View source for Fermi gas

{{Short description|Physical model of non-interacting fermions}}
{{Use American English|date = February 2019}}
{{Condensed matter physics|expanded=States of matter}}
A '''Fermi gas''' is an idealized model, an ensemble of many non-interacting [[fermion]]s. Fermions are [[subatomic particle|particles]] that obey [[Fermi–Dirac statistics]], like [[electron]]s, [[proton]]s, and [[neutron]]s, and, in general, particles with [[half-integer]] [[Spin (physics)|spin]]. 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. The model is named after the Italian physicist [[Enrico Fermi]].<ref>{{cite journal |last=Fermi |first=Enrico |author-link=Enrico Fermi |date=1926-11-01 |title=Zur Quantelung des idealen einatomigen Gases |url=https://ethw.org/w/images/e/e5/P1_1926_Zur_Quantelung_des_idealen_einatomigen_Gases.pdf |url-status=dead |journal=Zeitschrift für Physik |language=de |volume=36 |issue=11–12 |pages=902–912 |bibcode=1926ZPhy...36..902F |doi=10.1007/BF01400221 |issn=0044-3328 |s2cid=123334672 |archive-url=https://web.archive.org/web/20190406201635/https://ethw.org/w/images/e/e5/P1_1926_Zur_Quantelung_des_idealen_einatomigen_Gases.pdf |archive-date=2019-04-06}}</ref><ref>{{Cite arXiv |last=Zannoni |first=Alberto |date=1999 |title=On the Quantization of the Monoatomic Ideal Gas |eprint=cond-mat/9912229 |quote=An english translation of the original work of Enrico Fermi on the quantization of the monoatomic ideal gas, is given in this paper}}</ref>

This physical model is useful for certain systems with many fermions. Some key examples are the behaviour of [[Free electron model|charge carriers in a metal]], [[nucleon]]s in an [[atomic nucleus]], neutrons in a [[neutron star]], and electrons in a [[white dwarf]].

==Description==
An ideal Fermi gas or free Fermi gas is a [[mathematical model|physical model]] assuming a collection of non-interacting fermions in a constant [[potential well]]. Fermions are elementary or composite particles with [[half-integer]] spin, thus follow [[Fermi–Dirac statistics]]. The equivalent model for integer spin particles is called the [[Bose gas]] (an ensemble of non-interacting [[boson]]s). At low enough particle [[number density]] and high temperature, both the Fermi gas and the Bose gas behave like a classical [[ideal gas]].<ref>{{Cite book|last=Schwabl|first=Franz| url=https://books.google.com/books?id=kWjwCAAAQBAJ&q=classical+limit+fermi+gas| title=Statistical Mechanics| date=2013-03-09| publisher=Springer Science & Business Media| isbn=978-3-662-04702-6| language=en}}</ref>

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 non-interacting Fermi gas, unlike a Bose gas, concentrates a small number of particles per energy. Thus a Fermi gas is prohibited from condensing into a [[Bose–Einstein condensate]], although weakly-interacting Fermi gases might form a [[Cooper pair]] and condensate (also known as [[BCS theory|BCS]]-BEC crossover regime).<ref>{{cite journal|last1=Regal|first1=C. A.|last2=Greiner|first2=M.|last3=Jin|first3=D. S.|date=2004-01-28|title=Observation of Resonance Condensation of Fermionic Atom Pairs|journal=Physical Review Letters|volume=92|issue=4|pages=040403|doi=10.1103/PhysRevLett.92.040403|pmid=14995356|arxiv=cond-mat/0401554| bibcode=2004PhRvL..92d0403R|s2cid=10799388}}</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. For example, 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]].

The main assumption of the [[free electron model]] to describe the delocalized electrons in a metal can be derived from the Fermi gas. Since interactions are neglected due to [[screening effect]], the problem of treating the equilibrium properties and dynamics of an ideal Fermi gas reduces to the study of the behaviour of single independent particles. In these systems the Fermi temperature is generally many thousands of [[kelvin]]s, so in human applications the electron gas can be considered degenerate. The maximum energy of the fermions at zero temperature is called the '''Fermi energy'''. The Fermi energy surface in [[Reciprocal lattice|reciprocal space]] is known as the [[Fermi surface]].

The [[nearly free electron model]] adapts the Fermi gas model to consider the [[crystal structure]] of [[metal]]s and [[semiconductor]]s, where electrons in a crystal lattice are substituted by [[Bloch electron]]s with a corresponding [[crystal momentum]]. As such, periodic systems are still relatively tractable and the model forms the starting point for more advanced theories that deal with interactions, e.g. using the [[perturbation theory (quantum mechanics)|perturbation theory]].

==1D uniform gas==
The one-dimensional [[particle in a box|infinite square well]] of length ''L'' is a model for a one-dimensional box with the potential energy:
<math display="block">V(x) = \begin{cases}
0, & x_c-\tfrac{L}{2} < x <x_c+\tfrac{L}{2},\\
\infty, & \text{otherwise.}
\end{cases}</math>

It is a standard model-system in quantum mechanics for which the solution for a single particle is well known. Since the potential inside the box is uniform, this model is referred to as 1D uniform gas,<ref name=":0">{{Cite journal| last1=Giorgini|first1=Stefano |last2=Pitaevskii|first2=Lev P. |last3=Stringari|first3=Sandro| date=2008-10-02 |title=Theory of ultracold atomic Fermi gases |url=https://link.aps.org/doi/10.1103/RevModPhys.80.1215 |journal=Reviews of Modern Physics|volume=80|issue=4 |pages=1215–1274 |doi=10.1103/RevModPhys.80.1215 | arxiv=0706.3360 | bibcode=2008RvMP...80.1215G |s2cid=117755089}}</ref> even though the actual number density profile of the gas can have nodes and anti-nodes when the total number of particles is small.

The levels are labelled by a single quantum number ''n'' and the energies are given by:

<math display="block">E_n = E_0 + \frac{\hbar^2 \pi^2}{2 m L^2} n^2. </math>
where <math>E_0</math> is the zero-point energy (which can be chosen arbitrarily as a form of [[gauge fixing]]), <math>m</math> the mass of a single fermion, and <math>\hbar</math> is the reduced [[Planck constant]].

For ''N'' fermions with [[spin-1/2|spin-{{1/2}}]] in the box, no more than two particles can have the same energy, i.e., two particles can have the energy of <math display="inline">E_1</math>, two other particles can have energy <math display="inline">E_2</math> and so forth. The two particles of the same energy have spin {{1/2}} (spin up) or −{{1/2}} (spin down), leading to two states for each energy level. In the configuration for which the total energy is lowest (the ground state), all the energy levels up to ''n''&nbsp;=&nbsp;''N''/2 are occupied and all the higher levels are empty.

Defining the reference for the Fermi energy to be <math>E_0</math>, the Fermi energy is therefore given by
<math display="block">E_{\mathrm{F}}^{(\text{1D})}=E_{n}-E_0=\frac{\hbar^2 \pi^2}{2 m L^2} \left(\left\lfloor \frac{N}{2} \right\rfloor\right)^2,</math>
where <math display="inline">\left\lfloor \frac{N}{2} \right\rfloor</math> is the [[Floor and ceiling functions|floor function]] evaluated at ''n''&nbsp;=&nbsp;''N''/2.

=== Thermodynamic limit ===
In the [[thermodynamic limit]], the total number of particles ''N'' are so large that the quantum number ''n'' may be treated as a continuous variable. In this case, the overall number density profile in the box is indeed uniform.

The number of [[quantum state]]s in the range <math>n_1 < n < n_1 + dn</math> is:
<math display="block">D_n(n_1)\, dn = 2 \, dn\,.</math>

[[Without loss of generality]], the zero-point energy is chosen to be zero, with the following result:

<math display="block">E_n = \frac{\hbar^2 \pi^2}{2 m L^2} n^2 \implies dE = \frac{\hbar^2 \pi^2}{m L^2} n \, dn
= \frac{\hbar \pi}{L}\sqrt{\frac{2E}{m}} dn \,.</math>

Therefore, in the range:
<math display="block">E_1=\frac{\hbar^2 \pi^2}{2 m L^2} n^2_1< E < E_1 + dE\,,</math>
the number of quantum states is:
<math display="block">D_n(n_1) \, dn = 2\frac{dE}{dE/dn} = \frac{2}{\frac{\hbar^2 \pi^2}{m L^2}n} \, dE
\equiv D(E_1) \, dE\,.</math>

Here, the [[Density of states|degree of degeneracy]] is:

<math display="block">D(E)=\frac{2}{dE/dn}
=\frac{2L}{\hbar \pi}\sqrt{\frac{m}{2E}}
\,.</math>

And the [[density of states]] is:

<math display="block">g(E)\equiv \frac{1}{L}D(E)=\frac{2}{\hbar \pi}\sqrt{\frac{m}{2E}}\,.</math>

In modern literature,<ref name=":0" /> the above <math>D(E)</math> is sometimes also called the "density of states". However, <math>g(E)</math> differs from <math>D(E)</math> by a factor of the system's volume (which is <math>L</math> in this 1D case).

Based on the following formula:

<math display="block">\int^{E_\mathrm{F}}_{0} D(E) \, dE = N \,,</math>

the Fermi energy in the thermodynamic limit can be calculated to be:

<math display="block">E_{\mathrm{F}}^{(\text{1D})}=\frac{\hbar^2 \pi^2}{2 m L^2} \left(\frac{N}{2}\right)^2\,.</math>

==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, a metal can be considered as a Fermi gas at zero temperature as a first 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]].

== Arbitrary-dimensional uniform gas ==

=== Density of states ===
Using a volume integral on <math display="inline">d</math> dimensions, the density of states is:

<math display="block">g^{(d)}(E)=g_s \int\frac{\mathrm{d}^d\mathbf{k}}{(2\pi)^d}\delta\left(E-E_0-\frac{\hbar^2|\mathbf{k}|^2}{2m}\right)=g_s\ \left(\frac{m}{2\pi\hbar^2}\right)^{d/2} \frac{(E-E_0)^{d/2-1}}{\Gamma(d/2)} </math>

The Fermi energy is obtained by looking for the [[number density]] of particles:
<math display="block">\rho = \frac{N}{V} = \int_{E_0}^{E_0 + E_{\mathrm{F}}^{(d)}} g^{(d)}(E) \, dE</math>

To get:
<math display="block">E_{\mathrm{F}}^{(d)}=\frac{2\pi\hbar^2}{m}\left(\tfrac{1}{g_s}\Gamma\left(\tfrac{d}{2}+1\right)\frac{N}{V}\right)^{2/d}</math>
where <math display="inline">V</math> is the corresponding ''d''-dimensional volume, <math display="inline">g_s</math> is the dimension for the internal Hilbert space. For the case of spin-{{sfrac|1|2}}, every energy is twice-degenerate, so in this case <math display="inline">g_{s}=2</math>.

A particular result is obtained for <math>d=2</math>, where the density of states becomes a constant (does not depend on the energy):
<math display="block">g^{(2\mathrm{D})}(E) = \frac{g_s}{2}\frac{m}{\pi \hbar^2}. </math>

== Fermi gas in harmonic trap ==
{{Main|Gas in a harmonic trap}}
The [[Quantum harmonic oscillator|harmonic trap potential]]:

<math display="block">V(x,y,z) = \frac{1}{2}m\omega_x^2x^2+\frac{1}{2}m\omega_y^2y^2+\frac{1}{2}m\omega_z^2z^2</math>

is a model system with many applications<ref name=":0" /> in modern physics. The density of states (or more accurately, the degree of degeneracy) for a given spin species is:

<math display="block">g(E) = \frac{E^2}{2(\hbar\omega_\text{ho})^3}\,,</math>

where <math>\omega_\text{ho}=\sqrt[3]{\omega_x\omega_y\omega_z}</math> is the harmonic oscillation frequency.

The Fermi energy for a given spin species is:

<math display="block">E_{\rm F}=(6N)^{1/3}\hbar\omega_\text{ho}\,.</math>

==Related Fermi quantities==
Related to the Fermi energy, a few useful quantities also occur often in modern literature.

The '''Fermi temperature''' is defined as <math display="inline">T_{\mathrm{F}} = \frac{E_{\mathrm{F}}}{k_{\rm B}} </math>, where <math>k_{\rm B}</math> is the [[Boltzmann constant]]. The Fermi temperature can be thought of as the temperature at which thermal effects are comparable to quantum effects associated with Fermi statistics.<ref>{{cite web|title=PHYS 3700: Introduction to Quantum Statistical Thermodynamics|url=http://www.physics.usu.edu/torre/3700_Spring_2015/Lectures/08.pdf|last=Torre|first=Charles|date=2015-04-21|website=Utah State University|access-date=2018-03-21}}</ref> The Fermi temperature for a metal is a couple of orders of magnitude above room temperature. Other quantities defined in this context are '''Fermi momentum''' <math display="inline"> p_{\mathrm{F}} = \sqrt{2 m E_{\mathrm{F}}} </math>, and '''Fermi velocity'''<ref>{{cite web|title=Fermi level and Fermi function|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/Fermi.html|last=Nave|first=Rod|publisher=[[HyperPhysics]]|access-date=2018-03-21}}</ref> <math display="inline"> v_{\mathrm{F}} = \frac{p_{\mathrm{F}}}{m}</math>, which are the [[momentum]] and [[group velocity]], respectively, of a [[fermion]] at the [[Fermi surface]]. The Fermi momentum can also be described as <math>p_{\mathrm{F}} = \hbar k_{\mathrm{F}} </math>, where <math>k_{\mathrm{F}}</math> is the radius of the Fermi sphere and is called the '''Fermi wave vector'''.<ref>{{cite book|last1=Ashcroft|first1=Neil W.|url=https://archive.org/details/solidstatephysic00ashc|title=Solid State Physics|last2=Mermin|first2=N. David|publisher=[[Henry Holt and Company|Holt, Rinehart and Winston]]|year=1976|isbn=978-0-03-083993-1|url-access=registration}}</ref>

Note that these quantities are ''not'' well-defined in cases where the Fermi surface is non-spherical.

==Treatment at finite temperature==

===Grand canonical ensemble===
Most of the calculations above are exact at zero temperature, yet remain as good approximations for temperatures lower than the Fermi temperature. For other thermodynamics variables it is necessary to write a [[thermodynamic potential]]. For an ensemble of [[identical particles|identical fermions]], the best way to derive a potential is from the [[grand canonical ensemble]] with fixed temperature, volume and [[chemical potential]] ''μ''. The reason is due to Pauli exclusion principle, as the occupation numbers of each quantum state are given by either 1 or 0 (either there is an electron occupying the state or not), so the (grand) [[Partition function (statistical mechanics)|partition function]] <math>\mathcal{Z} </math> can be written as
:<math>\mathcal{Z}(T,V,\mu)=\sum_{\{q\}}e^{-\beta (E_q-\mu N_q)}=\prod_{q}\sum_{n_q=0}^1e^{-\beta(\varepsilon_q-\mu )n_q}=\prod_q\left(1+e^{-\beta(\varepsilon_q-\mu )}\right),</math>
where <math>\beta^{-1}=k_{\rm B}T </math>, <math>\{q\} </math> indexes the ensembles of all possible microstates that give the same total energy <math display="inline">E_q = \sum_{q} \varepsilon_q n_q </math> and number of particles <math display="inline">N_q=\sum_{q} n_q </math>, <math display="inline">\varepsilon_q </math> is the single particle energy of the state <math display="inline">q </math> (it counts twice if the energy of the state is degenerate) and <math display="inline">n_q=0,1</math>, its occupancy. Thus the [[grand potential]] is written as
:<math>\Omega(T,V,\mu)=-k_{\rm B}T\ln\left(\mathcal{Z}\right)=-k_{\rm B}T\sum_q\ln\left(1+e^{\beta(\mu-\varepsilon_q)}\right). </math>
The same result can be obtained in the [[canonical ensemble|canonical]] and [[microcanonical ensemble]], as the result of every ensemble must give the same value at [[thermodynamic limit]] <math display="inline">(N/V\rightarrow\infty) </math>. The [[grand canonical ensemble]] is recommended here as it avoids the use of [[combinatorics]] and [[factorial]]s.

As explored in previous sections, in the macroscopic limit we may use a continuous approximation ([[Thomas–Fermi approximation]]) to convert this sum to an integral:
<math display="block">\Omega(T,V,\mu) = -k_{\rm B} T \int_{-\infty}^\infty D(\varepsilon) \ln \left(1 + e^{\beta(\mu-\varepsilon)} \right) \, d\varepsilon </math>
where {{math|''D''(''ε'')}} is the total density of states.

===Relation to Fermi–Dirac distribution===
The grand potential is related to the number of particles at finite temperature in the following way
<math display="block">N=-\left(\frac{\partial \Omega}{\partial \mu}\right)_{T,V}=\int_{-\infty}^\infty D(\varepsilon)\mathcal{f}\left(\frac{\varepsilon-\mu}{k_{\rm B}T}\right)\,\mathrm{d}\varepsilon </math>
where the derivative is taken at fixed temperature and volume, and it appears
<math display="block">\mathcal{f}(x)=\frac{1}{e^x+1} </math>
also known as the [[Fermi–Dirac statistics|Fermi–Dirac distribution]].

Similarly, the total internal energy is 
<math display="block">U = \Omega - T\left(\frac{\partial \Omega}{\partial T}\right)_{V,\mu} - \mu\left(\frac{\partial \Omega}{\partial \mu}\right)_{T,V} = \int_{-\infty}^\infty D(\varepsilon) \mathcal{f} \!\left(\frac{\varepsilon-\mu}{k_{\rm B}T}\right) \varepsilon\, d\varepsilon. </math>

===Exact solution for power-law density-of-states===
{{Cite check section|date=November 2020|reason=equations are inconsistent with other parts of the article.}}

Many systems of interest have a total density of states with the power-law form:
<math display="block">D(\varepsilon) = V g(\varepsilon) = \frac{V g_0}{\Gamma(\alpha)} (\varepsilon - \varepsilon_0)^{\alpha-1}, \qquad \varepsilon \geq \varepsilon_0</math>
for some values of {{math|''g''<sub>0</sub>}}, {{math|''α''}}, {{math|''ε''<sub>0</sub>}}. The results of preceding sections generalize to {{math|''d''}} dimensions, giving a power law with:
* {{math|1=''α'' = ''d''/2}} for non-relativistic particles in a {{math|''d''}}-dimensional box,
* {{math|1=''α'' = ''d''}} for non-relativistic particles in a {{math|''d''}}-dimensional harmonic potential well,
* {{math|1=''α'' = ''d''}} for hyper-relativistic particles in a {{math|''d''}}-dimensional box.
For such a power-law density of states, the grand potential integral evaluates exactly to:<ref>{{cite book |last1=Blundell |title=Concepts in Thermal Physics |date=2006 |publisher=Oxford University Press |isbn=9780198567707 |chapter=Chapter 30: Quantum gases and condensates}}</ref>
<math display="block">\Omega(T,V,\mu) = - V g_0 (k_{\rm B}T)^{\alpha+1} F_{\alpha} \left( \frac{\mu - \varepsilon_0}{k_{\rm B}T} \right),</math>
where <math>F_{\alpha}(x)</math> is the [[complete Fermi–Dirac integral]] (related to the [[polylogarithm]]). From this grand potential and its derivatives, all thermodynamic quantities of interest can be recovered.

==Extensions to the model==

===Relativistic Fermi gas===
[[File:ChandrasekharLimitGraph.svg|right|thumb|upright=1.5|Radius–mass relations for a model white dwarf, relativistic relation vs non-relativistic. The [[Chandrasekhar limit]] is indicated as ''M''<sub>Ch</sub>.]]
The article has only treated the case in which particles have a parabolic relation between energy and momentum, as is the case in non-relativistic mechanics. For particles with energies close to their respective [[rest mass]], the equations of [[special relativity]] are applicable. Where single-particle energy is given by:
<math display="block">E=\sqrt{(pc)^2 + (mc^2)^2}.</math>

For this system, the Fermi energy is given by:
<math display="block">E_{\mathrm{F}}=\sqrt{(p_{\mathrm{F}}c)^2+(mc^2)^2}-mc^2\approx p_{\mathrm{F}}c,</math>
where the <math>\approx</math> equality is only valid in the [[ultrarelativistic limit]], and<ref>{{cite book|title=Thermodynamics and Statistical Mechanics|last1=Greiner|first1=Walter |last2=Neise |first2=Ludwig |last3=Stöcker|first3=Horst|date=1995|publisher=Springer, New York, NY|isbn=9780387942995|series=Classical Theoretical Physics|pages=[https://archive.org/details/thermodynamicsst0000grei/page/341 341–386] |language=en|doi=10.1007/978-1-4612-0827-3_14|author-link=Walter Greiner|author-link3=Horst Stöcker |url=https://archive.org/details/thermodynamicsst0000grei/page/341}}</ref>
<math display="block">p_{\mathrm{F}} = \hbar\left(\frac{1}{g_s} 6\pi^2 \frac{N}{V}\right)^{1/3}.</math>

The relativistic Fermi gas model is also used for the description of massive white dwarfs which are close to the [[Chandrasekhar limit]]. For the ultrarelativistic case, the degeneracy pressure is proportional to <math>(N/V)^{4/3}</math>.

===Fermi liquid===
In 1956, [[Lev Landau]] developed the [[Fermi liquid theory]], where he treated the case of a Fermi liquid, i.e., a system with repulsive, not necessarily small, interactions between fermions. The theory shows that the thermodynamic properties of an ideal Fermi gas and a Fermi liquid do not differ that much. It can be shown that the Fermi liquid is equivalent to a Fermi gas composed of collective excitations or [[quasiparticle]]s, each with a different [[effective mass (solid-state physics)|effective mass]] and [[magnetic moment]].

==See also==
*[[Bose gas]]
*[[Fermionic condensate]]
*[[Gas in a box]]
*[[Jellium]]
*[[Two-dimensional electron gas]]

==References==
{{Reflist}}

==Further reading==
*[[Neil Ashcroft|Neil W. Ashcroft]] and [[N. David Mermin]], ''Solid State Physics'' (Harcourt: Orlando, 1976)
*[[Charles Kittel]], ''[[Introduction to Solid State Physics]]'', 1st ed. 1953 – 8th ed. 2005, {{ISBN|0-471-41526-X}}

{{Authority control}}

[[Category:Quantum models]]
[[Category:Fermi–Dirac statistics]]
[[Category:Ideal gas]]
[[Category:Phases of matter]]