We consider N particles, but for simplicity work in the grand canonical ensemble. The spatial states are characterized by the wave vector~k of the wave function. The spin of each particle can be up or down (mS = ±12). Considering no additional interaction apart from the intrinsic one due to the particles being fermions, this implies:
ZG =
∏
~k,mS
z~k,m
S
z~k,m
S =1+e−β(e~k,mS−µ) n~k,m
S = 1
eβ(e~k,mS−µ)+1
For the dispersion relation, we have the classical relation for a massive particle com- bined with the de Broglie relation:
e~k,m
S = p
2
2m = h¯
2k2 2m
In the following we will use a factor 2 for the spin degrees and replace the sums by integrals:
~k,m
∑
S...=2 V h3
Z
d~p...=2 V h3
Z
dpp24π...
=2 V h3
Z ∞
0 de4πm32 (2e)12 ...
= N Z ∞
0 de V
2π2N 2m
¯ h2
32 √ e
| {z }
density of statesD(e)
...
The concepts used here are the same ones as used before for the Debye solid and the black body radiation. While hereD∝√
e, for the phonons and photons we hadD∝e2 due to the linear dispersion relation.
Fermi energy
For given particle number N, the chemical potentialµhas to be determined from:
N =
∑
~k,mS
n~k,m
S = N
Z ∞
0 deD(e) 1
eβ(e−µ)+1
| {z }
n(e)
We first consider the limitT→0:
n(e)→1−Θ(e−µ) =Θ(µ−e) The value ofµatT=0 is called‘Fermi energy’:
eF= p
2F
2m =µ
T =0,v= V N
⇒ N=
∑
~k,m
S f or p≤pF
1=2V h3
Z
p≤pF d~p= 2V h3
4π 3 p3F Here we integrated over the ‘Fermi sphere’.
⇒ eF= 3π223 h¯2
2mv23 = 3π223 ¯h2ρ23 2m Typical values for the Fermi energy:
eF=
10−4eV 3He
10eV electrons in metal 1MeV electrons in white dwarf 35MeV neutrons in atomic nucleus
Since kBTR = eV40, for electrons in metals at room temperature TR we typically have eF kBTR. We evaluate the occupancy around the Fermi-edge (compare Figure 5.5):
n(e) =0.5±0.23 fore= µ∓kBT n(e) =0.5±0.45 fore= µ∓3kBT
We see that the width of the step at finite T is only a fewkBT. Therefore only a few of the N electrons in the‘Fermi sea’are thermally excited aboveeF.
−0.5 0 0.5 1
n()
µ
Figure 5.5: The occupation numbernas a function ofeat finite temperature (blue) and the difference of the curve with respect to the one at T = 0 (red and red- dashed above blue).
Specific heat
We use this result to calculate the specific heat based on the‘Sommerfeld method’. We consider an arbitrary function f(e)(eg f(e) =e12):
I =
Z ∞
0 de f(e)n(e) =
Z µ
0 de f(e) +
Z ∞
0 de f(e) [n(e)−Θ(µ−e)]
| {z }
6=0 only in small region aroundµ
Expansion of f(e)around the Fermi edge:
f(e) = f(µ) + f0(µ)(e−µ) + ...
We introducex= β(e−µ):
⇒ η(x) =n(e)−Θ(µ−e) = 1
ex+1 −Θ(−x)
= 1
−(1−Θ(x))
=− 1
−Θ(x))
=−η(−x)
η(x)being odd in x implies that all even terms on the Taylor expansion vanish.
⇒ I =
Z µ
0 de f(e) + 1 β
Z ∞
−βµ
dx
f(µ) + f0(µ)x β+...
η(x) For low temperatures:βµ→∞ :
=
Z µ
0 de f(e) + f
0(µ) β2
Z ∞
−∞dx xη(x)
| {z }
=2R∞
0 dx xη(x)=2R∞
0 dx exx+1=π62
=
Z µ
0 de f(e) + π
2
6β2 f0(µ)
We now apply this result to our normalization condition:
1=
Z ∞
0 deD(e)n(e) withD(e)∝e
1
2 (compare Figure 5.6) to determine the chemical potentialµ(T,v).
0 0.5 1 1.5 2 2.5
ǫ
n(ǫ)
n(ε) D ∝ ε0.5
µ
Figure 5.6: The average occupancyn(blue) as a function ofenext to the density of states D∝e12 (red).
1=
Z µ
0 deD(e)
| {z }
=
Z eF
0 deD(e) +
Z µ
eF
deD(e)
=1+ (µ−eF)D(ee)
+π
2
6β2 D0(µ)
Here 1 is the result forT =0 andeesome value betweenµandeFaccording to the mean value theorem.
⇒ µ−eF =− π
2
6β2
D0(µ)
D(ee) ≈ − π
2
6β2
D0(eF) D(eF)
| {z }
=12e1
F
usingµ−eF ∝T2
⇒ µ=eF
"
1−π
2
12 kBT
eF 2#
forT eF
kB (Figure 5.7)
0
T
µ(T,v)
ǫF
ǫF kB
Figure 5.7: The chemical potentialµ(T,v)decreases with increasing temperature. For T ekF
B it can be taken to beµ=const=eF.
We note that in general the chemical potential µ has to go down with temperature because for fixed particle number the joint integral with the density of states has to stay constant, compare Figure 5.8. Therefore higher order term in this expansion are not expected to change the general picture.
We next evaluate the energy:
E N =
Z ∞
0 de D(e)e
| {z }
f(e)∝e
32
n(e)
=
Z eF
0 deD(e)e+ (µ−eF)eeD(ee) + π
2
6β2
µD0(µ) +D(µ)
≈ E0
N + (µ−eF)
| {z }
−π2
6β2 D0(eF)
D(eF)
eFD(eF) + π
2
6β2
eFD0(eF) +D(eF)
⇒ E= E0+Nπ
2
6 D(eF) (kBT)2
⇒ c = ∂E
= Nπ2
k2D( )T
ǫ nF,D
µ(T1) µ(T2)
T2> T1 nF(T1)
nF(T2)
D∝ǫ
1 2
Figure 5.8: The fermionic occupation numbernFas a function ofefor different temper- atures (blue and red curves) next to the density of statesD(green).
The specific heat of an electron gas atT ekF
B is linear in temperature T.
We useD(e) = Ae12 to write 1=
Z eF
0 deD(e) = A Z eF
0 de√
e
= 2
3AeF32 = 2
3D(eF)eF
⇒ D(eF) = 3 2eF
⇒ cV =Nπ2 2
kBT eF kB
Disregarding the numerical prefactor, this result is easy to understand: a fraction keBT
F of
the electrons from the Fermi sea is thermally excited, each contributing aroundkB. Our calculation is only valid forT keF
BT. At high temperature, we have to recover the classical limit:
cV = 3 2NkB
Therefore the complete result schematically has to look like shown in Figure 5.9.
We also comment on the role of lattice vibrations. From the Debye model we know that lattice vibrations contribute a term∝T3.
⇒ cV = a T+b T3
One can measure a and b experimentally and thus extract the Fermi energyeFand the Debye frequencyωD. With these two values, we know the most important numbers for a given solid.
~ T
constant
T cV
3/2
T0
Figure 5.9: Two regime behaviour of the specific heat at constant volume: While for TT0 = ekF
B cV ∝T,cVis approximately constant forT T0. Full solution
Until now we have worked in an expansion around theT = 0-case. We can also write the full solution for arbitraryT, however we will end up with integrals that cannot be solved but rather lead to definitions of new functions.
We start with the grandcanonical potential and use the same concepts as above:
Ψ(T,V,µ) =−kBTlnZG
= −kBT
h3 2V(4π)
Z ∞
0 p2dpln
1+eβ(e−µ)
= −2kBTV
λ3 f5/2(z) where we have defined a new function
f5/2(z):= √4 π
Z ∞
0 x2dxln
1+ze−x2
=
∑
∞ 1(−1)α+1 z
α
α5/2
and where we have used the dimensionless momentumxdefined byx2= βp2/2mand fugacityz =eβµ.
Particle number can be written in a similar manner:
N = 2V(4π) h3
Z ∞
0 p2dp
1 eβ(e−µ)+1
= 2V λ3
√4 π
Z ∞
0 x2dx z
ex2+z
= 2V
λ3 f3/2(z) with another new function
f3/2(z):= √4 π
Z ∞
0 x2dx z
ex2 +z
=
∑
∞ 1(−1)α+1 z
α
α3/2
As a function of z, both functions increase monotonously from 0 with a decreasing
One can easily check that the two formula are consistent:
N= 1
β∂µlnZG= 1
β(βz)∂zlnZG= 2V
λ3(z∂z)f5/2(z) = 2V
λ3 f3/2(z)
One can also calculate the variance as σN2 = (1/β)∂N/∂µ. For low temperature, we would get the same results as above.
Fermi pressure
We consider the definition of pressure:
p= − ∂E
∂V T,N
=−
∑
~k,m
S
∂e~k,m
S
∂V n~k,m
S
where in the last step we have neglected any temperature-dependent change in the occupation level (second order effect, a more rigorous treatment would again start from the grandcanoncial ensemble). Sincee~k = (¯hk2m)2 andki ∝ 1
V13, we have e~k ∝ 1
V23 ⇒ ∂e~k,mS
∂V =−2 3
e~k,m
S
V
⇒ p= 2 3
E
V = 2E0
3V + π
2
6 NkBT
V kBT
eF
| {z }
→0 forT→0 like for the ideal gas
Interestingly, this contribution to the pressure is always positive, showing that the Fermi gas is effectively like a gas with repulsive interactions. There is also a temperature- independent term:
E0 N =
Z eF
0 deD(e)e= 3 5eF
⇒ p T→→0 2 5
N
VeF = 3π
223 5
¯ h2 mv53
The ‘Fermi pressure’in a Fermi fluid at very low temperature accounts for the incom- pressibility of matter and essentially results from the Pauli principle. For example, it prevents that the earth collapses under gravitation. This is also true for white dwarfs (electrons) or neutron stars, but not for the sun. In the latter case classical ideal gas pressure atT = 5·107 K(temperature in the center of the sun) balances gravitational attraction.