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{{Refimprove|date=January 2013}}
{{Condensed matter physics|expanded=States of matter}}
A '''Fermi gas''' is an ensemble of a large number of [[fermions]]. Fermions, named after [[Enrico Fermi]], are [[subatomic particle|particles]] that obey [[Fermi–Dirac statistics]]. These statistics determine the energy distribution of fermions in a Fermi gas in [[thermal equilibrium]], and is characterized by their [[number density]], [[temperature]], and the set of available energy states. 

By the [[Pauli exclusion principle]], no [[quantum state]] can be occupied by more than one fermion with an identical set of [[quantum number]]s. Thus a noninteracting Fermi gas, unlike a [[Bose gas]], is prohibited from condensing into a [[Bose-Einstein condensate]], although interacting Fermi gases might.<ref>[http://www.conferences.uiuc.edu/bcs50/PDF/Jin.pdf]</ref> The total energy of the Fermi gas at [[absolute zero]] is larger than the sum of the single-particle [[ground state]]s because the Pauli principle implies a sort of interaction or pressure that keeps fermions separated and moving. For this reason, the [[pressure]] of a Fermi gas is non-zero even at zero temperature, in contrast to that of a classical ideal gas. This so-called [[degeneracy pressure]] stabilizes a [[neutron star]] (a Fermi gas of neutrons) or a [[white dwarf]] star (a Fermi gas of electrons) against the inward pull of [[gravity]], which would ostensibly collapse the star into a [[Black Hole]]. Only when a star is sufficiently massive to overcome the degeneracy pressure can it collapse into a singularity.

It is possible to define a [[Fermi temperature]] below which the gas can be considered degenerate (its pressure derives almost exclusively from the Pauli principle). This temperature depends on the mass of the fermions and the [[density of states|density of energy states]]. For metals, the electron gas's Fermi temperature is generally many thousands of [[kelvin]]s, so in human applications they can be considered degenerate. The maximum energy of the fermions at zero temperature is called the [[Fermi energy]]. The Fermi energy surface in [[momentum space]] is known as the [[Fermi surface]].

==Ideal Fermi gas==
{{seealso|free electron model}}
An ideal Fermi gas or free Fermi gas is a [[Mathematical model|physical model]] assuming a collection of non-interacting fermions. It is the [[quantum mechanics|quantum mechanical]] version of an [[ideal gas]], for the case of fermionic particles. The behavior of electrons in a [[white dwarf]] or [[neutron]]s in a [[neutron star]] can be approximated by treating them as an ideal Fermi gas. Something similar can be done for periodic systems, such as electrons moving in the [[crystal structure|crystal lattice]] of [[metal]]s and [[semiconductor]]s, using the so-called ''quasi-momentum'' or ''crystal momentum'' ([[Bloch wave]]). Since interactions are neglected by definition, the problem of treating the equilibrium properties and dynamics of an ideal Fermi gas reduces to the study of the behavior of single independent particles. As such, it is still relatively tractable and forms the starting point for more advanced theories that deal with interactions, e.g., using the [[perturbation theory (quantum mechanics)|perturbation theory]].

Assuming that the concentration of fermions does not change with temperature, then the [[total chemical potential]] ''µ'' ([[Fermi level]]) of the three dimensional ideal Fermi gas is related to the zero temperature [[Fermi energy]] ''E''<sub>F</sub> by the following expansion (assuming <math>kT \ll E_F</math>):
:<math>\mu = E_0 + E_F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{E_F}\right) ^2 - \frac{\pi^4}{80} \left(\frac{kT}{E_F}\right)^4 + \cdots \right] </math>
where ''E''<sub>0</sub> is the potential energy per particle, ''k'' is the [[Boltzmann constant]] and ''T'' is [[temperature]].

Hence, the [[internal chemical potential]], ''µ''-''E''<sub>0</sub>,  is approximately equal to the Fermi energy at temperatures that are much lower than the characteristic Fermi temperature ''E<sub>F</sub>''/''k''. The characteristic temperature is on the order of 10<sup>5</sup> [[kelvin|K]] for a metal, hence at room temperature (300 K), the Fermi energy and internal chemical potential are essentially equivalent.

==References==
{{reflist}}

==See also==
* [[Fermi liquid]]
* [[Bose gas]]
* [[Free electron model]]
* [[Gas in a box]]

{{DEFAULTSORT:Fermi Gas}}
[[Category:Ideal gas]]
[[Category:Fermi–Dirac statistics]]