Short Notes K73

phys. stat. sol. (b) 162, K73 (1990) Subject classification: 63.20; S1.2

Department of Physics, D . A. V. College, Kanpur )

New Approach to the Lattice Dynamical Study of Cubic Metals

BY M.K. MISHRA

Introduction Recently Singh and Rathore 111 have studied the lattice dynamics of some cubic metals based on the generalised Morse 121 potential. According to this study, cohesive energy, lattice constant, and compressibility are the input data for the empirical Morse potential 121. As we know, compressibility and cohesive energy are the sum of ionic interaction and interaction due to electrons. Mishra and Rathore 131 have separated the two- and three-body part of the compressibility on the lines of Mohammad et al. I41 and then have used the Morse potential for a lattice dynamical study of some metals. The most significant contribution to the binding energy which arises f rom the interaction between the metal ions and- the electrons is not included in the potential 1 3 1 , even though the potential is fitted to the total cohesive energy.

In this note 1 have developed a three-body potential based on a two-body Morse potential to predict the lattice dynamical behaviour of f.c.c. cobalt. The ionic part of the compressibility and of the cohesive energy have been used as inputsfor the evaluation of parameters of the two-body potential.

Theory The response of the loosely coupled electrons towards the ionic displacement necessitates the inclusion of a three-body force into the system. For this purpose the effect of the third common nearest (1" , ) neighbour into the coupling of the atom (1' , 1: ) with the atom (1, k) is considered. The trio considered here is located on the vertices of an equilateral triangle. The Morse I21 potential modified to represent the three-body interaction among the said atoms (1,

k ) , (1' , $ ), and (l", ) assumes the form

where r1 and r2 are the respective separations of the atoms ( l 'g ) and (1" ,< )

from the atom (1,k). A(K) is the three-body parameter, a measures the hardness of the potential, and p is a parameter depending upon the equilibrium separation rn, which may be written as

l) Kanpur, India.

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8 = exp(a ro) . The prime at the first summation in (1) means

l’rt # t’ll .

The two-body Morse potential may be expressed as

physica status solidi (b) 162

(2)

where D represents the dissociation energy of the pair, j goes over 140 atoms (including neighbouring atoms) in f.c.c. metals, and r. is the distance of the j-th atom from the origin, i.e.

1

where (11,12,13) are integers representing the co-ordinates of the j-th atom and a is the semi-lattice constant.

The elements of the diagonal and off-diagonal matrix may be given, after solving the usual secular determinant, as

2 2 - 4a C I C I + 48, Sa, t 4a2 (Spt + Spl ) I P r

a a2 are the first and p,, 8, are the second derivatives of the potential while B3 is the second derivative of Q ( 3 )

(r1,r2)’

Computation and discussions Mishra and Rathore /3 / have used the total cohesive energy for the evaluation of the two-body parameters which means that the whole cohesion comes from the two-body interaction only. It is now required to elaborate paired and unpaired parts in terms of the cohesive energy. The following procedure has been adopted for this purpose, i.e.

Q = Q i + Q e , (6)

where Q is the total cohesive energy, Qi is the energy due to ions, 0, is the

Short Notes K75

3 . 1 c - z

6

5

4

3

2

1

I

Fig.

l l l l I 1 l 1 ,

02 04 06 0.8 I I I I I 1 I l y

0.8 OK Q4 02 ,

9- ' -9 0.2 0.4 9-

1. Phonon dispersion curve for cobalt; present study, - 0 - /6 / , 0 , A , , A experimental /5/

energy due to electrons. Further

$e = Ef + Ex + Ec

2 where Ryd, Ex (exchange energy) = -0.9161r Ryd, Ec (correlation energy) = (0.0622 In r - 0.096)Ryd. (1 Ryd = 1 3 . 0 ~ 1 . 6 ~ 1 0 - ~ ~ erg.)

Ef (Fermi energy) = 2.211r

Hence the energy due to electrons is

+ (0.0622 In r - 0.0961) (7 1

here r is a dimensionless quantity and may be valued 2 , 3, 4 , or 5. The three parameters defining the two-body potential $(2) are evaluated by the

knowledge of the equilibrium lattice constant, the ionic part of the compressibility as well as by the cohesive energy of the solid by the procedure laid down by

(r j )

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Girifalco and Weizer 1 7 1 . Finally the three-body parameter A(k) is evaluated from the knowledge of the measured Cauchi discrepancy in the second-order elastic constants.

Fig. 1 shows the dispersion curve for the f.c.c. metal cobalt. My prediction in cobalt follows closely the experimental data 151. The theoretical findings of Gupta 161, however, exhibit deviations from the experimental findings I 5 I .

) are a1 = -0.249, 6, = 2.784,

p3 = 0.032, a2 = 0.047, f3, = -0.034, and the three-body parameter A(k) = - 0 . 3 4 2 ~ 1 0 - ~ ~ erg.

physica status solidi (b) 162

4 -1 dyn/cm The computed force constants (in 10

References /1/ G . SINGH and R.P.S. RATHORE,

phys. stat. sol. (b) 135, 513 (1986); 136, 57 (1986);

Indian J. pure Phys. 24, 303 (1986).

/ 2 / P.M. MORSE, Phys. Rev. 34, 57 (1929).

131 M.K. MISHRA and R.P.S. RATHORE, Acta phys. Polon. 75, 525 (1989).

Phys. Rev. B 29, 3117 (1984);

Solid State Commun. 48, 147 (1983).

/ 4 / K. MOHAMMAD, M.M. SHUKLA, F. MILSTEIN, and J.L. MERG,

151 S.M. SAPIRO and S.C. MOSS, Phys. Rev. B l5, 2726 (1977).

/6/ O.P. GUPTA, Solid State Communn. 42, 31 (1982).

/ 7 / L.A. GIRIFA.T,CO and V.G. WEIZER, Phys. Rev. 114, 687 (1959).

(Received March’ 1, 1990; in revised form September 7, 1990)