Editing Fermi gas (section)

==3D uniform gas==
[[File:Nucleus_drawing.svg|thumb|A model of the atomic nucleus showing it as a compact bundle of the two types of [[nucleon]]s: protons (red) and neutrons (blue). As a first approximation, the nucleus can be treated as composed of non-interacting proton and neutron gases.]]
The three-dimensional [[isotropic]] and non-[[special relativity|relativistic]] uniform Fermi gas case is known as the ''Fermi sphere''.

A three-dimensional infinite square well, (i.e. a cubical box that has a side length ''L'') has the potential energy
<math display="block">V(x,y,z) = \begin{cases}
0, & -\frac{L}{2}<x,y,z<\frac{L}{2},\\
\infty, & \text{otherwise.}
\end{cases}</math>

The states are now labelled by three quantum numbers ''n''<sub>''x''</sub>, ''n''<sub>''y''</sub>, and ''n''<sub>''z''</sub>. The single particle energies are
<math display="block">E_{n_x,n_y,n_z} = E_0 + \frac{\hbar^2 \pi^2}{2m L^2} \left( n_x^2 + n_y^2 + n_z^2\right) \,,</math>
where ''n''<sub>''x''</sub>, ''n''<sub>''y''</sub>, ''n''<sub>''z''</sub> are positive integers. In this case, multiple states have the same energy (known as [[degenerate energy levels]]), for example <math>E_{211}=E_{121}=E_{112}</math>.

=== Thermodynamic limit ===
When the box contains ''N'' non-interacting fermions of spin-{{sfrac|1|2}}, it is interesting to calculate the energy in the thermodynamic limit, where ''N'' is so large that the quantum numbers ''n''<sub>''x''</sub>, ''n''<sub>''y''</sub>, ''n''<sub>''z''</sub> can be treated as continuous variables.

With the vector <math>\mathbf{n}=(n_x,n_y,n_z)</math>, each quantum state corresponds to a point in 'n-space' with energy
<math display="block">E_{\mathbf{n}} = E_0 + \frac{\hbar^2 \pi^2}{2m L^2} |\mathbf{n}|^2 \,</math>

With <math> |\mathbf{n}|^2 </math>denoting the square of the usual Euclidean length <math> |\mathbf{n}|=\sqrt{n_x^2+n_y^2+n_z^2} </math>.
The number of states with energy less than ''E''<sub>F</sub>&nbsp;+ ''E''<sub>0</sub> is equal to the number of states that lie within a sphere of radius <math>|\mathbf{n}_{\mathrm{F}}|</math> in the region of n-space where ''n''<sub>''x''</sub>, ''n''<sub>''y''</sub>, ''n''<sub>''z''</sub> are positive. In the ground state this number equals the number of fermions in the system:

<math display="block">N =2\times\frac{1}{8}\times\frac{4}{3} \pi n_{\mathrm{F}}^3 </math>

[[File:K-space.JPG|thumb|The free fermions that occupy the lowest energy states form a [[sphere]] in [[reciprocal lattice|reciprocal space]]. The surface of this sphere is the [[Fermi surface]].]]
The factor of two expresses the two spin states, and the factor of 1/8 expresses the fraction of the sphere that lies in the region where all ''n'' are positive.
<math display="block">n_{\mathrm{F}}=\left(\frac{3 N}{\pi}\right)^{1/3} </math>
The '''Fermi energy''' is given by
<math display="block">E_{\mathrm{F}} = \frac{\hbar^2 \pi^2}{2m L^2} n_{\mathrm{F}}^2 = \frac{\hbar^2 \pi^2}{2m L^2} \left( \frac{3 N}{\pi} \right)^{2/3}</math>

Which results in a relationship between the Fermi energy and the [[particle number density|number of particles per volume]] (when ''L''<sup>2</sup> is replaced with ''V''<sup>2/3</sup>):
:{{box|border color=#ccccff|<math>E_{\mathrm{F}} = \frac{\hbar^2}{2m} \left( \frac{3 \pi^2 N}{V} \right)^{2/3} </math>}}

This is also the energy of the highest-energy particle (the <math>N</math>th particle), above the zero point energy <math>E_0</math>. The <math>N'</math>th particle has an energy of
<math display="block"> E_{N'} = E_0 + \frac{\hbar^2}{2m} \left( \frac{3 \pi^2 N'}{V} \right)^{2/3} \,=E_0 + E_{\mathrm{F}} \big |_{N'} </math>

The total energy of a Fermi sphere of <math>N</math> fermions (which occupy all <math>N</math> energy states within the Fermi sphere) is given by:

<math display="block">E_{\rm T} = N E_0 + \int_0^N E_{\mathrm{F}}\big |_{N'} \, dN' = \left(\frac{3}{5} E_{\mathrm{F}} + E_0\right)N</math>

Therefore, the average energy per particle is given by:
<math display="block"> E_\mathrm{av} = E_0 + \frac{3}{5} E_{\mathrm{F}} </math>

=== Density of states ===
[[File:Free-electron_DOS.svg|thumb|upright=1.4|Density of states (DOS) of a Fermi gas in 3-dimensions]]
For the 3D uniform Fermi gas, with fermions of spin-{{sfrac|1|2}}, the number of particles as a function of the energy <math display="inline">N(E)</math> is obtained by substituting the Fermi energy by a variable energy <math display="inline">(E-E_0)</math>:

<math display="block">N(E)=\frac{V}{3\pi^2}\left[\frac{2m}{\hbar^2}(E-E_0)\right]^{3/2},</math>

from which the [[density of states]] (number of energy states per energy per volume) <math>g(E)</math> can be obtained. It can be calculated by differentiating the number of particles with respect to the energy:

<math display="block">g(E) =\frac{1}{V}\frac{\partial N(E)}{\partial E}= \frac {1}{2\pi^2} \left(\frac {2m}{\hbar^2}\right)^{3/2}\sqrt{E-E_0}.</math>

This result provides an alternative way to calculate the total energy of a Fermi sphere of <math>N</math> fermions (which occupy all <math>N</math> energy states within the Fermi sphere):

<math display="block">\begin{align}
E_T&=\int_0^N E \mathrm{d} N(E)=EN(E)\big |_0^N-\int_{E_0}^{E_0+E_F} N(E) \mathrm{d} E \\
&=(E_0+E_F)N-\int_{0}^{E_F} N(E) \mathrm{d} (E-E_0) \\
&=(E_0+E_F)N- \frac{2}{5}E_FN(E_F)
= \left(E_0+\frac{3}{5} E_{\mathrm{F}}\right)N
\end{align}</math>

===Thermodynamic quantities===
====Degeneracy pressure====
[[File:Quantum ideal gas pressure 3d.svg|thumb|Pressure vs temperature curves of classical and quantum ideal gases (Fermi gas, [[Bose gas]]) in three dimensions. Pauli repulsion in fermions (such as electrons) gives them an additional pressure over an equivalent classical gas, most significantly at low temperature.]]
By using the [[first law of thermodynamics]], this internal energy can be expressed as a pressure, that is
<math display="block">P = -\frac{\partial E_{\rm T}}{\partial V} = \frac{2}{5}\frac{N}{V}E_{\mathrm{F}}= \frac{(3\pi^2)^{2/3}\hbar^2}{5m}\left(\frac{N}{V}\right)^{5/3},</math>
where this expression remains valid for temperatures much smaller than the Fermi temperature. This pressure is known as the '''degeneracy pressure'''. In this sense, systems composed of fermions are also referred as [[degenerate matter]].

Standard [[star]]s avoid collapse by balancing thermal pressure ([[plasma (physics)|plasma]] and radiation) against gravitational forces. At the end of the star lifetime, when thermal processes are weaker, some stars may become white dwarfs, which are only sustained against gravity by [[electron degeneracy pressure]]. Using the Fermi gas as a model, it is possible to calculate the [[Chandrasekhar limit]], i.e. the maximum mass any star may acquire (without significant thermally generated pressure) before collapsing into a black hole or a neutron star. The latter, is a star mainly composed of neutrons, where the collapse is also avoided by neutron degeneracy pressure.

For the case of metals, the electron degeneracy pressure contributes to the compressibility or [[bulk modulus]] of the material.

====Chemical potential====
{{See also|Fermi level}}
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 a [[Sommerfeld expansion]] (assuming <math>k_{\rm B}T \ll E_{\mathrm{F}}</math>):
<math display="block">\mu(T) = E_0 + E_{\mathrm{F}} \left[ 1- \frac{\pi ^2}{12} \left(\frac{k_{\rm B}T}{E_{\mathrm{F}}}\right) ^2 - \frac{\pi^4}{80} \left(\frac{k_{\rm B}T}{E_{\mathrm{F}}}\right)^4 + \cdots \right], </math>
where ''T'' is the [[temperature]].<ref>{{cite web|title=Statistical Mechanics of Ideal Fermi Systems |url=http://www.uam.es:80/personal_pdi/ciencias/jgr/pdfs/fermi.pdf| last=Kelly|first=James J.|date=1996| website=Universidad Autónoma de Madrid|url-status=dead |archive-url=https://web.archive.org/web/20180412225816/http://www.uam.es/personal_pdi/ciencias/jgr/pdfs/fermi.pdf| archive-date=2018-04-12| access-date=2018-03-15}}</ref><ref>{{cite web |title=Degenerate Ideal Fermi Gases |url=http://www.physics.usyd.edu.au/ugrad/sphys_old/sphys_webct/PHYS3905_SM/TSM12.pdf |url-status=dead|archive-url=https://web.archive.org/web/20080919073627/http://www.physics.usyd.edu.au/ugrad/sphys_old/sphys_webct/PHYS3905_SM/TSM12.pdf| archive-date=2008-09-19 |access-date=2014-04-13}}</ref>

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 ''T''<sub>F</sub>. This 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.

===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]].