1. Lecture 27
Electron Gas
The model, the partition function and the Fermi function
Thermodynamic functions at T = 0
Thermodynamic functions at T > 0

2. Number of free electrons and number of states
The model assumes ideal gas of non-interacting electrons. The only constrain is that no two electrons can occupy the same quantum state, since electrons are fermions
Number of states per unit volume between energy = 0 to energy = kT for an ideal gas
This number is about 4 orders of magnitude larger than number of electrons is a typical metal. The reason is two fold. (i) The electron density is very high (~ electron per 10Å3) and (ii) electron mass is small, thus G(kT) is small
Consequently a classical model of an ideal gas is not operational

3. Partition Function It is convenient to use grand canonical ensemble
The total number of electrons
is the sum of number of electrons in single electron quantum states (each quantum state has either zero or one electron)
Since the electrons are not interacting

4. Partition Function - II
In term of single energy levels
Which is the same as
For example with just 2 single electron states the top formula gives
which is the same as the bottom formula

5. Partition Function - III
For Nkmax =1 (fermions)
Thus the partition function
And the logarithm of the partition function

6. Average occupation number
For Nkmax =1 (fermions)
Which implies that the average number of electrons in a single electron state is
This is called the Fermi function

7. Fermi function
T = T > 0 Nk Nk kT k k

8. Energy Density of states = 2 times density of state of an ideal gas
where is the zero temperature Fermi level

9. Fermi level (T=0)
Number of electrons
From which

10. T > 0 Number of electrons Energy Integrating by parts and expanding
around  =  gives

11. Energy Heat Capacity Nk kkT k
The energy ~ T2 can be also seen in the following: Number of excited electrons ~ kT, and the energy of each excited electrons ~ kT, thus the total energy ~ kT x kT ~ (kT)2