1. Degenerate Fermi Gas

2. Fermi gas at low T
Most applications are to electons, assume degeneracy g= 2 s +1 = 2
Increase with decreasing T
Small enough T, wave functions overlap, quantum statistics becomes important

3. Fermi gas at T=0 No more than 1 (2 for g=2) electrons in each state
First “e” goes to lowest state, the second must occupy higher energy state
And so on until all the “e”s are put it
There “motion” (non-zero energy) of electrons even at T=0

4. Fermi gas at T=0
T=0, beta = infinity

5. Distribution in Fermi gas at T=0
Occupation number <n> 1
energy μ

6. What is μ?
Chemical potential of
Fermi gas determined
by density only

7. Total energy EoS of the type P ~ n^\gamma are called polytropic EoS
gamma =5/3 for Fermi gas

8. Applicability T <<
Recall, we talked about de Broglie wave length comparable to inter-particle distance. This is exactly the condition. Maxwell-Boltzmann is applicable for opposite inequality.

9. The Fermi Gas of Nucleons in a Nucleus
Let’s apply these results to the system of nucleons in a large nucleus (both protons and neutrons are fermions). In heavy elements, the number of nucleons in the nucleus is large and statistical treatment is a reasonable approximation. We need to estimate the density of protons/neutrons in the nucleus. The radius of the nucleus that contains A nucleons:
Thus, the density of nucleons is:
For simplicity, we assume that the # of protons = the # of neutrons, hence their density is
The Fermi energy
EF >>> kBT – the system is strongly degenerate. The nucleons are very “cold” – they are all in their ground state!
The average kinetic energy in a degenerate Fermi gas = 0.6 of the Fermi energy
- the nucleons are non-relativistic

10. Finite T<< εF Only electrons near FERMI SURFACE
What happens as we raise T, but keep kBT<<EF so that   EF?
Empty states are available only above (or within ~ kBT ) of the Fermi energy, thus a very small fraction of electrons can be excited
The electrons with energies  < EF - (few) kBT cannot interact with anything unless this excitation is capable of
T =0
~ kBT
(E-EF)
occupancy
raising them all the way to the Fermi energy.
Only electrons near FERMI SURFACE
participate in motion (thermal or, e.g.,
due to electric field