Solid State Communications, Vol. 42, No. 1, pp. 31-32. 1982. Printed in Great Britain.

0038 - 1098/82/130031-02 $03.00/0. Pergamon Press Ltd.

CRYSTAL DYNAMICS OF FACE CENTERED CUBIC COBALT

O.P. Gupta

Physics Department, J. Christian College, Allahabad, India

(Received 30 October 1981 by G.S. Zhdanov)

An equilibrium condition which preserves the crystal stability is derived. This condition in conjunction with recent dynamical model is employed to study the crystal dynamics of f.c.c. Co. The agreement between theory and experiment is good.

1. INTRODUCTION

ELEMENTAL Co undergoes martensitic phase trans- formation from f.c.c, to h.c.p, structure at 693 K. Single crystal of f~e.c. Co has not been studied to date because of the facts (a) 693 K phase transformation is unstable and (b) Co is a strong neutron absorber. Since Fe and Co have similar atomic masses and sizes, the alloy Coo.92Feo.oa may be assumed as an equivalent of pure f.c~. Co. The experimental results on phonon energies [1] and phonon frequencies [2] can be used to test the reasonableness of the interpretation in terms of pure f~.c. Co.

In this paper, we report the phonon dispersion, fre- quency spectrum, and temperature dependence of equivalent Debye temperature O, and Debye-Waller (DW) factor of f.c.c. Co on the basis of our recent dynamical model [3, 4] in conjunction with an equilibrium condition. The condition, which preserves the crystal stability, is derived by considering Fermi, exchange, and correlation energy of conduction electrons.

2. THE MODEL

For equilibrium, the first derivative of the potential energy E (= E ~ + E e) of the crystal must vanish, i.e.

( d E i / d I 2 ) a - - e e = 0, where--PC = dEe/d[2, (1)

where [2 is crystal volume, pe the electron pressure, E t the potential energy due to ion-ion interaction, and E e the electron energy. Following Gellman and Brueckner [5], the Fermi, exchange, and correlation parts o fE e can be written as:

E e = [2.21Z2/a(ro/ao) - 2 - 0 . 9 1 6 Z l / a ( r o / a o ) - '

+ 0.622 In (ro/ao Z l / a ) -- 0.096]. (2)

The equilibrium condition equation (1) with the aid of equation (2) takes the form:

13.6a [ 4.41Z 2/3 0.916Z 1/3 A1 +A2 - [- t-

1fro (ro/ao) 2 (ro/ao)

+ 0.622Z-1/3], (3)

where a0 is the Bohr radius, Z the chemical valence, ro the interelectron spacing, and Aj ( / = 1, 2) are tangential pair potential force constants. Three equations of elastic constants [4], one ZB frequency relation [4], and equation (3) form necessary set of equations to deter- mine force constants.

3. NUMERICAL COMPUTATION

Experimental values of elastic constants and other parameters used in calculations are: Cn = 22.1, Ct2 = 14.7, C44 = 12A (all in units of l0 II dyncm -2 [1]), a = 3.55A, VT(100) = 5.87 THz [1],Z = 2, density = 8.7148 gm cm -3, and (m*/m) = 1. The calculated force constants are: A1 = -- 10.639,A2 = 2.717, B1 = 65.067, B2 = 6.202, and ake = -- 55.208 (all in units of 103 dyn cm-l). These force constants are used to calcu- late frequency/wavevector dispersion relations along the principal symmetry directions. The frequency spectrum, lattice specific heat, and DW factor are calculated as described earlier [3, 4].

4. RESULTS AND DISCUSSION

We display our theoretical results for phonon disper- sion along with the experimental values [1, 2] in Fig. 1. The agreement between theory and experiment is good. The frequency distribution of Shapiro and Moss [1 ], Svensson et al. [2], and our computed values are plotted in Fig. 2. Our theoretical values compare well with the experiments. Our theoretical results for lattice specific heat expressed in terms of O are depicted in Fig. 3 along with the values of Shapiro and Moss [1], and Svensson et al. [2]. The general behaviour and the

31

32 CRYSTAL DYNAMICS OF FACE CENTERED CUBIC COBALT Vol. 42, No. 1

8

6

~5

,,=,~

LU ==3

2

I

o!

A~ X K ~I: i i t i , i i

I I I I I I I 0 . 2 0 . z 0 ,6 0 ,8 1.0 0 . B 0 . 6 0 . 4 0 .2

REDUCED WAVE V E C T O R

i A ~

i i i l

y //

I I 0.1 0'.2 0'.3 0.4 0.5

Fig. 1. Phonon dispersion curves of f.c.c. Co. - - , Present calculations. Experimental points: o, Shapiro and Moss; o, Svensson et al.

0.2

I 0 T(K) --" 200 800

.-< - 0.2

-0 .4

- 0.6 To = 298K \ ~ X

-0.8 ~ ~~" \ ~~,,

- 1.0

Fig. 4. Debye-Waller factor of f.c.c. Co as a function of temperature. - - , Present calculation, Experimental curve: . . . . , Svensson et al.

m

2

.r,

<

g

I I I I I I I I I

3 .,.: / A - . "

2 'ii

l!i , ..." ~ .... I.- L -

0 1 2 3 4 5 6 7 }) (THz) ,-

Fig. 2. F r e q u e n c y d i s t r i b u t i o n curves o f f .c .c. Co. Present ca l cu la t i on . E x p e r i m e n t a l curves: . . . . . , Shap i ro and Moss; - - - - - - , Svensson et al.

450

430

4~0 ®

39O

370

350

I I I I

i I i i 60 120 180 240 300

T(KJ .-

Fig. 3. Equivalent Debye temperature of f.c.c. Co as a function of temperature. , Present calculation. Experimental curves: . . . . . , Shapiro and Moss, Svensson et al.

position of minimum of these curves show closer resemblance. The variation of our ® values with T follows the general prediction of Blackman [6]. In Fig. 4, the calculated values of DW factor agree satisfactorily with the reported values of Svensson et al. [2].

The contribution to Cauchy discrepancy CD by the present model can be written as Cm -- C44 = ICe -- 2P e, i.e. CD is associated with the pressure and bulk modulus of the electron gas. This lends support to Fuchs' [7] prediction.

The present scheme is found to provide a reasonably good explanation for the physical interaction, lattice stability, Cauchy discrepancy, phonon dispersion, fre- quency spectrum, equivalent Debye temperature, and DW factor of f.c.c. Co.

REFERENCES

1. S.M. Shapiro & S.C. Moss, Phys. Rev. B15, 2726 (1977).

2. E.C. Svensson, B.M. Powell, A.D.B. Woods & W.D. Teuehert, Can. Z Phys. 57,253 (1979).

3. O.P. Gupta, H.L. Kharoo & M.P. Hemkar, Can. Z Phys. 56,447 (1978).

4. O.P. Gupta, Lattice Dynamical Study o f Solids, D. Phil. Thesis, Physics Department, Allahabad University (1978).

5. M. Gellman & K.A. Brueekener, Phys. Rev. 106, 364 (1957).

G. M. Blackman, Rep. Prog. Phys. 8, 11 (1941). 7. K. Fuchs, Proc. R. Soc. A153, 634 (1936).