258 Problems d Solutions on Thermodynamics 6' Statistical Mechanics(c) The variation of Ni versus Ei is as shown in Fig. 2.18.(a) T=K (b) J=$-K
Fig. 2.18.2087
Suppose that in some sample the density of states of the electrons D(E)
is a constant Do for energy E > 0 (D(s) = 0 for E < 0) and that the total
number of electrons is equal to N.
(a) Calculate the Fermi potential po at 0 K.
(b) For non-zero temperatures, derive the condition that the system is
non-degenerate.
(c) Show that the electronic specific heat is proportional to the tem-(UC, Berkeley)perature, T, when the system is highly degenerate.Solution:
(a) When T = 0 K, all the low lying energy levels are occupied, while
those levels whose energies E are greater than PO are all vacant. Taking the
1/2 spin of electrons into consideration, every state can accomodate two
electrons, and hence 2DopoV = N, or
N
Po = -
2VD0 'where V is the volume of the sample.
(b) The non-degeneracy condition requires that exp (fi) << 1, then