IL NUOVO CIMENTO VOL. 11 D, N. 7 Luglio 1989

Dielectric and Anharmonic Behaviour

of Mixed Alkali Halides (*).

SHOBHA SHARMA and R. P. GOYAL

Department of Physics, St. John's College - Agra-282002, India

B. R. K. GUPTA

Department of Physics, St. John's College - Agra-282002, India Department of Physics, G. B. Pant University - Pantnagar-263145, India

(ricevuto 1'11 Aprile 1988; manoscritto revisionato ricevuto il 20 Febbraio 1989)

Summary. - - In the present paper an analysis of the dielectric behaviour and anharmonic contribution to the dielectric constants has been made of KC1-KBr mixed ionic crystals. The temperature and volume derivatives of static (s0), electronic (5| dielectric constants and energy gap parameter (Eg) have been calculated using the Szigeti and Havinga and Bosman dielectric theories. The anharmonic contributions have been estimated in terms of temperature derivatives of dielectric constants at constant volumes. The modified form of Clausius-Mossotti theory of dielectric polarization has been used for the mixed crystal under study. The calculated quantities have been compared with the available experimental data. A good agreement has been obtained.

PACS 77.90 - Other topics in dielectric properties and materials.

1. - I n t r o d u c t i o n .

In past years the studies on the t empera tu re and volume derivat ives of dielectric constants of mixed ionic crystals have been found to be very useful in view of the i r optical applications part icularly in the fields of high-power laser

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.

6 5 - I1 N u o v o C i m e n t o D 963

964 SHOBHA SHARMA, R. P. GOYAL and B. R. K. GUPTA

beam and storage cellsC.2). The temperature and volume dependence of important parameters such as the energy gap parameter appearing in the dielectric theory of chemical bonding can easily be estimated from the knowledge of temperature and volume derivatives of dielectric constants. In recent papers we have studied the dielectric and anharmonic properties of mixed systems (3.4). In the present paper, we have extended this analysis incorporating the anharmonic contributions to dielectric constants of KC1-KBr mixed crystals. The theories of Szigeti (5) and Havinga and Bosman (6) have been used to calculate the anharmonic contributions. The modified form of Clausius-Mossotti relation in view of three-body interactions has also been followed to calculate the static and electronic dielectric constants of the mixed-crystal systems under study.

The purpose of the present investigation is to replace all the empirical formulation of dielectric constants (7.8) by the knowledge achieved from the interionic potential model developed for mixed crystals. In the present study we have calculated the dielectric constants, temperature and volume derivatives of dielectric constants and energy gap parameter for the entire range of composition in mixed KC1-KBr crystals. The theory and method of computation are given in sect. 2. The results are discussed and compared with the available experimental data in sect. 3.

2. - T h e o r y and m e t h o d o f c o m p u t a t i o n .

In view of three-body interactions(9,1~ the static or electronic dielectric constant and the corresponding polarizability can be related to each other by the modified Clausius-Mossotti or the Lorentz-Lorenz relation as follows:

(1) ~ - 1 4 = ~ V + 2 3(1 - ~a/V) '

where a and V are, respectively, the polarizability and volume per ion pair. ~ is the three-body parameters defined as

2 = - ~ e ~ Z a

(1) j . (2) j . (3) B. (9 S. (5) B. (6) E. (7) K. (8) p. (9) S. (,o) B.

A. VAN VECHTEN: Phys. Rev., 182, 891 (1969). C. PHILLIPS: Rev. Mod. Phys., 42, 317 (1970). R. K. GUPTA and R. P. GOYAL: Nuovo Cimento D, 3, 331 (1984).

K. RAWAT, R. P. GOYAL and B. R. K. GUPTA: Czech. J. Phys. B, 37, I288 (1987). SZIGETI: Proc. R. Soc. London, Set. A, 252, 217 (1958). E. HAVINGA and A. J. BOSMAN: Phys. Rev., 129, 1593 (1963). KAMYOSHI and Y. NIGARA: Phys. Status Solidi A, 6, 223 (1971). VAROTSOS: Phys. Star. Sol. (a), 100, K133 (1980). K. RAWAT and B. R. K. GUPTA: Proc. Solid State Phys. C, 28 (1985). R. K. GUPTA and M. P. VERMA: J. Phys. Chem. Solids, 38, 929 (1977).

DIELECTRIC AND ANHARMONIC BEHAVIOUR OF MIXED ALKALI HALIDES 965

where

(2) e = [Z(Z + 12f(a) + 4adf/da)] 1~.

In eq. (2), Z is the valence of ion and df/da is the space derivative of three-body charge transfer parameter f(a).

For mixed crystal systems, two types of assumptions have been considered. According to the first assumption, the polarizability per unit volume depends linearly on the concentration of component crystals, i.e. we can write eq. (1) in the following way:

(3) ~ - 1 $1 -- 1 ~2- 1 e + 2 - ~1 - ~ - ~ ~- ]~2 - - ~ 2 -[- 2

and

(4) ~: =/~1~:1 "[- ~252,

where �9 is the dielectric constant of mixed crystal, ~1 and ~2 are the dielectric constants of pure crystal mixed in the proportions of/~1 and/~e. Relation (4) has also been used by Farraro et al. (11), following the study on mixed crystals made by Rao and Smakula (i2). The values of static and electronic dielectric constants of mixed crystals have been calculated from eq. (3) by taking the values of dielectric constants for pure alkali halide crystals from the work of Lownden and Martin (13). The temperature and volume derivatives of dielectric constants and energy gap parameter are calculated using the expressions as described by Shanker et al. (14,15) in the case of alkali halides and are, therefore, not repeated here. The input data used in calculations have been compiled from ('.15). According to the theory of anharmonic effects given by Szigeti (~) we can write

(5) e~ - ~ = v + G,

where ~ is the quasi-harmonic contribution and G represents purely the anharmonic contribution which is a function of the phonon frequencies and density of phonons.

Assuming ~ and G to be unique functions of temperature and volume, respectively, Szigeti found that

(11) j . R. FERRARO, C. POSTMUS, S. S. MITRA and C. J. HOSKn~S: Appl. Opt., 9, 5 (1970). (~) K. RAO and A. SMAKULA: J. Appl. Phys., 36, 2031 (1965). (13) R. P. LOWNDES and D. H. MARTIN: Proc. R. Soc. London, 4308, 473 (1969). (14) j . SHANKER and R. SUNDARIJ: Phys. Status Solidi A, 101, 303 (1980); 115, 67 (1983). (19 J. SHANKER and G. D. JAIN: Phys. Status Solidi A, 119, 185 (1983).

966 SHOBHA SHARMA, R. P. GOYAL and B. R. K. GUPTA

TABLE I. - Calculated values of the temperature and volume derivatives of energy gap Eg in KCI-KBr mixed crystals.

Crystal ~gl (~Tg) -~v[dEg\ 5 % of KBr ~ (10-6K-1) ~ ~ - ~ - ) r (10- K-l)

(a) (b) (*) (a)

0 1.38 1.9 1.85 16.8 1.36 - - 1.81 38.2 1.34 - - 1.78 59.8 1.33 1.72 80.0 1.32 1.67

100.0 1.30 1.30 1.65

(*) Deduced from the temperature dependence of the reflectance peaks (se).

TABLE II . - Calculated values of temperature and volume derivatives of ~o and ~| of KC1- KBr mixed crystals. Available experimental values are given in parenthesis.

~EO - 4 - 1 (~-~)V (10-4K-1) \SV] Crystal ( ~ ) v ( 1 0 K ) lV(8~o~ 1V[8~| % of KBr \3V]

0 6.75 8.43 (8.45)* 3.75 - 0.91 ( - 0.93) 16.8 6.06 8.42 4.05 - 0.95 38.2 5.76 8.41 4.89 - 0.99 59.8 5.10 8.38 5.26 - 1.03 80.0 4.62 8.36 5.68 - 1.07

100.0 3.97 8.35 (8.35) 5.96 (6.25) - 1.10 ( - 1.10)

TABLE I I I . - Values of anharmonic parameters (G, hVr) defined by Szigeti and calculated in the present study. Experimental values are given in parenthesis.

G A~TT Crystal So ~| % of KBr eo - ~ ~o - ~|

0 4.84 (4.84)1 2.168 (2.17) 0.09 0.654 16.8 4.86 (4.87) 2.20 0.088 0.660 38.2 4.89 (4.90) 2.28 0.082 0.665 59.8 4.90 (4.91) 2.32 - 0.079 0.671 80.0 4.91 (4.92) 2.35 - 0.078 0.705

100.0 4.90 (4.90) 2.36 (2.358) - 0.076 0.761

(1~) G. W. RUBLOFP: Phys. Rev. B, 5, 622 (1972).

DIELECTRIC AND ANHARMONIC BEHAVIOUR OF MIXED ALKALI HALIDES 967

and

(7) G = T - ~ v \ OT/vJ"

Thus the anharmonic parameters 5~r and G can be determined using the calculated value of temperature and volume derivatives of static and electronic dielectric constants. The values of temperature and volume derivatives of ~s and ~| and Eg calculated in the present study are given in table I and II, respectively, along with the available experimental data. The calculated values of anharmonic parameters are shown in table III.

3. - R e s u l t s a n d d i s c u s s i o n .

The earlier studies on dielectric and anharmonic behaviour have been concentrated on pure alkali halides which are ideally simple and highly ionic solids. The mixed crystals with varying composition investigated in the present study show significantly deviations from the ionic character which is clear from table I. Moreover, the values of the temperature derivatives of Eg have been obtained along with their implicit and explicit contributions. The implicit term is directly related to the dielectric behaviour whereas the explicit term provides valuable information about the anharmonic behaviour of mixed crystals. The direct experimental determination of the explicit term which represents the temperature dependence of Eg at constant volume is not easily possible. The values calculated in the present paper may therefore be quite useful.

In table II, we have listed the values of temperature and volume derivatives of static and electronic dielectric constants. The volume derivatives of ~| in KC1- KBr mixed crystals for all missing ratio are negative whereas those of ~0 are all positive. However, the temperature derivatives of ~0 and ~ are found to be positive. The quasi-harmonic and anharmonic parameters v and G obtained from (6) and (7) are reported in table III. The magnitudes of G, representing the anharmonic contribution, are about (4 + 6)% of the lattice contribution to dielectric polarization, i.e. (~0 - ~) in case of the mixed KC1-KBr halide crystal. The sign of G can be explained in terms of different contribution to G (5.17). It is found that G consists of two terms. The first is a negative contribution arising from quartic anharmonicities and the second is positive at low frequencies associated with cubic anharmonicities. In fact, the positive contribution arises from the interaction between acoustic and optical modes. Thus, a low value of G is indicative of a small interaction between the acoustic and optical modes. At

(x7) G. A. SAMARA: Phys. Rev., 13, 4529 (1976).

968 SHOBHA SHARMA, R. P. GOYAL and B. R. K. GUPTA

presen t it seems more appropr ia te to explain the sign of G in t e r m s of an in te rpre ta t ion given by Fuchs (18) based on the model of Szigeti. I t has been

shown mathemat ica l ly (~8) tha t the mater ia ls with higher values of dielectric constant ~o should have lower value of G. We have listed the value of ~0 in table I I I in order to demons t r a t e the validity of the Fuchs criterion. In fact, these correlat ions provide a link be tween classical and quantum theories of dielectric constant .

To summarize , the analysis p resen ted in this pape r provides the theoret ical values of t e m p e r a t u r e der iva t ives of Eg at constant values and gives a deeper unders tand ing of the dielectric and anharmonic behaviour of the mixed crys ta l sys tems.

(~8) R. FucHs: M I T Labor Instrum. Tech. Rep., No. 167 (1961).

�9 R I A S S U N T O (*)

Nel presente lavoro si fa un'analisi del comportamento dielettrico e del contributo anarmonico alle costanti dielettriche di cristalli ionici misti di KCt-KBr. Si calcolano le derivate della temperatura e del volume di costanti dielettriche statiche (~o) ed elettroniche (~| e del parametro di gap energetico (Eg) usando le teorie dielettriche di Szigeti e di Havinga e Bosman. Si sono stimati i contributi anarmonici nei termini delle derivate della temperatura di costanti dielettriche a volumi costanti, l~ stata usata la forma modificata della teoria di Clausius-Mossotti della polarizzazione dielettrica per il cristallo misto allo studio. Si sono confrontate le quantit~ calcolate con i dati sperimentali disponibili. E stato ottenuto un buon accordo.

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