Short Notes K9

phys. stat. sol. (b) - 123, K9 (1984)

Subject classification: 6; 4; 21.3

Applied Physics Section, Institute of Technology, Banaras Hindu University, Varanasi l )

Phonon Frequencies and Einding Energy of F . C . C . Calcium

BY E. PRASAD

Several theoretical calculations of the phonon frequencies and the binding energy

of calcium have been performed within the framework of the pseudopotential

theory of metals. Pseudopotential calculations of the phonon frequencies of

calcium had been done by Animalu /1/ using the Heine-Abarenkov form factors

/1, 21. Taut andEschrig /3/ have calculated the phonon frequencies of this

metal using a non-local pseudopotential. Moriarty /4/ calculated the phonon

spectra and the binding energy of this metal employing a psrudopotential ap-

proach which takes into account the effect of hybridization with the d-states.

Recently, we investigated the phonon frequencies and binding energies of

several metals 1 5 to 7/ using the local pseudopotential proposed by Gurskii and

Krasko / 8 / . The theoretical results are found to be in good agreement with ex-

perimental data. More recently, the neutron scattering data on calcium have

been made available by Stassis et al . / 9 / and it is worthwhile t o compare the

above model with these neutron data.

Following Gurskii and Krasko /8/, a local model potential may be written

as

(1 )

ar: I , 87c ze2 [- -b

1

' 0 2 q 2 [ ( q r c ) 2 + ~ ~ + l q r c ) 2 +112 W O ( d =

where q = 16 + 21 with 4 and h the phonon wave vector and reciprocal lattice

vector, respectively; a and r a r e the model parameters . Proceeding in usual

way the secular determinant for calculating phonon frequencies is written as C

(2 1 2 ID,@) - mIw I = 0 ,

where m denotes the mass of ion, I the unitary matrix of order three,and w

the c i rcular frequency. The main contributions t o the dynamical matrix D 4 Ca a r e the electrostatic and band s t ructure energies. The electrostatic con-

in

1 ) Varanasi 221 005, India.

K10 physica status solidi (b) 123

experimental

1.458

6

- 1. ‘ “ 3 d

5 2

3 1

0 05 70 u5 0 05

theoretical Moriarty /13/

1.466 1.478

Fig. 1. Dispersion curves for Ca; - this work; ---1Moriarty; 0, 0 T branch, respectively, experimental; x T branch, experimental

L and

1

tributions have been taken directly from / lo / . The electronic contribution t o the

dynamical matrix has been calculated using the following expression:

(3) 2 z [h h*O 1 E

D,p(q) == C F ( q ) ( Q + h&(Q+h)o - > F(h)hoLhg ,

where F(q) is called the energy wave number characterist ic defined in / 8 / . All

other symbols have their usual meaning as given in /5, 6/. The total binding

energy (-E ) has been calculated a s outlined in /ll/. b In the case of Ca, the following data have been used in atomic units:

8 =293.5, a =2.745, r = 0.571, Z = 2 . 0 C

Computations of the phonon frequencies along three symmetry directions have

been carr ied out by solving the secular determinant for 48 points. The cal-

culated phonon dispersion curves in three principal symmetry directions have

been drawn and compared in Fig. 1 along with the experimental data due to

Stassis et al. /9/ and the theoretical data due to Moriarty /4/. It is c l ea r from

Fig. 1 that the present results a r e in bet ter agreement with experiment, es-

pecially for the L branches than those obtained by Moriarty. The calculated

value of binding energy along with the experimental value is given in Table 1.

T a b l e 1

Short Notes K11

The theoretical result obtained by Moriarty is also given in the table. It is

c l ea r from Table 1 that the present value ag rees well with the experiment,

bet ter than that obtained by Moriarty. These resul ts indicate that the model

potential /8/ is fairly realist ic.

The resul ts may b e improved by taking a new form of the screening /12/, higher o r d e r pseudopotential t e rms , and Born-Mayer exchange repulsion con-

tributions. However, these have not been tried in the present work.

References

/1/ A. 0. E. ANIMALU, Phys. Rev. - 161, 445 (1967).

/2/ A O.E. ANIMALU, Proc. Roy. SOC. (London) - A294, 376 (1966).

/3/ M. TAUT and H. ESCHFUG, phys. stat. sol. (b) 73, 151 (1976).

/4/ J . A MORIARTY, Phys. Rev. €3 - 6, 4445 (1972). /5/ B. PRASAO and R.S. SRIVASTAVA, Phys. Let ters A - 38, 527 (19'72). /6/ B. PRASAi) and R.S. SRJYASTAVA, J. Phys. F - 3, 18 (1973). / 7 / B. PRASAD and R.S. SRIVASTAVA, phys. s ta t . sol. (b) - 87, 771 (1978). /8/ Z.A. GURSKII and G.L. KRASKO, Soviet Phys. - Solid State 11, 2447

-

- (1970).

/ 9 / C. STASSIS, J. ZERETSKY, D.K. MISEMER, H.L. SKRIVER, B.N.

HAEiMON, and R.M. NICKLOVJ, Phys. Rev. B - 27, 3303 (1983). /lo/ E.W. KELLERMANN, Phil. T rans . Roy. SOC. - A238, 513 (1940).

/11/ V.K. SAXENA, Q.S. KAPOOR, and D .L . EHATTACHARYA, phys. stat. sol. - 34, 145 (1960).

/12/ D . J . W . GELDART and R. TAYLOR, Canad. J. Phys. 48, 155 (1970). /13/ J . A . MORIARTY,

- phys. Rev. B 16, 2537 (1977). -

(Received February 9, 1984)