Fermi gas

A Fermi gas is an ensemble of a large number of fermions (named after Enrico Fermi). Fermions are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density, temperature, and the set of available energy states.

By the Pauli exclusion principle, no quantum state can be occupied by more than one fermion with an identical set of quantum numbers. Thus a non-interacting Fermi gas, unlike a Bose gas, is prohibited from condensing into a Bose–Einstein condensate, although interacting Fermi gases might.^[1] The total energy of the Fermi gas at absolute zero is larger than the sum of the single-particle ground states because the Pauli principle implies a sort of interaction or pressure that keeps fermions separated and moving. For this reason, the pressure of a Fermi gas is non-zero even at zero temperature, in contrast to that of a classical ideal gas. This so-called degeneracy pressure stabilizes a neutron star (a Fermi gas of neutrons) or a white dwarf star (a Fermi gas of electrons) against the inward pull of gravity, which would ostensibly collapse the star into a black hole. Only when a star is sufficiently massive to overcome the degeneracy pressure can it collapse into a singularity.

It is possible to define a Fermi temperature below which the gas can be considered degenerate (its pressure derives almost exclusively from the Pauli principle). This temperature depends on the mass of the fermions and the density of energy states. For metals, the electron gas's Fermi temperature is generally many thousands of kelvins, so in human applications they can be considered degenerate. The maximum energy of the fermions at zero temperature is called the Fermi energy. The Fermi energy surface in momentum space is known as the Fermi surface.

Ideal Fermi gas

An ideal Fermi gas or free Fermi gas is a physical model assuming a collection of non-interacting fermions. It is the quantum mechanical version of an ideal gas, for the case of fermionic particles. The behavior of electrons in a white dwarf or neutrons in a neutron star can be approximated by treating them as an ideal Fermi gas. Something similar can be done for periodic systems, such as electrons moving in the crystal lattice of metals and semiconductors, using the so-called quasi-momentum or crystal momentum (Bloch wave). Since interactions are neglected by definition, the problem of treating the equilibrium properties and dynamics of an ideal Fermi gas reduces to the study of the behavior of single independent particles. As such, it is still relatively tractable and forms the starting point for more advanced theories that deal with interactions, e.g., using the perturbation theory.

Assuming that the concentration of fermions does not change with temperature, then the total chemical potential µ (Fermi level) of the three-dimensional ideal Fermi gas is related to the zero temperature Fermi energy E_F by a Sommerfeld expansion (assuming $kT\ll E_{F}$):

$\mu =E_{0}+E_{F}\left[1-{\frac {\pi ^{2}}{12}}\left({\frac {kT}{E_{F}}}\right)^{2}-{\frac {\pi ^{4}}{80}}\left({\frac {kT}{E_{F}}}\right)^{4}+\cdots \right]$

where E₀ is the potential energy per particle, k is the Boltzmann constant and T is temperature.^[2][3]

Hence, the internal chemical potential, µ-E₀, is approximately equal to the Fermi energy at temperatures that are much lower than the characteristic Fermi temperature E_F/k. The characteristic temperature is on the order of 10⁵ K for a metal, hence at room temperature (300 K), the Fermi energy and internal chemical potential are essentially equivalent.

References

1. http://www.conferences.uiuc.edu/bcs50/PDF/Jin.pdf
2. http://www.uam.es/personal_pdi/ciencias/jgr/pdfs/fermi.pdf page 10
3. http://www.physics.usyd.edu.au/ugrad/sphys_old/sphys_webct/PHYS3905_SM/TSM12.pdf

See also

• Fermi liquid
• Bose gas
• Free electron model
• Gas in a box