Physica 92B (1977) 249-252 © North-Holland Publishing Company

THEORY OF THE STRAIN DERIVATIVES OF THE STATIC DIELECTRIC CONSTANT AND POLARIZABILITY OF ALKALI HALIDES

O. P. SHARMA*, R. NIWAS, H. P. SHARMA and J. SHANKER Department of Physics, Agra College, Agra-282002, India

Received 4 February 1977

The strain derivatives of the static dielectric constant and the polarizability of NaC1 structure alkali halides have been evaluated on the basis of the shell model incorporating the effect of exchange charge polarizations. Values of the strain derivative of the static dielectric constant and the polarizability calculated in the present study are found to be in close agreement with experimental data. The volume dependence of the static polarizability has also been discussed in the light of experimental data on f'trst and second order strain derivatives of the static dielectric constant of alkali halides.

1. Introduction

A theory of the dielectric constant for ionic crystals was proposed by Dick and Overhauser [1] (hereafter referred to as DO) proposing the shell model for ions and incorporating the effect of exchange charge polar- izations. The theory was initially intended to make an interpretation of the effective ionic charge parameter in alkali halides introduced by Szigeti [2]. Recently it has been pointed out by Jones [3] that the dielectric theory of DO can be used to study the pressure dependence of the static or low frequency dielectric constant %. However, Jones did not make a detailed quantitative investigation on this line but fitted his experimental data on the pressure dependence of es and obtained indirectly the values of the derivatives of the exchange charge polarization parameter introduced by DO. The experimental data on the pressure depend- ence of es of alkali ha[ides have been reported recently by Lowndes and Martin and by Fontanella et al. [5]. It is desirable to make a theoretical analysis of the strain derivative of es. Studies on the pressure depend- ence of the static dielectric constant are useful for understanding the nature of interatomic forces and thereby the constitution of solids. The strain deriva- tives of esis much more affected by the short range

*P.G. College, AMBAH (M.P.) India.

249

interaction than es itself. Therefore the studies on this subject can provide a powerful test of the theoretical models for the dielectric constant.

In the present paper we have developed a method based on the dielectric theory of DO to evaluate the strain derivative of es and the static polarizability c~ s of NaC1 type alkali halides. Values of the strain derivative of es and o~ s calculated in the present study are found to be in close agreement with experimental data. The volume dependence of as has been discussed in the light of first and second order strain derivatives of es.

2. Calculation of the strain derivative of ct s and es

Following the dielectric theory of DO, the Clausius-Mossotti relation for the static dielectric constant es of alkali halides can be written

1 1 1 ) e s - 1 3 V _ ( e + D ) 2 - + - - + e s + 2 4rt A k+ k_

+ 2(e + D)[(n+k+ D ) (n_e + D)

+ (n+e + D) 2 (n_e - D) 2

+ k+ k_

(i)

250 O. P. Sharma et al./Strain derivatives of the static dielectric constant

where V is the volume per ion pair, e the electronic charge, D the exchange charge polarization parameter, and A the force constant between nearest neighbours. n+, k+ and n_ , k_ are the shell model parameters for the cations and the anions, respectively, n and k repre- sent respectively, the shell charge and the force constant between the shell and the core of the same ion.

Differentiating eq. (1)wi th respect to V we fred

3 . a e s 3 V e s - 1 3 aot s

(e s + 2) 2 a V 47r + - - - es + 2 4~r aV

Table I Values of A (104 erg/cm2), aA/aV (1028 erg/cm 5) D (10 -10 e.s.u.), aD/aV (1012 e.s.u./cm 3) and ~s/aV

aA aD aa s Crystal A D - - aV av aV

LiF 9.02 -1.25 -0.29 4.27 0.36 LiCI 5.56 -0.44 -0.28 2.24 0.33 LiBr 4.86 -0.33 -0.58 4.03 0.32

NaF 6.25 -0.59 -0.38 3.79 0.35 NaC1 4.92 -0.33 -0.50 3.50 0.32 NaBr 4.04 -0.23 -0.63 3.66 0.32 NaI 3.14 -0.14 -0.62 2.76 0.31

e + D _ _ a n a A , KF 5.15 -0.36 -0.21 1.51 0.32 - A2 [2A~v - (e + D) ~V]" (2) KC1 3.86 -0.19 -0.50 2.54 0.30

KBr 3.56 -0.16 -0.62 2.90 0.29 KI 2.97 -0.11 -0.75 2.91 0.29

In deriving eq. (2), it has been assumed that the shell model parameters n and k do not change with a varia- tion in the volume under hydrostatic pressure. This assumption can be supported by the fact that n and k are determined from the free ion polarizabilities and thus are the characteristic parameters of the ions. Therefore, it is to be expected [6] that their depend- ence on the volume will be much smaller as compared to other crystal parameters like A and D. The force constant A between nearest neighbours can be ex- pressed as [7]

2 , r ] A = 2[q~"(r) + r ~b ( )1, (3)

where ~(r) is the short range repulsive potential between the ions separated by a distance r, and can be repre- sented as a function of r in the following form:

~b(r) = B exp ( - r /o ) , (4)

RbF 5.19 -0.34 -0.11 0.77 0.30 RbC1 3.97 -0.19 -0.34 1.69 0.28 RbBr 3.46 -0.14 -0.56 2.36 0.28 RbI 3.09 -0.12 -0.82 3.04 0.28

The exchange charge polarization parameter D was introduced by DO in order to make an interpretation of the deviation of the effective charge parameter from its nominal value of 1. In addition to the exchange charge polarization effect there is another effect, namely, the short range interaction polarization which also contributes to 1 - (e*[e). Considering these two effects, DO have derived the following relation for e*/e:

e" e+O[n e O n ; O ] e - e k+ _

1 ~ _ ) ] . (6) l i e ( 1 + kZ +

in which B and p are known as the Born repulsive para- meters. Differentiating eq: (3) and remembering that V = 2r 3 for the NaC1 structure, we get

aA 1 OA B e x p ( - r / p ) [ 2 2 " ]

a---V : 6r ---~ Or - ' 3r2p [rp + r-2 - #2] • (5)

With the help of eqs. (3 ) - (5 ) one can evaluate A and a A / a V taking B and p from Tosi [8]. Values of A and aA/O V for alkali halides thus calculated are listed in table I.

Values of D were calculated by DO considering the various approximations about the magnitudes and situations of exchange charges between the ions. Havinga [6] has criticized these approximations and has pointed out that the values of D obtained by DO are highly uncertain. It should also be mentioned that using the values of D as obtained by DO, the magni- tudes of e*/e in alkali halides estimated from eq. (6) do not present a good agreement with the experimental values of e*/e reported by Lowndes and Martin [9]. In order to circumvent all these objections we have calcu-

0. P. Shanna et aLlStrain derivatives of the static dielectric constant 251

lated D from eq. (6) taking experimental values of e*/e and using revised data of the shell model para- meters [lo]. Following DO and Hardy [l 1 ] one can write approximately

D a exp (-r/p), (7)

which yields

aD D -=__ -* aV 6r2p (8)

Values of i3D/a V can be calculated from eq. (8). The values of D and aD/W calculated following the pro- cedure described above are listed in table I. Substitu- ting in eq. (2) the values for A, &l/W, D and aD/W

from table I, we have calculated acu,/W and &,/av: Values of &x,/W thus calculated are given in table I and values of &s/W are compared with experimental data in table II.

3. Comparison of results with experimental data

The pressure dependence of es for alkali halides has been studied experimentally by Jones [3], Lowndes

Table II Values of v(aes/av)

Crystals Calculated Experimental from eq. (2)

Jones [3] Lowndes and Fontanella Martin [4] et al. [5]

LiF 30.86 29.41 22.95 30.89 LiCl 37.14 30.36 LiBr 40.80 40.32

NaF 14.75 11.97 12.52 12.76 NaCl 14.62 14.31 13.02 14.69 NaBr 15.84 15.49 14.86 15.87 NaI 18.25 18.11 16.55

KF 13.53 12.74 KC1 10.51 8.91 8.45 9.27 KBr 10.45 8.61 8.35 9.03 KI 10.16 8.91 8.16

RbF 14.12 14.46 RbCl 9.55 8.63 7.79 RbBr 9.26 7.93 7.46 RbI 9.41 6.83 6.94

and Martin [4] and by Fontanella et al. [S] . The strain

derivatives of es from experimental data can be ob- tained using the following relation:

v@ = L(2), (9)

where XT is the isothermal compressibility. Values of XT have been taken from Tosi [8]. It is interesting to

observe from table II that values of V@e,/W) calcu-

lated from eq. (2) present a good comparison with those corresponding to experimental data. The most accurate and recent values among all the experimental data are those of Fontanella et al. known for six alkali halides only. Our calculated values of V(ae,/iW) for IiF, NaCl and NaBr are in remarkably close agreement with the corresponding values obtained from the experimental data reported by Fontanella et al.

A remarkable feature of the present analysis, how- ever, is the prediction that iJa,/aV remains nearly constant in most of the alkali halides (table I). This implies that the interactions operative in these crystals are qualitatively similar. If we assume that the variation of os with V under hydrostatic pressure is given by the relation

(IL, = KVq, (10)

where K and q are constants for a given crystal, eqs. (2) and (10) yield

ae s 4) av =&- 1)(es+2)(es-1). (11)

The validity of eq. (10) can be tested by calculating the second order strain derivative of the static dielec-

tric constant of alkali halides. On differentiating eq. (2) and making use of eq. (10) we obtain

V2 ( )

3 z3es+ 2)@,- 1)

(12)

Fontanella et al. [S] have recently reported the values of V2(a2es/W2) for LiF, NaF, NaCl, NaBr, KC1 and

252 O. P. Sharma et al./Strain derivatives of the static dielectric constant

Table III Values of q

Crystal q from eq. (11) q from eq. (12)

LiF 2.01 1.94 NaF 2.32 2.33 NaCI 2.12 2.39 NaBr 2.05 2.72 KC1 2.05 2.54 KBr 2.00 2.64

KBr crystals. Thus, we can understand the role of volume dependence of the polarizabili ty in deriving the first and second order strain derivatives o f the static dielectric constant. The values of q correspond- ing to eqs. (11) and (12) are given in table III. In LiF, NaF and NaC1 the values o f q corresponding to ~es/~V and a2es/aV 2 are quite close. In NaBr, KC1 and KBr the values of q corresponding to a2es/aV 2 are some- what higher than the values corresponding to aes/aV. The experimental data on a2es/~V 2 for other alkali halides will be valuable in order to reach a conclusion about the implication of the relation between polar- izabili ty and volume [eq. (10)]. It should be men- t ioned here that the importance of the parameter

q [= (V/as)(aas/aV)] has already been emphasised by Jones [3]. His prediction based on the first order strain derivative o f the dielectric constant suggests a value 2 for q. However, in view of the analysis pre- sented above based on the second order strain deriva- tive of the dielectric constant the value o f q generally lies between 2 and 3.

References

[1] B. G. Dick and A. W. Overhauser, Phys. Rev. 112 (1958) 90.

[2] B. Szigeti, Proc. Roy. Soc. A204 (1950) 51. [3] B. W. Jones, Phil. Mag. 16 (1967) 1085. [4] R. P. Lowndes and D. H. Martin, Proc. Roy. Soc. A316

(1970) 351. [5] J. FontaneUa, C. Andeen and D. Schuele, Phys. Rev. B6

(1972) 582. [6] E. E. Havinga, Phys. Rev. 119 (1960) 1193. [7] N. F. Mott and M. J. Littleton, Trans. Faraday Soc. 34

(1938) 485. [8] M. P. Tosi, Solid State Phys. 16 (1964) 1. [9] R. P. Lowndes and D. H. Martin, Proc. Roy. Soc. A308

(1969) 473. [10] B. G. Dick, Phys. Rev. 145 (1966) 609. [11] J. R. Hardy, Phil. Mag. 7 (1962) 663.