.

7. 8.

J. Manning, Kinetics of Atomic Diffusion in Crystals [Russian translation], Mir, Moscow (1971). J. Mikusinski, Operational Calculation, Pergamon (1969). G. B. Fedorov and E. A. Smirnov, Metallurgy and Thermal Processing [in Russian], Vol. 8, Itogi Nauki Tekh., Moscow (1974).

CALCULATION OF GRUNEISEN PARAMETER AND TEMPERATURE

DEPENDENCE OF ELASTIC CONSTANTS IN ALKALI METALS

L. V. Belan-Gaiko, V. I. Bogdanov, and D. L. Fuks UDC 539.21

The pseudopotential method isused to determine the GrUneisen constant and the temperature dependence of the elastic constants for T > % in alkali metals. An analysis is made of the influence of exchange-correlational effects in the elec- tronic gas on the quantities mentioned above. The calculated values are ingood agreement with experimental data in Na, K, Rb, and Cs. The results of the paper indicate that the considered anharmonic characteristics of alkali metals are de- termined primarily by the interaction in the first two coordination spheres.

The harmonic approximation in lattice dynamics is inadequate for explaining thermal ex- pansion and the temperature dependence of the elastic constants and the heat capacity. At the same time, it is known [i] that the calculations of the vibrational part of the respective thermodynamic functions with allowance for the anharmonicity prove to be hopelessly complex if no simplifying assumptions are made about the dependence of the frequencies of the phonon spectrum on the temperature and the volume. As a first step, use is ordinarily made of a

quasiharmonic approximation in which the relations of anharmonictheory are preservedSut with volume-dependent force constants and, hence, frequencies. This approximation has been substantiated for temperatures at which the anharmonic contributions are not too great for the vibration spectrum to be still described adequately in terms of noninteracting phonons. In the quasiharmonic approximation the parameters which characterize the anharmonicity prove to be the microscopic GrOneisen constants [i] determining the shift in phonon frequencies under a change in volume. If in the harmonic approximation the thermodynamic functions:en- tropy, free energy, and heat capacity are expressed in terms of the frequencies of the phonon spectrum, then the quasiharmonic contribution to these functions should be defined in terms of the microscopic GrUneisen constants, averaged over the Brillouin zone with appropriate weights. Direct solution of the problem of taking account of the anharmonlc contribution to the thermodynamic properties runs into considerable difficulties; it is necessary to know the frequencies of the spectrum and the microscopic GrUneisen constants at all points of the Brillouin zone. Such step-by-step microscopic consideration can he accomplished by the pseudo- potential method in which the phonon frequencies and the microscopic GrUneisen constants are calculated in terms of the pseudopotential form factor modeling the probability of electron scat- tering from a metal ion. In this case all the calculations are carried out in reciprocal space and this worsens the convergence of the series in the vectors K. However, the afore- mentioned difficulties, due to the need to calculate the phonon frequencies and the micro- scopic Gr~neisen constants in all directions and subsequent summation of these constants over the Brillouin zone, remain.

On the other hand, the thermodynamic properties of bodies can be expressed over the en- tire temperature range in terms of the moments of the frequencies of the phonon spectra, and the anharmonic corrections to these properties can be related to the macroscopic GrUneisen constants. Unlike the microscopic constants, the macroscopic constant characterizes the changes in the moments of the frequency spectrum under a change in volume.

The moments of the frequency of the phonon spectrum and the macroscopic GrUneisen con- stants may be calculated on the basis of the pseudopotential method. In view of this, it is

Vologodsk Polytechnic Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 36-39, April, 1979. Original article submitted February 6, 1978.

370 0038-5697/79/2204-0370507.50 �9 1979 Plenum Publishing Corporation

TABLE I. Values of y (2) in Alkali Metals

[ Na K [ R b [ C s

�9 1, (2) (L) 0,~23 0,918 0,949 0.935

}, (2) (GV) 1,015 t,024 1,102 ],086 0,94 1,159 1,28 1,28

Y (2)exP [5] [6] [51 [5]

expedient to calculate the thermodynamic functions precisely in terms of the moments of the frequencies of the pbonon spectrum and the macrosaopic Gruneisen constants since this allows the calnulations to be confined to only several frequency moments and one or two Gruneisen

constants.

In the present paper we de=ermine the GrUneisen constant in the thermodynamic equation of state as well as the temperature dependence of the elastic constants of alkali metals above the Debye temperature. In this case, as in [2], use is made of the calculated pair inter- atomic potential in R space.

The dependence of the elastic constants and thermal expansion of the temperature can be explained in the qu~siharmonic approximation. one macroscopic Gr~neisen constant y (2),

In order to do this, it i~ necessary to find

v (2) ~--- - - , ( z ) 2 aln(2

where ~= is the second moment of the frequency spectrum and ~ is the volume per unit cell.

In the case of a pair central interaction between atoms, it is not difficult to show that this quantity can he expressed in terms of the derivatives of the interatomic interac- tion potential ~R)

(2) =

z~ (R) + 2?" (R) 1 - . R=R d .

here z i is the number of atoms per coordinate sphere of radius R i. The value of ~(R) and its derivatives can be calculated on the basis of the pseudopotential method. In this case, the influence of exchange-correlational effects in the electronic gas can be studied and the long-range character of the interatomic interaction in metals can be taken into account.

The results of numerical calculations of the GrUneisen constant for ha, K, Rb, and Cs are listed in Table i. As in [2], we used the model pseudopotential from [3]; screening was taken into account after Lindhart (L) and Geldart-Vosko (GV) [4]. It follows from Table 1 that the calculated values are in good agreement with experimental data. Summation of the series (2) was carried out up to the eighth coordination sphere. Inclusion of exchange-cor- relational effects leads to better conformity with experiment. If follows that both in the harmonic approximation [2] and when 7(2), describing the anharmonicity, is calculated in the quasiharmonic approximation, the main contribution to Eq. (2) is made by the first two coordi- nation spheres. This is due to the oscillating character of the interatomic interaction potential, resulting in the compensation of successive terms of the series.

The values of y (2) obtained were used to determine the temperature dependence of the elastic constants. The expression for Cxl is of the form

' ~ R ~ a 2 o ~ 2 372 (2) / (h) + + - - R 2 § R ~/R=e~ -~-aR ~- 2

(3)

371

Here, C** is the elastic constant C** in the harmonic approximation, a is the lattice con-

stant, ~a xT(2) KT --= �9 ~ is the compressibility, Rx(h) is the projection of the vector R h on a Q '

the x axis, ~ is Boltzmann's constant, and T is the temperature. The expression for C12, which in our case is equal to C44, is found from Eq. (3) by replacing R](h) with R~(h).R~(h).

The second term in Eq. (3) describes the contribution to the elastic constants from the change in the elastic part of the free energy under thermal strain whereas the third term is determined directly by the change in the vibrational free energy owing to anharmonicity. It will be the same for both Cxx and Ca2.

The theoretical results are in good agreement with the experimental data. Thus, e.g., for Rb we have d in C**/dT =--8.6.10 -4 deg -I, while the experimental value is --7.9.10 -~ deg-* [7]. For Cs we have d in C44/dT = --1.44.10-" deg-* as compared to an experimental value of --1.30.10 -s deg-* and we have d in (C,~ -- C,=)/dT = -1.27"10 -3 deg -lwhile the experi- mental value Is--0.98.10-" deg-* [8]. Note that the contribution to the dependence of Cij(T) on thermal expansion is somewhat larger than the purely anharmonic contribution. The theo- retical values given here were obtained with GV screening. We studied the problem of the in- fluence of exchange-correlatlonal effects in the electronic gas on the behavior of Cij(T). It turned out that failure to take these effects into account (i.e., using the Lindhart screening function) results in the false behavior of the quantity d In Cij/dT; in these metals this quantity turns out to be positive.

Thus, employing the interatomlc interaction potential in R space, calculated on the basis of the pseudopotential method, makes it possible to ascertain the thermodynamic prop- erties (harmonic and anharmonic) of alkali metals quite simply and graphically. These prop- erties are found to be affected quite substantially by the choice of the type of screening of the electronic gas. As a result of the oscillating character ef~the interatomlc potential it is sufficient to confine oneself to the first two coordination spheres in calculating the properties of these metals.

i.

.

.

4. 5. 6. 7. 8.

LITERATURE CITED

L. G. Lelbfrled and W. Ludwig, Theory of Anharmonic Effects in Crystals [Russian transla- tion], IL, Moscow (1963). L. V. Belan-Gaiko, V. I. Bogdanov, and D. L. Fuks, Izv. Vyssh. Uchebn. Zaved. Fiz. (in

press). G. L. Krasko and Z. A. Gurskil, Pisma Zh. Eksp. Teor. Fiz., 9, 596 (1969). I. W. Geldart and S. H. Vosko, Can. J. Phys., 44, 2137 (1966). D. G. Martin, Phys. Rev., 139A, 150 (1965). D. R. Schouten and C. A. Swenson, Phys. Rev., 108, 2175 (1974). C. A. Roberts and R. Meister, J. Phys. Chem. Solids, 27, 1401 (1965). F. I. Kollarlts and I. Trlvlsonno, J. Phys. Chem. Solids, 29, 2133 (1968).

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