Dynamical study of the temperature dependence of grüneisen parameters in alkali metals

V. P. SINGH and M. P. HEMEAR: Temperature Dependence of Griineisen Parameters 381

phys. stat. sol. (b) 78, 381 (1976)

Subject classification: 6; 8; 12.1; 21.2

Department of Physics, Allahabad University

Dynamical Study of the Temperature Dependence of Griineisen Parameters in Alkali Metals

BY V. P. SINGH and M. P. HEMICAR

Using a non-central force model the microscopic Griineisen parameters yq, j and the effec- tivevalue ye a t different temperatures are calculated for the alkali metals in terms of elastic constants and their pressure derivatives. The microscopic Griineisen parameters yq, j are calculated as a function of wave vector q along the principal symmetry directions [loo], [IlO], [lll], [210], [211], and [221] and the values are averaged over the wave vector space in the first Brillouin zone using the modified Houston spherical six term integration method. The results obtained for y~ show a fairly good agreement with the experimental observations:

Mit einem nicht-zentralen Kraftmodell werden die mikroskopischen Griineisenparameter yq, j und der effektive Wert yG bei verschiedenen Temperaturen fiir die Alkalimetalle durch die elastischen Konstanten und ihre Druckableitungen ausgedriickt. Die mikroskopischen Griineisenparameter yq, j werden als Funktion des Wellenvektors q in den Hauptsymme- trierichtungen [loo], [llo], [lll], [210], [211] und [221] berechnet und die Werte iiber den Raum der Wellenvektoren in der ersten Brillouinschen Zone mit Hilfe der modifizierten sphiirischen Sechsterm-Integralmethode nach Houston gemittelt. Die fiir ye erhaltenen Er- gebnisse zeigen weitgehende Ubereinstimmung mit den experimentellen Beobachtungen.

1. Introduction The study of the Griineisen parameters, ye, for a solid has its fundamental

importance to the equation of state of a system and the related thermodynamic properties of solids. It enables one to describe and discuss the various physical properties of a system such as the high-temperature specific heat of the lattice, the temperature variation of the elastic constants, thermal expansion, and thermal conductivity. The parameter yo plays also a significant role in the study of thermoelastic properties. Griineisen [l] first tried to derive an extensive theory for the thermal expansion and gave an explicit relation for the thermal volume expansion coefficient a in terms of a dimensionless parameter yo, referred to as Griineisen parameter, given by

(1) aV

YO=-, X G where C, is the heat capacity of the solid a t constant volume, x the compressi- bility, and V the crystal volume. Being directly related to the thermal expansion coefficients, the parameter ye provides a measure for the crystal anhar- monicity and accounts for bulk of the side-effects of anharmonic behaviour in the crystal lattice. The parameters yo are related to the changes in frequency modes with the changes in volume [ Z ] . Griineisen, however, assumed in his theory the parameter y o to be a constant, independent of temperature and lattice frequencies.

382 V. P. SINGH and M. P. HEMKAR

During the past decades, considerable attention has been paid to the experimental and theoretical study of thermal expansion of solids and the Griineisen parameters. The conclusion drawn from the experiments [3 to 71 does not justify the constancy of the parameter yG as assumed in the Griineisen picture. The observed variation of the parameter ya with temperature shows the in- adequacy of such an assumption. There is also no obvious justification for taking the same values of all yq,j corresponding to a particular wave vector q and polarization j . Barron [8 ] and Blackman [9] have made a lattice-dynamics1 study of thermal expansion by using certain idealized models and have shown that a considerable change in ya values occurs a t temperatures of about 0.20 (0 Debye temperature). By taking into account the temperature variation of the elastic constants, Horton [lo] has calculated the ya values for Cu a t different temperatures on the basis of a central-nearest and next-nearest force model. By extending the Hartree-Fock self-consistent approach, Toya [ 111 also attempted to compute the values of yo for alkali and noble metals from first principles. In one of his rigorous approaches, Sheard [ 121 has derived a relation between thermal expansion and pressure dependence of the elastic constants on an anisotropic elastic continuum picture of the solid. Following the same approach, Collins [13] also used the anisotropic continuum model for the temperature variation study of yo for a number of f.c.c. and b.c.c. structures. The temperature variation of yo for copper has also been studied by Sharma and Joshi [14] on the basis of an anisotropic dispersive continuum model, and by Gupta et al. 1151 on the basis of the Sharma and Joshi model [16].

However, all the lattice-dynamical models used so far suffer from one draw- back or another. By making use of the pressure derivative of the elastic constants, Sharma and Singh [ 171 have tried to study the yo values on the basis of Cheveau's model [18], a more realistic one in the sense that the model satisfies the symmetry requirements of the lattice and preserves the internal equilibrium without recourse to any external force. The Cheveau model, however, overlooks the ion-ion angular forces [ 19,201 which play an important role in lattice vibrations [Zl]. By taking into account the effect of angular forces in the lattice vibration, the authors [22] have recently modified the Cheveau model in order to make it more in accord with the physical situation of the crystal lattice.

Here we have used the modified Cheveau model [22] for the study of ya values of alkali metals at different temperatures using the observed values of elastic constants and their pressure derivatives. The model is found to give a fairly reasonable explanation of the lattice-dynamics1 behaviour of platinum and other b.c.c. metals [23]. It was thought worthwhile to study the thermal expansion and Griineisen parameter yc on the basis of a more realistic model. An independent stimulus for such a study was also imparted by the recently available experimental data for the parameter ye.

2. Theory

In the harmonic approximation the unit crystal containing N atoms can be represented by a set of independent linear oscillators with 3N normal modes of vibration, and the total energy for the system can be given by

3N

j=1 '%, % . . . I n3N = uC f .z nfhop#j 9 (2)

Temperature Dependence of Griineisen Parameters in Alkali Metals 383

where U, is the internal energy of the system at absolute zero including the zero point energy. The second term corresponds to the vibrations of atoms about their equilibrium position. Using the partition function

the free energy for the system is obtained:

where wq,j is the angular frequency of a normal mode of wave vector q and polarization j . Equation (4) with the use of the thermodynamic relation for the volume expansion coefficient u of a solid given by

gives the Griineisen relation (l), where the Griineisen parameter yQ representing the mean value of the microscopic Griineisen parameters yq,j with the specific heat contributions from the respective vibrational modes as weighting factors, can be given by

Here E ( x ) is the Einstein specific heat function given by

x2 ex h%,j with x = - k T E ( x ) = (ex - 1)2 and

With the introduction of volume-dependent frequencies [24] in (8), the theory moves from the harmonic to the quasi-harmonic approximation for the lattice. Replacing the summation over q by an integration over allowed values of q within the first Brillouin zone, equation (6) reduces to

where f2 is the solid angle in wave vector space. It is clear from (6) that a t high temperatures ya is simply the arithmetical mean of yq,j, but a t low temperatures yq,j values for lower frequencies give a major contribution to the average value so that the parameter yG in general becomes a temperature-dependent factor.

3. Numerical Calculation

The Griineisen parameters yQ for the alkali metals lithium, sodium, potassium, and rubidium are calculated at different temperatures from equation (9) by using the modified Houston method. The integration over q is performed

384 V. P. SINGH and M. P. HEMKAR

I m i [J71] I ! I l l I l l 1

numerically and the integration over D is evaluated by using the modified Houston spherical six-term integration method as developed by Betts et al. [25]. The applicability of this procedure for such calculations particularly a t lower temperatures is widely discussed by several workers [26]. This method gives proper weight to each reciprocal lattice point and is inherently preferable to the sampling technique which relies merely on taking a very large number of points in the reciprocal space. The six directions used for the calculation are [loo], [110], [lll], [210], [211], and [221]. The dynamical matrix giving the angular frequencies for the normal mode of lattice vibrations is solved for these symmetry directions and the solutions are used for the evaluation of the integral over q.

The microscopic Griineisen parameters yq,j in (8) are calculated in terms of observed values of the elastic constants C,, C,, and C, and their pressure derivatives by using the phonon dispersion relations along the principal symmetry directions. In the calculation we have used the temperature-dependent values of the elastic constants and the lattice parameter. The experimental values for the pressure derivatives are taken at room temperature as no data are available for them over a wide temperature range.

4. Results and Discussion

Fig. 1 shows the computed values of the microscopic Griineisen parameters yq,j for the alkali metals lithium, sodium, potassium, and rubidium as a function of the reduced wave vector along the three symmetry directions [loo], [110], and [l l l] . The results shown in the figure correspond to the room temperature values of the pressure derivatives of the elastic constants experimentally mea-

r I I

DW 05

Fig. 1. Microscopic Griineisen parameters yq,j for Li, Na, K, and Rb along the principal symmetry directions [IOO], [110], and [lll] at room temperature. Pressure derivative data refer to the measurements of Daniels [27] for sodium, of Smithand Smith [28] for potassium, and to theoretical calculations of Suzuki et al. [29] for lithium and rubidium. “L” and “T” stand for the longitudinal and transverse modes. Theoretical curves: - Li; - - - Na; .-.-. K; -..-..- Rb


__/--

700 vs, -

Fig. 3

Fig. 2. The effective Griineisen parameters ture T/O, . Experimental points (0) are taken from Gray 1351

Fig. 3. The effective Griineisen parameters 70 for sodium as a function of T/O,. The solid curve refers to the experimental pressure derivative data, the dashed curve to the theoretical pressure derivative data. Experimental points (0) are taken from Siege1 and Quimby [36]

for lithium as a function of reduced tempera-

sured by Daniels 1271 and Smith and Smith 1281 for sodium and potassium, respectively. The pressure derivative data of lithium and rubidium are taken from the theoretical calculations of Suzuki et al. [29] for lithium, sodium, potassium, and rubidium on the basis of the model local pseudopotential of Ashcroft [30]. The calculation for the mean values of the Griineisen parameters

shown in Fig. 2 to 5 for lithium, sodium, potassium, and rubidium, respectively, makes use of the temperature-dependent values for the elastic constants [31 to 341. In the calculation of the parameter yo for sodium and potassium we have used both the theoretical [29] and the experimental [27,28] values of the pressure derivatives. For comparison we have also shown in the figures the experimental results for ya derived from measurements on thermal expansion and specific heat. The values for the pressure derivatives of the elastic constants and the temperature range and sources for thermal expansion data are given in Table 1.

The microscopic Griineisen parameters yg,j shown in Fig. 1 give a picture of the relative contributions of longitudinal and transverse modes of lattice vibrations to the effective values of the parameter yo. It shows that the yq,j values

/-- --- ___---- .

rreo - Tree - Fig. 4 Fig. 6

Fig. 4. The effective Griineisen'parameters YQ for potassium as a function of T/OW The details are the same as in Pig. 3. Experimental points (0 ) are taken from Monfort and Swenson [37]

Fig. 5. The effective Griineisen parameters 70 for rubidium as a function of T/O,. Experimen- tal points (0) are taken from Kelly and Pearson [38]

26 physica (b) 78/1

386 V. P. SINQH and M. P. HEMU

metal

Table 1 Pressure derivatives of elastic constants and the experimental data of thermal expansion

measurements

temperature range (K) ref.

3.1163 3.4493 3.4783

3.8026 3.6603 3.6633

Li Na

K

Rb

1.034 1.630 1.667

1.629 1.639 1.666

76to300 80 to 290

20 to 320

77 to 297

~351 ~361

pressure derivatives

c;,

3.2863 3.9013 0

3.9303

4.3046 4.1663 4.1693

ref.

for the longitudinal modes are comparatively higher than those due to the transverse modes. Thus the longitudinal branches give a major contribution to the effective Gruneisen parameters ya and hence account for the bulk of anharmonic effects in the crystal lattice.

The mean values of the Griineisen parameters ya show a fairly good agreement with the experiments for all metals investigated. The computed values for lithium are lower than those expected from the experiments. Due to the large scattering in the experimental results, i t is difficult to estimate quantitatively the discrepancies between the theoretical and experimental results for potassium. At low temperatures we see a considerable difference between the computed values obtained from the two sets of pressure derivative data for the elastic constants. However, both results are almost the same in the case of sodium.

As things stand, it is clear from the present study that the modified Cheveau model gives an adequate interpretation for the temperature dependence of the Gruneisen parameter yo for all alkali metals. It is expected that a consideration of temperature-dependent values of the pressure derivatives of the elastic constants may improve the results further.

Acknowledgements

The authors are grateful to Prof. Vachaspati, Head of the Physics Depart- ment, for providing facilities and continuous encouragement, and Dr. s. Chan- dra, Dr. J. Prakash, Mr. L. P. Pathak, and Mr. H. L. Kharoo for helpful dis- cussions. The authors are also thankful to the C.S.I.R., India, for financial support.

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Dynamical study of the temperature dependence of grüneisen parameters in alkali metals

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