Degenerate Fermi Gas

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

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

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

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

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

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

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.

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

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