B. PRASAD : Thermal Expansion and Mode Gruneisen Parameters in Lithium 91

phys. stat. sol. (b) 122, 91 (1984)

Subject classification: 8; 21.2

Applied Physics Bection, Institute of Technology, Banaras Hindu University, Varanasi')

Thermal Expansion and Mode Gruneisen Parameters in Lithium BY B. PRASAD

The mode Gruneisen parameters yqj, the mean value of Gruneisen parameter Y M , and the thermal expansion coefficient @ are computed for b.c.c. lithium employing the model potential proposed by Krasko and Gurskii. The present results are compared with available experimental data. Rea- sonable agreement between theory and experiment is obtained.

Die Gruneisenparameter yqj , der Mittelwert des Gruneisenparameters YM und der thermische Aus- dehnungskoeffizient @ werden fur k.r.z.-Lithium mit dem von Krasko und Gurskii vorgeschlagenen Modellpotential numerisch berechnet. Die erhaltenen Ergebnisse werden mit vorhandenen experi- mentellen Werten verglichen. Es wird eine vernunftige ubereinatimmung zwischen Theorie und Ex- periment erhalten.

1. Introduction

Several theories have been developed for the Griineisen spectra and the thermodynam- ic properties of simple metals using model potentials and pseudopotentials, and reasonably good agreement with experiment has been obtained. Wallace [l, 21 has calculated the mode Griineisen parameters and thermal expansion coefficients for sodium, potassium, and lithium using Harrison's [3] modified point-ion pseudopoten- tial with two adjustable parameters and has obtained reasonable agreement with experiment. Taylor and Glyde [4] have computed the mode Griineisen parameters in potassium using the model potential of Dagens et al. [5] with considerable success. Very recently, Soma et al. [S] have investigated the thermal expansion and Griineisen spectra of alkali metals employing the local Heine-Abarenkov model potential [7] in the perturbation method and they have compared their computed results with experi- mental data. Prasad and Srivastava [8 to 111 have computed the lattice dynamical and thermal properties of several metals using a local model potential proposed by Krasko and Gurskii [12]. The same model is used here to study the similar properties of lithium, a metal which shows rather anomolous properties (e.g. the crossing in the (100) branches, etc.) as compared with those in other alkali metals.

2. Theory and Method of Calculations

The angular frequency wnj is obtained by solving the secular determinant in the usual way as described in our previous paper [13]. Similarly the Griineisen parameter Y q j is obtained by differentiating the secular determinant and solving it for d(wqj)/dV as outlined in papers [9, lo]. The mean value of Griineisen parameter yM can be calculated by

..

l) Varanasi 221 005, India.

92 B. PRASAD

All these terms are defined in [9]. The thermal expansion coefficient #I has keen com- puted in the manner of Wallace [l]. We define

p = - - ( - ) 1 82F . BT ? V a T VT

Neglecting the anharmonic contribution to the free energy and the electronic excitation contribution, as this is negligibly small for lithium, one can write the quasiharmonic lattice dynamical contribution in a dimensionless form as

where all the terms have their usual meaning as obtained in [lo].

3. Comparison of Results with the Experimental Data

The data used in the present calculations (all in atomic units) have already been given in [13]. The mode Griineisen parameters yqj along the three principal symmetry directions have been calculated and shown in Fig. 1. No comparison is made for the spectrum of Griineisen parameters because the neutron data are not available in this laboratory. However, it is observed that the present theoretical values of yqj are comparable to those obtained in recent papers [2, 61.

The temperature variation of y M has been investigated and the computed points are shown in Fig. 2 along with the experimental points taken from the hand book edited by Gray [la]. It is seen that there is a qualitative agreement between theory

05 - Fig. 1. Calculated mode Griineisen parameters ygj

I l / I I l l 1 I for Li 0 0.2 0.6 7.0 0.6 02 0 0.2 05

4- -q-

T(KI - Fig. 2 Fig. 3

Fig. 2. Mean value of Gruneisen parameter yH for Li, ~ calculated, o experimental

Fig. 3. Thermal expansion coefficient /? for Li, __ calculated, o experimental

Thermal Expansion and Mode Griineisen Parameters in Lithium 93

and experiment. The calculated quasiharmonic curve of BBTV/3kB as a function of temperature T is shown in Fig. 3 with the available experimental results as outlined in [I]. It is evident from Fig. 2 and 3 that the present theoretical results are somewhat higher than the experimental data.

4. Conclusions

These results indicate that (i) the assumption of the spherically symmetric Fermi surface of lithium is not accurate enough, though it is approximately correct. The effective mass of the electrons on the Fermi surface is larger than the true electron mass and moreover, it is considerably anisotropic. (ii) It does not seem appropriate to represent the wave functions of the conduction electron by the single plane wave. (iii) The mean free path of the conduction electrons is found to be highly anisotropic, so that the single variational approach seems to be inadequate. (iv) The neglect of the short-range Born-Mayer exchange repulsion contribution and the harmonic contribution to the free energy is not accurate enough. Thus it is concluded that perhaps due to lack of p-shell in the core lithium shows some anomolous properties j13] as compared with those in other alkali metals.

Acknowledgements

The author is grateful to Prof. R. N. Singh and Dr. 0. N. Singh for their constant help and encouragement in the present work and to Mr. G . La1 for his help in the preparation of manuscript.

References [l] D. C. WALLACE, Phys. Rev. 176, 832 (1968). [2] D. C. WALLACE, Phys. Rev. 178, 900 (1969). [3] W. A. HARRISON, Pseudopotentials in the Theory of Metals, W. A. Benjamin, Inc., New York

[4] R. TAYLOR and H. R. GLYDE, J. Phys. F 6, 1915 (1976). [5] L. DAUENS, M. RASOLT, and R. TAYLOR, Phys. Rev. B 11, 2726 (1975). [6] T. SOMA, H.-M. KAUAYA, and Y. KIMURA, phys. stat. sol. (b) 116, 57 (1983). [7] V. HEINE and I. V. ABARENKOV, Phil. Mag. 9, 451 (1964). [8] B. PRASAD and R. S. SRIVASTAVA, Phil. Mag. 28, 203 (1973). [9] B. PRASAD and R. S. SRIVASTAVA, phys. stat. sol. (b) 80, 379 (1977).

[lo] B. PRASAD and R. S. SRIVASTAVA, phys. stat. sol. (b) 87, 771 (1978). [ll] B. PRASAD, Ph. D. Thesis, Banaras Hindu University, Varanasi (India) 1972. [l2] G. L. KRASKO and Z. A. GURSKII, Zh. eksper. teor. Fiz., Pisma 9, 596 (1969). [13] B. PRASAD and R. S. SRIVASTAVA, Phys. Rev. B 6, 2192 (1972). [14] D. E. GRAY, (Ed.), American Institute of Physics Hand Book, McGraw-Hill Publ. Co., New

(Received November 10, 1983)

1966.

York 1963.