3.3 Fermions

Electrons, along with protons and neutrons, are fermions, which means they have particle spin . The occupation of quantum states by fermions is restricted by the Pauli exclusion principle, which says that no more than one identical fermion can occupy a given quantum state. The only way to distinguish one fermion from another of the same particle is its spin, and because this has only two possible values, +1/2 or –1/2, at most two fermions of opposite spin can occupy a given quantum state, so that the number of fermions per quantum state is g_s = 2.

At low temperatures, all quantum states with energy less than their chemical potential are filled, and all quantum states with energy greater than their chemical potential are empty. In a cold electron gas, the energy of the most energetic degenerate electron is called the Fermi energy, E_F, and the momentum of particles with this energy is called the Fermi momentum, p_F. The Fermi energy is the sum of a particle’s kinetic energy and rest-mass energy. At low speeds, in the so-called non-relativistic limit, the Fermi kinetic energy and Fermi momentum for electrons are related by

where m_e is the electron mass. The total number of degenerate electrons in the gas is evaluated as

Because the number density of degenerate electrons is simply the total number of electrons per unit volume, , and there are two spin states for electrons (g_s = 2), this equation can be rearranged to express the magnitude of the Fermi momentum as

Fermi energies are conveniently expressed in units of eV, keV or MeV. For comparison, the gap between atomic energy levels in a hydrogen atom is a few eV, the rest mass energy of an electron is 511 keV and that of a proton or neutron is about 940 MeV.