Editing Fermi–Dirac statistics (section)

===Distribution of particles over energy===
[[File:Fermi-Dirac distribution.gif|thumb|500px|Fermi function <math>F(\epsilon)</math> with <math>\mu = 0.55~\text{eV}</math> for various temperatures in the range <math>2~\text{K} \leq T \leq 375~\text{K}</math>]]

From the Fermi–Dirac distribution of particles over states, one can find the distribution of particles over energy.{{refn|group=nb|These distributions over energies, rather than states, are sometimes called the Fermi–Dirac distribution too, but that terminology will not be used in this article.}} The average number of fermions with energy <math>\varepsilon_i</math> can be found by multiplying the Fermi–Dirac distribution <math>\bar{n}_i</math> by the [[degenerate energy level|degeneracy]] <math>g_i</math> (i.e. the number of states with energy <math>\varepsilon_i</math>),<ref name="Leighton1959">{{cite book |last=Leighton |first=Robert B. |author-link=Robert B. Leighton |title=Principles of Modern Physics |url=https://archive.org/details/principlesofmode00leig |url-access=registration |publisher=McGraw-Hill |year=1959 |page=[https://archive.org/details/principlesofmode00leig/page/340 340] |isbn=978-0-07-037130-9}} Note that in Eq.&nbsp;(1), <math>n(\varepsilon)</math> and <math>n_s</math> correspond respectively to <math>\bar{n}_i</math> and <math>\bar{n}(\varepsilon_i)</math> in this article. See also Eq.&nbsp;(32) on p.&nbsp;339.</ref>

: <math>\begin{align}
 \bar{n}(\varepsilon_i) &= g_i \bar{n}_i \\
      &= \frac{g_i}{e^{(\varepsilon_i - \mu) / k_\text{B} T} + 1}.
\end{align}</math>

When <math>g_i \ge 2</math>, it is possible that <math>\bar{n}(\varepsilon_i) > 1</math>, since there is more than one state that can be occupied by fermions with the same energy <math>\varepsilon_i</math>.

When a quasi-continuum of energies <math>\varepsilon</math> has an associated [[density of states]] <math>g(\varepsilon)</math> (i.e. the number of states per unit energy range per unit volume<ref name=Blakemore2002p8>{{harvnb|Blakemore|2002|p=8}}.</ref>), the average number of fermions per unit energy range per unit volume is

: <math>\bar{\mathcal{N}}(\varepsilon) = g(\varepsilon) F(\varepsilon),</math>

where <math>F(\varepsilon)</math> is called the '''Fermi function''' and is the same [[function (mathematics)|function]] that is used for the Fermi–Dirac distribution <math>\bar{n}_i</math>:<ref name='Reif1965FermiFnc'>{{harvnb|Reif|1965|p=389}}.</ref>

: <math>F(\varepsilon) = \frac{1}{e^{(\varepsilon - \mu) / k_\text{B}T} + 1},</math>

so that

: <math>\bar{\mathcal{N}}(\varepsilon) = \frac{g(\varepsilon)}{e^{(\varepsilon - \mu) / k_\text{B} T} + 1}.</math>