL, K. K. SARKAR, and S. C. GOYAL, phys. stat. sol. (b) 75, 101 (1976). [17] V. K. GARG and M. P. VERMA, phys. stat. sol. (b) 83, 139 (1977). [l8] Z. P. CHANG and G. R.. BARSCH, phys. stat. sol. 23, 577 (1967). [19] V. K. GARG, D. S. PURI, and M. P. VERMA, phys. stat. sol. (b) 80, 63 (1977). [20] Z. P. CHANG, G. R. BARSCH, and D. L. MILLER, Phys. Rev. Letters 19, 1381 (1967). [21] V. K. GARG, D. S. PURI, and M. P. VERMA, phys. stat. sol. (b) 82, 481 (1977). [22] J. SHANKER, J. P. SINGH, and G. D. JAIN, phys. stat. sol. (b) 105, 385 (1981).

(Received October 4 , 1985, in revised form June 18 , 1986)