Fermi energy ($E_F$)

The Fermi energy is the energy of the highest occupied one-particle state of a noninteracting Fermi gas at zero temperature. Equivalently, it is the zero-temperature chemical potential.

The kinetic electron Fermi temperature is $T_F=[\sqrt{m_e^2{c}^4+p_F^2{c}^2}-m_e{c}^2]/k_B$, where the Fermi momentum $p_F=\hslash (3{\pi }^2{n}_e{)}^{1/3}$ counts the net Electrons. Strong degeneracy requires $T\ll T_F$. With mean molecular weight per electron ${\mu }_e$ and $n_e=\rho /({\mu }_e{m}_u)$, $T_F=(m_e{c}^2/k_B)[\sqrt{1+(\rho /{\rho }_{\ast }{)}^{2/3}}-1]$. Its nonrelativistic and ultrarelativistic limits scale as ${\rho }^{2/3}$ and ${\rho }^{1/3}$, respectively.

Fermi momentum is the momentum at the boundary of occupied one-particle states in a zero-temperature Fermi gas. For spin-one-half particles in three dimensions, counting the two spin states gives $p_F=\hslash (3{\pi }^{2}n{)}^{1/3}$, where $n$ is number density.

The Fermi sea is the zero-temperature many-fermion state in which every one-particle state below the Fermi energy is occupied and every state above it is empty.