Electron-ion interaction in metals

Solid State Communications,

Vol. 11, PP. 1223—1225, 1972.Pergamon Press.

Printed in Great Britain

ELECTRON—ION INTERACTION IN METALS B. Sharan,* Ashok Kumar and K. Neelakandan Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi-29, India (Received in revised form 3 August 1972 by A.R. Verma)

A semi-phenomenological approach has been made to treat the effect of the presence of electrons in a metal on its lattice vibrations, by averaging the compressive strain over the actual shape of Wigner— Seitz polyhedron.

m(r)can be written as

IN GENERAL, the lattice dynamical study of metals involves the total interaction expressed as a sum of two parts: ion—ion and electron—ion. The ion—ion interaction is evaluated by standard methods. To evaluate the electron—ion interaction an expression is derived for the force on an ion in the metallic crystal, arising from certain ener-

the mean value of >~

(e q)A exp z(q R

=

~ .

—

-

I

k)e

+

i(e. q)A exp i(q. Rm

dT

q —

wt)

(4)

(D —

gies which are due to the compressibility of the conduction electrons and their interaction with ions. In the harmonic approximation the change in the potential energy of the crystal due to the 1 is presence of electrons ~Ke f~2(r)dr, (1)

v

-r q~)

.

+ —

±

ci(q~+

+

~ ‘-‘2 +

q2)

B 3

where B,, B2 and B3 are the Cartesian compon~ f en s o 1 -exp(iq . r)nds, (5)

j

.

.

S

n being the unit vector. where Ke is the bulk modulus of electron gas. Thus the force on the mth ion which is due For a plane wave vector q, frequency w and polarization e the compressive strain Xm(r) associated with rnth ion at position Rtmis given by =

—

i(e. q)A exp i[q

.

(Rm

to electrons is (Ftm)

r) — wt]. (2)

=

where ~

—AK~c1q(e.q) S2exp i(q. R~—wt), =

(B,

±

B 2

Using the trigonometrical identity exp(iq. r)

V ~(i +

+

+

B3)/ci(q~+ q~.- q~)

,

integral (5) can be evaluated easily over the actual shape of the Wigner—Seitz polyhedron for any type of crystal lattice

(3)

+ q~) V

For a body-centred tetragonal crystal, the *

(7)

and ci is the volume of atomic polyhedron. The

k)exp (iq . r)}

z(q~÷q

(6)

Present address: Professor of Physics, Department of Science Education, N.C.E.R.T., New Delhi-16, India.

expres~ionfor S takes the following forms:

1223

1224

ELECTRON—ION INTERACTION IN METALS

For q~~ 0.

S

1

s

256 c(q,— q~--q~)

C1

S’

1(2a

Sin aq~/2, COS

S,

aq~’2,2)q~ 4c, C, c’

sin aq~2, cos aq, 2a2)q~/4c. 2, cos(c’—

sin(c’— 2a

c) sin a, sin a 4 sin(~x q~—

a

—

Ca, a

a

‘2a— c\

—

Vol. 11, No. 9

.

The model is applied to indium to calculate

/

the Debye characteristic temperature. Fig. 1 gives the 0~ vs. T curve obtained by considering

C

(

caa,, — )sin a1 sin a, sin q~--q~—a_~z) a 4 —1 sin . a1 sin . a, sin a.a,

(~r

Ea qz)

qy

a/4

four-neighbour central interaction and two-neighbour angular interaction of de Launay’s type ~ for the ion—ion interaction and the present approach for electron—ion interaction. Experimentdl curve ~ and theoretical ooints of Slutsky and

sin a, sin a, sin(~x

—~

a, a,

~~_~~z)a/4 /40

}

a, a4q~ S1 (oiS2S’_~~z(c2c’_ S2 — {qrsl aa,q~

~1

a ——ci~(cc c

A

130

~,1a

cos cq~/4)~J (8)

rb

For q,

0, but q~= 0, q~,~ 0,

t20

[/ 2 2 16 c — 0) S1S,q~q~(q~ - qy) ac’cq~q,’(q, q~) 4a

S

_____________________

o

20

~o

~o

ao

/00

1C~<) —

4q2q~ sin(q~-~ q1)a/4 . sin q~a/4- sin q~a/4

-

S1(1

c2)q~’± S,(1

-

For q~ q,~, S where 4a

c1)q~]

0, but q, ~ 2S 1q,(c’ a’)2q,’ -~ 8a(1 ac a

a(~r~_ qzj

4a 2

C

4a

a(

(9)

a C

q~—qz),

4a~

( (

a q,

a q1

c)

‘

a _~Z)~

C

a

(10)

FIG. 1. 0~ — T curve for indium. 6 for 11D T have also been shown Livingstone for comparison purpose. The agreement between experimental and theoretical values is found to be within 7 per cent and could be improved further by cons idering long range interaction which seems difficult ~‘nth only six parameters, the elastic constants, known.

q~ .4Cknowledgements — Financial assistance from CSIR (India) is gratefully acknowledged. REFERENCES

1.

SHARMA P.K. and JOSHI S.K., J. Chem. Phys. 39, 2633 (1963).

2.

SHARAN B. and BAJPAI R.P., Phys. Lett. 31A, 120 (1970).

3.

SHARAN B. and BAIPAI R.P., .J. Phys. Soc. Japan 26, 1359 (1969).

4.

SHARAN B. and BAJPAI R.P., Indian I. Pure Appi. Phys. 8, 331 (1970).

5.

CLEMENT JR. and QUINNEL E.H., Phys. Rev. 92, 258 (1953).

6.

SLUTSKY L.J. and LIVINGSTONE A., J. Chem. Phys. 32, 1093 (1960).

Vol. 11, No. 9

ELECTRON—ION INTERACTION IN METALS

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