Fermi-Dirac Statistics and the Degenerate Fermi Gas

Fermi-Dirac statistics describes identical fermions constrained by the Pauli exclusion principle, producing the filled Fermi sea that governs electrons in metals and degenerate stars.

Definition

Fermi-Dirac statistics is the occupation rule for identical fermions, which permits at most one particle per quantum state, and the degenerate Fermi gas is the low-temperature state in which states are filled up to the Fermi energy in accordance with this rule.

Scope

This topic covers the Fermi-Dirac distribution, the Fermi energy and Fermi surface, the ideal degenerate Fermi gas, the Sommerfeld expansion giving the linear low-temperature electronic heat capacity and Pauli paramagnetism, and applications to electrons in metals and to degeneracy pressure in compact stars. The connection to band structure is left to condensed-matter physics.

Core questions

• How does the Pauli exclusion principle produce the Fermi-Dirac distribution?
• What are the Fermi energy and Fermi surface, and why do they dominate low-temperature behavior?
• Why is the electronic heat capacity linear in temperature at low temperature?
• How does degeneracy pressure support white dwarf and neutron stars?

Key concepts

• Fermi-Dirac distribution and the Pauli principle
• Fermi energy and Fermi surface
• Degenerate Fermi gas
• Sommerfeld expansion and electronic heat capacity
• Degeneracy pressure

Key theories

Degenerate Fermi gas

At low temperature fermions fill all single-particle states up to the Fermi energy; only states near the Fermi surface respond to temperature, giving a heat capacity linear in T and a nonzero degeneracy pressure even at absolute zero.

Clinical relevance

The degenerate Fermi gas explains the heat capacity and magnetic response of metals, the behavior of electrons in semiconductors, and the degeneracy pressure that stabilizes white dwarfs against gravitational collapse up to the Chandrasekhar limit.

History

Fermi and Dirac formulated the statistics of exclusion-obeying particles in 1926, and Sommerfeld soon applied it to the electron gas in metals, resolving the longstanding puzzle of why electrons contribute so little to the heat capacity.

Key figures

• Enrico Fermi
• Paul Dirac
• Arnold Sommerfeld

Seminal works

• fermi1926
• pathria2011

Frequently asked questions

Why do metals have such a small electronic heat capacity?

Because the Pauli principle blocks most electrons from changing state; only those within about kT of the Fermi energy can be thermally excited, so only a small fraction of electrons contribute, giving a heat capacity far below the classical equipartition prediction.