= Fermi energy
{c}
{title2=$E_F$}
{wiki}
The Fermi energy is the highest occupied one-particle <energy> in the <ground state> of a noninteracting <Fermi gas> at zero <temperature>, measured here relative to the lowest one-particle <energy>. The <Pauli exclusion principle> allows only one <fermion> in each complete one-particle <quantum state>, including its <spin>, so adding more particles fills progressively higher <energy eigenstates>. A cold gas can therefore have substantial <kinetic energy> even when its <temperature> is zero. In the <thermodynamic limit>, with the same <energy> reference, $E_F=\mu(T=0)$ where $\mu$ is the <chemical potential>.<Statistical Physics by David Tong/3 Quantum Gases>[Statistical Physics by David Tong; §3.6.2, equation (3.130), zero-temperature filling]{ref}
For a uniform, three-dimensional gas of noninteracting <electrons>, with <spin one-half>, equal populations of the two <spin> states, no applied <magnetic field>, and <nonrelativistic particles> of <mass> $m=m_e$,
$$
n=\frac NV,\qquad
p_F=\hbar(3\pi^2n)^{1/3},\qquad
E_F=\frac{p_F^2}{2m}=\frac{\hbar^2}{2m}(3\pi^2n)^{2/3},\qquad
T_F=\frac{E_F}{k_B}.
$$
Here $n$ is the total <number density>, $V$ the <volume>, $p_F$ the <Fermi momentum>, $\hbar=h/(2\pi)$ the reduced <Planck constant>, and $k_B$ the <Boltzmann constant>. The <electron Fermi temperature> $T_F$ expresses the same <energy> scale in <kelvin>; it is not the actual gas <temperature>. Strong degeneracy of a <degenerate electron gas> means $T\ll T_F$. The <momentum> formula counts states and survives changing the <energy–momentum relation>, whereas $E_F=p_F^2/(2m)$ requires $p_F\ll mc$.<Degenerate Electron Gases by Richard Fitzpatrick>[Richard Fitzpatrick, Degenerate Electron Gases, equations (506)–(509), two spin states and density dependence]{ref}
The reference matters: the displayed $E_F$ excludes rest <energy> and sets the zero at $p=0$. Including rest <energy> shifts both the one-particle <energy> and the zero-temperature <chemical potential> by $mc^2$, leaving occupations unchanged. More generally, adding a constant $C$ to every one-particle <energy> also adds $C$ to $\mu$; the <Fermi-Dirac distribution> depends only on their difference. In a finite <quantum system> the <chemical potential> at zero <temperature> can lie between the last occupied and first empty <energy eigenstates>; its equality with the filling edge is a continuum statement.
At positive <temperature>, $E_F$ remains the zero-temperature scale associated with the specified <number density>. The <Fermi level> $\mu(T)$ must instead satisfy the particle-number condition
$$
N=\int_0^\infty\frac{g(E)}{\exp[(E-\mu(T))/(k_BT)]+1}\,dE,
$$
where $g(E)$ is the extensive <density of states>, including both <spin> states. Smearing the <Fermi-Dirac distribution> while arbitrarily holding $\mu=E_F$ generally changes $N$. This distinction matters when interpreting <electron heat capacity> and any experiment that infers <temperature> from an occupation profile.<Statistical Physics by David Tong/3 Quantum Gases>[Statistical Physics by David Tong; §§3.6.1–3.6.4, equation (3.127) fixes particle number, and finite-temperature chemical potential is determined separately]{ref}
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