P. S. MAHESH and B. DAYAL: Krebs’s Model for the Alkali Metals 351

phys. stat. sol. 9, 351 (1965)

Banaras Hindu University, Department of Physics

Krebs’s Model for the Alkali Metals and the Screening Parameter BY

P. S . MAEESH~) and B. DAYAL

The specific heats of sodium, potassium, rubidium, and caesium are calculated on the basis of Krebs’s model (1). It is found that the theoretical 8- T curves agree with experi- ment if the screening parameter is taken as an adjustable constant. The deduced values of this parameter show a gradual change from the Bohm-Pines to the Thomas-Fermi form during the transition from sodium to caesium. The model, however, fails t o give agreement in the case of lithium. Since the measured elastic constants of this metal do not give the correct value of O,,, the discrepancy for this case may be due to the errors in measurement.

Die spezifischen Warme von Natrium, Kalium, Rubidium und Casium wurden in dem von Krebs vorgeschlagenen Modell berechnet. Es wurde gefunden, daB die theoretischen 0 - T-Kurven fast mit den experimentellen Kurven zusammenfallen, wenn der Abschirm- parameter eine Konstante ist, die an die Kurven anzupassen ist. Die Werte dieser Kon- stanten, die zur Anpassung a n die experimentell bestimmten spezifischen Wiirmen not- wendig sind, zeigen eineii allmiihlichen ubergang von der Bohm-Pines-Theorie zu der von Thomas und Fermi, wenn man vom Natrium zum Casium fortschreitet. Das Modell kann jedoch keine ubereinstimmung fur Lithium ergeben. Da die gemessenen elastischen Kanstanten fur dieses Metal1 nicht die korrekten Werte fur 0, ergeben, kann die Dis- krepanz in diesem Falle auf Menfehlern beruhen.

1. Introduction

Quite recently a new elastic force model has been proposed by Krebs [l] for the study of the lattice dynamice of metals. This model is superior to other force models in the sense that the frequencies given by it are periodic in reciprocal space and thus satisfy the symmetry requirements. He applied this model t o sodium and obtained a very good agreement between the calculated frequencies and the observed dispersion curves of this metal. Shukla [2, 31 from this labo- ratory has applied it to the case of copper and has found that it is able to explain the observed dispersion curves as well as the specific heat data very satisfactorily

In Krebs’s model the influence of the conduction electrons on the equation of motion is considered through the screening of long-range Coulomb inter- action between the ions. In the metallic theory a screened Coulomb potential of this type is found both in the Thomas-Fermi method and in the collective approach of Bohm and Pines. However, the screening parameter given by the latter is about half of that occuring in the Thomas-Fermi theory. Krebs used the Bohm and Pines parameter because the frequencies calculated by its use agreed well with those observed in neutron scattering. The choice is quite reasonable because it is generally expected that the free electron plasma

l) On leave from D.A.V. College, Muzaffarnagar.

352 P. S. NAEESE and R. DAYAL

theory would have its greatest validity for this metal. There is, however, no reason t o suppose that this theory would be applicable t o the heavier atoms. Shukla and Dayal [4] have calculated the specific hcats of the three noble metals copper, silvcr, and gold from Krcbs's model and have found that a very good agreement between the thcortical and experimental data is possiblc if the screening parameter is considered to be an adjustable one. Their studics bring out the important fact that the value of the empirical screening parametcr shows a gradual transition from the Bohm and Pines value t o that of Thomas- Fcrmi as one goes from lighter atoms t o heavier ones. In the present paper, the authors have taken up the cases of alkali metals and have shown that the same type of variation in the values of screening parameter is discernable in the casc of these metals also.

2. Secular Determinant

The following determinant given by Krebs [l] for a body-centered cubic lattice has been adopted for the present computation

IJZij - . 2 2 M I va I = 0,

where i, j = 1, 2, 3 and I is the unit matrix of the order three, M is thc mass of the atom and u is the Hooke's constant for the nearest neighbours given by 112 a C,, a being the lattice constant and C44, the elastic constant. Each element Mi, is the sum of two coupling Coefficients (M,j)i and Iij contributed by the ions and electrons, respectively. There is it slight error in the expressione for coupling coefficients given by Krebs. Thc corrected values for typical diagonal and non-diagonal ionic coupling coefficients for the body-centred cubic metal are of the form

[MI,]' = 1 - cos (n k, a) cos (n k, a ) cos (n k, a) -+ c ! sin2 (n k, a ) ,

[Af,,]' = sin (n k, a) sin (n k, a) cos (n k, a) . 2 c44

The electronic coupling coefficient Iij is given by

(3)

where Cij are the clastic constants, 2 n ki are the components of the phonon

wave vcctor 2 n k (k = l / A ) a n d qi are cqual to -' . The vector h = (h, , h,. h,)

is defined by -h- , kh being the reciprocal lattice vector. A is the screening para-

meter (generally written as k,) related to the Fermi radius kF by the relation

a k. 2 n a k

2 n

3. = A T,'/' kF . (4)

r, is thc interelectronic distance which is equal to the radius of the atomic sphere in monoatomic metals and is given in Bohr units. A is a numerical

Krebs’s Model for the Alkali Metals and the Screening Parameter 353

Na K Rb cs

constant, the value of which was taken by Krebs as 0.353 from the Bohm and Pines plasma theory. I n this paper this is considered as an empirical parameter adjusted to give the best fit with the specific heat data.

0.945 0.779 0.554 0.459 0.372 0.263 0.330 0.286 0.196 0.245 0.208 0.159

The functions f ( t ) and g(u) are defined below:

and 3 (sin u - u cos u) 2 n r * ICI + hl du) = , u1 = 9 a

n h t - - )

a kF 9 2 -

u, =

3. Evaluation of Frequency Spectrum and Specific Heats

For the putpose of frequency computation, the. reciprocal space has been divided into miniature cells whose axes are one-tenth of the axes of the ordinary reciprocal cell. This gives three thousand frequencies corresponding to the 1000 points of the miniature lattice lying within the first BrilEouin zone. From the symmetry requirements these 1000 points are reduced to only 47 non-equi- valent points lying within the 1/4Sti1 part of the Brillouin zone which is irredu- cible under the symmetry operations that leave the roots of the secular deter- minant unchanged. The calculations of frequencies have been made for these non-equivalent points from the secular determinant (1) . Each frequency has been weighted according to the symmetrically equivalent points.

At very lowtemperatures, this mesh of points becomes too coarse for accurate evaluation of specific heats. The main contributions in this range arise from the central part of the Brillouin zone. We, therefore, took a finer mesh for the central part. The central portion consisting of eight miniature cells was sepa- rated and further sub-divided into smaller units so that it contained 8 x 8 = 64 smaller cells. The frequencies for the additional points were also calculated and the statistical weights were adjusted accordingly.

Even this mesh is too coarse for temperatures below 5 OK. For this reason the specific heats were not calculated below this temperature. 8 - T curves have been extraljolated to give the value of 8, as given by de Launay’s formula

The elastic constants used in the calculation have been given in Table 1. The constants for sodium are the same as used by Krebs. The values for the other three metals have been taken from the review article of Huntington [6].

r51.

Table 1

Elastic constanta of the alkali metals in units of 1011 dyn/cm2

354 P. S. MAHESH and B. DAYAL

30 t

Fig. 1. 0 - T curves for aodium, potasslum, rubi-

0 Filby and Martin x Martin 0 Krier et al. dium, and caesium. Experimental data are:

0 I0 20 30 40 50 7PK)-

For rubidium and caesium the experimental data are not available and the figures given in Table 1 are the theoretical values.2)

The C, have been calculated by numerical computation from histograms. For this purpose the frequencies have been divided into intervals of 0.2 x 10l2 5-1

in case of sodium, 0.1 x 1012 s-1 for potassium, and 0.05 x 1012 s-1 fur rubidium and caesium. The constant A has been empirically taken to be 0.353 for sodium, 0.45 for potassium, and 0.814 in the case of rubidium and,caesium.

The calculated 0-T curves for the four alkali metals Na, K, Rb, and Cs are plotted in Fig. 1. The experimental points are shown as solid circles, crosses, and circles. The sdid circles represent the data of Filby and Martin [7, 81 and the crosses those of Martin 1191. The circles represent the measurements of Krier, Craig, and Wallace [lo].

The values of 7, the coefficient of electronic specific heat,, have been taken from Filby and Martin as 390, 497, 576, and 764 in units of pcal/"K2 g-atom for sodium, potassium, rubidium, and caesium, respectively.

4. Discussion

Fig. 1 shows that there is an excellent agreement between the calculated and experimental values of 0 for the four alkali metals a t all temperatures provided we used the empirical values of t,he screening constant A given above. The very small discrepancy for Rb a t low temperatures is also liable t o disappear com- pletely if A is taken to have a slightly lower value of 0.75. As it is, the agree-

2, The theoretical elastic constants are unpublished results of Bailyn privately commun- icated to Dr. Huntington [6].

Krebs’s Model for t h e Alkali Metals and the Screening Parameter 355

Na K Rb cs

ment in Fig. 1 is so satisfactory t,hat it was not considered necessary to repeat the calculations for this value.

An attempt was also made to study the specific heats of lithium by means of this model. Various values of the constant A were tried but no agreement was found between the calculated and the experimental values of 8. The only measurements of the elastic constants of this metal are by Nash and Smith [ll]. Alers and Neighbours [12] have calculated 8,, the characteristic temperature for T = 0 from these constats and have obtained a value 334.6 which is signi- ficantly different from 369.0 obtained from the specific heat data. This dis- agreement throws some doubt on the accuracy of measurement of the elastic constants. This is particularly so because the discrepancy observed in the case of our calculated values of 6 for higber temperatures is of the same order and in the same direction as observed by Alers and Neighbours.

The problem of screening in metals is highly complex. According to the free electron plasma theory of Bohm and Pines, the screening constant A in (4) should have a value 0.353 while its value in the Thomas and Fermi theory is 0.814 which is more than twice the former. There are difficulties in reconciling these values. Ziman [13] has discussed this point and considers the value of Bohm and Pines to be more reliable for those metals where the density of states is near the free electron value. However, there is no guidance about its value in the other cases and as pointed out by him the experimental information6 is inconclusive on this point.

Krebs’s theory has been able to explain the neutron scattering results on the dispersion curves of sodium (Krebs [l]) and copper (Shukla [2]) very satisfac- torily. Amongst monoatomic metals, these are the only ones for which such data are available. The theory has also been able to give a very accurate explanation of the specific heats of the noble metals (Shukla and Dagal [4]) and of the alka- lies. We can, therefore, draw some valid conclusions about the screening para- meter from it. These results show that in both the groups of metals there is a gradual transition from the Bohm and Pines theory to that of Thomas and Fermi as we go from the lower members to those higher up in the series.

I n the case of the alkalimetals, the increase in A follows the order of ?. It is, therefore, possible to connect it with the density of states. However, it is not possible to do so in the case of the noble metals where the variation in

m m

0.95 1.33 1.48 1.69

m -from one member to another is negligible. This can be seen from Table 2 m* where we have given the - and A for both the groups of metals. The values m

m* of A for the noble metals have been taken from the paper of Shukla and

m Dayal [4], and those of from Ziman [13] and Kittel [15].

m

T a b l e 2

m

m* -

1.02 1.08 1.13 1.21

A 1 1 Element Ionic radii (4 0.96 1.26 1.37

m m*

0.988 1.008 1.006

- A

0.353 0.50 0.814

356 P. S. MAHESH and B. DAYAL: Krebs’s Model for the Alkali Metals

Raimes [14] has pointed out that in the atoms where the ion cores are large, the distinction between the valence and the core electrons tends to break down and we should not expect the free electron plasma theory to apply to such cases. The present study ;shows that the metals with very large ionic sizes seem t o obey the Thomas-Permi theory. I n Table 2 we have also given the ionic radii of the various metals taken from Kittel’s book [15]. We find that the transition from Bohm and Pines theory to that of Thomas and Permi is represented by the gradual increase in the sizes of ions.

The authors thank Dr. A. R. Verma for the research facilities in the depart- ment. The younger author (P.S.M.) also thanks the authorities of the D.A.V. College, Muzaffarnagar, for leave facilities.

References [I] K. KREBS, Phys. Letters (Netherlands) 10, 12 (1964). [a] M. M. SHUHLA, phys. stat. sol. 7, K11 (1964). [3] M. M. SHUKLA, phys. stat. sol. 8, 475 (1965). [4] M. M. SHUKLA and B. DAYAL, under publication in J. Phys. Chem. Solids. [S] J. DE LAUNAY, Solid State Physics, Vol. 2, Academic Press Inc., New York 1956

[6] H. B. HUNTINGTON, Solid State Physics, Vol. 7, Academic Press Inc., New York 1958

[7] J. D. FILBY and D. L. MARTIN, Proc. Roy. SOC. A 276, 187 (1963). [8] J. D. FILBY and D. L. MARTIN (private communication). [9] D. L. MARTIN, Proc. Roy. SOC. A 254, 433 (1960).

(p. 285).

(p. 288).

[lo] C. A. KRIER, R. S. CRAIG, and W. E. WALLACE, J. phys. Chem. 61, 522 (1957). [I I ] H. C. NASH and C. S. SMITH, Bull. Amer. Phys. SOC. (Ser. 11) 3, 123 (1958). [la] G. A. ALERS and J. R. NEIQHBOURS, Rev. mod. Phys. 31, 3, 675 (1959). [13] J. M. ZIMAN, Electrons and Phonons, Chap. V, Oxford University Press, 1963. [I41 S. RAIMES, The Wave Mechanics of Electrons in Metals, Chap. X, North Holland

[I51 C. KITTEL, Introduction t o Solid State Physics, Asia Publ. House, Bombay 1960

(Received January 27,1965)

Publishing Company, Amsterdam 1961.

(P. 82).