B. PRASAD and R. S. SRIVASTAVA: Lattice Dynamics and Thermal Properties 379

phys. stat. sol. (b) 80, 379 (1977)

Subject classification: 6 and 8; 21.2

Department of Physics, Banaraa Hindu Univereity, Varanaail)

Lattice Dynamics and Thermal Properties of Simple Metals Based on Model Pseudopotential Calculations

BY B. PRASAD and R. S. SRIVASTAVA

The phonon frequencies and Griineisen parameters are calculated for different values of the phonon wave vector in the three principal symmetry directions for b.c.c. rubidium and cesium using a new form of the local model potential proposed by Krasko and Gurskii. The average values of the Griineisen parameter y y are also calculated for different temperatures and the results are compared with available experimental and theoretical results.1

Phononenfrequenzen und Griineisenparameter werden fur verschiedene Werte des Phononen- Wellenvektors in den drei Hauptsymmetrierichtungen fur k.r.2.-Rubidium und -Caesium mit einer neuen Form des lokalen Modellpotentials berechnet, das von Krasko und Gurskii vorgeschla- gen wurde. Der mittlere Wert des Gruneisenparameters yM wird ebenfalls fiir verschiedene Tem- peraturen berechnet und die Ergebnisse mit experimentellen und anderen theoretischen Ergeb- nissen verglichen.

1. Introduction Recently [l to 51 we have used the Krasko-Gurskii model pseudopotential t o cal-

culate the lattice dynamical properties of some simple metals. The same model is used here to investigate these properties of rubidium and cesium. Like other alkali metals, these two metals crystallize in b.c.c. structure and retain this structure down to 12 K. No spontaneous phase transformations were observed. However, some specific heat anomalies have been observed by Filby and Martin [6] a t very low temperatures. The Fermi surfaces of these metals indicate that the surface of constant energy approximates a sphere, i.e. the conduction electron may be treated as a plane wave. Using neutron scattering techniques, Copley et al. 171 have measured the phonon frequencies for rubidium, a metal having a large thermal expansion coefficient. However, no neutron scattering results are available for the phonon frequencies of cesium, the heaviest of all alkali metals. Cesium is very reactive and extremely sensitive to compression of the electronic band structure IS] that is why neutron data are not available for it a t present.

Very little theoretical work has been done for these two metals. Theoretical calcula- tions of phonon frequencies and Griineisen parameters of these metals have been made by Toya 19, 101 using a first-principle method. A significant improvement of the results has been made by Srivastava and Prasad 111, 121 using Toya’s theory. HO [13] calculated the phonon frequencies of R b and Cs using a linearly screened Heine- Abarenkov type ion potential. The model parameters were determined according to the experimental elastic constants. More recently Rosengren and Johansson [ 141 have studied the lattice dynamics of rubidium using the Krasko-Gurskii model potential with modified parameters. The temperature variation of yy for these metals has been calculated by Sharma and Singh 1151 using the Chevean model.

l) Varanasi-221006, India.

380

Following Gurskii and

wok) =

B. PRASAD and R. S. SRIVASTAVA

2. Theory Krasko [lG] the local model potential may be written as

( 1) 1 +-

where no represents the atomic volume and a and rc denote the niodel parameters. The energy-wave nuniber characteristics F(q) is written as

The modified Hartree dielectric function &*(q) niay be written as

where

and the function f ( q ) corrects for exchange and correlation effects of the conduction electrons. Several approximate fornis of f (q ) have been proposed [ 17 to 191. All these involve interpolation between small and large values of q. Following Kleinnian [20] f ( q ) takes the form

where the exchange and correlation parameter 2 has been calculated aft,er [21, 241.

2.1 Evaluation of Grfineisen constants

According to Griineisen [22] the thermal expansion coefficient /3 is written as

where C, is the specific heat a t constant volume, and y the Griineisen parameter as defined by

the compressibility of t.he volume,

d In wPp d In v y = - - - . (7)

mqp represents the angular frequency of the lattice vibrations. Griineisen assumed that y is constant and does not depend on lattice frequency and temperature. However, this assumption is not correct and due account has to be taken of its variation with ogp and temperature T. Taking this variation into account, (6) can be written as

where yM denotes the mean values of all yqp and given as

Lattice Dynamics and Thermal Properties of Simple Metals 381

with x2 e"

(eZ - 1)2' E(x) =

The temperature dependence of yM can be studied with the help of (9). The angular frequency oqP is obtained by solving the secular determinant in the

usual way as described in our previous papers [2 to 51. The value d(coqp)/dv is ob- tained by differentiating the secular determinant and solving it for d(oqp)/du.

Table 1

of NoziBres-Pines (NP) [21] and Geldart-Vosko (GV) [24] Data used in the calculation (at. units) along with the values of calculated from the expressions

E E I

Rb 587.0 2.293 0.779 5.196 0.369 I 1.715 1 1.761 1 2.32 cs 1 744.5 I 2.214 I 0.871 1 1.744 2.35 I 5.585 1 0.346 I 1.701

3. Numerical Computation and Results

The phonon frequencies wqp and Griineisen parameters yqP have been calculated for 1000 points uniformly distributed in the first Brillouin zone with parameters given in Table 1. These 1000 points, however, are reduced t o only 47 points from synmietry considerations. The numerical results for phonon frequencies in three principal sym- metry directions are compared in Fig. la with the experimental results of Copley et al. 171 for R b and in Fig. 1 b with the theoretical results of Ho [ 131 for Cs. Similarly the numerical results for Griineisen parameters are drawn in Pig. 2 a and b for rubid-

t l4 7 10 P s \

0.6

02

05 10 05 0 0.5 "O 06 4- c- q- 4

b I 06 0 122

t- 4

Fig. 1. Phonon frequencies of a) Rb : __ cal- culated, 0, experiment; b) Cs: - calcu-

lated by Ho [13]

Fig. 2. Calculated Griineisen parameters for a) Rb, b) Cs

382 B. PRASAD end R. S. SRIVASTAVA

Fig. 3. Mean values of Griineisen parameters YM versus temperature T for a) Rb: ~ calculated, o experi- ' 13 ment and b) Cs

a t 735E 125

LTz3 15!i0 10 40 T(K) -- 60

iuin and cesium. The value of ylf a t a particular temperature has been estimated by (9) assigning a proper weight factor to ypp. yM at different temperatures has been com- puted by Blackman's sampling technique and is shown in Fig. 3 a and b for R b and cs.

4. Summary and Discussion

Fig. l a compares the calculated phonon frequencies for R b with the 120 K meas- urement of Copley et al. [7]. The agreement between theory and experiment is qualita- tively good. I n Fig. 1 b, the calculated phonon dispersion curves are compared with the theoretical values of Ho [13]. No experimental results are available for comparison.

We have plotted our theoretical results for ypp in Fig. 2a and b for R b and Cs, respectively. It is observed that the nature of y p p in the three principal symmetry directions is similar t o that calculated by Toya [ 101, but our values are slightly lower than those obtained by Toya. I n the case of Rb, the variation of yM with temperature has been shown in Fig. 3a and compared with experimental results of Kelley and Pearson [23]. No comparison has been made for cesium due to the lack of experimental data. It is seen from Fig. 3a that the calculated results for yar for R b are in fair agree- ment with experiment.

The discrepancies between the theoretical and experimental results may probably be due to the neglect of higher-order pseudopotential terms and to the Born-Mayer exchange repulsion contribution.

Acknowledgements

The authors wish to thank Prof. P. C. Sood and Prof. P. Krishna for their interest in the present work. One of the authors (B.P.) is grateful to Mr. G . La1 for assistance in the preparation of the manuscript.

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Elastic Scattering, Vol. 1, I.A.E.A., Vienna 1968 (p. 209). [S] S. D. MAHANTI and T. P. DAS, Phys. Rev. 183, 674 (1969). [9] T. TOYA, 5. Res. Inst. Catalysis, Hokkaido Univ. 7, 60 (1959).

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Lattice Dynamics and Thermal Properties of Simple Metals 383

[ll] R. S. SRNASTAVA and B. PRASAD, Indian J. pure appl. Phys. 8,594 (1970). [12] R. S. SRWASTAVA and B. PRASAD, Indian J. pure appl. Phys. 10,205 (1972). [13] P. S. Ho, Phys. Rev. 169,523 (1968). [la] A. ROSENQREN and B. JOHANSSON, J. Phys. F 5,629 (1975). [15] P. K. SHARMA and N. SINGE, Phys. Rev. B 1,4635 (1970). El61 Z. A. GURSKII and G. L. KRASEO, Soviet Phys. - Solid State 11,2447 (1969). [17] K. S. SINQWI, A. SJOLANDER, M. P. TON, and R. H. LAND, Phys. Rev. B 1, 1044, (1970). [l8] F. TOIW and T. 0. WOODRUFF, Phys. Rev. B 2, 3959 (1970). [19] D. J. W. GELDART and R. TAYLOR, Canad. J. Phys. 48, 155 (1970). [20] L. KLEINMAN, Phys. Rev. 160, 585 (1967). [21] P. NOWERES and D. PINES, Phys. Rev. 111,441 (1958). [22] E. GRUNEISEN, Hdb. Phys., Vol. 10, Springer-Verlag, 1926 (p. 27). [23] F. M. KELLEY and W. B. PEARSON, Canad. J. Phys. 33,17 (1955). [24] D. J. W. GELDART and S. H. VOSKO, Canad. J. Phys. 44,213 (1966).

(Received November 3, 1976)