Volume dependence of gruneisen constant

Solid State Communications,Vol. 15, PP. 249—250, 1974.

Pergamon Press.

Printed in Great Britain

VOLUME DEPENDENCE OF GRUNEISEN CONSTANT S. Tolpadi Department of Physics, Indian Institute of Technology, Kharagpur, India (Received 11 February 1974 by A.R. Verma)

An expression is derived for a new isothermal parameter A = ~-(~) which gives the volume dependence of Gruneisen constant of cubic crystals and its value is calculated in the case of NaCI, KCI, Cu, Ag, Al and Au. It is shown that the A parameter modifies the thermal contribution to the isothermal compressibility of crystals.

compressibility 13T and it is given by,

GRUNEISEN’ has investigated in detail the various conclusions deducible from the equatior. of state of crystalline solids and has established satisfactory experimental confirmations of the theoretical results. Using Gruneisen’s method, we shall modify the expres-

-~_

~T

-~-L;-- + dUL

(1)

\aVJT

=

dv2 —~~(~~)T _E1(~2~ ~

(3)

FJV/TJ

(~‘)_~.(TC_E)

(4)

where C~~is specific heat for a given lattice frequency. We now define a new isothermal parameter A1 which is given by, =

Multiplying equation (1) by v and differentiating the equation with respect to v at any given temperature, we get,

(~

3N

Differentiating the thermal energy expression with respect to v at the given temperature T, we get,

where p and v are the pressure and volume of the crystal at a given temperature T of the crystal, UL is the lattice potential, y, and E 1 are the Gruneisen constant and thermal energy for a given lattice frequency. The summation in equation (1) is over all the 3N lattice frequencies.

~ + ~

\aV T

where the quantities (aEf/aV)T and (~,I/aV)T are isothermal parameters.

The starting point in our investigation is the general 2 equation of state of cubic crystals 3N which is given by, =

+

=

‘cc’ fyjE 1 L~ k v 1=1

sion for the thermal variation of the compressibility by introducing a new parameter which represents the volume dependence of Gruneisen constant.

p

=

(5)

v(a~1) 7j

~V

T

Introducing the values of A. and (aE,/av)T from equation (4) and (5) in equation (3) we get, 2 UL + 7 v = ~ (.-~— (6) d T~ 1=1 1E1 J

‘cc’

2 + —~ d~U~ dv — V dv

~

— ~

.—

—

3N

~

+

(2)

It is seen that the compressibility of the crystal at any temperature T depends on two terms of which the first one depends on lattice potential and the second on thermal vibrations. To evaluate the thermal

1=1

Substituting the value ofp from equation (1) in equation (2), we get the expression for the isothermal 249

250

VOLUME DEPENDENCE OF GRUNEISEN CONSTANT

contribution to the compressibility, it is necessary to calculate A, ~,, C1~,and E1 as a function of the lattice frequency and then determine the sum over the 3N lattice frequencies as that indicated in equation 2 assumed the ‘y~are the same(6). for all Gruneisen the lattice frequencies. If we further assume that A are also the same, we get from equation (6) a simple expression for the isothermal compressibility and it is given by,

_i.~= v

+~

~

dV2

(1

—

TT (~‘

i)

—

AT)

(7)

potential in the case of metals and alkali halide crystals. Table 1. Calculated values of Gruneisen constant and A parameter ___________________________________________________

Substance

Gruneisen

NaCI KCI Cu

1.61 1.41 2.02

1.48 .2.27 2.65

Ag Al Au

2.34 2.17 2.94

0.98 1.47 3.72

constant

X Parameter

V

where the Gruneisen constant ~(T~the parameter AT, specific heat C~and the thermal energy E refer to the crystal temperature Twhen its volume is v under normal atmospheric pressure. If we put AT = 0 in equation (7) we get the compressibility equation used by Born eta!.,2

The Gruneisen constant ‘IT is calculated by using the relation, v aT ‘IT = ~ (10)

d2ULYTEJ

1 —

Vol. 15, No. 2

=

dV2 V—

—

7T(~

—

i)}

(8)

Rearranging equation (7), the expression for the new isothermal parameter AT is given by, l

AT =

)

7T( —

E

—

1 +~

V

(d2UL ‘¼dv2

1)

(9)

~

We have calculated the parameter AT using equation (9) in the neighbourbood of the Debye temperature in the case of NaCl, KCI, Cu, Ag, Al and Au. The second derivative of the lattice potential appearing equation (9) has been determined by using separate and appropriate forms of the

where aTexperimental is the thermal expansion coefficient. The relevant data used in our calculations are taken from a compilation by Wallace.3 The calculated results on ‘IT and AT are given in Table 1. It may be noted that the main assumption considered in our investigation is that the parameters ‘y~ A~are the same for all the lattice frequencies. Using this simple assumption it is shown that the new parameter A modifies the thermal contribution to the compressibility. We are also calculating the frequency dependence of the parameters A 1 and ‘I~by using a model for the lattice vibration frequencies and these results will be published elsewhere.

REFERENCES 1.

BORN M. and HUANG K.,Dynamica! Theory of Crystal Lattices p.38, Oxford University Press, London (1954)

2.

GRUNEISEN E., Handb. derPhys. Vol. 10, p.22, Springer (1926)

3.

WALLACE D.C., Thermodynamics of Crystals, p.463 John Wiley, New York, (1972).

Em Ausdruck wird für einen neuen Isothermalan Parameter A = V abgeleitet, der die Volumenabhingigkeit der GRUNEISENschen7~’/T Konstante der kubischen Kristallen liefert. Sein Wert wird bei NaCl, KCI, Cu, Ag, Al und Au bestimmt. Es wird hur gezeigt, dass der Parameter A den thermischen Beitrag zur Isothermalan Kompressibiitat der Kristallen modifiziert.