550
10 Quantum Statistics
to obtain an expression for M ce in terms of the 0 K result (10.4.56b) plus a correction
term for the temperature dependence. In particular, we can see from Eq. (10.4.49)
that we may safely approximate μ by F in the second term on the right-hand side
of Eq. (10.4.59), but we must be more careful with the μ
1
2 term, by approximating
it as
μ
1
2
1
2
F
1 −
π 2
12
k B T
F
2
+ . . .
1
2
≈
1
2
F
1 −
π 2
24
k B T
F
2
,
in order not to neglect an additional contribution of order T 2 to M ce . Thus, we obtain
the leading temperature dependence of the magnetic moment of the conduction
electrons as
M ce (T ) 2μ
2
0 BB 0 (V ))
1
2
F
1 −
π 2
24
k B T
F
2
−
π 2
24
k B T
F
2
=
3N ce μ 2
0 B
2 F
1 −
π 2
12
k B T
F
2
,
(10.4.60)
in which the leading term is the T = 0 K magnetic moment. Note also that the
leading temperature-dependent term is indeed very small.
10.5 Electrons in Semiconductors
We have seen that the density of states for a free electron gas can be written as
=
8πV
h 3
2m
3
1
2 .
(10.5.1)
When atoms form a crystal (solid), some of the electrons no longer belong to
particular atoms, but are shared in some sense by all atoms in the solid, giving
those electrons a dual nature of atomic and free electrons; there is a broadening
of the discrete atomic orbitals to form bands in the solid. These two situations are
illustrated in Fig. 10.7. See also Appendix F.
The probability that a one-particle state i is not occupied can be represented by
n
i , with
n
i ≡ 1 − n i =
1
e −β(( i −μ) + 1
;
(10.5.2)
if we take the condition in which all the electronic states are occupied as the
standard, then we say that the state i is occupied by a hole when it is not occupied by