Advanced Solid State Physics

Figure 21: Plot of the specific heat to get the constants

Sommerfeld Expansion: Entropy, Free Energy, Pressure
We can also calculate the entropy for the system. The specific heat is divided by temperatureTand
we get the derivation of the entropy (with constantNandV). So we get an entropy densityswhich
is linear with the temperature

s=

3 π^2

4

k^2 BT

n

EF^3 /^2

μ^1 /^2 +...

The Helmholtz free energy can be calculated with:

f=u−T·s

Also other thermodynamic quantities: The pressurePcan be calculated with:

P=−

(

∂U

∂V

)

N

=

2

5

nEF with EF=

~^2

2 m

3 π^2 N

V

This pressure comes from the kinetic energy of the electrons, not from the e-e interaction, which
was neglected in this calculation. The bulk modulusB(how much pressure is needed to change the
volume) is:

B=−V

∂P

∂V

=

5

3

P=

10

9

U

V

=

2

3

n·EF

We can put all the calculated thermodynamic quantities into a table for free electrons (1d-, 2d-
, 3d-Schrödinger equation - see table 1). We started our calculation with the eigenfunction solution
(Ak·ei(kx−wt)), which we put into the Schrödinger equation to get the dispersion relationship(E=~ω).
To get the density of statesD(k)we have to know how many waves fit in a box with periodic boundary
conditions. After that we calculate the density of states for energyD(E) =D(k)dEdk from that we
can get the Fermi energy. Until now everything can be calculated analytically, so now we go over to
the Sommerfeld equation to get the chemical potential (and internal energy, specific heat, entropy,
Helmholtz free energy, bulk modulus, etc. like seen before).