Temperature dependence of Szigeti effective charge of alkali halides

Solid State Communications, Vol. 25 pp. 397-399, 1978.

Pergamon Press.

Printed in Great Britain

TEMPERATURE DEPENDENCE OF SZIGETI EFFECTIVE CHARGE OF ALKALI HALIDES* C.K. Kim,** A. Feldman, D. Horowitz and R.M. Waxler Institute

f o r M a t e r i a l s R e s e a r c h , N a t i o n a l Bureau o f S t a n d a r d s , Washington, DC 20234, USA ( R e c e i v e d 11 November 1977 by A. G. Chynoweth) The second S z i g e t i r e l a t i o n was used t o o b t a i n t h e t e m p e r a t u r e d e p e n d e n c e o f t h e S z i g e t i e f f e c t i v e c h a r g e , e s . The r e s u l t s a r e d i s c u s s e d i n t h e framework o f t h e d e f o r m a t i o n d i p o l e model. Recent e x p e r i m e n t a l d a t a a r e u s e d t o show t h a t t h e volume d e r i v a t i v e s o f e s o f most i o n i c s o l i d s a r e p o s i t i v e , t h u s p r o v i d i n g e v i d e n c e t h a t t h e d e f o r m a t i o n d i p o l e model ~is qualitatively valid.

16

The Szigeti relations I-3 have been widely used to understand the dielectric properties of ionic solids. %-8 Of particular interest is the dependence of the Szigeti charge e s on the volume and the temperature. In this paper the total temperature derivative of the Szigeti charge (I/es)(des/dT) has been derived for four alkali halides. Two terms contribute to (i/es)× (de~/dT), a t e r m that depends on the partial derlvative with temperature and a t e r m that depends on tke thermal expansion. In order to acertain the relative importance of these terms, we make use of the deformation dipole model.9, I0 However, questions have been raised about the validity of this model because certain experimental data had contradicted the expected pressure dependence of e s based on this model. 7,8 In this paper we show that on the basis of more recent data, II the deformation dipole model is qualitatively correct, and hence can be used to explain the different contributions to (I/es)× (des/dT). We are not concerned with the actual value of e s. An expression for (I/es)(des/dT) is derived from the second Szigeti relation 3

4~ eo-e~

= ~

(¢+2)2

es

de

de

1

I

-8

I

-18 16

0

8 0

,

,

,

I

(a)

"•

-8

KCI

-16 100

I

I

I

I

NaF

I

(b)

I

I

160

I

(c)

"~ I

240

I

i

I

I

320 100 160 TEMPERATURE (K)

I

I

I

240

I

320

Temperature dependence of (I/es)(des/dT) of (a) LiF, (b) NaF, (c) KCI, and

(d)

(1)

KBr.

o b t a i n e d from r e c e n t measurements o f dn/dT. 13 The r e s u l t s o f our c a l c u l a t i o n s f o r f o u r i o n i c s o l i d s a r e shown i n F i g . 1. The c h a r a c t e r i s t i c f e a t u r e o f t h e c u r v e s i s w e l l r e p r e s e n t e d by LiF. At low t e m p e r a t u r e , say T ~ 100K f o r LiF, ( 1 / e s ) ( d e s / d T ) i s n e g a t i v e . With i n c r e a s i n g t e m p e r a t u r e , t h e v a l u e i n c r e a s e s t o become p o s i t i v e , p a s s e s a peak and becomes n e g a t i v e again. The same t r e n d s a r e c o n f i r m e d f o r t h e other t h r e e m a t e r i a l s through approximate c a l c u l a t i o n s , even though t h e c o m p l e t e f e a t u r e s a r e n o t shown on t h e f i g u r e due t o t h e l a c k o f low temperature experimental data. To u n d e r s t a n d the main features of the curves we write

where e o = low f r e q u e n c y d i e l e c t r i c c o n s t a n t , e® = h i g h f r e q u e n c y d i e l e c t r i c c o n s t a n t , ~ = r e d u c e d mass 6 f an i o n p a i r , ~TO = t r a n s v e r s e o p t i c mode f r e q u e n c y , and V = volume p e r ion pair. We o b t a i n 1 des 5 1 d~To 1 ~s dT = ~ a + mTO dT + 2 ( e o - e = )

,

of__.....

Fig. I. '

i LiF

8

2

2 V~TO

I

(2)

de®

where a = t h e coefficient o f linear t h e r m a l e x p a n s i o n . To o b t a i n a number f o r ( 1 / e s ) ( d e s / d T ) , we need t h e e x p e r i m e n t a l v a l u e s f o r t h e q u a n t i t i e s listed in eq. (2). Values of a, 12 (d~T0/dT),ll and (deo/dT)6 were obtained from the literature, and values of (de®/dT) were

1

des 1 Des V Des d--T-- = ~-- ~-T-- ÷ 3a ~-- ~--~-S

S

(3)

S

According to the deformation d i p o l e model9,1° ~es/~V i s always p o s i t i v e ; h e n c e ; we r e q u i r e Work s u p p o r t e d i n p a r t by t h e D e f e n s e Advacned Research P r o j e c t s Agency. On l e a v e from Korea S t a n d a r d s R e s e a r c h I n s t i t u t e , S e o u l , Korea. 397

398

Vol. 25, No. 6

TEMPERATURE DEPENDENCE OF SZIGETI EFFECTIVE CHARGE OF ALKALI IIALIDES

TABLE I.

Values of ______(V/es]f~esl~V)

T r a n s v e r s e o p t i c mode Gruneisen parameter YTO =

CVles) (aes/~V)

(V/~TO) C8wTO/8V)

Theoretical values from deformation dipole model a

Values used by Singh et al. a

Values obtained by. Lowndes and Rastogi b

Singh et al. a

Present c

LiF

2.6, 2.15

2.35±0.16

-0.55, +0. i0

-0.10±0.19

+0.56

NaF

3.0,

2.08±0.18

-0.49,

+0.43±0.21

+0.50 +0.96

Crystal

2.80

0.29

NaC1

2.4

2.55±0.16

+0.24

+0.29±0.19

NaBr

3.0

2.37±0.20

0.13

+0.50±0.25

+1.07

KC1

2.9,

2.46

2.28±0.18

-0.46,

-0.02

+0.16±0.20

+0.82

KBr

2.6,

2.83

2.06±0.13

-0.02,

-0.25

+0.52±0.16

+1.09

aFrom r e f e r e n c e 8. bFrom r e f e r e n c e 12. C c o r r e s p o n d i n g to t h e v a l u e s by Lowndes and R a s t o g i . @es/@ T t o be n e g a t i v e ~n o r d e r t o have a n e g a t i v e d e s / d T . However, t h e d e f o r m a t i o n d i p o l e has been q u e s t i o n e d by B a r t o n and Batana 7 and a l s o by S i n g h e t a l . B They u s e d e a r l i e r e x p e r i mental values o f the Gruneisen parameter to o b t a i n ~es/SV , which t u r n e d out t o be n e g a t i v e for most materials. However, if we use the most recent experimental results of the Gruneisen parameter by Lowndes and Rastogi II we obtain positive values for all of the materials except LiP. These results are shown in Table i. Even though t h e v a l u e we o b t a i n f o r LiF i s n e g a t i v e , the limits of error extend into the positive r e g i o n ; h e n c e , more a c c u r a t e e x p e r i m e n t a l measurements could lead to a p o s i t i v e value f o r LiF. We t h e r e f o r e s u g g e s t that t h e anomaly p o i n t e d out by Barron and Batana does n o t i n f a c t e x i s t and t h a t t h e u s u a l o v e r l a p and d i s t o r t i o n picture of the deformation dipole model is essentially correct. However, even though the signs o f the theoretical values tend to agree with the signs of the experimental values, the magnitude of the theoretical values are significantly lar~br than the experimental values. Still, we expect the deformation dipole model to he qualitatively correct. We can show independently that Bes/BT is negative. We follow a procedure similar to that used to obtain the Debye-Waller factor. I~ The deviation of e s from the electron charge, e, can be ascribed to a distortion dipole moment, mfr), along each nearest neighbor bond, 3 where es-e = 1/3 N[m'(ro)+2m(ro)/ro]

,

(4)

and N is the number of nearest neighbors, r o is the equilibrium spacing, m'(r) is the first derivative of m(r) with respect to r and the effects of thermal vibrations are ignored. According to the deformation dipole theoryg, I0 m(r) varies exponentially with r, thus

re(r) = m ° exp{-B[(r/ro)-I ])

(5)

Values of B have been calculated by Hardy 9 and Mitskevitch. I0 In order to obtain the effects of thermal vibrations on eq. (4) we must obtain the thermal average of m(r) which can be shown to be 14 T = m O exp{-B[(r/ro)-l]} 8 2 ( T + T ) × exp[~2 ] r

'

(6)

o

where T and T a r e t h e mean s q u a r e d i s p l a c e m e n t s o f t h e p o s i t i v e and t h e n e g a t i v e atom r e s p e c t i v e l y a t t e m p e r a t u r e T. Hardy ~ has p o s t u l a t e d t h a t B i s p r o p o r t i o n a l t o r o where t h e p r o p o r t i o n a l i t y c o n s t a n t has a small tempe r a t u r e d e p e n d e n c e . In t h i s c a s e the deformat i o n d i p o l e would be e x p e c t e d t o i n c r e a s e w i t h i n c r e a s i n g t e m p e r a t u r e b e c a u s e and are expected to increase significantly with t e m p e r a t u r e . T h i s r e s u l t c o r r e s p o n d s t o an i n c r e a s i n g o v e r l a p o f t h e e l e c t r o n i c wave functions with temperature. Substituting T f o r m(ro) i n eq. (4) we o b t a i n es(T) = e + 1/5 N [ m ' ( r o ) + 2 m ( r o ) / r o ]
× exp[ F1 82(.

2 > + % r

We c a n s e e f r o m e q .

+


2 > - ~)]

o

(7) t h a t

since [m'(ro)+2mCr)/r

(7)

Ses/~T

is negative

] has been shown t o be

n e g a t i v e . 3 We hav~ t h e r e f o r e shown t h a t t h e two t e r m s i n eq. (3) a r e o f o p p o s i t e signs. The experimental data suggest there is a balance b e t w e e n t h e s e two t e r m s ; t h e volume t e r m domin a t e s i n an i n t e r m e d i a t e t e m p e r a t u r e r a n g e and t h e p u r e l y t e m p e r a t u r e term d o m i n a t e s a t h i g h

Vol. 25, No. 6

TEMPERATURE DEPENDENCE OF SZIGETI EFFECTIVE CHARGE OF ALKALI HALIDES

and low temperature extremes. Because of uncertainties in the values of.the temperature derivatives of 8, T and T, we were unable to obtain quantitative results from eq. (7). However, an approximate calculation predicts the correct order of magnitude for (i/es)(des/dT) (~I0-SK-I). In conclusion, we have obtained for the

first time the temperature derivative of the Szigeti effective charge. The result indicates that two competing mechanisms account for the effect, a direct temperature contribution and a contribution due to thermal expansion. We have also shown that the anomaly pointed out by Barton and Batana probably does not exist and that the deformation dipole model is qualitatively valid.

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. i0. Ii. 12. 13. 14.

SZIGETI B., Proceedings of Royal Society, A204, 51 (1950). SZIGETI B., Proceedings of Royal Society, A261, 274 (1961). BORN M. ~ HUANG K., D}rnamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954). DICK B.G. ~ OVE~{AUSER A.W., Physical Review, 112, 90 (1958). LO|qNDES R.P. ~ MARTIN D.H., Proceedings of Royal Society, A308, 473 (1969). LOWNDES R.P. ~ MARTIN D.H., Proceedings of Royal Society, A316, $51 (1970). BARRON T.H.K. ~ BATANA A., Philosophical Magazine, 20, 619 (1969). SINGH A.V., SHARMA H.P. ~ SHPuNKER J., Solid State Communication, 21, 64S (1977). HARDY J.R., Philosophical Magazine, 6, 27 (1960). MITSKEVITCH V.V., Soviet Physics-Solid State, 5, 2568 (1964). LOWNDES R.P. ~ RASTOGI A., Physical Review, BI4, $598 (1976). American Institute of Physics Handbook, Third Edition (McGraw-Hill, New York, 1972). FELDMAN A., HOROI~ITZ D., WAXLER R.M. ~ KIM C.K. (to be published). KITTEL C., Introduction to Solid State Physics, Fourth Edition (John Wiley ~ Sons, New York, 1971), p. 84.

399