Fermi-Dirac Distribution

Equivalent forms

\langle n_i \rangle = \frac{1}{e^{\beta(\varepsilon_i - \mu)} + 1}

E_F = \frac{\hbar^2}{2m_e}\left(3\pi^2 n\right)^{2/3}

C_V^{\text{el}} = \frac{\pi^2}{2} N k_B \frac{T}{T_F}

The +1 in the denominator encodes Pauli exclusion; swap it to −1 and you get bosons.

Symbol	Name	SI unit	Dimension	Range
$f(E)$	Occupation probabilityoutput Mean number of fermions in a state of energy E	dimensionless	1	0 – 1
$E$	Energy Energy of the single-particle state	eV	ML²T⁻²	0 – 15
$μ$	Chemical potential (Fermi energy at T=0) Energy at which f = 0.5; equals Fermi energy E_F at T=0	eV	ML²T⁻²	1 – 10
$T$	Temperature Absolute temperature	K	Θ	1 – 50000

Unit systems

SI:: energies in J, $k_B = 1.380649e-23\,\mathrm{J/K}$
eV:: $k_B T \approx 0.02585\,\mathrm{eV}$ at 300 K; E_F of copper $\approx 7\,\mathrm{eV}$
natural:: $k_B = 1$ , energies in temperature units

Where it holds

Applies to any system of identical spin-1/2 (or half-integer spin) particles in thermal equilibrium; relativistic extension used for white dwarfs and neutron stars

Discovery

Enrico Fermi & Paul Dirac · 1926

Fermi and Dirac independently derived the distribution in 1926 by applying Pauli's exclusion principle to quantum gases, predicting the anomalously small heat capacity of electrons in metals.

Try this

Why don't all electrons collapse to the lowest energy level?

Electrons are fermions — no two can occupy the same quantum state. This Pauli exclusion fills levels up to the Fermi energy even at absolute zero, making metals conduct and white dwarfs resist gravitational collapse.

Research status: stable

Common misconceptions

\mu is

NOT fixed at E_F for T > 0; it shifts slightly downward in 3D metals as T rises (Sommerfeld expansion). In semiconductors

\mu

sits in the gap.

Derivation

Using the grand canonical ensemble, the mean occupation of state i is ⟨n_i⟩

= -(1/\beta ) \partial \ln \Xi /\partial \varepsilon _i

.

For fermions the state has n_i ∈ {0,1},

so \Xi _i = 1 + e^{-\beta (\varepsilon _i - \mu )}

, giving ⟨n_i⟩

= 1/(e^{\beta (\varepsilon _i - \mu )} + 1)

.

Limiting cases

T \to 0

⟶

f = 1

for E <

\mu

,

f = 0

for E >

\mu

(sharp step = Heaviside)

E - \mu

≫ k_B T⟶

f \approx e^{-\beta (E-\mu )}

→ Maxwell-Boltzmann tail

E = \mu (any T

> 0)⟶

f = 1/2

exactly; defines chemical potential

Canonical Partition Function→Bose-Einstein Distribution→Grand Canonical Potential (Landau Free Energy)→