B. PRASAD and R. S. SRWASTAVA: Thermal Properties and Griineisen Parameters 771

phys. stat. sol. (b) 87, 771 (1978)

Subject classification: 6; 8; 21.2

Department of Physics, Banaras Hindu University, Varanasil)

Thermal Properties and Gruneisen Parameters of Alkali Metals Employing the Model Potential of Krasko and Gurskii

BY B. PRASAD and R. S. SRNASTAVA

A new form of the local model potential proposed by Krasko and Gurskii is used for computa- tion of phonon frequencies, specific heat, and Griineisen parameters for the two alkali metals, potassium and sodium. The Brillouin zone is divided into lo00 miniature cells. Corresponding to the centre of each miniature cell, phonon frequencies and Griineisen parameters are computed and used for studying the variation of the mean value of the Griineisen parameter with temper- ature. Variation of the thermal expansion coefficient with temperature is also studied in the manner of Wallace. The computed results are discussed in the light of available experimental and theoret- ical results. There is over all qualitative agreement between theory and experiment.

Eine neue Form des von Krasko und Gurskii vorgeschlagenen lokalen Modellpotentials wird zur Berechnung der Phononenfrequenzen, spezifischen Wiirme und Griineisenparameter fur die beiden Alkalimetalle K a h m und Natrium benutzt. Die Brillouinzone wird dam in 1000 Miniaturzellen geteilt. Entsprechend dem Zentrum jeder Miniaturzelle werden Phononenfrequenzen und Griin- eisenparameter berechnet und fiir die Untersuchung der Anderung des Mittelwertes des Griineisen- parameters mit der Temperatur benutzt. Die hde rung des thermischen Ausdehnungskoeffizienten mit der Temperatur wird ebenfalls ngch der Methode von Wallace untersucht. Die berechneten Werte werden mit verfiigbaren experimentellen und theoretischen Ergebnissen diskutiert. Es wird eine insgesamt qualitative ffbereinstimmung zwischen Theorie und Experiment gefunden.

1. Introduction In our recent paper [l to 31 we have Calculated the lattice dynamical and thermal

properties of several simple metals. The computed results were found to be in good agreement with the experimental values. The same model is used here to study the similar properties for sodium and potassium. Among all the alkali metals, the study of potassium is the simplest from theoretical and experimental points of view. No phase transformation or specific heat anomalies have been observed in the case of potassium, but sodium [4] undergoes a transition of a martensitic type at about 36 K as Barret [5] found. The b.c.c. structure persists below the transition point in distorted form, dis- rupted by small inclusions of imperfect h.c.p. type. That is why Dagens et al. [6] point- ed out that neither Li nor to a lesser degree Na are entirely simple metals. Potassium is more favourable than sodium from the experimental point of view as it has a higher ratio of coherent to incoherent scattering cross-sections [7]. The phonon frequencies and Griineisen parameters in three symmetry directions have been measured for K by IS, 71 and for Na by [9, 101 using neutron diffraction techniques. Several experimental measurements have been made [ll to 151 to study the variation of mean value of the Griineisen parameter and thermal expansion coefficient with temperature.

Theoretical studies pertaining to the thermal properties of sodium and potassium have received comparatively greater attention. Toya [ 161 calculated the Griineisen parameters of alkali metals and of Cu. Using Toya's theory Srivastava and Singh [ 171 and Prakash and Joshi [18] calculated the Griineisen parameter of Na, K, and Al,

1) Varanasi-221006, India.

772 B. PRASAD and R. S. SRIVASTAVA

respectively. Wallace [ 19 to 211 computed the spectrum of the Griineisen parameter for Na, K, Li, and A1 using Harrison’s [22] modified point-ion model potential. The Griineisen parameters of K have been calculated by Taylor and Glyde [23] using the model potential of Dagens et al. [6]. But the comparison with the observed Griin- eisen parameters due to Mayer et al. [7] shows some significant discrepancies between theory and experiment.

2. Computation of Phonon Frequencies, Specific Heat, and Gruneisen Parameters

The angular frequency uqj is obtained by solving the secular determinant in the usual way as described in our previous paper [26]. For computing the Griineisen par- ameters yqj, the value of d(w,j)/dv is obtained by differentiating the secular deter- minant and solving it for d(oqj)/dv as given in paper [l].

2.1 Computation of thermal expansion coefficient

We have computed the thermal expansion coefficient B for K and Na in the manner of Wallace [191. We define

where BT is the isothermal bulk modulus and F the total Helmholtz free energy. Neg- lecting the anharmonic and the electronic excitation contributions as being small for Na and K, we can write the quasiharmonic lattice dynamical contribution in a dimen- sionless form as

3. Numerical Computation

The calculation of phonon frequencies uej and Griineisen parameters yqj has been made for 1000 points uniformly distributed in the first Brillouin zone with the par- ameters given in Table 1. These 1000 points, however, are reduced to only 47 points from symmetry considerations. The numerical results of phonon frequencies for longi- tudinal and transverse modes along three principal symmetry directions are compared in Fig. l a with the experimental results of Cowley et al. [8] for K and in Fig. l b with the experimental results of Woods et al. [9] for Na. Similarly the numerical results for Griineisen parameters are drawn in Fig. 2 a and b for K and Na, respec- tively. The Griineisen parameters of K have been compared with recent neutron data due to Mayer et al. [7]. No coniparison is made for Ka because paper [lo] is not available in this laboratory.

Table 1 Data used in the calculations (at. units) along with the values of 7 calculated from expressions

of Nozieres-Pines (NP) and Geldart-Vosko (GV)

I DO I a I rC I ’ I ‘s I ~~theoretica1~ 7(NP) I q(0V)

K 481.2 0.689 4.862 0.394 Nr& 1 254.5 1 i:::: I 0.487 1 i 1 3.931 1 0.488 1 i:;:: 1 ::::: 1

The specific heats of these two metals have been calculated by Blackman’s sampling technique 1271. For this purpose the frequencies were divided into intervals of Am = = 0.2 x 10l2 He and the specific heats were evaluated from Einstein’s function cor-

Thermal Properties and Griineisen Parameters of Alkali Metals 773

Fig. 1. Phonon frequencies of a) K and b) Na. - calculated; 0, x experimental

a I 1 I 1 I

0 02 a6 %O 126 02 0 a2 aS 4- -4-

b

Fig. 2. Griineisen parameter y of a) K and b) Na. ~ calculated; 0, experimental

responding to the mid-point of each interval. These values of specific heats were converted into the corresponding Debye temperature 8,. The theoretical 8,-T curve is compared in Fig. 3a with the experimental results of Roberts [28] and Simon and Zeidler [29] for K and in Fig. 3b with the experimental measurements of Martin [30] and Parkinson and Quarrington [31] for Na.

The value of 7th at a particular temperature has been estimated on the basis of [l], assigning a proper weight factor to yqj. 7th at different temperatures has been computed by Blackman’s sampling techniques and is shown in Fig. 4 a and b for K and Na, respectively. It has also been compared with the experimentally measured values.

We have calculated the thermal expansion coefficient in a dimensionless form using equation (2). The calculations were done a t the volume which corresponds to zero temperature and pressure and is given in Table 1. These calculations are compared with the corresponding experimentally measured quantities and with those of Wallace in Fig. 5 a and b.

Fig. 3. eD-T curve for a) K and b) Na. - calculated; 0, experimental

774 B. PRASAD and R. S. SRIVASTAVA

\l3-1 &* 7 3 i;r/F’. 0 b

TfKl - TIKI - 70 20 z 40 N 60 a 725 1 loo a 70 zo 30 40 M 60

Fig. 4. Mean value of Griineisen parameter, yth, versus temperature T for a) K and b) Na. calculated ; o experimental

a b a 100 200 300 0 700 zoo 300

TfKl - TfKl - Fig. 5. Thermal expansion coefficient versus temperature for a) K and b) Na. ___ calculated;

o Wallace [19], - - - - experimental

4. Results and Discussions 4.1 Potassium

Fig. 1 a and 2a show the phonon frequency and Griineisen dispersion curves, respec- tively. As can be seen from Fig. l a , the phonon frequencies computed from the present model potential, agree fairly well with the observed values of Cowley et al. [8]. However, from Fig. 2a, we see that the computed Griineisen parameters are not in fair agreement with the observed values due to Mayer et el. [7]. This shows that our results are similar to those of very recent calculations made by Taylor and Glyde [23].

The theoretical 8,-T curve for K is shown in Fig. 3a along with the experimental data due to [28, 291. It is clear from the figure that there is a qualitative agreement between theoretical and experimental points. All the calculated values are slightly lower than those of experimental values.

The temperature variation of 7th has been studied and the results are shown in Fig. 4a along with experimental measurements due to Monfort and Swenson [12]. A qualitative reasonable agreement is observed.

The calculated quasiharmonic curve of /3BTV7/3kB as a function of temperature T is shown in Fig. 5a along with the experimental curve [19]. The agreement found is fairly satisfactory and compares well with Wallace’s results [ 191. The experimental values of the thermal expansion coefficient /3 and BT for K have been taken from [12, 151. Buyers and Cowley [32] also calculated the thermal expansion coefficient for K on the basis of anharmonic effects. But their results deviate [17] nearly by 25% from the experimental values due to Monfort and Swenson [8].

4 2 Sodium

Fig. l b and 2b show the phonon frequency and Griineisen dispersion curves, respectively. It is clear from Fig. lb that our calculated phonon frequencies compare well with experimentally measured values found by Woods et al. 191. No comparison

Thermal Properties and Griineisen Parameters of Alkali Metals 775

is made for the spectrum of Griineisen parameters because the neutron data found by Ernst [lo] are not available in this laboratory.

The theoretical Debye characteristic temperature 8, against temperature T has been plotted in Fig. 3 b. For comparison the experimental points are also shown. Fig. 3 b shows that there is a good agreement between the experimental and the theoretical values up to about 35 K. Below this temperature there is a discrepancy between the cal- culated and experimental values. In the low-temperature region a rigorous comparison between theory and experiment is not meaningful due to phase transformations of Ka from b.c.c. to h.c.p.

The temperature variation of Yth is presented in Fig. 4b along with the experimental measurements due to Siege1 andQuimby [ll]. It is seen that the agreement improves as the temperature increases.

Fig. 5 b shows the temperature variation of the thermal expansion coefficient in the form of pBT v/3kBT. In comparing theory and experiment, the function p & v / 3 k ~ T was chosen because it is theoretically simple; according to equation (2) it is just a weighted average of the phonon Griineisen parameters. The experimental values for p and BT are taken from [ll, 13, la]. As is evident from the figure, a fair agreement with the experimental results is observed.

6. Summary and Conclusions It is worth remarking that the model potential used in the present work represents

a better qualitative agreement with the experimental data on phonon frequencies, specific heat, Griineisen parameters, and thermal expansion coefficients of alkali metals. However, a few comments should be offered about the discrepancies between theory and experiment. We conclude by considering the following steps that could taken to bring the theory into bett.er agreement with experiment.

(i) One can improve the agreement by considering higher-order pseudopotential terms in the dynamical matrix [22].

(ii) The neglect of the short-range Born-Mayer exchange repulsion term between ion cores may have affected the results slightly.

(iii) The deviation of the Fermi surface and the electron wave functions from their true behaviour are more important.

(iv) The neglect of the anharmonic contribution to the free energy and the elec- tronic excitation contribution, may affect our result. But we have not taken this into consideration.

Acknowledgements

The authors express their sincere thanks to Prof. Roger Taylor, N.R.C. Canada, for his useful comments and suggestions throughout the work. Thanks are also due to Mr. G. La1 for his assistance in the preparation of this manuscript.

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776 B. PRASAD and R. S. SRIVASTAVA: Thermal Properties and Griineisen Parameters

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York 1966.

(Received March 7, 1978)