R. P. SINGH and J. SHANKER: Evaluation of Third-Order Elastic Constants 373

phys. stat. sol. (b) 93, 373 (1979)

Subject classification: 12.2; 22.5.2; 22.6

Department of Physics, Agra College')

Evaluation of Third-Order Elastic Constants and Pressure Dependence of Second-Order Elastic Constants of Ionic Crystals BY R. P. SINGH~) and J. SHANKER

The third order elastic constants and the pressure dependence of second-order elastic constants are evaluated for alkali halides with NaCl and CsCl structures, and for alkaline earth oxides with NaCl structure on the basis of the Lowdin-Lundqvist three-body potential t,heory for ionic solids. The potential parameters are derived from the knowledge of the overlap int,egrals. An exponential form suggested by Cochran is used for the variation of the three-body overlap parameter. It is considered that the second-neighbour repulsive as well as the van der Waals interactions contri- bute to the short-range energy. The analysis presented uses only two parameters and determines six third-order elastic constants, three pressure derivatives of second-order elastic constants, and the cohesive energy of 22 crystals. The results obtained are in good agreement with experimental data.

Auf der Grundlage der Drei-Korper-Potentialtheorie fur Ionenfestkorper nach Lowdin-Lundqvist werden die elastischen Konstanten dritter Ordnung und die Druckabhangigkeit der elastischen Konstanten zweiter Ordnung fur Alkalihalogenide mit NaCI- und CsCI-Struktur und fur Erdalkali- oxide mit NaCI-Struktur berechnet. Die Potentialparameter werden aus der Kenntnis der tfber- lappungsintegrale abgeleitet. Ein von Cochran vergeschlagener Exponentialausdruck wird fur die Variation der Drei-Korper-Uberlappungsparameter benutzt. Dabei wird berucksichtigt, daB sowohl die abstol3enden Krafte zweiter Nachbarn als auch die van-der-Waalsschen anziehenden Krafte zur kurzreichweitigen Energie beitragen. Die vorgelegte Analyse benutzt nur zwei Para- meter und bestimmt sechs elastische Konstanten dritter Ordnung, drei Druckableitungen der elastischen Konstanten zweiter Ordnung und die Kohasionsenergie von 22 Kristallen. Die Ergeb- nisse befinden sich in guter ubereinstimmung mit experimentellen Werten.

1. Introduction The consideration of the many-body potential in the theory of cohesion of ionic solids has been very useful in explaining the elastic properties of these solids. Recently Puri and Verma [l] have derived expressions for the third-order elastic (TOE) con- stants and the pressure dependence of second-order elast'ic (SOE) constants of NaCl structure solids using the Lowdin-Lundqvist, three-body potent'ial [ 2 , 31 and following the well-known method of homogeneous deformation [4]. The work of Puri and Verma has been extended in two direct'ions viz. for CsCl structure solids by Puri et al. [5 ] and by including the effect of next-nearest-neighbour short-range interactions by Tripathi and Goyal [6] and also by Garg et al. [ 7 ] . Tripathi and Goyal considered the second-neighbour interactions as arising solely from the van der Waals int'eractions estimated by Mayer [8 ] . Their method of calculation is not applicable and does not yield good results even for all the alkali halides. This may be attributed to the fact that Tripathi and Goyal did not consider the second-neighbour repulsive interactions. Also the van der Waals potentials used by them are somewhat underestimated

1) Agra-282002, India. 2, Permanent address: Department of Physics, Narain College, Shikohabad, India.

374 R. P. SINGH and J. SHANKER

[9 to 111. On the other hand, Garg et al. [7] considered the second-neighbour repulsive interactions and did not take into account the van der Waals potentials. Since the pressure dependence of elastic constants is much more sensitive to the short-range interactions, therefore one should take into account the second-neighbour repulsive as well as van der Waals interactions. This is why the values of the repulsive hardness parameter obtained by Garg et al. are substantially lower than the experimental values [12]. The previous workers [l, 5 to 71 have used the data in calculations cor- responding to room temperature. This is not justified because their expressions are valid only a t 0 K and particularly because the thermal effects contribute significantly to the breakdown of the Cauchy relations. The phenomenological theory of Tripathi and Goyal [6] and Garg et al. [7] contains too many parameters to be derived from the second-order elastic constants and the crystal equilibrium condition. These in- vestigators therefore use one of the TOE or the pressure dependence of SOE constants as input data to estimate the potential parameters. I n view of this fact the agreement between theory and experiment for TOE constants and the pressure dependence of SOE constants obtained earlier [l, 5 to 71 is rather forced. Moreover the choice of input data is different in different papers. Thus Puri and Verma use ClI2, Tripathi and Goyal use the pressure dependence of bulk modulus dK’ldp, whereas Garg et al. have taken the pressure dependence of shear modulus dS’/dp as input data. It seems therefore that no single choice is capable of yielding good results in all the cases.

I n the present paper we perform an analysis of the TOE! constants and pressure derivatives of SOE constants on the basis of the Lowdin-Lundqvist three-body poten- tial. The Lundqvist overlap parameter and its variation with interionic distance are evaluated from the overlap integrals using an exponential law suggested hy Cochran [13]. This method uses only two parameters and provides an adequate explanation for the two-body and three-body short-range interactions separately. On the other hand, Puri and Verma [l] have absorbed the three-body potential in the two-body potential which is not justified. We have considered the second-neighbour repulsive as well as the van der Waals interactions. For alkali halides we have used low-temperature data in calculations. Thus in the present study we rectify the various shortcomings and inconsistencies of the calculations performed previously [5 to 71. Theory and method of calculations are given in Section 2. Results are discussed and compared with experimental data in Section 3.

2. Theory and Method of Calculation The potential energy of an ionic crystal including the effect of three-body potential [3] can be expressed as

M and M’ are the numbers related to the nearest and next-nearest neighbours. Con- ventionally we take M = M‘ = 12 for the NaCl structure, and M = 16, M‘ = 6 for the CsCl structure. The first term on the right hand of (1) represents the long-range electrostatic interaction, is the Madelung constant, E the valency, and f the three- body overlap parameter which is a function of r. Vl(r) and V,(r) are the short-range nearest-neighbour and next-nearest-neighbour potentials related to Lundqvist’s potential as follows [l] :

Vi(r ) = V2(r) (3)

Evaluation of Third-Order Elastic Constants of Ionic Crystals 375

T a b l e 1 Values of input data

crystals

LiF LiCl LiBr LiI

NaF NaCl NaBr NaI

K F KCl KBr KI

RbF RbCl RbBr RbI

MgO CaO SrO

CSCl CsBr CSI

U

1.996 2.539 2.713 2.951

2.295 2.789 2.954 3.194

2.648 3.116 3.262 3.489

2.789 3.259 3.410 3.638

2.107 2.406 2.580

2.055 2.145 2.280

? _ _ ~

0.256 0.302 0.321 0.343

0.264 0.306 0.323 0.346

0.294 0.329 0.340 0.360

0.306 0.341 0.353 0.371

0.333 0.333 0.333

0.359 0.376 0.401

0.924 0.924 0.924 0.924

1.181 1.181 1.181 1.181

1.503 1.503 1.503 1.503

1.645 1.645 1.645 1.645

0.950 1.250 1.390

1.903 1.903 1.903

1.133 1.648 1.809 2.048

1.133 1.648 1.809 2.048

1.133 1.648 1.809 2.048

1.133 1.648 1.809 2.048

1.150 1.150 1.150

1.648 1.809 2.048

4.24 2.27 1.590 1.320

2.290 1.123 0.986 0.798

1.35 0.54 0.56 0.22

1.255 0.649 0.474 0.36

9.248 5.900 4.496

0.756 0.702 0.589

6.49 2.692 2.052 1.350

2.899 1.331 1.070 0.781

1.336 0.663 0.52 0.368

0.952 0.493 0.4085 0.2920

15.581 8.100 5.600

0.817 0.760 0.644

Lattice parameter a from [5, 7, 101, repulsive hardness parameter Q from and r2 from [7] (all in A), elastic constants C12 and C,, (in loll dyn/cm2) from [

151, ionic radii r, 3 7, 101.

It should be emphasized that V,(r) and V,(r) are the two-body short-range interac- tions and can be expressed in terms of the Born-Mayer exponential law and the van der Waals potentials. Earlier workers [l, 5 to 71 for the sake of simplicity have re- presented V;( r ) by the Born-Mayer exponential law. This implies that f exponentially depends on r which is in contradiction with the values of the derivatives of f which they have obtained phenomenologically. We remove this inconsistency by following the suggestion of Cochran that

where r is the nearest-neighbour separation, fo a constant, and e the overlap repulsive hardness parameter which we will take from the overlap integrals.

Using the Born-Mayer exponential law, the van der Waals potentials, and (4), we can write

376 R. P. SINGR and J. SHANKER

where Bij are Pauling’s coefficients, rl and r, the ionic radii, cij and dij are the dipole- dipole and the dipole-quadrupole coefficients. It is evident from ( 5 ) and (6) that V;(r) and Vh(r) contain only two unknown parameters fo and b. We do not treat Q as a parameter, instead its values for alkali halides are derived from overlap integrals [14, 151. For alkaline earth oxides the overlap integrals are not available, we take e = 0.333 A following Tosi [16]. Garg et al. [ 7 ] treated @ as a parameter and deter- mined its values from the method of successive approximation. Values of @ so obtained by them are substantially lower than the usually accepted values [16]. This is probable because they did not consider the contribution arising from the van der Waals inter- actions.

We calculate the parameter b froin the equilibriuix condition

Bl + B, = - 1 . 1 6 5 ~ ( ~ + 12f) for NaCl structure,

B, + B, = -0.3392&(~ + l6f) for CsCl structure,

(7 a)

(7 bl

Tab le 2 Values of the potential parameters

-

crys- A, tals

LiF 10.16 LiCl 12.01 LiBr 11.86 LiI 13.28

NaF 11.02 NaCl 13.55 NaBr 14.43 NaI 14.81

KF 10.98 KC1 15.15 KBr 16.21 KI 17.58

RbF 11.26 RbCl 15.28 RbBr 16.47 RbI 17.97

MgO 27.09 CaO 30.10 SrO 32.59

CsCl 3.85 CsBr 4.50 CsI 4.92

-1.30 - 79.15 -1.41 -100.6 -1.40 -100.9 -1.53 -114.7

-1.20 - 97.83 -1.38 -127.1 -1.46 -134.5 -1.50 -140.6

-1.08 -104.2 -1.36 -152.8 -1.42 -166.6 -1.53 -182.3

-1.06 -108.9 -1.32 -157.7 -1.39 -172.9 -1.50 -191.8

-4.29 -168.0 -4.10 -207.5 -4.00 -249.8

-0.27 - 43.7 -0.31 - 51.1 -0.36 - 54.7

0.83 1.15 2.70 2.36

1.01 0.23 0.03 0.92

2.21 -0.24 -0.95 -0.14

2.17 -0.85 -0.96 - 1.45

10.22 10.04 11.09

1.51 1.13 0.71

0.016 0.188 0.155 0.361

-0.018 0.184 0.276 0.339

-0.087 0.164 0.269 0.301

-0.051 0.204 0.250 0.381

-0.880 -0.827 -0.818

-0.079 -0.040

0.011

- 14.5 0.247 - 32.6 0.173 - 57.9 0.183 - 66.9 0.130

- 17.0 0.280 - 18.2 0.219 - 21.5 0.203 - 43.4 0.171

- 34.5 0.227 - 8.31 0.232 - 3.23 0.187 - 22.0 0.229

- 36.8 0.166 0.130 0.166

- 1.46 0.183 - 1.12 0.169

- 82.84 1.734 -111 1.313 - 133 1.162

- 20.5 0.173 - 16.5 0.195 - 12.6 0.192

20.68 17.47 25.76 2.73

21.46 23.62 12.18

- 3.06

- 0.82 29.85

67.96 -13.21

-33.61 -48.10 - 26.65 -38.85

10.79 13.79 14.42

21.03 23.47 27.25

377 Evaluation of Third-Order Elastic Constants of Ionic Crystals

T a b l e 3 Calculated values of TOE constants dynjcmz)

-

crystals -__

LiF LiCl LiBr LiI

NaF

NaCl

NaBr NaI

K F KCI

KBr KI

RbF RbCl RbBr RbI

MgO CaO SrO

CsCl CsBr CSI

-21.57 - 13.87 -11.04 - 10.27

-18.72 ( - 14.8) - 12.30 (-8.8) - 10.83 - 8.90

-12.90 - 9.61

(- 7.01) - 9.20 - 7.49

-11.90 - 8.92 - 8.09 - 7.13

- 19.76 -25.16 -27.86

- 6.42 - 4.83 - 3.29

-1.38 - 1.06

-1.04

-1.18 (-2.7) -0.556

-0.513

-1.13

(-0.571)

-0.590

-1.181 -0.275

(-0.244) -0.269 -0.197

-1.155 -0.321 -0.235 -0.185

-7.236

-4.709 -5.358

-0.734 -0.756 -0.646

-2.84 -1.34 - 1.45 - 1.06

-1.61 (-1.14) -0.706

(-0.611) -0.573 -0.579

-1.17 -0.366

(-0.245) -0.238 -0.307

-0.940 -0.207 -0.187 -0.135

- 10.9 -6.73 -5.43

--0.756 -0.777 -0.666

0.685 0.319 0.226 0.202

0.480

0.230 (0.284) 0.194 0.153

0.321 0.149 (0.133) 0.144 0.085

0.302 0.161 0.125 0.100

2.39 1.58 1.27

(2.8)

-0.551 -0.599 -0.520

and fo from the Cauchy deviation

0.954 0.370 0.281 0.206

0.553 (0.46) 0.255 (0.258) 0.204 0.151

0.319 0.164

(0.127) 0.139 0.102

0.266 0.143 0.118 0.092

3.15 1.84 1.40

-0.592

-0.557 -0.638

1.09 0.395 0.309 0.208

0.589

0.267 (0.271) 0.209 0.150

0.318 0.171

0.137 0.111

0.247 0.133 0.114 0.088

3.53 1.98 1.46

-0.613 -0.658 -0.575

(0.118)

-

Experimental values given within parentheses are those cited in [7].

e2 C,, - C,, = 9.3204&(af’) ~ 4a4

for NaCl structure,

e2 4a4 C,, - C,, = 3.1336~(af’) __for CsCl structure.

The general expression for the TOE constants and the pressure dependence of SOE constants have been reported in [7] and therefore are not repeated in this paper. In Table 1 we have listed the input data used in calculations. The various derivatives of the short-range potential represented by A,, B,, C,, 4, B,, and C, defined in [7] are calculat,ed from the potential parameters using (5) and (6) . These are given in Table 2

378 R. P. SINCH and J. SHANKER

Tab le 4 Values of the pressure derivatives of SOE constants

crystals dC,&dp dS’/dp _ . .~

calculated

LiF 0.714 LiCl 0.486 LiBr 1.08 LiI 0.775

NaF 0.249 NaCl -0.003 NaBr -0.057 NaI 0.232

KF 0.405 KC1 -0.332 KBr -0.579 K I -0.148

RbF 0.239 RbCl -0.677 R.bBr -0.637 RbI -0.703

1.68 1.23 CaO

Sr 0 1.03

CsCl 1.79 CsBr 2.24 CSI 2.55

MgO

experimental

1.38 1.70 1.80 1.96

0.205 0.37 0.46 0.61

-0.43 -0.39 -0.33 -0.24

-0.70 -0.56 -0.55 -0.51

1.12 0.6

-0.2

3.56 3.68 3.72

calculated

3.41 3.92 3.64 3.94

3.72 4.80 4.85 4.72

3.94 5.34 5.59 5.63

4.04 5.44 5.64 5.92

0.841 1.85 2.42

2.58 1.89 1.41

dK’ldp ~ _ _ ~~

experimental calculated

3.60 3.98 3.70 4.79 3.75 5.05 4.00 5.37

4.79 4.21 4.79 4.95 4.83 5.06 4.80 5.41

5.25 5.04 5.61 5.02 5.68 5.22 6.03 5.26

4.93 5.24 5.88 5.24 6.03 5.26 6.26 5.42

3.67 3.25 3.40 3.96 4.00 4.35

0.89 5.40 0.68 5.38 0.84 5.35

experimental

5.30 5.63 5.68 6.1:

5.18 5.27 5.29 5.40

5.26 5.34 5.38 5.47

5.69 5.62 5.59 5.60

4.29 6.0 & 1.3 6.0 0.7

5.89 5.71 5.79

Experimental values are those cited in [5, 71.

along with the potential parameters. Calculated values of TOE constants and the pressure derivatives of SOE constants are given in Tables 3 and 4.

3. Results and Discussion

Experimental values of TOE constants, available only in few crystals are given within parentheses in Table 3. Experimental data on the pressure derivatives of SOE constants have been reported for all the solids under study and are included in Table 4 for the sake of comparison. Our calculated values of TOE constants and the pressure derivatives of SOE constants are a t least in as good agreement with experiment as those obtained by Garg et al. [7] and certainly better than those obtained by Ghate [17] and Narnyan [18]. It should be kept in mind that Garg et al. used a model with five parameters, viz. 71, Q, f , df ldr , and d2f/dr2. The last parameter was derived by them so as to satisfy the experimental values of dX’/dp = f d(C,, - C,,)/dp. In view of this fact the agreement obtained by Garg et al. with the help of five parameters is rather forced. On the other hand, we have removed much of the parametric nature of the theory by using (4) and values of p based on the overlap integrals. The agreement between theory and experiment obtained in the present study is remarkable partic-

Evaluation of Third-Order Elastic Constants of Ionic Crystals 379

Tab le 5 Cohesive energies (in kcal/mol)

crystals

____

LiF LiCl LiBr LiI

NaF NaCl NaBr NaI

K F KCI KBr K I

RbF RbCl RbBr RbI

MgO CaO SrO

CSCl CsBr CSI

-

calculated experimental

(c)

226.4 255.5 116.9 210.9 94.7 200.9 61.4 190.8

194.3 228.2 135.7 195.4 115.4 186.7 91.5 177.1

134.5 206.0 120.6 178.9 108.4 173.8 92.5 164.5

111.7 198.6 105.9 173.6 99.1 166.8 86.0 160.4

689.3 925.9 616.1 847.9 531.5 806.5

97.8 164.6 96.2 160.4 88.9 150.8

246.7 203.2 194.2 180.3

219.5 187.1 178.5 167.0

194.3 170.2 163.2 153.6

185.8 163.6 157.2 148.5

932 839 796

159.8 154.1 146.1

(a) Calculated using the potential parameters of Garg e t al. [7] and Puri et al. [5]. (b) Calculated in the present study using our own potential parameters. (c) Experimental values of cohesive energy for alkali halides from [19] and for alkaline earth oxides from [20].

ularly as it has been achieved with the help of only two parameters viz. b and fo. Moreover the inconsistencies in the treatment of Garg et al., mentioned in the Intro- duction, are now rectified.

Finally it should be stressed that the inclusion of the effect of the three-body poten- tial modifies the binding energy of the solid (see (1)). We have calculated the cohesive energies of the crystals under study using the potential parameters reported by Garg et al. [7] and Puri et al. [5 ] as well as those obtained in the present study. These are compared in Table 5 with experimental values based on thermodynamic data [19, 201. We note from there that the cohesive energies calculated from the potential param- eters reported by Garg et al. and Puri et al. are much lower than the experimental values. For example in case of LiI the calculated cohesive energy is only one third of the experimental value. I n contrast our calculated values of cohesive energies are in close agreement with experiment for all 22 crystals.

380 R. P. SINGH and J. SHANKER: Evaluation of Third-Order Elastic Constants

References [l] D. S. PURI and M. P. VERRIA, Solid State Commun. 18, 1295 (1976); Phys. Rev. B 15, 2337

[2] P. 0. LOWDIN, Ark. Mat. Astr. Fys. (Sweden) 36h, 30 (1947). [3] S. 0. LUNDQVIST, Ark. Fys. (Sweden) 9, 435 (1955). [4] D. C. WALLACE, Thermodynamics of Crystals, John Wiley & Sons, New York 1972. [5] D. S. PURI, V. K. GARG, and M. P. VERMA, phys. stat. sol. (b) i8, 113 (1976). [6] S. P. TRIPATHI and S. C. GOYAL, J. Phys. Chem. Solids 38, 1367 (1977). [7] V. K. GARG, D. S. PURI, and M. P. VERMA, phys. stat. sol. (b) 80, 63 (1977). [8] J. E. MAYER, J. chem. Phys. 1, 270 (1933). [9] J. K. JAIN, J. SHANKER, and D. P. KHANDELWAL, Phys. Rev. B 13, 5613 (1976).

[lo] C. R. A. CATLOW, K. M. DILLER, and M. J. NORGETT, J. Phys. C 10, 1395 (1977). 1111 J. SHANKER, G. G. AGRAWAL, and R. P. SINCH, J. chem. Phys. 69, 670 (1978). [I21 C. S. SMITH and L. S. CAIN, J. Phys. Chem. Solids 36, 205 (1975). [13] W. COCHRAN, CRC Crit. Rev. Solid State Sci. 2, 1 (1971). [I41 D. W. HAFEMEISTER and W. H. FLYGARE, J. chem. Phys. 43, 795 (1965). [15] D. W. HABEMEISTER and J. D. ZAHRT, J . chem. Phys. 47, 1428 (1967). [l6] M. P. TOM, Solid State Phys. 16, 1 (1964). [17] P. B. GHATE, Phys. Rev. 139, A1666 (1965). [18] A. A. NARNYAN, Soviet Phys. - Solid State 5, 129 (1963). [I91 M. F. C. LADD, J. chem. Phys. 60, 1954 (1974). [20] K. P. THAHUR, J. inorg. nuclear Chem. 36, 2171 (1974).

(1977).

(Received November 27, 1978)