Fermi Gas
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A Fermi gas is an idealized model, an ensemble of many non-interacting
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s. Fermions are particles that obey Fermi–Dirac statistics, like
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s,
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
s, and
neutron The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
s, and, in general, particles with half-integer 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 t ...
, and is characterized by their number density,
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
, and the set of available energy states. The model is named after the Italian physicist Enrico Fermi. This physical model is useful for certain 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. Until the 1960s, nucleons were thought to be ele ...
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 at the Department_of_Physics_and_Astronomy,_University_of_Manchester , University of Manchester ...
, neutrons in a
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
, and electrons in a
white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
.


Description

An ideal Fermi gas or free Fermi gas is a physical model assuming a collection of non-interacting fermions in a constant potential well. Fermions are elementary or composite particles with half-integer spin, thus follow Fermi–Dirac statistics. The equivalent model for integer spin particles is called the Bose gas (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 half odd-intege ...
s). At low enough particle number density 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 ...
. By the Pauli exclusion principle, no
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
can be occupied by more than one fermion with an identical set of
quantum number In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...
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 Density, densities is cooled to temperatures very close to absolute zero#Relation with Bose–Einste ...
, although weakly-interacting Fermi gases might form a Cooper pair and condensate (also known as BCS-BEC crossover regime). The total energy of the Fermi gas at
absolute zero Absolute zero is the lowest possible temperature, a state at which a system's internal energy, and in ideal cases entropy, reach their minimum values. The absolute zero is defined as 0 K on the Kelvin scale, equivalent to −273.15 ° ...
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 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 eve ...
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 stabilizes a
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
(a Fermi gas of neutrons) or a
white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
star (a Fermi gas of electrons) against the inward pull of
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
, which would ostensibly collapse the star into a
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
. 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 to describe the delocalized electrons in a metal can be derived from the Fermi gas. Since interactions are neglected due to screening effect, 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 base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
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 Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray diffraction, X-ray and Electron diffraction, electron diffraction as well as the Electronic band structure, e ...
is known as the Fermi surface. The nearly free electron model adapts the Fermi gas model to consider the
crystal structure In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat ...
of
metal A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, electricity and thermal conductivity, heat relatively well. These properties are all associated wit ...
s and
semiconductor A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
s, where electrons in a crystal lattice are substituted by Bloch electrons 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 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), m the mass of a single fermion, and \hbar is the reduced
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
. 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, 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 integer greater than or eq ...
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 (mathematics), 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 ...
, 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 embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
s in the range n_1 < n < n_1 + dn is: D_n(n_1)\, dn = 2 \, dn\,. Without loss of generality, 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 condensed matter physics, the density of states (DOS) of a system describes the number of allowed modes or quantum state, states per unit energy range. The density of states is defined as where N(E)\delta E is the number of states in the syste ...
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 In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
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), for example E_=E_=E_.


Thermodynamic limit

When the box contains ''N'' non-interacting fermions of spin-, 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-, 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 condensed matter physics, the density of states (DOS) of a system describes the number of allowed modes or quantum state, states per unit energy range. The density of states is defined as where N(E)\delta E is the number of states in the syste ...
(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 in the context of thermodynamic processes. For a thermodynamic process affecting a thermodynamic system without transfer of matter, the law distinguishes two ...
, 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. Standard
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s avoid collapse by balancing thermal pressure ( plasma 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. Using the Fermi gas as a model, it is possible to calculate the Chandrasekhar limit, 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 or k) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other mo ...
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 (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 quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
. 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, 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. This Fermi energy corresponds to a Fermi temperature of the order of 106 kelvins, much higher than the temperature of the Sun's surface. Any metal will boil before reaching this temperature under atmospheric pressure. Thus for any practical purpose, free electrons in a metal can be considered as a Fermi gas at zero temperature as an approximation (normal temperatures are small compared to ''T''F).


White dwarfs

Stars known as
white dwarf A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
s have mass comparable to the Sun, 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. Until the 1960s, nucleons were thought to be ele ...
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 , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
s does not affect the Fermi energy of the
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
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.


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 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-, 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 thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
. 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 (: momenta or momentums; 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. ...
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 (waves), envelope'' of the wave—propagates through space. For example, if a stone is thro ...
, respectively, of a
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
at the Fermi surface. 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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
. 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 equilibri ...
with fixed temperature, volume and
chemical potential In thermodynamics, the chemical potential of a Chemical specie, 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 potent ...
''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 or Landau potential or Landau free energy 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 ens ...
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 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 (mathematics), 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 ...
(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 equilibri ...
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 as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
and
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), 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 ...
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. 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 (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, ...
, the equations of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
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, and p_ = \hbar\left(\frac 6\pi^2 \frac\right)^. The relativistic Fermi gas model is also used for the description of massive white dwarfs which are close to the Chandrasekhar limit. For the ultrarelativistic case, the degeneracy pressure is proportional to (N/V)^.


Fermi liquid

In 1956,
Lev Landau Lev Davidovich Landau (; 22 January 1908 – 1 April 1968) was a Soviet physicist who made fundamental contributions to many areas of theoretical physics. He was considered as one of the last scientists who were universally well-versed and ma ...
developed the Fermi liquid theory, 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 quasiparticles, each with a different effective mass and
magnetic moment In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
.


See also

* Bose gas * Fermionic condensate * Gas in a box * Jellium * Two-dimensional electron gas


References


Further reading

* Neil W. Ashcroft and N. David Mermin, ''Solid State Physics'' (Harcourt: Orlando, 1976) * Charles Kittel, '' Introduction to Solid State Physics'', 1st ed. 1953 – 8th ed. 2005, {{Authority control Quantum models Fermi–Dirac statistics Ideal gas Phases of matter