Quantum Statistics
233
h i =
3/2
1/2
3
4 (2 )
V m
d
h
π
ε
ε
(7.86)
The density of electrons as a function of energy, is then given by
( )
dN
d
ε
ε
=
3/2
1/2
3
( ε ε )/
4
(2 )
1
f kT
V m
h
e
−
π
ε
+
(7.87)
and is shown in Fig. (7.8). The Fermi energy is determined from the condition
that the total number of particles is N, i.e.
0
0
e
e f
(a)
(b)
dN/de
kT
Fig. 7.8 Density of levels as function of ε, (a) for T = 0, (b) kT = 0.1 ε f (0).
N =
3/2
1/2
3
0
4
(2 )
exp [(
)/ ] 1
f
V m
d
kT
h
∞
π
ε ε
ε − ε
+
∫
(7.88)
At T = 0,
N =
3/2
1/2
3
0
4
(2 )
3
f
V m
d
h
ε
π
ε ε
∫
(7.89)
=
3/2
3/2
3
8 (2 )
3
f
V m
h
π
ε
which gives
ε f (0) =
2/3
2
3
2
8
h
N
m
V
π
(7.90)
A slightly involved calculation gives for kT < < ε f (0)
ε f (T) ≈
2
2
(0) 1 12
(0)
f
f
kT
π
ε
−
ε
(7.91)
233
h i =
3/2
1/2
3
4 (2 )
V m
d
h
π
ε
ε
(7.86)
The density of electrons as a function of energy, is then given by
( )
dN
d
ε
ε
=
3/2
1/2
3
( ε ε )/
4
(2 )
1
f kT
V m
h
e
−
π
ε
+
(7.87)
and is shown in Fig. (7.8). The Fermi energy is determined from the condition
that the total number of particles is N, i.e.
0
0
e
e f
(a)
(b)
dN/de
kT
Fig. 7.8 Density of levels as function of ε, (a) for T = 0, (b) kT = 0.1 ε f (0).
N =
3/2
1/2
3
0
4
(2 )
exp [(
)/ ] 1
f
V m
d
kT
h
∞
π
ε ε
ε − ε
+
∫
(7.88)
At T = 0,
N =
3/2
1/2
3
0
4
(2 )
3
f
V m
d
h
ε
π
ε ε
∫
(7.89)
=
3/2
3/2
3
8 (2 )
3
f
V m
h
π
ε
which gives
ε f (0) =
2/3
2
3
2
8
h
N
m
V
π
(7.90)
A slightly involved calculation gives for kT < < ε f (0)
ε f (T) ≈
2
2
(0) 1 12
(0)
f
f
kT
π
ε
−
ε
(7.91)