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'''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron|electrons]], [[Physics:Quantum neutrino|neutrinos]], [[Physics:Quantum quark|quarks]], and composite fermions such as many atoms and nuclei.
'''Fermi–Dirac statistics''' describe the occupation of quantum states by identical [[Physics:Quantum fermion|fermions]]. They apply to particles with half-integer spin, including [[Physics:Quantum electron|electrons]], [[Physics:Quantum neutrino|neutrinos]], [[Physics:Quantum quark|quarks]], and composite fermions such as many atoms and nuclei.<ref>{{Cite journal |last=Dirac |first=Paul A. M. |title=On the Theory of Quantum Mechanics |journal=Proceedings of the Royal Society A |year=1926 |volume=112 |issue=762 |pages=661-677 |doi=10.1098/rspa.1926.0133 |jstor=94692}}</ref>
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== Description ==
== Description ==
Fermions obey the [[Physics:Quantum Pauli exclusion principle|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 <math>E</math> is given by the Fermi-Dirac distribution:
Fermions obey the [[Physics:Quantum Pauli exclusion principle|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 <math>E</math> is given by the Fermi-Dirac distribution:<ref>{{Cite journal |last=Fermi |first=Enrico |title=Sulla quantizzazione del gas perfetto monoatomico |journal=Rendiconti Lincei |year=1926 |volume=3 |pages=145-149}}</ref>


<math>f(E)=\frac{1}{\exp\left((E-\mu)/k_{\mathrm B}T\right)+1}</math>
<math>f(E)=\frac{1}{\exp\left((E-\mu)/k_{\mathrm B}T\right)+1}</math>
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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.
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 [[Physics:Quantum Fermi surfaces|Fermi surfaces]] in condensed matter systems.
This statistical rule explains the structure of electron shells in atoms<ref>{{Cite journal |last=Sommerfeld |first=Arnold |title=Zur Elektronentheorie der Metalle |journal=Naturwissenschaften |year=1927 |volume=15 |issue=41 |pages=824-832 |doi=10.1007/BF01505083}}</ref>, the behavior of electrons in metals and semiconductors, degeneracy pressure in white dwarfs and neutron stars, and the existence of [[Physics:Quantum Fermi surfaces|Fermi surfaces]] in condensed matter systems.


== Historical names ==
== Historical names ==
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== References ==
== References ==
{{reflist|3}}
{{reflist|3}}
* {{Cite journal |last=Fermi |first=Enrico |title=Sulla quantizzazione del gas perfetto monoatomico |journal=Rendiconti Lincei |year=1926 |volume=3 |pages=145-149}}
* {{Cite journal |last=Dirac |first=Paul A. M. |title=On the Theory of Quantum Mechanics |journal=Proceedings of the Royal Society A |year=1926 |volume=112 |issue=762 |pages=661-677 |doi=10.1098/rspa.1926.0133 |jstor=94692}}
* {{Cite book |last=Dirac |first=Paul A. M. |title=Principles of Quantum Mechanics |edition=4th revised |location=London |publisher=Oxford University Press |year=1967 |isbn=978-0-19-852011-5}}
* {{Cite journal |last=Sommerfeld |first=Arnold |title=Zur Elektronentheorie der Metalle |journal=Naturwissenschaften |year=1927 |volume=15 |issue=41 |pages=824-832 |doi=10.1007/BF01505083}}


{{Author|Harold Foppele}}
{{Author|Harold Foppele}}

Revision as of 22:15, 23 May 2026

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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.[1]

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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:[2]

f(E)=1exp⁡((E−μ)/kBT)+1

where μ is the chemical potential, kB 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[3], 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

Table of contents (84 articles)

Index

Full contents

References

  1. ↑ Dirac, Paul 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. 
  2. ↑ Fermi, Enrico (1926). "Sulla quantizzazione del gas perfetto monoatomico". Rendiconti Lincei 3: 145-149. 
  3. ↑ Sommerfeld, Arnold (1927). "Zur Elektronentheorie der Metalle". Naturwissenschaften 15 (41): 824-832. doi:10.1007/BF01505083. 


Author: Harold Foppele


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