Short Notes

det (ti /2m)f - z)2 - &(k) Shd + U( I 2 - if'l)F(S - 2') I[ 1

K99

= 0 , (1)

phys. stat. sol. (b) g , K29 (1977)

Subject classification: 13.2; 21; 21.2

Department of Physics, University of Allahabad

Density of States and Fermi Energy of Simple Metals

BY MAHENDRA KUMAR and M.P. HEMKAR

K30

reciprocal lattice vector, and F(h - h ) the structure factor. If we know the solution

of this secular equation we would know the energy bands in the metal and could then

compute the Fermi surface, the density of states, and other properties.

physica status solidi (b) 82

* + I

A t the melting point, the above expression reduces to /ll/*

where A(Q) is the Fourier transform of the pair distribution function and can be ob-

tained from X-ray o r neutron diffraction data. On the basis of the isotropic properties

of A (Q) and U(Q), the energy at the Fermi level can be written a s

where

and n is the number of atoms per unit volume related to the Fermi wave number k

and valence Z by the relation F

2 1/3 k = ( 3 r nZ) . F

Also, from (2) the density of states on the Fermi surface is given by

neta

l

Li

Na

K

Rb cs

A1 Pb

Tab

le 1

Fer

mi

ener

gy o

f met

als

I H

eine

mod

el

1.11

70

0.92

30

0.74

60

0.69

80

0.64

50

1.74

00

1.57

10

7.06

96

5.14

49

3.39

19

2.97

16

2.53

67

18,8

090

15.8

870

App

apill

ai -

Wil

liam

mod

el

7.03

10

5.16

68

3 + 39

91

2.99

04

2.55

98

18.9

340

15.8

060

Ani

mal

u-

Hei

ne m

odel

expe

rim

enta

l A@

) co

mbi

ned

with

App

apill

ai-

7.08

61

4.95

78

3.20

81

2.76

97

2.32

93

17.2

461

14.6

172

7.11

52

5.15

33

3.41

47

2.96

04

2.54

37

18.4

310

13.4

570

5.35

60

5.22

55

3.55

85

3.36

70

3.01

30

18.2

000

14.8

400

free

elec

tron

valu

e)

7.49

69

5.11

89

3.34

39

2.92

74

2.50

80

18,1

920

14.8

300

Tab

le 2

kF

Den

sity

of

stat

es fo

r m

etal

s

erg-

')

met

al

Li

Na

K

Rb cs

A1

Pb

Ani

mal

u-

Hei

ne m

odel

A

ppap

illai

- W

illia

m m

odel

1.1170

0.9230

0.7460

0.6980

0.6450

1.7400

1.5710

0.2372

0.2975

0.4501

0.5134

0.6034

0.2429

0.4031

0.2462

0.2968

0.4501

0.5089

0.5959

0.2418

0.4073

~~ ex

peri

men

tal A

(&) c

ombi

ned

with

Ani

mal

u-

Hei

ne m

odel

0.2364

0.2970

0.4485

0.5140

0.6039

0.2461

0.4008

App

apill

ai-

Will

iam

mod

el

0.2451

0.2959

0.4470

0.5087

0.5980

0.2466

0.4062

WF

)

erg-

')

0.2801

0.2870

0.4215

0.4455

0.4978

0.2472

0.4063

No&)

(fre

e ele

ctro

nval

ue) s 'c P E. i

0.2001

0.2930

m

0

0.4486

0.5124

c E 3

OD

EJ

0.6000

0.2474

0.4064

Short Notes K33

From (3) and (7) the Fermi energy and the density of states of seven simple metals

a r e calculated using for U(Q) the model potentials of Animalu and Heine /4/ and

Appapillai and Williams /lo/. For A(&) we have used the theoretical structure factors

of Ashcroft and Lekner /12/, and the experimental structure factors a r e taken from

Gingrich and Heaton /13/ for alkali metals, from Gamertsfelder /14/ for Al, and

from North e t al. /15/ for Pb. The experimental determination of A(Q) for alkali

metals in the region of low Q is not sufficiently accurated due to low intensity of

scattering. However , these a r e the only comprehensive data available in the l i tera-

ture. The calculated results a r e listed in Table l and 2. A comparison with the cor-

responding values of the solid /16/ shows that there is , a t the melting point, an

insignificant change in the density of states on the Fermi surface. This conclusion

i s in conformity with the spin susceptibility measurements on lithium by Enderby

et al. /l/.

Acknowledgements

The authors would like to thank the State Council of Science and Technology for

a research grant, and Prof. Vachaspati, Head, Department of Physics, University

of Allahabad, India, for providing research facilities in the department.

References

/I/ J.E. ENDERBY, J.M. TITMAN, andG.D. WIGNALL, Phil. Mag. 1% 633

(1964).

/2/ W.A. HARRISON, Pseudopotentials in the Theory of Metals, Benjamin, Inc. , New York 1966.

/3/ V . HEINE and I.V. ABARENKOV, Phil. Mag. 9 , 451 (1964).

/4/ A.O.E. ANIMALU and V . HEINE, Phil. Mag. 12, 1249 (1965).

/5/ W.A. HARRISON, Phys. Rev. 136, A1107 (1964); 139, A139 (1965).

/6/ S.H. VOSKO, R. TAYLOR, and G.H. KEECH, Canad. J. Phys. 43, 1187 (1965).

/7/ R.W. SHAW, Phys. Rev. 174, 769 (1968).

/8/ P.S. HO, Phys. Rev. 169, 523 (1968).

/9/ D.C. WALLACE, Phys. Rev. E, 832 (1968).

/ lo/ M. APPAPILLAI andA.R. WILLIAMS, J. Phys. F_3, 759 (1973).

/11/ T . SCHNEIDER and E. STOLL, Adv. Phys. l.6, 731 (1967).

/12/ N.W. ASHCROFT and J. LEKNER, Phys. Rev. 145, 83 (1966).

3 physica (a)

K34 physica status solidi (b) 82

/13/ N.S. GINGRICH and L. HEATON, J . chem. Phys. 34, 873 (1961). /14/ G. GAMERTSFELDER, J. chem. Phys. 2, 450 (1941). /15/ D.M. NORTH, J . E . ENDERBY, and P . A . EGELSTAFF, J . Phys. C A , 1075

(1968).

/16/ T . SCHNEIDER and E . STOLL, Phys . kondens . Materie 2, 331, 364 (1966).

(Received May 4,1977)