B. PRASAD: First-Principles Thermal Expansion in Rb and Cs 497

phys. stat. sol. (b) 124, 497 (1984)

Subject classification: 6 and 8; q1.2

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

First-Principles Thermal Expansion in Rb and Cs

BY B. PRASAD

The thermal expansion of Rb and Cs is investigated using Toya's first-principle approach with modified g-function. The mode Griineisen parameters yq3 and the temperature dependence of both, the mean Griineisen parameter yT and the thermal expansion coefficient BT are calculated and compared with the experimental data available up to the present time. The agreement obtained between theory and experiment is satisfactory.

Die thermische Ausdehnung von Rb und Cs wird nach dem ,,first-principle"-Verfahren von Toya mit modifizierten g-Funktionen untersucht. Die Griineisen-Modenparameter yqj und die Tem- peraturabhangigkeit sowohl des mittleren Griineisenparameters yT als auch des thermischen Aus- dehnungskoeffizienten BT werden berechnet und mit gegenwiirtig verfiigbaren experimentellen Werten verglichen. Die zwischen Theorie und Experiment erzielte Ubereinstimmung ist be- friedigend.

1. Introduction

In our previous papers [l, 21, we calculated the phonon frequencies, the elastic con- stants, and the specific heatis of Rb and Cs using Toya's first-principle approach [3 to 51 with modified g-function. The computed results were found to be in good agreement with the experimental data. Recently, the anharmonic effect on the thermal and elastic properties of solids has become an interesting subject in the field of solid state physics. Theoretically, many studies about the mode Griineisen parameters ypj and the temperature dependence of both the mean Griineisen constant yT and thermal expansion coefficient PT have been performed for Rb and Cs. Toya [4, 51 used his method for the calculation of Griineisen parameters yqJ but only in the three symmetry directions. He estimated the mean valve of yT at various temperatures by using the values of yqj in the three symmetry direct ions only excluding the other directions of the Brillouin zone. Prasad and Srivastava [S] have calculated the mode Griineisen parameters yq3 and mean Griineisen parameters y T for these metals using a model potential technique. Vaks et al. [7] have also calculated ypj, yT, and BT for all five alkali metals. Anharmonic phonons in Cs have been investigated by Glyde and Taylor [S] using the pseudopotential method [9]. Quite recently, Soma et al. [lo] have studied the Griineisen spectra and thermal expansion of all alkali metals using the local Heine- Abarenkov model potential [ll] in the perturbation method and they have compared their results with experimental data. It seems, therefore, worthwhile to examine the variation of the Griineisen parameter of individual modes with the phonon wave vector q and to study the temperature dependence of the average Griineisen parameter YT and the thermal expansion coefficient PT to test the reliability and suitability of the modification made in [1, 21.

I) Varanasi 221 005, India.

32'

498 B. PRASAD

2. Toya's Modified g-Function

The phonon frequencies oqj are obtained by solving the secular determinant

where I is the unit matrix of order 3 and m the mass of the ion. The elements of the dynamical matrix M are obtained from

[ M - mcozII = 0 , (1)

[XYI = -C w(l&l)zy exp (GI%) , (2) 1

where w(l &I) is the total interaction potential between the ions. The total interaction potential w can be divided into three parts, i.e.

Accordingly the coupling coefficient [XY] also splits up into three parts, i.e., 2, = WC + + WE. (3 )

[XY] = [XYIC + [XYIR + [XYIE ,

[XYIC = - c %(I RI )zy exp (iqR2) where

1

and similar expressions for [XYIR and [XYIE. In (4) the first two terms represent the contributions from Coulomb and non-

Coulomb ion-ion interaction, while the third one is due to the conduction electrons. The expressions for [XYIc have been derived by Kellermann [12] and are also given

by Srivastava and Dayal[13] and by Toya [3]. The numerical values for b.c.c. crystals have been taken from Srivastava and Srivastava [14]. The Born-Mayer exchange repulsive coupling coefficient [XYIR has been taken from Srivastava and Dayal [13], defining OR([ &[ ) as the two-parameter Born-Mayer [15] potential, i.e.,

where LS, r,, and e are the constants taken from Toya [3]. The explicit expressions for [XYIE have been derived by Toay and also given by Dayal and Srivastava [16] and were taken as such. The summation over the reciprocal lattice vectors was limited up to (2/a) 13, a being the lattice constant.

Toya calculated the vibration frequencies of Rb with poor agreement. In order ti0 improve the agreement, the g-function of the electronic coupling coefficient [XYJE was modified [l, 21 in the manner of [16]. The argument of the g-function was multi- plied by a factor A exp (BU4) . The constant A was chosen so that the theoretical elastic constant C,, agrees well with the experimental value of GI,. For both rubidium and cesium A was found to be 1.10 and B was found to be 0.10. The function exp (BU4) was chosen so as to obtain a reasonable agreement between the calculated phonon frequencies of Rb and its measured value v L [17] a t the zone boundary for the longitu- dinal branch of [Ol l ] .

3. Evaluation of Griineisen Parameters yqj and yT The Griineisen parameters yqj are obtained by differentiating the above secular determinant and solving it for d(oqj)/dV as outlined in [18]. The mean value of the Griineisen parameter yT can be calculated using the equation

C yqj E( l zwq j / kBT) P j

E ( h a q j / k B T ) * Y T =

n j

All the terms are defined already in [18].

First-Principles Thermal Expansion in Rb and Cs 499

4. Evaluation of the Thermal Expansion Coefficient PT In order to provide an independent check on Toya’s theory the author has computed the thermal expansion coefficient for Rb and Cs in the manner of Wallace [19].

From thermodynamics one can define

where BT is the isothermal bulk modulus, F the total Helmholtz free energy, T the temperature, and V the volume. Neglecting the anharmonic contribution to the free energy and the electronic excitation contribution, one can write the above equation as

where kB is the Boltzmann constant.

5. Computational Results of Phonon Frequency and Griineisen Parameters

The data used in the present calculations (all in at. units) have already been given in [l, 21. The phonon frequencies wqj and Griineisen parameters ynj for the normal mode of wave vector q and polarization j have been calculated for 1000 points uniformly distributed over the entire Brillouin zone. These 1000 points, however, are reduced to only 47 points from symmetry considerations. The numerical results for yqj in three principal symmetry directions are drawn in Fig. l a and b for Rb and Cs, respectively. The yqj for Rb have been compared with experimental results of Copley et al. [20]. No comparison is made for Cs due to lack of experimental data on it. The comparison with observed yqj due to Copley et al. shows some significant discrepancies between theory and experiment. However, it is observed that the present calculated values of y4j are comparable to these obtained in recent papers [7, 101.

6. Temperature Dependence of the Mean Griineisen Constant yT

The value of yn. a t a particular temperature has been estimated by (7), assigning a proper weight factor to yqj. This has been done on the basis of [21]. yT a t different temperatures has been calculated by Blackman’s sampling technique. The numerical results are drawn and shown in Fig. 2a and b for Rb and Cs, respectively. Fig. 2a has

0 06 7 0 0.6 0 0.50 0.6 7.0 06 0 0.5 9 4

Fig. 1. a) Mode Griineisen parameters y,,j for a) Rb and b) Cs. ~ calculated, 0 I,, o T experi- mental after Copley et al. [ZO]

500 B. E’RASAD

Fig. 2. a) Mean value of Griineisen parameters yT for a) Rb and b) Cs. ~ calculated, o ex- perimental, - - - - Soma et al. [lo]

also been compared with the experimental results of Kelley and Pearson [22]. No comparison is made for Cs due to lack of experimental data. It is clear from Pig. 2a that theoretical results for yT of Rb are in fair agreement with measured values and its nature is similar to those obtained in [7, lo].

7. Temperature Dependence of the Thermal Expansion Coefficient PT The value of BT at a particular temperature has been estimated by (9) at the volume which corresponds to zero temperature and pressure. The numerical results for pT are drawn and shown in Pig. 3a and b for Rb and Cs, respectively. No comparison is made due to lack of experimental data on it for both metals. However, the nature of the curves is similar to those obtained in [7, 101.

8. Conclusions

In this way it is seen thab the obtained results for yqj, y T , and BT are in fair agreement with the experimental data available up to the present time. However, it is noted that the experimental errors for the mode Griineisen parameters yqj are considerably large. These discrepancies between theory and experiment may probably be offered to the plane wave approximation and the assumption Ug(r,) = 1. In Toya’s theory the constant Bl has been computed from the compressibility expression of Wigner and Seitz and is assumed to be independent of Iq + ZI . This seems to be somewhat question- able.

Fig. 8. a) Thermal expansion coefficient PT for Rb and b) Cs. ___ calculated

First-Principles Thermal Expansion in Rb and Cs 501

Acknowledgements

The author is grateful to Prof. T. R. Anantharanian, Director, Institute of Technology and Prof. R. S. Srivastava, Physics Department, Banaras Hindu University for their help in many ways. Thanks are also due to Mr. G. La1 for his assistance in the prepara- tion of this manuscript.

References

[l] R. S. SRIVASTAVA and B. PRASAD, Indian J. pure appl. Phys. 8, 594 (1970). [2] R. S. SRNASTAVA and B. PRASAD, Indian J. pure appl. Phys. 10,205 (1972). 133 T. TOYA, J. Res. Inst. Catalysis, Hokkaido Univ. 6, 16 (1958). [4] T. TOYA, J. Res. Inst. Catalysis, Hokkaido Univ. 7, 60 (1959). [5] T. TOYA, J. Res. Inst. Catalysis, Hokkaido Univ. 9, 178 (1961). [6] B. PRASAD and R. S. SRIVASTAVA, phys. stat. sol. (b) 80, 379 (1977). [7] V. G. VAKS, E. V. ZAROCHENTSEV, S. P. KRAVCHUK, V. P. SAFRONOV, and A. V. TREFILOV,

[ti] H. R. GLYDE and R. TAYLOR, Phys. Rev. B 10, 6500 (1982). [9] R. TAYLOR and A. H. MACDONALD, J. Phys. F 10, 2387 (1980).

phys. stat. sol. (b) 85, 749 (1978).

[lo] T. SOMA, H.-M. KAGAYA, and Y. KIMURA, phys. stat. sol. (b) 116, 57 (1983). [ I l l V. HEINE and I. V. ABARENKOV, Phil. Mag. 9, 451 (1964). [12] E. W. KELLARMANN, Phil. Trans. Roy. SOC. (London) A238, 513 (1940). [13] R. S. SRIVASTAVA and B. DAYAL, Progr. theor. Phys. (Kyoto) 31, 167 (1964). [14] P. L. SRIVASTAVA and R. S. SRIVASTAVA, Proc. Phys. SOC. 82,304 (1963). [15] M. BORN and J. MAYER, Z. Phys. 75, 1 (1932). [16] B. DAYAL and P. L. SRIVASTAVA, Proc. Roy. Soc. A283,394 (1965). [17] J. R. D. COPLEY, B. N. BROCKHOUSE, and S. H. CHEN, Proc. Internat. Symp. Neutron In-

[I81 B. PRASAD and R. S. SRIVASTAVA, phys. stat. sol. (b)86, 789 (1978). [19] D. C. WALLACE, Phys. Rev. 176, 832 (1968). [20] J. R. D. COPLEY, C. A. ROTTER, H. G . SMITH, and W. A. MAMITAKAHARA, Phys. Rev. Letters

[all B. SHARAN, Ph. D. Thesis, Banaras Hindu University, Varanasi 1960. [22] F. M. KELLEY and W. B. PEARSON, Canad. J. Phys. 33,17 (1955).

elastic Scattering, Vol. 1, I.A.E.A., Vienna 1968 (p. 209).

33,365 (1975).

(Received February 28,1984)