Editing Fermi gas (section)

===Typical values===

====Metals====
Under the [[free electron model]], the electrons in a metal can be considered to form a uniform Fermi gas. The number density <math>N/V</math> of conduction electrons in metals ranges between approximately 10<sup>28</sup> and 10<sup>29</sup> electrons per m<sup>3</sup>, which is also the typical density of atoms in ordinary solid matter. This number density produces a Fermi energy of the order:
<math display="block">E_{\mathrm{F}} = \frac{\hbar^2}{2m_e} \left( 3 \pi^2 \ 10^{28 \ \sim \ 29} \ \mathrm{m^{-3}} \right)^{2/3} \approx 2 \ \sim \ 10 \ \mathrm{eV}, </math>
where ''m<sub>e</sub>'' is the [[electron rest mass]].<ref>{{cite web|title=Fermi Energies, Fermi Temperatures, and Fermi Velocities|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/fermi.html |last=Nave |first=Rod|publisher=[[HyperPhysics]] |access-date=2018-03-21}}</ref> This Fermi energy corresponds to a Fermi temperature of the order of 10<sup>6</sup> kelvins, much higher than the temperature of the [[Sun]]'s surface. Any metal will boil before reaching this temperature under atmospheric pressure. Thus for any practical purpose, free electrons in a metal can be considered as a Fermi gas at zero temperature as an approximation (normal temperatures are small compared to ''T''<sub>F</sub>).

====White dwarfs====
Stars known as [[white dwarf]]s have mass comparable to the [[Sun]], but have about a hundredth of its radius. The high densities mean that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. The number density of electrons in a white dwarf is of the order of 10<sup>36</sup> electrons/m<sup>3</sup>. This means their Fermi energy is:

<math display="block">E_{\mathrm{F}} = \frac{\hbar^2}{2m_e} \left( \frac{3 \pi^2 (10^{36})}{1 \ \mathrm{m^3}} \right)^{2/3} \approx 3 \times 10^5 \ \mathrm{eV} = 0.3 \ \mathrm{MeV}</math>

====Nucleus====
Another typical example is that of the particles in a nucleus of an atom. The [[nuclear radius|radius of the nucleus]] is roughly:
<math display="block">R = \left(1.25 \times 10^{-15} \mathrm{m} \right) \times A^{1/3}</math>
where ''A'' is the number of [[nucleon]]s.

The number density of nucleons in a nucleus is therefore:

<math display="block">\rho = \frac{A}{ \frac{4}{3} \pi R^3} \approx 1.2 \times 10^{44} \ \mathrm{m^{-3}} </math>

This density must be divided by two, because the Fermi energy only applies to fermions of the same type. The presence of [[neutron]]s does not affect the Fermi energy of the [[proton]]s in the nucleus, and vice versa.

The Fermi energy of a nucleus is approximately:
<math display="block">E_{\mathrm{F}} = \frac{\hbar^2}{2m_{\rm p}} \left( \frac{3 \pi^2 (6 \times 10^{43})}{1 \ \mathrm{m}^3} \right)^{2/3} \approx 3 \times 10^7 \ \mathrm{eV} = 30 \ \mathrm{MeV} ,</math>
where ''m''<sub>p</sub> is the proton mass.

The [[nuclear radius|radius of the nucleus]] admits deviations around the value mentioned above, so a typical value for the Fermi energy is usually given as 38 [[MeV]].