Fermi gas
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An ideal Fermi gas is a
state of matter In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. Many intermediate states are known to exist, such as liquid crystal ...
which is an ensemble of many non-interacting
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s. Fermions are
particles In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
that obey
Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac di ...
, like
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s,
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s, and
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
s, and, in general, particles with
half-integer In mathematics, a half-integer is a number of the form :n + \tfrac, where n is an whole number. For example, :, , , 8.5 are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
spin. These statistics determine the energy distribution of fermions in a Fermi gas in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
, and is characterized by their
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
,
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
, and the set of available energy states. The model is named after the Italian physicist
Enrico Fermi Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...
. This physical model can be accurately applied to many systems with many fermions. Some key examples are the behaviour of charge carriers in a metal,
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons were ...
s in an
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron i ...
, neutrons in a
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. white ...
, and electrons in a
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
.


Description

An ideal Fermi gas or free Fermi gas is a
physical model A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
assuming a collection of non-interacting fermions in a constant
potential well A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is captur ...
. Fermions are elementary or composite particles with
half-integer In mathematics, a half-integer is a number of the form :n + \tfrac, where n is an whole number. For example, :, , , 8.5 are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
spin, thus follow Fermi-Dirac statistics. The equivalent model for integer spin particles is called the
Bose gas An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were deve ...
(an ensemble of non-interacting
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s). At low enough particle
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
and high temperature, both the Fermi gas and the Bose gas behave like a classical
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
. By the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
, no
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
can be occupied by more than one fermion with an identical set of
quantum number In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
s. Thus a non-interacting Fermi gas, unlike a Bose gas, concentrates a small number of particles per energy. Thus a Fermi gas is prohibited from condensing into a
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
, although weakly-interacting Fermi gases might form a
Cooper pair In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Coope ...
and condensate (also known as BCS-BEC crossover regime). The total energy of the Fermi gas at
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...
is larger than the sum of the single-particle
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
s because the Pauli principle implies a sort of interaction or pressure that keeps fermions separated and moving. For this reason, the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
of a Fermi gas is non-zero even at zero temperature, in contrast to that of a classical ideal gas. For example, this so-called
degeneracy pressure Degenerate matter is a highly dense state of fermionic matter in which the Pauli exclusion principle exerts significant pressure in addition to, or in lieu of, thermal pressure. The description applies to matter composed of electrons, protons, ne ...
stabilizes a
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. white ...
(a Fermi gas of neutrons) or a
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
star (a Fermi gas of electrons) against the inward pull of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, which would ostensibly collapse the star into a
black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
. 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. The main assumption of the
free electron model In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantu ...
to describe the delocalized electrons in a metal can be derived from the Fermi gas. Since interactions are neglected due to
screening effect In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases (classical plasmas), electrolytes, and charge c ...
, the problem of treating the equilibrium properties and dynamics of an ideal Fermi gas reduces to the study of the behaviour of single independent particles. In these systems the Fermi temperature is generally many thousands of
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
s, so in human applications the electron gas can be considered degenerate. The maximum energy of the fermions at zero temperature is called the Fermi energy. The Fermi energy surface in
reciprocal space In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial fu ...
is known as the
Fermi surface In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the cryst ...
. The
nearly free electron model In solid-state physics, the nearly free electron model (or NFE model) or quasi-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model i ...
adapts the Fermi gas model to consider the
crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
of
metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
s and
semiconductor A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
s, where electrons in a crystal lattice are substituted by
Bloch electron In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
s with a corresponding crystal momentum. As such, periodic systems are still relatively tractable and the model forms the starting point for more advanced theories that deal with interactions, e.g. using the
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
.


1D uniform gas

The one-dimensional
infinite square well In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
of length ''L'' is a model for a one-dimensional box with the potential energy: V(x) = \begin 0, & x_c-\tfrac < x It is a standard model-system in quantum mechanics for which the solution for a single particle is well known. Since the potential inside the box is uniform, this model is referred to as 1D uniform gas, even though the actual number density profile of the gas can have nodes and anti-nodes when the total number of particles is small. The levels are labelled by a single quantum number ''n'' and the energies are given by: E_n = E_0 + \frac n^2. where E_0 is the zero-point energy (which can be chosen arbitrarily as a form of
gauge fixing In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct c ...
), m the mass of a single fermion, and \hbar is the reduced
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. For ''N'' fermions with spin- in the box, no more than two particles can have the same energy, i.e., two particles can have the energy of E_1, two other particles can have energy E_2 and so forth. The two particles of the same energy have spin (spin up) or − (spin down), leading to two states for each energy level. In the configuration for which the total energy is lowest (the ground state), all the energy levels up to ''n'' = ''N''/2 are occupied and all the higher levels are empty. Defining the reference for the Fermi energy to be E_0, the Fermi energy is therefore given by E_^=E_-E_0=\frac \left(\left\lfloor \frac \right\rfloor\right)^2, where \left\lfloor \frac \right\rfloor is the
floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
evaluated at ''n'' = ''N''/2.


Thermodynamic limit

In the
thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
, the total number of particles ''N'' are so large that the quantum number ''n'' may be treated as a continuous variable. In this case, the overall number density profile in the box is indeed uniform. The number of
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s in the range n_1 < n < n_1 + dn is: D_n(n_1)\, dn = 2 \, dn\,.
Without loss of generality ''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
, the zero-point energy is chosen to be zero, with the following result: E_n = \frac n^2 \implies dE = \frac n \, dn = \frac\sqrt dn \,. Therefore, in the range: E_1=\frac n^2_1< E < E_1 + dE\,, the number of quantum states is: D_n(n_1) \, dn = 2\frac = \frac \, dE \equiv D(E_1) \, dE\,. Here, the degree of degeneracy is: D(E)=\frac =\frac\sqrt \,. And the
density of states In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
is: g(E)\equiv \fracD(E)=\frac\sqrt\,. In modern literature, the above D(E) is sometimes also called the "density of states". However, g(E) differs from D(E) by a factor of the system's volume (which is L in this 1D case). Based on the following formula: \int^_ D(E) \, dE = N \,, the Fermi energy in the thermodynamic limit can be calculated to be: E_^=\frac \left(\frac\right)^2\,.


3D uniform gas

The three-dimensional
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
and non- relativistic uniform Fermi gas case is known as the ''Fermi sphere''. A three-dimensional infinite square well, (i.e. a cubical box that has a side length ''L'') has the potential energy V(x,y,z) = \begin 0, & -\frac The states are now labelled by three quantum numbers ''n''''x'', ''n''''y'', and ''n''''z''. The single particle energies are E_ = E_0 + \frac \left( n_x^2 + n_y^2 + n_z^2\right) \,, where ''n''''x'', ''n''''y'', ''n''''z'' are positive integers. In this case, multiple states have the same energy (known as
degenerate energy levels In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give th ...
), for example E_=E_=E_.


Thermodynamic limit

When the box contains ''N'' non-interacting fermions of spin ½, it is interesting to calculate the energy in the thermodynamic limit, where ''N'' is so large that the quantum numbers ''n''''x'', ''n''''y'', ''n''''z'' can be treated as continuous variables. With the vector \mathbf=(n_x,n_y,n_z), each quantum state corresponds to a point in 'n-space' with energy E_ = E_0 + \frac , \mathbf, ^2 \, With , \mathbf, ^2 denoting the square of the usual Euclidean length , \mathbf, =\sqrt . The number of states with energy less than ''E''F +  ''E''0 is equal to the number of states that lie within a sphere of radius , \mathbf_, in the region of n-space where ''n''''x'', ''n''''y'', ''n''''z'' are positive. In the ground state this number equals the number of fermions in the system: N =2\times\frac\times\frac \pi n_^3 The factor of two expresses the two spin states, and the factor of 1/8 expresses the fraction of the sphere that lies in the region where all ''n'' are positive. n_=\left(\frac\right)^ The Fermi energy is given by E_ = \frac n_^2 = \frac \left( \frac \right)^ Which results in a relationship between the Fermi energy and the number of particles per volume (when ''L''2 is replaced with ''V''2/3): : This is also the energy of the highest-energy particle (the Nth particle), above the zero point energy E_0. The N'th particle has an energy of E_ = E_0 + \frac \left( \frac \right)^ \,=E_0 + E_ \big , _ The total energy of a Fermi sphere of N fermions (which occupy all N energy states within the Fermi sphere) is given by: E_ = N E_0 + \int_0^N E_\big , _ \, dN' = \left(\frac E_ + E_0\right)N Therefore, the average energy per particle is given by: E_\mathrm = E_0 + \frac E_


Density of states

For the 3D uniform Fermi gas, with fermions of spin-½, the number of particles as a function of the energy N(E) is obtained by substituting the Fermi energy by a variable energy (E-E_0): N(E)=\frac\left frac(E-E_0)\right, from which the
density of states In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
(number of energy states per energy per volume) g(E) can be obtained. It can be calculated by differentiating the number of particles with respect to the energy: g(E) =\frac\frac= \frac \left(\frac \right)^\sqrt. This result provides an alternative way to calculate the total energy of a Fermi sphere of N fermions (which occupy all N energy states within the Fermi sphere): \begin E_T&=\int_0^N E \mathrm N(E)=EN(E)\big , _0^N-\int_^ N(E) \mathrm E \\ &=(E_0+E_F)N-\int_^ N(E) \mathrm (E-E_0) \\ &=(E_0+E_F)N- \fracE_FN(E_F) = \left(E_0+\frac E_\right)N \end


Thermodynamic quantities


Degeneracy pressure

By using the
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amoun ...
, this internal energy can be expressed as a pressure, that is P = -\frac = \frac\fracE_= \frac\left(\frac\right)^, where this expression remains valid for temperatures much smaller than the Fermi temperature. This pressure is known as the degeneracy pressure. In this sense, systems composed of fermions are also referred as
degenerate matter Degenerate matter is a highly dense state of fermionic matter in which the Pauli exclusion principle exerts significant pressure in addition to, or in lieu of, thermal pressure. The description applies to matter composed of electrons, protons, neu ...
. Standard
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s avoid collapse by balancing thermal pressure (
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...
and radiation) against gravitational forces. At the end of the star lifetime, when thermal processes are weaker, some stars may become white dwarfs, which are only sustained against gravity by
electron degeneracy pressure Electron degeneracy pressure is a particular manifestation of the more general phenomenon of quantum degeneracy pressure. The Pauli exclusion principle disallows two identical half-integer spin particles (electrons and all other fermions) from si ...
. Using the Fermi gas as a model, it is possible to calculate the
Chandrasekhar limit The Chandrasekhar limit () is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about (). White dwarfs resist gravitational collapse primarily through electron degeneracy pressure, compared ...
, i.e. the maximum mass any star may acquire (without significant thermally generated pressure) before collapsing into a black hole or a neutron star. The latter, is a star mainly composed of neutrons, where the collapse is also avoided by neutron degeneracy pressure. For the case of metals, the electron degeneracy pressure contributes to the compressibility or
bulk modulus The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other moduli describe ...
of the material.


Chemical potential

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 A Sommerfeld expansion is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics. Physically, the integrals represent statistical averages using the Fe ...
(assuming k_T \ll E_): \mu(T) = E_0 + E_ \left 1- \frac \left(\frac\right) ^2 - \frac \left(\frac\right)^4 + \cdots \right where ''T'' is the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
. Hence, the internal chemical potential, ''µ''-''E''0, is approximately equal to the Fermi energy at temperatures that are much lower than the characteristic Fermi temperature ''T''F. This 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.


Typical values


Metals

Under the
free electron model In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantu ...
, the electrons in a metal can be considered to form a uniform Fermi gas. The number density N/V of conduction electrons in metals ranges between approximately 1028 and 1029 electrons per m3, which is also the typical density of atoms in ordinary solid matter. This number density produces a Fermi energy of the order: E_ = \frac \left( 3 \pi^2 \ 10^ \ \mathrm \right)^ \approx 2 \ \sim \ 10 \ \mathrm, where ''me'' is the
electron rest mass The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent of a ...
. This Fermi energy corresponds to a Fermi temperature of the order of 106 kelvins, much higher than the temperature of the
sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
surface. Any metal will boil before reaching this temperature under atmospheric pressure. Thus for any practical purpose, a metal can be considered as a Fermi gas at zero temperature as a first approximation (normal temperatures are small compared to ''T''F).


White dwarfs

Stars known as
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
s have mass comparable to our
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, but have about a hundredth of its radius. The high densities mean that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. The number density of electrons in a white dwarf is of the order of 1036 electrons/m3. This means their Fermi energy is: E_ = \frac \left( \frac \right)^ \approx 3 \times 10^5 \ \mathrm = 0.3 \ \mathrm


Nucleus

Another typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly: R = \left(1.25 \times 10^ \mathrm \right) \times A^ where ''A'' is the number of
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons were ...
s. The number density of nucleons in a nucleus is therefore: \rho = \frac \approx 1.2 \times 10^ \ \mathrm This density must be divided by two, because the Fermi energy only applies to fermions of the same type. The presence of
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
s does not affect the Fermi energy of the
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s in the nucleus, and vice versa. The Fermi energy of a nucleus is approximately: E_ = \frac \left( \frac \right)^ \approx 3 \times 10^7 \ \mathrm = 30 \ \mathrm , where ''m''p is the proton mass. The radius of the nucleus admits deviations around the value mentioned above, so a typical value for the Fermi energy is usually given as 38
MeV In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacu ...
.


Arbitrary-dimensional uniform gas


Density of states

Using a volume integral on d dimensions, the density of states is: g^(E)=g_s \int\frac\delta\left(E-E_0-\frac\right)=g_s\ \left(\frac\right)^ \frac The Fermi energy is obtained by looking for the
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
of particles: \rho = \frac = \int_^ g^(E) \, dE To get: E_^=\frac\left(\tfrac\Gamma\left(\tfrac+1\right)\frac\right)^ where V is the corresponding ''d''-dimensional volume, g_s is the dimension for the internal Hilbert space. For the case of spin-½, every energy is twice-degenerate, so in this case g_=2. A particular result is obtained for d=2, where the density of states becomes a constant (does not depend on the energy): g^(E) = \frac\frac.


Fermi gas in harmonic trap

The harmonic trap potential: V(x,y,z) = \fracm\omega_x^2x^2+\fracm\omega_y^2y^2+\fracm\omega_z^2z^2 is a model system with many applications in modern physics. The density of states (or more accurately, the degree of degeneracy) for a given spin species is: g(E) = \frac\,, where \omega_\text=\sqrt /math> is the harmonic oscillation frequency. The Fermi energy for a given spin species is: E_=(6N)^\hbar\omega_\text\,.


Related Fermi quantities

Related to the Fermi energy, a few useful quantities also occur often in modern literature. The Fermi temperature is defined as T_ = \frac , where k_ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
. The Fermi temperature can be thought of as the temperature at which thermal effects are comparable to quantum effects associated with Fermi statistics. The Fermi temperature for a metal is a couple of orders of magnitude above room temperature. Other quantities defined in this context are Fermi momentum p_ = \sqrt , and Fermi velocity v_ = \frac, which are the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
and
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
, respectively, of a
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
at the
Fermi surface In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the cryst ...
. The Fermi momentum can also be described as p_ = \hbar k_ , where k_ is the radius of the Fermi sphere and is called the Fermi wave vector. Note that these quantities are ''not'' well-defined in cases where the Fermi surface is non-spherical.


Treatment at finite temperature


Grand canonical ensemble

Most of the calculations above are exact at zero temperature, yet remain as good approximations for temperatures lower than the Fermi temperature. For other thermodynamics variables it is necessary to write a thermodynamic potential. For an ensemble of identical fermions, the best way to derive a potential is from the
grand canonical ensemble In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium ...
with fixed temperature, volume and
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
''µ''. The reason is due to Pauli exclusion principle, as the occupation numbers of each quantum state are given by either 1 or 0 (either there is an electron occupying the state or not), so the (grand) partition function \mathcal can be written as :\mathcal(T,V,\mu)=\sum_e^=\prod_\sum_^1e^=\prod_q\left(1+e^\right), where \beta^=k_T , \ indexes the ensembles of all possible microstates that give the same total energy E_q = \sum_ \varepsilon_q n_q and number of particles N_q=\sum_ n_q , \varepsilon_q is the single particle energy of the state q (it counts twice if the energy of the state is degenerate) and n_q=0,1, its occupancy. Thus the
grand potential The grand potential is a quantity used in statistical mechanics, especially for irreversible processes in open systems. The grand potential is the characteristic state function for the grand canonical ensemble. Definition Grand potential is defi ...
is written as :\Omega(T,V,\mu)=-k_T\ln\left(\mathcal\right)=-k_T\sum_q\ln\left(1+e^\right). The same result can be obtained in the
canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...
and
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
, as the result of every ensemble must give the same value at
thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
(N/V\rightarrow\infty) . The
grand canonical ensemble In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium ...
is recommended here as it avoids the use of
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
and
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
s. As explored in previous sections, in the macroscopic limit we may use a continuous approximation ( Thomas–Fermi approximation) to convert this sum to an integral: \Omega(T,V,\mu) = -k_ T \int_^\infty D(\varepsilon) \ln \left(1 + e^ \right) \, d\varepsilon where is the total density of states.


Relation to Fermi-Dirac distribution

The grand potential is related to the number of particles at finite temperature in the following way N=-\left(\frac\right)_=\int_^\infty D(\varepsilon)\mathcal\left(\frac\right)\,\mathrm\varepsilon where the derivative is taken at fixed temperature and volume, and it appears \mathcal(x)=\frac also known as the
Fermi–Dirac distribution Fermi–Dirac may refer to: * Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
. Similarly, the total internal energy is U = \Omega - T\left(\frac\right)_ - \mu\left(\frac\right)_ = \int_^\infty D(\varepsilon) \mathcal \!\left(\frac\right) \varepsilon\, d\varepsilon.


Exact solution for power-law density-of-states

Many systems of interest have a total density of states with the power-law form: D(\varepsilon) = V g(\varepsilon) = \frac (\varepsilon - \varepsilon_0)^, \qquad \varepsilon \geq \varepsilon_0 for some values of , , . The results of preceding sections generalize to dimensions, giving a power law with: * for non-relativistic particles in a -dimensional box, * for non-relativistic particles in a -dimensional harmonic potential well, * for hyper-relativistic particles in a -dimensional box. For such a power-law density of states, the grand potential integral evaluates exactly to: \Omega(T,V,\mu) = - V g_0 (k_T)^ F_ \left( \frac \right), where F_(x) is the
complete Fermi–Dirac integral In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index ''j '' is defined by :F_j(x) = \frac \int_0^\infty \frac\,dt, \qquad (j > -1) This equals :-\operatorname_(-e^x), where \operatornam ...
(related to the
polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...
). From this grand potential and its derivatives, all thermodynamic quantities of interest can be recovered.


Extensions to the model


Relativistic Fermi gas

The article has only treated the case in which particles have a parabolic relation between energy and momentum, as is the case in non-relativistic mechanics. For particles with energies close to their respective
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
, the equations of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
are applicable. Where single-particle energy is given by: E=\sqrt. For this system, the Fermi energy is given by: E_=\sqrt-mc^2\approx p_c, where the \approx equality is only valid in the
ultrarelativistic limit In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light . The expression for the relativistic energy of a particle with rest mass and momentum is given by :E^2 = m^2 c^4 + p^2 c^2. The energy of ...
, and p_ = \hbar\left(\frac 6\pi^2 \frac\right)^. The relativistic Fermi gas model is also used for the description of large white dwarfs which are close to the Chandresekhar limit. For the ultrarelativistic case, the degeneracy pressure is proportional to (N/V)^.


Fermi liquid

In 1956,
Lev Landau Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet- Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics. His a ...
developed the
Fermi liquid theory Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interactions among the particles of the many-body ...
, where he treated the case of a Fermi liquid, i.e., a system with repulsive, not necessarily small, interactions between fermions. The theory shows that the thermodynamic properties of an ideal Fermi gas and a Fermi liquid do not differ that much. It can be shown that the Fermi liquid is equivalent to a Fermi gas composed of collective excitations or
quasiparticle In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum. For exam ...
s, each with a different effective mass and
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
.


See also

*
Bose gas An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were deve ...
*
Fermionic condensate A fermionic condensate or Fermi–Dirac condensate is a superfluid phase formed by fermionic particles at low temperatures. It is closely related to the Bose–Einstein condensate, a superfluid phase formed by bosonic atoms under similar condit ...
*
Gas in a box In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other exce ...
*
Jellium Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), is a quantum mechanical model of interacting electrons in a solid where the positive charges (i.e. atomic nuclei) are assumed to be uniformly distributed in ...
*
Two-dimensional electron gas A two-dimensional electron gas (2DEG) is a scientific model in solid-state physics. It is an electron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion ...


References


Further reading

* Neil W. Ashcroft and
N. David Mermin Nathaniel David Mermin (; born 30 March 1935) is a solid-state physicist at Cornell University best known for the eponymous Mermin–Wagner theorem, his application of the term " boojum" to superfluidity, his textbook with Neil Ashcroft on sol ...
, ''Solid State Physics'' (Harcourt: Orlando, 1976). *
Charles Kittel Charles Kittel (July 18, 1916 – May 15, 2019) was an American physicist. He was a professor at University of California, Berkeley from 1951 and was professor emeritus from 1978 until his death. Life and work Charles Kittel was born in New Yo ...
, ''
Introduction to Solid State Physics ''Introduction to Solid State Physics'', known colloquially as ''Kittel'', is a classic condensed matter physics textbook written by American physicist Charles Kittel in 1953. The book has been highly influential and has seen widespread adoption ...
'', 1st ed. 1953 - 8th ed. 2005, {{Authority control Quantum models Fermi–Dirac statistics Ideal gas Phases of matter