The ideal fermion gas 431due to thermal fluctuations. These excitations will give rise to smallfinite-temperature
corrections to the thermodynamic quantities we derived above atT= 0. Although we
will not give all of the copious mathematical details of how to obtain these corrections,
we will outline how they are derived. First, consider the density equation obtained in
eqn. (11.5.25) with the lowest nonvanishing temperature-dependent term:
ρλ^3 =4 g
3√
π[
(lnζ)^3 /^2 +π^2
8
(lnζ)−^1 /^2 +···]
. (11.5.73)
If (μ/kT)^3 /^2 is factored out, we obtain
ρλ^3 =
4 g
3√
π[(
μ
kT) 3 / 2
+
π^2
8(μ
kT)− 1 / 2
+···
]
=
4 g
3√
π[
(μ
kT) 3 / 2
(
1 +
π^2
8(
kT
μ) 2
+···
)]
. (11.5.74)
The term proportional toT^2 is a small thermal correction to theT= 0 limit. Working
only to orderT^2 , we can replace theμappearing in this term withμ 0 =εF, which
yields
ρλ^3 =4 g
3√
π[
(μ
kT) 3 / 2
(
1 +
π^2
8(
kT
εF) 2
+···
)]
. (11.5.75)
Solving eqn. (11.5.75) forμ(which is equivalent to solving forζ) gives
μ≈kT[
3 ρλ^3√
π
4 g] 2 / 3 [
1 + (
π^2
8(
kT
εF) 2 ]−^2 /^3
≈εF[
1 −
π^2
12(
kT
εF) 2
+···
]
, (11.5.76)
where the second line in eqn. (11.5.76) is obtained by expanding 1/(1 +x)^2 /^3 about
x= 0.
In order to obtain the thermal corrections, we must expand the average occupation
number formula about theμ 0 =εFvalue using eqn. (11.5.76) and then carry out the
subsequent integrations. Skipping the details, it can be shown thatto orderT^2 , the
total energy is given by
E=
3
5
NεF[
1 +
5
12
π^2(
kT
εF) 2
+···
]
. (11.5.77)
This thermal correction is necessary in order to obtain the heat capacity at constant
volume (which is zero atT= 0):
CV=
(
∂E
∂T
)
V