Physics:Quantum Fermi–Dirac statistics

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Fermi–Dirac statistics describe the occupation of quantum states by identical fermions. They apply to particles with half-integer spin, including electrons, neutrinos, quarks, and composite fermions such as many atoms and nuclei.

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Fermi-Dirac statistics fill one-particle states subject to the Pauli exclusion principle.

Description

Fermions obey the Pauli exclusion principle: no two identical fermions may occupy the same quantum state. For a system in thermal equilibrium, the average occupation of a one-particle state with energy $E$ is given by the Fermi-Dirac distribution:

$f(E)=\frac{1}{\exp ((E-\mu )/k_{\mathrm{B}}T)+1}$

where $\mu$ is the chemical potential, $k_{\mathrm{B}}$ is the Boltzmann constant, and $T$ is temperature.

Physical meaning

At absolute zero, states below the Fermi energy are filled and states above it are empty. At nonzero temperature the boundary is smoothed, but the exclusion principle still limits each available state to one fermion of a given quantum state.

This statistical rule explains the structure of electron shells in atoms, the behavior of electrons in metals and semiconductors, degeneracy pressure in white dwarfs and neutron stars, and the existence of Fermi surfaces in condensed matter systems.

Historical names

• Enrico Fermi developed the statistical treatment of particles obeying the exclusion principle.
• Paul Dirac independently developed the same statistics in quantum theory.
• Wolfgang Pauli formulated the exclusion principle on which the statistics depends.

See also

• Physics:Quantum fermion
• Physics:Quantum Pauli exclusion principle
• Physics:Quantum electron
• Physics:Quantum Fermi surfaces
• Physics:Quantum Bose–Einstein statistics

References

• Fermi, E. (1926). "Sulla quantizzazione del gas perfetto monoatomico". Rendiconti Lincei 3: 145-149. 
• Dirac, P. A. M. (1926). "On the Theory of Quantum Mechanics". Proceedings of the Royal Society A 112 (762): 661-677. doi:10.1098/rspa.1926.0133. 
• "Fermi-Dirac statistics". https://www.britannica.com/science/Fermi-Dirac-statistics. 

Author: Harold Foppele