Fermi gas is a
physical modelA mathematical model uses mathematical language to describe a system. Mathematical models are used not only in the natural sciences and engineering disciplines but also in the social sciences ; physicists, engineers, computer scientists, and economists use...
assuming a collection of non-interacting fermions. It is the
quantum mechanicalQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
version of an
ideal gasAn ideal gas is a theoretical gas composed of a set of randomly-moving point particles that interact only through elastic collisions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At...
, for the case of fermionic particles. The behavior of
ElectronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s in
metalA metal is a chemical element that is a good conductor of both electricity and heat, forms cations and ionic bonds with non-metals. In chemistry, a metal is an element, compound, or alloy characterized by high electrical conductivity. In a metal, atoms readily lose electrons to form positive ions...
s and
semiconductorA semiconductor is a material that has an electrical resistivity between that of a conductor and an insulator, that is, generally in the range 10
3 Siemens/cm to 10
−8 S/cm. Devices made from semiconductor materials are the foundation of modern electronics, including radio,...
s and
neutronThe neutron is a subatomic particle with no net electric charge and a mass slightly larger than that of a proton.Neutron are usually found in atomic nuclei. The nuclei of most atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of protons in a...
s in a
neutron starA neutron star is a type of remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and roughly the same mass as protons...
can be approximated by treating them as Fermi gases. The energy distribution of the fermions in a Fermi gas in thermal equilibrium is determined by their
densityIn physics, astronomy, and chemistry, number density is an intensive quantity used to describe the degree of concentration of countable objects in the three-dimensional physical space...
,
temperatureIn physics, temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics...
, and the set of available energy states, via
Fermi-Dirac statisticsFermi-Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
.
By the Pauli principle, no
quantum stateIn quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be...
can be occupied by more than one fermion (with identical properties); thus a Fermi gas, unlike a
Bose gasAn ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose-Einstein statistics...
, is prohibited from condensing into a Bose-Einstein condensate.
Fermi gas is a
physical modelA mathematical model uses mathematical language to describe a system. Mathematical models are used not only in the natural sciences and engineering disciplines but also in the social sciences ; physicists, engineers, computer scientists, and economists use...
assuming a collection of non-interacting fermions. It is the
quantum mechanicalQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
version of an
ideal gasAn ideal gas is a theoretical gas composed of a set of randomly-moving point particles that interact only through elastic collisions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At...
, for the case of fermionic particles. The behavior of
ElectronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s in
metalA metal is a chemical element that is a good conductor of both electricity and heat, forms cations and ionic bonds with non-metals. In chemistry, a metal is an element, compound, or alloy characterized by high electrical conductivity. In a metal, atoms readily lose electrons to form positive ions...
s and
semiconductorA semiconductor is a material that has an electrical resistivity between that of a conductor and an insulator, that is, generally in the range 10
3 Siemens/cm to 10
−8 S/cm. Devices made from semiconductor materials are the foundation of modern electronics, including radio,...
s and
neutronThe neutron is a subatomic particle with no net electric charge and a mass slightly larger than that of a proton.Neutron are usually found in atomic nuclei. The nuclei of most atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of protons in a...
s in a
neutron starA neutron star is a type of remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and roughly the same mass as protons...
can be approximated by treating them as Fermi gases. The energy distribution of the fermions in a Fermi gas in thermal equilibrium is determined by their
densityIn physics, astronomy, and chemistry, number density is an intensive quantity used to describe the degree of concentration of countable objects in the three-dimensional physical space...
,
temperatureIn physics, temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics...
, and the set of available energy states, via
Fermi-Dirac statisticsFermi-Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
.
By the Pauli principle, no
quantum stateIn quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be...
can be occupied by more than one fermion (with identical properties); thus a Fermi gas, unlike a
Bose gasAn ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose-Einstein statistics...
, is prohibited from condensing into a Bose-Einstein condensate. Therefore the total energy of the Fermi gas at
Absolute zeroAbsolute zero is a temperature marked by a 0 entropy configuration. It is the coldest temperature theoretically possible and cannot be reached by artificial or natural means, because it is impossible to decouple a system fully from the rest of the universe...
is larger than the sum of the single-particle ground states because the Pauli principle acts as an sort of interaction/pressure that keeps the fermions sepearated and moving. For this reason, the
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
of a Fermi gas is nonzero even at zero temperature, in contrast to that of a classical ideal gas. This so-called degeneracy pressure stabilizes a
neutron starA neutron star is a type of remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and roughly the same mass as protons...
(a Fermi gas of neutrons) or a
white dwarfA white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...
star (a Fermi gas of electrons) against the inward pull of gravity, which would ostensibly collapse the star into a
Black HoleIn general relativity, a black hole is a region of space in which the gravitational field is so powerful that nothing, not even light, can escape. The black hole has a one-way surface, called an event horizon, into which objects can fall, but out of which nothing can come...
. 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 statesIn statistical and condensed matter physics, the density of states of a system describes the number of states at each energy level that are available to be occupied. A high DOS at a specific energy level means that there are many states available for occupation...
. For metals, the electron gas's Fermi temperature is generally many thousands of
kelvinThe kelvin is a unit increment of temperature and is one of the seven SI base units. The Kelvin scale is a thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero kelvin...
s, so in human applications they can be considered degenerate. The maximum energy of the fermions at zero temperature is called the
Fermi energyThe Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature...
. The Fermi energy surface in
momentum spaceThe Momentum space associated with a particle is a vector space in which every point {} corresponds to a possible value of the momentum vector . Representing a problem in terms of the momenta of the particles involved, rather than in terms of their positions, can greatly simplify some problems in...
is known as the
Fermi surfaceIn condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline...
.
Since interactions are neglected by definition, the problem of treating the equilibrium properties and dynamical behaviour of a Fermi gas reduces to the study of the behaviour of single independent particles. As such, it is still relatively tractable and forms the starting point for more advanced theories that deal with interatction(such as Fermi liquid theory or
perturbation theoryIn quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
).