My name is Alexander FufaeV and here I will explain the following topic:

# Fermi Distribution: The Occupation Probability of Fermions

## Formula

What do the formula symbols mean?

## Probability

Unit
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

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

## Chemical potential

Unit
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: $$\mu = W_{\text F}$$.

## Temperature

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

## Boltzmann Constant

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

## Video

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.

Formula anchor

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.