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Physica B 223&224 (1996) 625 627

Can there be a T 3 In T kind of behaviour of the low-temperature specific heat of a system of free electron gas? Subodha Mishra*, D.N. Tripathy Institute o f Physics, Sachivalaya Marg, Bhubaneswar 751005, Orissa, India

Abstract It is shown that the temperature dependence of the interparticle correlation in a system of free electron gas can give rise to a T 3In T kind of behaviour of its low-temperature specific heat. Comparing this with a similar behaviour observed in normal liquid 3He, one may conclude that the T 3 In T behaviour is universal to all Fermi gases at low temperatures.

I. Introduction For a system of electron gas which constitutes a Fermi system the specific heat at constant volume for low temperatures has been known to be given as C~. = a T ,

(1)

where a is a characteristic constant of the system that depends on the density of particles in the system. For the case of normal liquid aHe, which is also known to be a Fermi liquid, the specific heat data for temperatures below about 100 mK fits well with a formula [1] C,, = a T + bT31n T ,

(2)

where b is another characteristic constant of the system. The second term in the above equation is assumed to be arising because of the paramagnons present in liquid 3He. These paramagnons are the long-lived excitations associated with the fluctuations of spins of the neighbouring atoms. Theoretical derivation of the T 3In T term was first given by Birk and Schrieffer [2] and independently by Doniach and Engelsberg [3]. Considering the case of a free electron gas, since one cannot talk of the paramagnons being present there, a T 3 In T kind of behaviour of its low-temperature specific heat is not realizable. In order to see whether in an electron gas there can be a T 3In T kind of behaviour of its low-temperature * Corresponding author.

specific heat, we, in the present work, have tried to calculate the low-temperature Fermionic specific heat of the system by considering the effects of temperature on the self-energy contributions to the bare single-particle energy. In a recent paper I-4] it has been shown by us that by introducing the self-energy contributions to the free particle energy within the Hartre~Fock approximation (HF) we are able to get rid of the divergence encountered in the value of (l/m*), m* being effective mass of an electron, using a self-consistent method of calculation. By choosing the Hartree-Fock self-energy, it had only simplified our calculation to a great extent. Because of this, we, in this paper, have made a calculation of the lowtemperature specific heat of a system of free electron gas by looking at the effects of temperature over the Hartree-Fock self-energy contributions. It is interesting to note that by doing this, we not only encounter a correction term of the kind T 3In T to the low-temperature electronic specific heat of the system but also a term of the type T In T. Since the Fermi temperature of a system of electron gas is very high, the contributions to the specific heat due to both T 3 In T and T In T terms are found to be negligibly small for low temperatures.

2. Mathemaicai derivation of the theory The total energy of the system of electron gas for finite temperatures is given as

0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S092 1 - 4 5 2 6 ( 9 6 / 0 0 1 0 g - s

S. Mishra, D.N. Tripathy / Physica B 223&224 (1996) 625-627

626

E(T) = 2 ~ ~:(p) r7(p),

(3)

where

P

where g(p) denotes the effective single-particle energy, which is a function of the wave vectorp and r~(p) is the corresponding one-particle Fermi-Dirac (FD) distribution function. The factor of 2 in the above equation arises due to the spin summations. The expression for g(p) is written as

M, (p) = - (e2kF~j pl [{rh(O) -- p2}ln ~ +

q_l.

(10i

where e(p) = p2/2m (in the unit of h = 1) and M(p) denotes the self-energy contributions. The expression for the self-energy M(p) in the H F approximation is given as

The expressions for M2(p, T) and M3(p, T) are not relevant in the present discussion. Because of this they are not quoted here. Using M~ (p) as shown in Eq. (10), we now carry out, the integration over "p" to obtain the terms that contribute to the total energy of the system E(T). This is explicitly shown as follows:

M(p) = - 1_~ v(p - q)Fl(q)

Eim(T) = 2 ~ M ~ ( p ) r i ( p )

g(p) = e(p) + M ( p ) ,

(4)

(5)

2 q

P

where v(Q) = 4ne2/Q 2. In order to avoid the difficulty of using g(p) in the FD-distribution function, we, as we had done in our earlier paper [5], approximate ri(p) by the one under the effective-mass approximation. This enables us to evaluate the integral (4) for finite temperatures, without much difficulties. To do this, we use dimensionless units throughout our calculations. The chemical potential # that appears in M(p) through ri(q) is a temperature-dependent quantity whose expression is given in Ref. [4]. Since in the present case M(p) happens to be temperature dependent, one will have rh to be a function of the temperature T. Using the expression for g(p) one obtains rfi(T) using the definition

g=m2p ap ].=,

(6)

This ultimately enables us to write rh(T) as [6]

[

r~(T) - r~(0) 1 -- a

,

(7)

,re2k~

.~

×ln

Ff 1

e2k~' 2 4 ~8(T)(T) n ×

In

}

_

(11)

In the present case, we are not interested in the temperature-independent contributions to E(T). Collecting all the temperature-dependent terms that account for the contributions to E(T) for low temperatures, we obtain the expression for the low-temperature specific heat of the system as

where

1 +,/-~ --

-- r f i ( 0 ) J

{1 -- tfi(0)} 2

"

(8)

The physical meanings of the symbols like ~, and rs, etc., that appear in the above equations are same as mentioned in our earlier paper [4]. Thus, from Eq. (7) one finds that the temperature dependence of rh (T) is of the same form as that of p(T). Substituting the value of#(T) and rh(T) in the expression for M(p), we finally write it as

6n rE +~2

---(1

2

\ TF,]

+ In

~(2a+41n2--4)

+ lnla

+ r/I)

M(p) = M1 (p) + M2 (p, T) + Ma(p,

T)

+ O ( T 3) +

... ,

(9)

-m~t

-~

3

a+

S. Mishra, D.N. Tripathy/ Physica B 223&224 (1996) 625-627

+

3~TF

2

~

6rcTv

In

~F

for T ¢ 0 K . (12)

It is interesting to see from Eq. (12) that the specific heat of system of electron gas exhibits both T I n T and T 3 In T kinds of behaviour for low temperatures. Besides it is seen that, the contribution from T 3 In T term is of the same sign as that encountered in the case of normal 3He [3] for low temperatures. Quantitatively for the electron gas, the T I n T contribution to the specific heat is found to be larger than that of the T 3 In T term. We would like to mention here that for T = 0 K, the last term of Eq. (12) goes to zero as - (T/TO. Dividing the right-hand side of Eq. (12) by (C~) = {n 2 n kt~( 1/2) (T/Tv) }), the value of the electronic specific heat of the system of noninteracting electrons, we obtain the expression for the effective mass ~ ( T ) of an electron as

l

627

The expression for F{~(0) } is same as that given in our earlier paper [4]. In order to obtain the value of rh(0) for various electron densities, we have to solve Eq. (14) selfconsistently. This reproduces the experimental value of rh(0) = 1.27 for sodium corresponding to a 6 = 0.12377. Using this value of 6, we can calculate rh(0) for different electron densities. An evaluation of rfi(T) following Eq. (13) shows that rfi(T) decreases as T increases. This is what is expected to happen, physically.

3. Conclusion We, in the present work, have shown that even for a free electron gas, it is possible to obtain the T 3 1 n T kind of behaviour of its low-temperature specific heat by looking at the temperature dependence of the interparticle correlations. Besides we also encounter a T In T contribution to its low-temperature specific heat. The T 3In T contribution arises even without invoking concept like the p a r a m a g n o n as is the case with liquid 3He. Surprisingly, the contribution of the T 3 In T term is of the same sign as that of liquid 3He. For liquid 3He, the occurrence of the T In T behaviour has not been established, yet, although such a behaviour was speculated by Anderson long back [7]. Thus, it looks that T 3In T behaviour of the low-temperature specific heat could be universal to all Fermi gases.

= ff115/2)(0) + 3mr, fi [F{rfi(0)} -- 1.65197rh2(0)] 7~

References

\TrJ \ 4 + In ~-~ 8e~6r~r~2(O) f T'~ 2, f T'~

F r o m this we have for T = 0 K, rfi(0) = fits/2(0) +

3e r~ 6

[F{rfi(0)} -- 1.65197n~2(0)].

(14)

[1] W.R. Abel, A.C. Anderson, W.C. Black and J.C. Wheatley, Phys. Rev. 147 (1966) 111. [2] N.F. Berk and J.R. Schrieffer, Phys. Rev. Lett. 17 (1966) 433. I-3] S. Doniach and S. Engelsberg, Phys. Rev. Lett. 17 (1966) 750. [4] S. Mishra and D.N. Tripathy, Phys. Lett. A 199 (1995) 99. I-5] L.K. Mishra and D.N. Tripathy, Phys. Stat. Sol. (b) 148 (1988) 585. I-6] J.M. Ziman, Principles of the Theory of Solids (Vikas, New Delhi, 1972). 1-7] P.W. Anderson, Physics 2 (1965) 1.