Alexander Fufaev
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Fermi Distribution: The Occupation Probability of Fermions

Important Formula

Formula: Fermi Distribution Function
What do the formula symbols mean?


The occupation probability indicates the probability \(P\) that a state with energy \( W \) is occupied at temperature \( T \). At absolute zero (\(T=0 \, \text{K}\)), the probability that the state with energy \( W \) is occupied is exactly 50%: \( P(W) ~=~ \frac{1}{2}\).


Energy state which can be occupied by a fermion, for example by an electron.

Chemical potential

Chemical potential gives the change of the internal energy when the particle number of the Fermi gas (e.g. free electron gas) changes. At \( T=0 \, \text{K} \) the chemical potential correspons to the Fermi energy: \( \mu = W_{\text F} \).


Absolute temperature of the Fermi gas, for example a free electron gas in a metal.

Boltzmann Constant

Boltzmann constant is a physical constant from many-particle physics and has the following exact value: $$ k_{\text B} ~=~ 1.380 \, 649 ~\cdot~ 10^{-23} \, \frac{\mathrm{J}}{\mathrm{K}} $$
Fermi distribution graph at finite temperature

The Fermi distribution (also known as the Fermi-Dirac distribution) is particularly important in the description of electrons in solids, especially in metals.

The Fermi distribution indicates the occupation probability \( P(W) \) that a fermion in a system acquires the energy \( W \). The occupation probability depends on the temperature \( T \) of the system.

Fermi distribution graph at finite temperature
Fermi, Bose and Boltzmann distribution graphs

In contrast to the Bose distribution, the Fermi distribution contains a positive term in the denominator, which ensures that the probability of occupying a state does not approach zero when the temperature approaches zero. This means that fermions cannot occupy the same quantum mechanical state according to the Pauli principle.