FermiTemperatur

Fermi temperature, abbreviated T_F, is a temperature scale defined for a system of fermions in thermal equilibrium. It is given by T_F = E_F / k_B, where E_F is the Fermi energy. The Fermi energy is the energy of the highest occupied single-particle state at zero temperature and sets the boundary between quantum-degenerate and classical behavior.

In a nonrelativistic, ideal three-dimensional gas of spin-1/2 fermions with number density n, the Fermi energy is E_F = (ħ^2/2m) (3π^2 n)^{2/3}. Therefore T_F = (ħ^2/2m k_B) (3π^2 n)^{2/3}. For a gas with spin degeneracy g_s, E_F = (ħ^2/2m) [ (6π^2 n) / g_s ]^{2/3}. These expressions apply to electrons in metals and to ultracold atomic gases, among other systems.

Physically, T_F marks the transition between quantum and classical behavior. At temperatures well below T_F, the gas is quantum degenerate and the occupation of states is governed by the Fermi-Dirac distribution with a sharp Fermi surface at E_F; thermodynamic quantities often vary linearly with T. At temperatures well above T_F, the distribution approaches the classical Maxwell-Boltzmann form.

Typical values vary widely. For electrons in metals, n ≈ 10^{28} m^{-3} gives T_F on the order of 10^4–10^5 K (energies of a few electronvolts). In ultracold-atom experiments, T is kept in the nanokelvin to microkelvin range, making T/T_F a central diagnostic. The concept provides a convenient reference scale across condensed matter, atomic physics, and astrophysical contexts.