10.4 Fermi–Dirac Statistics
537
0.0
0.5
1.0
1.5
2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ε/μ
f(ε)
T = 0.1T F
T = 0.2T F
Fig. 10.4 Fermi–Dirac distribution for finite temperature
relating the thermodynamic pressure and the internal energy, to obtain an expression
for P ce ≡ P (T = 0). We begin with the defining relation,
U =
k
n k k ,
for the thermodynamic internal energy, and convert the summation over the discrete
energy states into an integration over continuum energy states via Eqs. (10.2.13)
with I a =
1
2 . The internal energy is then given in general as
U 4π
2m e
h 2
3
2
V
∞
0
f (())
3
2 d .
(10.4.18)
Because the Fermi–Dirac distribution for temperature T = 0 is a step function,
expression (10.4.18) simplifies considerably to yield for U 0 the result
U 0 4π
2m e
h 2
3
2
V
F
0
3
2 d
=
8π
5
2m e
h 2
3
2
V V
5
2
F .
(10.4.19)
537
0.0
0.5
1.0
1.5
2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ε/μ
f(ε)
T = 0.1T F
T = 0.2T F
Fig. 10.4 Fermi–Dirac distribution for finite temperature
relating the thermodynamic pressure and the internal energy, to obtain an expression
for P ce ≡ P (T = 0). We begin with the defining relation,
U =
k
n k k ,
for the thermodynamic internal energy, and convert the summation over the discrete
energy states into an integration over continuum energy states via Eqs. (10.2.13)
with I a =
1
2 . The internal energy is then given in general as
U 4π
2m e
h 2
3
2
V
∞
0
f (())
3
2 d .
(10.4.18)
Because the Fermi–Dirac distribution for temperature T = 0 is a step function,
expression (10.4.18) simplifies considerably to yield for U 0 the result
U 0 4π
2m e
h 2
3
2
V
F
0
3
2 d
=
8π
5
2m e
h 2
3
2
V V
5
2
F .
(10.4.19)