Problems and Solutions on Thermodynamics and Statistical Mechanics

2 70 Problems d Solutiotu on Thermodynamics d Statistical Mechanics

L

2lr

(b) dn, = -dk, ,

L r

dn, = -dk, ,

dn, = -dk,.

2T
L
2.n
Thus, in the volume V = L3, the number of quantum states of free
electrons in the region k, -+ k, + dk,, k, -+ k, + dk,, k, --t k, + dk, is
(considering the two directions of spin)

V

dn = dn,dn,dn, = 2 dk,dk,dk - -dk,dk,dk,.

I- 4n-3

At T = 0 K, the electrons occupy the lowest states. According to the Pauli
exclusion principle, there is at most one electron in a quantum state. Hence

so that

113

k,,,, = (3~~;).

The Fermi energy is

(c) At T = 0 K, the electrons occupy all the quantum states of energies
from 0 to EF. When the temperature is increased, some of the electrons
can be excited into states of higher energies that are not occupied, but they
must absorb much energy to do so, so that the probability is very small.
Thus the occupancy situation of most of the states do not change, except
those with kT near the Fermi energy EF. Therefore, only the electrons in
such states contribute to the specific heat. Let N,fi denote the number
of such electrons, we have N,R = kTN/EF. Thus the molar specific heat
contributed by the electrons is