Physica 106B (1981) 247-250 North-Holland Publishing Company

ANALYSIS OF INTERIONIC POTENTIALS IN ALKALINE EARTH FLUORIDE CRYSTALS

Jai SHANKER, J. P. SINGH and V. C. JAIN*

Physics Department, Agra College, Agra-282002, India

Received 17 December 1979 Received in revised form 9 October 1980

An analysis of interionic potentials in CaF2, SrF2 and BaFz crystals is performed by modifying the traditional Born model treatment. The short-range overlap repulsive parameters are related to overlap integrals. The Van der Waals dipole- dipole and dipole-quadrupole coefficients recently determined from the variational method are used. The cohesive energy, bulk modulus and its pressure derivative are calculated and found to be in good agreement with experimental data. A com- parison of short-range force constants with those available in the literature is presented.

1. Introduction

There have been several attempts [l-5] based on the Born model to study the interionic potentials in alkaline earth fluorides solidifying with fluorite crystal structure. In the Born model the short range repulsive parameters are determined empirically from the data on interionic separation and compressibility. In the present work we relate these parameters to overlap integrals. It has been evident from recent studies on ionic crystals [6-lo] that the Van der Waals poten- tials contribute significantly to the binding energies and play an important role in determining the dielectric and elastic properties of these crystals. The recently calculated values of Van der Waals coefficients [8] are substantially larger than the old values reported by Mayer [ 111 which have been generally used by previous investigators [ 121. In the present analysis we use new Van der Waals potentials derived from the variational method [8, 131. The cohesive energy, bulk modulus and its pressure derivative for CaF,, SrF2 and BaF, crystals are calculated and compared with experi- mental data. A comparison of short-range force con- stants with those obtained by other workers is also presented.

* Present address: P.D. College, Shikohabad (U.P.), India.

2. Theory and method of calculation

The crystals of CaF,, SrF, and BaF, are highly ionic solids [ 141 composed of closed shell ions. The crystal energy W for such solids can be written as follows

w=w,+w,+w,+w,, (1)

where We, the long-range electrostatic energy arising from the interaction of point-like charges, is

Z2,2 We=*M-----. (2)

r

Here czM is the Madelung constant, e is the electron charge, Z is the highest common factor of the valencies of ions. r is the nearest neighbour separation. WV is the Van der Waals energy which can be expressed as the sum of dipole-dipole and dipole-quadrupole inter- action energies. Thus

where C and D are linear combinations of the Van der Waals coefficients for interactions of the various ion pairs weighted by appropriate lattice sums. The con- stants C and D can be written as

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248 J. Shanker et al. /Interionic potentials in alkaline earth fluoride crystals

c= c+_s+_ + c--s__ + c+,s+,, (4) + fd++b exp Or+ - k2rYP++

D=D,_T+_ -+-D--T__ +D+,T+,, (5) + 1 X ._ b exp W - k&/p_ _ ,

where S and Tare appropriate lattice sums which for

CaF, type crystals have been reported by Tosi [ 121. Values of dipole-dipole coefficients Cij and dipole- quadrupole coefficients Dii have been determined

using the formulation based on variational theory [8, 131. These values are found to be significantly larger in magnitudes than those obtained from the formula- tion [ 1 l] based on perturbation theory [ 1.5, 161.

where Bij are the coefficients introduced by Pauling [ 171 in order to provide appropriate weightage for various pair interactions. b and pii are the strength and hardness parameters. I+ and r_ are the radii of cation and anion, respectively; k, = 1.1547 and k, = 1.6330. Values of pij are derived using the correlation between repulsive energy and overlap integrals for a pair of neighbouring ions [ 18, 191, viz.

The zero point energy can be estimated from the relation [ 21

w, =PN(n+ + n_)hv,,,

where N is the Avogadro number, n, and n_ are the number of cations and anions in one molecule of the crystal. h is Plan&s constant and v_ is the Debye limiting frequency.

The repulsive energy IV,, by taking into account the interactions between nearest and next nearest cation- anion, cation-cation and anion-anion, can be written as follows [2]

IV, = 80+-b exp (r+ + r_ - rYp+_

+ 68--b exp (2r_ - klr)l~__

(7)

where y is a dimensionless proportionality constant introduced by Dick and Overhauser [20] and Sij is the overlap integral. Values of Sij for various ion pairs in CaF,, SrF, and BaF, crystals have been reported by Ra [21]. Values of pij based on these overlap integrals are given in table I. The repulsive strength parameter b can be calculated from the crystal equilibrium condi- tion. The data on ionic radii, interionic separation and Van der Waals constants used in the present calculation are also included in table I.

The bulk modulus K and its pressure derivative can

be calculated from the relations

(9)

Table I

Values of input data used in calculation. The Madelung constant a~ = 5.038785

Ref. CaFZ* SrFz BaFz

r (A) [41 2.357 (2.352) 2.503 (2.495) 2.674 (2.668) r+ (A) [251 1.14 (0.99) 1.30 (1.13) 1.50 (1.35) r- (A) [261 1.16 (1.36 ) 1.16 (1.36) 1.16 (1.36)

C(lO+O erg cm6) I81 509 (208) 724 (296) 1196 (456) 0(10-76 erg cmg) 181 194 (228) 285 (407) 490 (802) P+- (‘Q 1211 0.274 (0.303) 0.290 (0.331) 0.310 (0.335) P++ Gv 1211 0.166 (0.303) 0.199 (0.331) 0.226 (0.335) P-- (‘9 [211 0.357 (0.303) 0.364 (0.331) 0.367 (0.335) b( lo-l2 erg) 0.912 (0.518) 0.871 (0.601) 0.811 (0.552)

calculated

* Values within parentheses are those used by Benson and Dempsey (ref. 2).

.L Shanker et al /Interionic potentials in alkaline earth fluoride crystals 249

and Table III An account of various contributions (in kcal/moI) to the lattice energy of alkaline earth fluorides

(10)

which are derived from the equation of state [ 121.

3. Results and discussion

Values of cohesive energy, bulk modulus and its pressure derivative calculated for CaF,, SrF, and BaF2 crystals from eqs. (l), (9) and (10) are reported in table II. For the sake of comparison the experimental values have also been included in table II. It is found that the calculated values agree fairly well with experi- mental data. It should, of course, be remembered that the calculation of K and dK/dP involves the higher order (second and third) derivatives of the potential energy. Therefore the calculation of these quantities provides a sensible test of theoretical models for inter- ionic potentials. The Van der Waals potentials calcu- lated from the variational method and used in the

present study are substantially larger than those

derived from the formulae [2] based on perturbation theory [ 111. Various contributions to the cohesive energy are listed in table III and compared with those obtained in the previous work [2]. The larger values of the Van der Waals energies are consistent with recent phenomenological [7,9] as well as quantum

mechanical [lo] studies of ionic crystals. The cohesive energies calculated from the new Van der Waals poten- tials agree with experimental values within 3%. The results of Benson and Dempsey [2], who have calcu-

we (a) (b)

Wr (a) (b)

I% (a) (b)

WZ

CaF, SrFz BaFz

-710.2 -668.8 -626.0 -711.5 -670.8 -627.4

120.8 112.5 109.0 102.4 102.3 94.3

-45.6 -45.0 -49.8 -21.2 -21.6 -22.7

3.2 2.4 1.7

(a): Calculated in the present study. (b): Calculated by Benson and Dempsey (ref. 2).

lated the cohesive energies only, are also included in table II. Values obtained by Benson and Dempsey are in slightly better agreement with experiment. However, if we reduce the ionic charge by about 1.5% our calculations reproduce exactly the observed cohesive energies. Such a reduction in the value of ionic charge is quite consistent with the lattice dynamical study of CaF, crystal [ 221.

An essential feature of the present calculations is that’the repulsive hardness parameters for interactions between cation-anion, cation-cation and anion- anion are unequal. This is in contrast with the assump-

tion P+- = p++ = p_ _ usually made in the Born model. Such an assumption is not justified in view of the fact that anions are much softer or more compress- ible than the cations [23, 241.

A comprehensive study of the lattice dynamics of CaF, crystal has been made by Elcombe and Pryor [22] using next nearest neighbour repulsive force

Table II Values of cohesive energy (in kcal/mol), bulk modulus (10’ 2 dyn/cm*) an d the pressure derivatives of bulk modulus

Crystal Cohesive energy (kcal/mol)

Calculated Experi-

Bulk modulus K(lOI* dyn/cm*)

Calcula- Experi-

dN@

calcula- Experi-

present study

Ref. mental ted mental ted mental

2 (Ref. 3) (Ref. 27) (Ref. 27)

CaF2 -632 -627 -617 0.954 0.814 4.41 4.96 SrFz -599 -588 -584 0.771 0.693 4.46 5.00 BaF2 -565 -554 -549 0.611 0.567 4.52 5.07

250 J. Shanker et al./Interionic potentials in alkaline earth fluoride crystals

models. They have used three models. In model I the

ions are considered to be rigid. Models II and III take into account the polarizability of ions and are based on the shell model theory. In model II ionic charge is allowed to vary, whereas in model III it is fixed at a nor.mal value. Elcombe and Pryor [22] have defined eight short range force constants A I, B, , A 2, B,, A 3, B3, A4 and B, related to nearest neighbour and next nearest neighbour interactions. We have calculated these force constants using our short-range potential parameters obtained in the present work. The short-

range force constants thus calculated are compared in table IV with those obtained from the three models of Elcombe and Pryor. It is evident from table IV that the short-range force constants arising from nearest and next nearest neighbour interactions (A 1, B, , A 2 and B2) are dominant whereas those related to third neigh- bour interactions (4 3, B3, A4 and B4) are quite small.

Table IV Calculated values of short-range force constants for CaF, crystal defined in ref. 22

Parameters Present Elcombe and Pryor [ 221 study

model I model II model III

Al 18.6 7.6 14.6 15.1

Br -2.06 0.33 -1.56 1.70

-42 2.66 2.97 1.39 1.30

B2 -0.346 -0.22 0.06 0.09

-43 -0.119 1.47 0.35 0.23

B3 0.018 -0.62 -0.26 -0.28

A4 -0.099 0 -0.16 -0.18

B4 -0.006 0 0.058 0.062

Acknowledgement

We are grateful to the referee for his valuable comments and suggestions.

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