On the pressure dependence of the effective ionic charge of cesium halides

I. Pkp. Chm Sdids Vd. 41. pp. M-%4 Pagmon Press Ltd.. 190. Printed in Great B&in

ON THE PRESSURE DEPENDENCE OF THE EFFECTIVE IONIC CHARGE OF CESIUM HALIDES A. BATANA and C. HENSE Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires, Argentina (Recehed

I I October 1979; occepfed 19 December 1979)

Abstract-The importance of temperatureeffects in the use of the second Szigeti relation to calculate the volume dependence of the effective ionic charge of cesium halide crystals is studied. The effect obtained is larger than the one for alkali halide crystals of the NaCl structure. Positive values of (d In s/d In o) are obtained using low temperature experimental data with the Griineisen parameter Y, (T-0 K) calculated using the generalized first Szigeti relation. Values of Y, are also obtained with a rigid ion model and they differ little from the previous ones.

The effective ionic charge Zes is defined by the second Szigeetirelation [ I] l/2

( >

Zes =w, y

3

c,+2

(Cc&Y2

(1)

where o, is the angular frequency of transverse optical waves at long wavelengths, E,, the static dielectric constant, G the dielectric constant at frequencies o % W, cc the reduced mass mAmB/(mA t ms), v. the volume per ion pair = (8/3d/j)i), r is the distance between nearest neighbours. A lattice model of non-vibrating spherical ions which are polarizable by the electric field at the centre of the ion but are not deformable otherwise, is characterized by a value of s equal to unity only if the effects of anharmanic interactions can be neglected. Therefore at low temperatures s f I can be interpreted as a deviation from that model, and is usually assumed to be due to ionic distortion caused by the. mutual interaction of neighbouring ions[2]. For most salts s < I. so that compression should decrease its value. (a In s/a In v)~ is obtained by differentiating eqn (I) with respect to volume (a In $a In vh = G -I --

kk[(~},-(~I,1 2 r,t2

aem -x4 xapT ( I

-2n)I

where y, = -(a In @,/aIn v) and xT is the isothermal compressibility. Previous calculations for sodium[3], potassium[4] and rubidium halides [3] show that for ionic crystals with the NaCl structure positive values of (a In s/a In v)~ are obtained using experimental data at low temperatures. Crystals with the CsCl structure are studied here, in particular CsBr for which the experimental data of all magnitudes used in the calculations are available. For this salt, which has a relatively large radius of the cation,

it is plausible to neglect short-range forces between second neighbours. The importance of temperature effects is analysed. As there are no experimental data of y, at low temperatures its value is calculated using the generalized first Szigeti relation(41; for the CsCl structure it gives 2_~ e,+2 @’ -/L (-I(--St2

8r 32 V/(3)x 3V3’p >

(3)

and is valid at all pressures. Differentiation of eqn (3) with respect to volume gives: -----I

de.=

c-,+2 dp

-;xp(I-;xp)-‘(~-~).

I

da

e,,t2dp

,l!V_5 I

2dp

6 (4)

This expression is valid both for NaCl and CsCl structures. In order to minimize the effect of anharmonic interactions, experimental data at T + 0 K must be usedin the calculation of a In s/a In v. For low temperatures, c0[5], l[[s] and xT.[6] were obtained from measurements at helium temperatures; (d43p)T at T-+0 K was obtained from room temperature experimental dataI7l assuming that (&.,/a In u)~ is independent of temperature (for CsI the room temperature value is obtained assuming the additivity rule for the electron polarizabilities and their pressure dependence); (a,&/#), is identical at T+OK to (a,y,-‘/$)T, this value was obtained by extrapolation to low temperatures of the ultrasonic data181 at 77 and 300K. (addp)T was obtained by extrapolation using the method suggested by Hardy and Karo [9]. In Table I are given, for CsBr and CsI, the values of a In s/a In v calculated with eqns (2) and (4) using data at T+OK and, for CsBr, the value obtained with eqn (2) using data at T= 3OOK (column 2; this last result, in parenthesis, was obtained for the study of temperature effects); y, obtained with eqn (4) and data at T+O K is given in column 5. The values of a In da In v are speci-

A. BATANA and C. HENSE

864

Table I. Calculatedvalues of (d In s/a In u); experimentaly, at room temperature; calculated values of y, salt

CeBr

01na/aw)

0.19

(alns/alnv)o

‘d’t

G

“+

experimentsl~ld

equation(4)

2.1

2.31

2.27

2.71

2.74

1.11

equation

(5)

(1.02) co1

0.2I

1.49

ally sensitive to variations in (&/a~)~ and y, as can be seen from the contributions of the different terms in eqn (2). For CsBr at low temperatures these are: In (&,JJp), = 1.43; In y, = -2.31; In (k/Jp)T = 0.57; a In da In 0 = 0.19. y, was also obtained from the Born and Huang theory of the long wavelength optical modes in ionic crystals[l], assuming the rigid ion model of Kellermann[lO], where the ionic polarizabilities a+ = a_ = 0. This gives

valid at all pressures. This equation was derived for an average potential energy per ion pair u(r) = -(a&‘/r) t 1+44(r),where aM is the Madelung constant, M the coordination number and d(r) is an overlap potential between nearest neighbours. The values of y, calculated with eqn (5) using data at T + 0 K are given in Table 1, column 6. Values of (8 In s/a In ok, were also obtained from the theoretical model of Hardy [ 1I], which is a deformation-dipole model and therefore consistent with the use of eqn (I). The model assumes that the deformation dipole associated with each ion pair of neighbouring ions varies exponentially with the distance R between them m(R) = m(ro)exp(- a(R/rO)). a was calculated as described previously[4]. (8 Ins/a In u) is therefore given by (alnslalnvh,=~(a-5)

(6)

where for the CsCl structure a = (Sg(3) ro’/e2XoaM)+ 2 and aM = 1.7627. The results are given in Table I, column 3. Although they are subject to the reliability of Hardy’s model, positive values of (a In s/a In o) are obtained, and the result for CsI is larger than for CsBr as could be expected due to the relative sizes of the anions. Similar

behaviour was obtained for alkali halides of the NaCl structure [3]. The importance of temperature effects in the use of the second Szigeti relation can now be extended to the CsCl structure (CsBr: Table 1, column 2). The effect is larger than the one obtained for alkali halide crystals of the NaCl structure [3]. The results show the need of experimental values of 7, at T+O K to calculate (8 In s/a In u) with the second Szigeti relation. Using experimental data at low temperatures the calculation of (d In daIn u) gives similar results to those of the NaCl .structure[3]. Positive values are obtained, as is expected on the basis of the physical interpretation of this derivative. If the model introduced by the use of the generalized first Szigeti relation to calculate yt( T+O K) is an adequate one, similar results might be expected using the low temperature experimental y, values when available. Acknowledgemen&-The authors thank Dr. R. W. MUM of UMIST, Manchester, England, for helpful comments on the

manuscript. I. Born M. and Huang K., Dynamical 7keory of Crystal Lattices, Clarendon Press, Oxford (1954). 2. Mitskevich V. V., Soa. Phys. Solid %te 5.2568 (1964). 3. Batma A. and Gonzalez E. R., Phys. StatusSolidi @I)76, KIOS(1976). 4. Barron T. H. K. and Batana A., PM. Msg. 20 (1651,619 (1969). 5. Lowndes R. P. and Martin D. H., Prvc. Roy. Sot. AM, 473 (1%9). 6. Vallin J., Beckman 0. and Salama K., 1. Appl. Phys. 35, 1222 (1964). 7. Kormer S. B., Yushko K. B. and Kirshkevich G. V., JETP r_M. 3.39 (1966). 8. Reddy P. J. and Ruoff A. L.. Pm. Int. Conf on the Physics of Solids at High Pnssure, Tucson. Academic Press (1965). 9. Hardy J. R. and Karo A. M.. Phys. Reu. B7(10),4696(1973). IO. Kellermann E. W., Phil. Trw~s.Roy. Sot. 2dl. 105(I948). Il. Hardy J. R., Phil. Msg. 7, 315 (1%2). 12. Postmus C., Ferraro J. R. and Mtra S. S., Inorg. Nucl. Chem. Letl. 4, 55 (1966).