Chapter 13
Ideal Fermi gas
The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We
shall consider the limit:
k
B
T µ, βµ 1 ,
which defines the degenerate Fermi gas. In this limit, the quantum mechanical nature of
the system becomes especially important, and the system has little to do with the classical
ideal gas.
Since this chapter is devoted to fermions, we shall omit in the following the subscrip t (−)
that we used for the fermionic statistical quantities in the previous chap ter.
13.1 Equation of state
Consider a gas of N non-interacting fermions, e.g., electrons, whose one-particle wave-
functions ϕ
r
(r) are plane-waves. In this case, a complete set of qua ntum numbers r is
given, for instance, by the three cartesian components of the wave vector
k and the z spin
projection m
s
of an electron:
r ≡ (k
x
, k
y
, k
z
, m
s
) .
Spin-independent Hamiltonians. We will consider only spin independent Hamiltonian
operator of the type
ˆ
H =
k
k
c
†
k
c
k
+
d
3
r V (r) c
†
r
c
r
,
where the first and the second terms are respectively the kinetic and th potential energy.
The summation over the states r (whenever it has to be p erformed) can then be reduced
to the summation over states with different wavevector k (p = ¯hk):
r
. . . ⇒ (2s + 1)
k
. . . ,
where the summation over the spin quantum number m
s
= −s, −s + 1, . . . , s has been
taken into account by the prefactor (2s + 1).
159
160 CHAPTER 13. IDEAL FERMI GAS
Wavefunctions in a box. We as-
sume that the electrons are in a vol-
ume defined by a cube with sides L
x
,
L
y
, L
z
and volume V = L
x
L
y
L
z
. For
the one-particle wavefun ction
r|k = ψ
k
(r) =
1
√
V
e
ik·r
we use periodicity condition, here
along the x-direction, at the cube’s
walls,
e
ik
x
x
= e
ik
x
x+ik
x
L
x
,
which is then translated into a condition for the allowed k-values:
e
ik
x
L
x
= e
i2πn
x
, k
x
=
2π
L
x
n
x
, n
x
∈ Z .
Analogously for the y- and for the z direction.
Summation over wavevectors. Each state has in k-space an average volume of
Δk =
(2π)
3
L
x
L
y
L
z
=
(2π)
3
V
. (13.1)
For large V → ∞ we can then replace the sum
r
over all quantum number by
r
→ (2s + 1)
1
Δk
d
3
k = (2s + 1)
V
(2π)
3
d
3
k
= (2s + 1)
V
h
3
d
3
p , (13.2)
where k = p/¯h has been used.
The factor
1
h
(
1
h
3N
for N particles) introduced “ad hoc” in clas-
sical statistical physics in Sect. 8.2 appears naturally in quan-
tum statistical physics. It is a direct conse quence of the fact
that particles correspond to wavefunctions.
13.1.1 Grand canonical potential
We consider now the expression (12.36) for the fermionic grand canonical potential Ω(T, V, µ)
that we derived in Sect. 12.5,
Ω(T, V, µ) = −k
B
T
r
ln
1 + e
−β(
r
−µ)
.
13.1. EQUATION OF STATE 161
Using the substitution (13.2) and
d
3
k = 4π
k
2
dk we write the grand canonical potential
as
− β Ω(T, V, µ) = (2s + 1)
V
(2π)
3
4π
∞
0
dk k
2
ln
1 + z e
−β¯h
2
k
2
/(2m)
, (13.3)
where we used the usual expr essions
z = e
βµ
,
r
→
k
=
¯h
2
k
2
2m
for the fugacity z and for the one-pa rticle dispersion and used an explicit expression for
the one-particle energies for free electrons
k
.
Dimensionless variables. Expression (13.3) is transformed further by introducing with
x = ¯hk
β
2m
, k
2
dk =
2m
β¯h
2
3/2
x
2
dx
a dimensionless variable x. One obtains
−β Ω(T, V, µ) = (2s + 1)
4V
√
π
m
2πβ¯h
2
3/2
∞
0
x
2
dx ln
1 + z e
−x
2
.
De Broglie wavelengths. By making use of the definition of the thermal de Broglie
wavelength λ,
λ =
2πβ¯h
2
m
,
we then get
− β Ω(T, V, µ) =
(2s + 1)
λ
3
4V
√
π
∞
0
dx x
2
ln
1 + z e
−x
2
. (13.4)
Term by term integra tion. We use the Taylor expansion of the logarithm,
ln(1 + y) =
∞
n=1
(−1)
n+1
y
n
n
,
in order to evaluate the integral
∞
0
x
2
dx ln
1 + z e
−x
2
=
∞
n=1
(−1)
n+1
z
n
n
∞
0
dx x
2
e
−nx
2
=
∞
n=1
(−1)
n+1
z
n
n
−
d
dn
∞
0
dx e
−nx
2
=
∞
n=1
(−1)
n+1
z
n
n
−
d
dn
1
2
√
π
1
√
n
162 CHAPTER 13. IDEAL FERMI GAS
term by term. The result is
∞
0
x
2
dx ln
1 + z e
−x
2
=
√
π
4
∞
n=1
(−1)
n+1
z
n
n
5/2
.
Grand canonical potential. Defining
f
5/2
(z) =
∞
n=1
(−1)
n+1
z
n
n
5/2
=
4
√
π
∞
0
dx x
2
ln
1 + z e
−x
2
(13.5)
we obtain
β Ω(T, V, µ) = −
2s + 1
λ
3
V f
5/2
(z) (13.6)
for the grand canonical potential for an ideal Fermi gas.
Pressure. Our result (13.6) reduces with Ω = −P V to
P
k
B
T
=
2s + 1
λ
3
f
5/2
(z) , λ =
2π¯h
2
k
B
T m
, z = e
µ/(k
B
T )
, (13.7)
which yields the pressure P = P (T, µ).
Density. With
Ω = −k
B
T ln Z = −P V
P V
k
B
T
= ln Z
we find, compare Eq. (10.14),
ˆ
N = z
∂
∂z
ln Z
T,V
= V z
∂
∂z
P
k
B
T
T,V
(13.8)
for the number of particles N. The density n(T, µ) =
ˆ
N/V is then given by
n =
ˆ
N
V
=
2s + 1
λ
3
(T )
f
3/2
(z) . (13.9)
where we have defined
f
3/2
(z) = z
d
dz
f
5/2
(z) =
∞
n=1
(−1)
n+1
z
n
n
3/2
(13.10)
Thermal equation of state. As a matter of principle one could solve (13.9) for the
fugacity z = z(T, n), which could then be used to substitute the fugacity in (13.7) for T
and n, yielding such the thermal equation of state. This procedure can however not be
performed in closed form.
13.2. CLASSICAL LIMIT 163
Rewriting the particle density. The function f
3/2
(z) entering the expression (13.9 )
for the part icle density may be cast into a different form. Helping is here the result (13.7)
for P (T, µ):
P
k
B
T
=
2s + 1
λ
3
f
5/2
(z), f
5/2
(z) =
λ
3
2s + 1
ln Z
V
,
P V
k
B
T
= ln Z ,
which leads to
f
3/2
(z) = z
d
dz
f
5/2
(z) = z
d
dz
λ
3
2s + 1
ln Z
V
With
d
dz
=
d
dβ
dβ
dz
=
1
µz
d
dβ
, β =
ln z
µ
(13.11)
and λ =
(2πβ¯h
2
)/m we then find
µV (2s + 1)f
3/2
(z) =
d
dβ
λ
3
ln Z
=
3λ
3
2β
ln Z + λ
3
d
dβ
ln Z .
For the particle density (13.9) we finally obtain
n =
2s + 1
λ
3
(T )
f
3/2
(z), µnV =
3
2
P V +
d
dβ
ln Z , nV = N , (13.12)
where we have used ln Z/β = P V .
Caloric equation of state. The expression for µnV = µN in (13.12) leads with
U = −
d
dβ
ln Z + µ
ˆ
N, Z =
r
e
−β(
r
−µ)
.
to the caloric equation of stat e
U =
3
2
P V . (13.13)
The equation U = 3P V/2 is also valid for the classical ideal gas, as discussed in Sect. 8.2,
but it is not anymore valid for relativistic fermions.
13.2 Classical limit
Starting from the general formulas (13.7) for P (T, µ) and ( 13.9) for n(T, µ), we first
investigate the classical limit (i.e. the non-degenerate Fermi gas), which corresponds, as
discussed in Chap. 11, to
nλ
3
1 , z = e
βµ
1 .
164 CHAPTER 13. IDEAL FERMI GAS
Under this condition, the Fermi-Dirac distribution function reduces to the Maxwell-
Boltzmann distribution function:
ˆn
r
=
1
z
−1
e
β
r
+ 1
≈ ze
−β
r
.
Expansion in the fugacity. For a small fugacity z we may retain in the series expansion
for f
5/2
(z) and f
3/2
(z), compare (13.5) and (13.10), the first terms:
f
5/2
(z) ≈ z −
z
2
2
5/2
βP λ
3
≈ (2s + 1)z
1 −
z
2
5/2
f
3/2
(z) ≈ z −
z
2
2
3/2
nλ
3
≈ (2s + 1)z
1 −
z
2
3/2
(13.14)
where we have used (13.7) and (13.9) respectively.
High-temperature limit. The expression for nλ
3
in (13.14) reduces in lowest approxi-
mation to
z
(0)
≈
nλ
3
2s + 1
, nλ
3
1 . (13.15)
The number of particles nλ
3
in the volume spanned by the Broglie wavelength λ ∼ 1/
√
T
is hence small are small. This is the case at elevated temperatures.
Fugacity expansion. Expanding in the fugacity z = exp(βµ)
z ≈
nλ
3
2s + 1
z
(0)
1
1 − z2
−3/2
≈
z
(0)
1 − z
(0)
2
−3/2
, z
(1)
≈ z
(0)
1 + z
(0)
2
−3/2
,
where 1/(1 − x) ≈ 1 + x for x 1 was used. The equa tion of state (13.14) for the
pressure, namely βP λ
3
≈ (2s + 1)(z − z
2
2
−5/2
), is then
βP λ
3
≈ (2s + 1)
z
(0)
1 + z
(0)
2
−3/2
− 2
−5/2
(z
(0)
)
2
= nλ
3
1 + 2
−5/2
nλ
3
2s + 1
,
when
1
√
2
3
−
1
√
2
5
=
2
√
2
5
−
1
√
2
5
=
1
√
2
5
is used. Altogether we then find
P V =
ˆ
Nk
B
T
1 +
nλ
3
4
√
2(2s + 1)
. (13.16)
In this expression, the first term corresponds to the equation of state for the classical ideal
gas, while the second term is the first quantum mechanical correction.
13.3. DEGENERATED FERMI GAS 165
13.3 Degenerated Fermi gas
In the low temperature limit, T → 0, the Fermi distribution function behaves like a step
function:
n
k
=
1
e
β(
k
−µ)
+ 1
T → 0
−−−→
0 if
k
> µ
1 if
k
< µ
i.e.,
lim
T →0
n
k
= θ(µ −
k
) .
Fermi energy. This means that all the states with energy below the Fermi energy
F
,
F
= µ(n, T = 0) ,
are occupied and all those above are empty.
Fermi sphere. In momentum space the occu-
pied states lie within the Fermi sphere of radius
p
F
. The system is then deep in the quantum
regime.
The Fermi energy is then be determined by the
condition that the the Fermi sphere contains the
correct number of states:
N =
states r
with
r
<
F
1 ,
which can be written for the case of free
fermions, and with (13.1), d
3
k/Δk
3
=
[V/(2π)
3
]d
3
k, as
N =
(2s + 1)V
(2π)
3
|k|<|k
F
|
d
3
k =
(2s + 1)V
(2π)
3
4
3
πk
3
F
. (13.17)
Here, k
F
=
p
F
¯h
is the Fermi wave number. We have
n =
N
V
=
2s + 1
6π
2
k
3
F
, k
F
=
6π
2
n
2s + 1
1/3
.
The Fermi energy is then
F
=
¯h
2
k
2
F
2m
,
F
=
¯h
2
2m
6π
2
n
2s + 1
2/3
.
166 CHAPTER 13. IDEAL FERMI GAS
13.3.1 Ground state properties
At T = 0, the system is in its ground state, with the internal energy U
0
given by
U
0
=
|
k|<k
F
k
=
(2s + 1)V
(2π)
3
k
F
0
dk (4πk
2
)
¯h
2
k
2
2m
=
(2s + 1)V
(2π)
3
¯h
2
2m
4π
5
k
5
F
.
Using the expression for total particle number N,
N =
V (2s + 1)
(2π)
3
4π
3
k
3
F
,
for k
5
F
= k
3
F
k
2
F
, one obtains
U
0
N
=
3
5
¯h
2
k
2
F
2m
,
U
0
N
=
3
5
F
(independent of s) .
for internal energy per particle at absolute zero.
Pressure. Since P V = 2U/3, we obtain now an expression for the zero-point pressure
P
0
:
P
0
≡ P
T =0
=
2
5
n
F
.
– The zero-point pressure arises from the fact that fermionic particles move even at
absolute zero. This is because the zero-momentum state can hold only one par ticle
of a given spin state.
– Taking a Fermi energy of typically
F
≈ 10 eV = 16 · 10
−19
J and an electron density
of n ≈ 10
22
· 100
3
m
−3
we find a zero-point pressure of
P
0
≈ 3.2 · 10
3
· 10
6
J
m
3
≈ 3.2 · 10
4
bar,
where we have used that 1 P = 1 J/m
3
= 10
−5
bar.
13.3.2 Fermi temperature
At low but a finite temperature, the Fermi distribution function ˆn
r
= n() for the
occupation number smooths out around the Fermi energy.
13.3. DEGENERATED FERMI GAS 167
Such an evolution of n() with increasing temperature is due to the excitation of fermions
within a layer beneath the Fermi surface to a layer above. “Holes” are left beneath the
Fermi surface.
Fermi temperature. We define the Fermi temperature T
F
as
F
= k
B
T
F
.
– T T
F
For low temperatures T T
F
, the Fermi distributi on deviates from that at T = 0
mainly in the neighborhood of
F
in a layer of thi ckness k
B
T . Parti cles at energies
of order k
B
T below the Fermi energy are excited to energies of order k
B
T above the
Fermi energy.
– T T
F
For T T
F
, the Fermi distribution approaches the Maxwell-Boltzmann distribu-
tion. The quantum nature of the constituent particles becomes irrelevant.
Frozen vs. active electrons. The typical magnitude,
F
≈ 2 eV, T
F
≈ 2 × 10
4
K ,
of the Fermi temperature in metals implies that room temperature electrons are frozen
mostly below the Fermi level. Only a fraction of the order of
T
T
F
≈ 0.015
of the electrons contributes to thermodynamic properties involving excited states.
Particles and holes. We can define that
the absence of a fermion of energy , mo-
mentum p and charge e corresp onds to the
presence of a hole with
energy = −
momentum = −p
charge = −e
The concept of a hole is useful only at low
temperatures T T
F
, when there are few holes b elow the Fermi surface. The Fermi
surface “disappears” when T T
F
, with the system approaching the Maxwell-Boltzmann
distribution function.
Specific heat. Since the average excitation energy per particle is k
B
T , the internal
energy of the system is of order
U ≈ U
0
+
T
T
F
Nk
B
T , (13.18)
168 CHAPTER 13. IDEAL FERMI GAS
where U
0
is the ground-state energy. The specific heat capacity C
V
is then of the order of
C
V
Nk
B
∼
T
T
F
, C
V
=
∂U
∂T
V
. (13.19)
The electronic contribution to the specific heat vanishes linearly with T → 0. The room-
temperature contribution of phonons (lattice vibrations) to C
V
is therefore in general
dominant.
13.4 Low temperature expansion
In this section we will deri ve the scaling relations (13.18) and (13.19) together with the
respective prefactors.
13.4.1 Density of states
We will work from now on with density of state per volume D(E) = Ω(E)/V , whi ch is
defined as the derivative of the integrated phase space per volume, φ(E) = Φ(E)/V :
D(E) =
∂φ(E)
∂E
, φ(E) =
1
V
k
≤E
d
3
k
Δk
3
, Δk
3
=
(2π)
3
V
.
The spin degeneracy factor 2s + 1 will be added furt her below.
Fermi sphere. With the phase space being isotropic we may write the volume of the
Fermi sphere as
V Δk
3
φ(E) =
4
3
πk
3
E
=
4π
3
2mE
¯h
2
3/2
, E =
¯h
2
k
2
E
2m
.
Introducing the spin degeneracy factor 2s + 1 we then obtain
D(E) =
A
√
E if E ≥ 0,
0 otherwise,
A =
2s + 1
(2π)
2
2m
¯h
2
3/2
(13.20)
Since we did not use any special properties of a fermionic system, this expression is also
valid for bosons. For both types of systems, D(E) shows a
√
E dependence.
Energy and particle density. The energy density U/V is given by
U(T, µ)
V
=
+∞
−∞
dE E n(E)D(E), n(E) =
1
e
β(E−µ)
+ 1
, (13.21)
13.4. LOW TEMPERATURE EXPANSION 169
where n(E) is the Fermi distribution as a function of the energy. The analogous expression
for the particle density is
n(T, µ) =
ˆ
N
V
=
+∞
−∞
dE n(E)D(E) . (13.22)
For a bosonic system one substitutes the Boson dist ribution function 1/(e
β(E−µ)
− 1) for
n(E).
13.4.2 Sommerfeld expansion
We are interested in the thermodynamic properties of a fermioni c system at small but
finite temperature, viz at a low temperature expansion of expectation values like (13.21)
and (13.22).
Sommerfeld expansion. We denote with H(E) a function depending exclusively on
the one-particle energy E. We will show that
H =
∞
−∞
dEH(E)n(E) ≈
µ
−∞
dEH(E) +
π
2
6
(k
B
T )
2
H
�
(µ) (13.23)
holds terms or order T
4
or higher.
– The first term on the r.h.s. of (13.23) survives when T → 0 and n(E) → θ(µ − E).
It represents the ground- state expectation value.
– The second term results from expan ding both H(E) and n(E) around the chemical
potential µ.
Partial integration. We start the derivation of the Sommerfeld expansion with the
definition
K(E) =
E
−∞
dE
�
H(E
�
), H(E) =
dK(E)
dE
,
which allows as to perform the partial integration
H =
∞
−∞
dE
dK(E)
dE
n(E) =
∞
−∞
dE K(E)
−
dn(E)
dE
. (13.24)
– For the integration we have used lim
E→∞
n(E) = 0, namely that the probability to
find particles at elevated energies E falls of exponentially.
– We have also assumed that lim
E→−∞
H(E) = 0.
Taylor expansion. Substituting the first two terms of the Taylor expansion
K(E) = K(µ) + (E −µ)K
�
(µ) +
(E − µ)
2
2
K
��
(µ) + O((E − µ)
3
)
170 CHAPTER 13. IDEAL FERMI GAS
of K(E) around the chemical potential µ into (13.24) leads to
H ≈
∞
−∞
dE
K(µ) + (E − µ)K
�
(µ) +
(E − µ)
2
2
K
��
(µ)
−
dn(E)
dE
. (13.25)
Ground state contribution. The first term in (13.25) is
K(µ)
∞
−∞
dE
−
dn(E)
dE
= K(µ)
n(−∞) − n(∞)
=
µ
−∞
dEH(E) ,
viz the T = 0 value of H.
Symmetry cancellation. The second term in (13.25) vanishes because
−
dn
dE
=
βe
β(E−µ)
[e
β(E−µ)
+ 1]
2
=
βe
−β(E−µ)
[1 + e
−β(E−µ)
]
2
, n(E) =
1
e
β(E−µ)
+ 1
,
is symmetric in E − µ.
Finite temperature correction. The third term in (13.25) yields the first non-trivial
correction
K
��
(µ)
∞
−∞
dE
(E − µ)
2
2
−
dn(E)
dE
=
π
2
6
(k
B
T )
2
H
�
(µ) ,
where the scaling with (k
B
T )
2
= 1/β
2
follows from a transfor mation to dimensionless
variables y = β(E − µ). The factor π
2
/6 results form the final dimensionless integral.
This concludes our derivation of the Sommerfeld expansion (13.23).
13.4.3 Internal energy at low temperatures
We start applying the Sommerfeld expansion (13.23) to the particle density (13.22):
n(T, µ) =
+∞
−∞
dE D(E)n(E) (13.26)
=
µ
−∞
dE D(E) +
π
2
6
(k
B
T )
2
D
�
(µ)
=
F
−∞
dE D(E) +
µ
F
dE D(E)
≈ (µ−
F
)D(
F
)
+
π
2
6
(k
B
T )
2
D
�
(µ)
,
where we have taken into account that µ = µ(T ) may be different from the Fermi energy
F
= lim
T →0
µ(T ). We have also assumed that the density of states D(E) is essential ly
constant around the Fermi energy.
Constant particle density. The terms inside the bracket in (13.26) need to cancel if
the particle density n is to be constant. The chemical potential µ varies hence as
µ =
F
−
π
2
6
(k
B
T )
2
D
�
(
F
)
D(
F
)
, µ =
F
−
π
2
12
(k
B
T )
2
F
(13.27)
as a function of temperature.
13.4. LOW TEMPERATURE EXPANSION 171
– We have approximated D
�
(µ) in (13.27) consistently by D
�
(
F
).
– The scaling D(E) ∼
√
E, as given by (13.20), leads to D
�
/D = 1/(2E).
Internal energy. The energy density (13.21) evaluated with the Sommerfeld expansion
is
u = U/V =
+∞
−∞
dE E n(E)D(E)
= u
0
+
µ
F
dE E D(E) +
π
2
6
(k
B
T )
2
d
dE
ED(E)
E=µ
3
2
D(
F
)
,
where u
0
=
0
−∞
ED(E)dE is the ground state energy. The substitution µ →
F
performed
for the argument of last term, for which we used D(E) ∼
√
E, is correct to order T
2
. The
second term is
µ
F
dE E D(E) ≈ (µ −
F
)
F
D(
F
) = −
π
2
(k
B
T )
2
12
D(
F
) .
when D(E) ∼
√
E. One finds hence with 3/12 − 1/12 = 1/6 that
U
V
= u
0
+
π
2
(k
B
T )
2
6
D(
F
) u − u
0
∝ (k
B
T )
2
D
0
. (13.28)
The internal energy increases quadratically with the temperature, being at the same time
proprotional to the density of states D
0
= D(
0
) at the Fermi level.
Specific heat. Assuming a constant density of states D(E) ≈ D(
F
) close to the Fermi
energy
F
we find
c
V
=
C
V
V
=
1
V
∂U
∂T
, c
V
≈
π
2
3
k
2
B
T
D(
F
) (13.29)
for the intensive specific heat. A trademark of a fermionic gas is that c
V
is linear in the
temperature.
The heat capacity per volume saturates however
for T → ∞, where it becomes identical with
the ideal gas value C
V
/V = 3nk
B
/2 derived in
Sec. 3.5.1.
172 CHAPTER 13. IDEAL FERMI GAS
