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Fermi gas

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Physical model of non-interacting fermions

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AFermi gas is an idealized model, an ensemble of many non-interactingfermions. Fermions areparticles that obeyFermi–Dirac statistics, likeelectrons,protons, andneutrons, and, in general, particles withhalf-integer spin. These statistics determine the energy distribution of fermions in a Fermi gas inthermal equilibrium, and is characterized by theirnumber density,temperature, and the set of available energy states. The model is named after the Italian physicistEnrico Fermi.^[1]^[2]

This physical model is useful for certain systems with many fermions. Some key examples are the behaviour ofcharge carriers in a metal,nucleons in anatomic nucleus, neutrons in aneutron star, and electrons in awhite dwarf.

Description

An ideal Fermi gas or free Fermi gas is aphysical model assuming a collection of non-interacting fermions in a constantpotential well. Fermions are elementary or composite particles withhalf-integer spin, thus followFermi–Dirac statistics. The equivalent model for integer spin particles is called theBose gas (an ensemble of non-interactingbosons). At low enough particlenumber density and high temperature, both the Fermi gas and the Bose gas behave like a classicalideal gas.^[3]

By thePauli exclusion principle, noquantum state can be occupied by more than one fermion with an identical set ofquantum numbers. 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 aBose–Einstein condensate, although weakly-interacting Fermi gases might form aCooper pair and condensate (also known asBCS-BEC crossover regime).^[4] The total energy of the Fermi gas atabsolute zero is larger than the sum of the single-particleground states because the Pauli principle implies a sort of interaction or pressure that keeps fermions separated and moving. For this reason, thepressure of a Fermi gas is non-zero even at zero temperature, in contrast to that of a classical ideal gas. For example, this so-calleddegeneracy pressure stabilizes aneutron star (a Fermi gas of neutrons) or awhite dwarf star (a Fermi gas of electrons) against the inward pull ofgravity, which would ostensibly collapse the star into ablack hole. Only when a star is sufficiently massive to overcome the degeneracy pressure can it collapse into a singularity.

It is possible to define a Fermi temperature below which the gas can be considered degenerate (its pressure derives almost exclusively from the Pauli principle). This temperature depends on the mass of the fermions and thedensity of energy states.

The main assumption of thefree electron model to describe the delocalized electrons in a metal can be derived from the Fermi gas. Since interactions are neglected due toscreening 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 ofkelvins, so in human applications the electron gas can be considered degenerate. The maximum energy of the fermions at zero temperature is called theFermi energy. The Fermi energy surface inreciprocal space is known as theFermi surface.

Thenearly free electron model adapts the Fermi gas model to consider thecrystal structure ofmetals andsemiconductors, where electrons in a crystal lattice are substituted byBloch electrons with a correspondingcrystal 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 theperturbation theory.

1D uniform gas

The one-dimensionalinfinite square well of lengthL is a model for a one-dimensional box with the potential energy: $V(x)={\begin{cases}0,&x_{c}-{\tfrac {L}{2}}<x<x_{c}+{\tfrac {L}{2}},\\\infty ,&{\text{otherwise.}}\end{cases}}$

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,^[5] 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 numbern and the energies are given by:

$E_{n}=E_{0}+{\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}n^{2}.$ where $E_{0}$ is the zero-point energy (which can be chosen arbitrarily as a form ofgauge fixing), $m {\displaystyle m}$ the mass of a single fermion, and $\hbar$ is the reducedPlanck constant.

ForN fermions withspin-1⁄2 in the box, no more than two particles can have the same energy, i.e., two particles can have the energy of ${\textstyle E_{1}}$ , two other particles can have energy ${\textstyle E_{2}}$ and so forth. The two particles of the same energy have spin1⁄2 (spin up) or −1⁄2 (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 ton = 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 $E_{\mathrm {F} }^{({\text{1D}})}=E_{n}-E_{0}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}\left(\left\lfloor {\frac {N}{2}}\right\rfloor \right)^{2},$ where ${\textstyle \left\lfloor {\frac {N}{2}}\right\rfloor }$ is thefloor function evaluated atn = N/2.

Thermodynamic limit

In thethermodynamic limit, the total number of particlesN are so large that the quantum numbern may be treated as a continuous variable. In this case, the overall number density profile in the box is indeed uniform.

The number ofquantum states 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:

$E_{n}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}n^{2}\implies dE={\frac {\hbar ^{2}\pi ^{2}}{mL^{2}}}n\,dn={\frac {\hbar \pi }{L}}{\sqrt {\frac {2E}{m}}}dn\,.$

Therefore, in the range: $E_{1}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}n_{1}^{2}<E<E_{1}+dE\,,$ the number of quantum states is: $D_{n}(n_{1})\,dn=2{\frac {dE}{dE/dn}}={\frac {2}{{\frac {\hbar ^{2}\pi ^{2}}{mL^{2}}}n}}\,dE\equiv D(E_{1})\,dE\,.$

Here, thedegree of degeneracy is:

$D(E)={\frac {2}{dE/dn}}={\frac {2L}{\hbar \pi }}{\sqrt {\frac {m}{2E}}}\,.$

And thedensity of states is:

$g(E)\equiv {\frac {1}{L}}D(E)={\frac {2}{\hbar \pi }}{\sqrt {\frac {m}{2E}}}\,.$

In modern literature,^[5] 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 $L {\displaystyle L}$ in this 1D case).

Based on the following formula:

$\int _{0}^{E_{\mathrm {F} }}D(E)\,dE=N\,,$

the Fermi energy in the thermodynamic limit can be calculated to be:

$E_{\mathrm {F} }^{({\text{1D}})}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}\left({\frac {N}{2}}\right)^{2}\,.$

3D uniform gas

A model of the atomic nucleus showing it as a compact bundle of the two types ofnucleons: protons (red) and neutrons (blue). As a first approximation, the nucleus can be treated as composed of non-interacting proton and neutron gases.

The three-dimensionalisotropic and non-relativistic uniform Fermi gas case is known as theFermi sphere.

A three-dimensional infinite square well, (i.e. a cubical box that has a side lengthL) has the potential energy $V(x,y,z)={\begin{cases}0,&-{\frac {L}{2}}<x,y,z<{\frac {L}{2}},\\\infty ,&{\text{otherwise.}}\end{cases}}$

The states are now labelled by three quantum numbersn_x,n_y, andn_z. The single particle energies are $E_{n_{x},n_{y},n_{z}}=E_{0}+{\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)\,,$ wheren_x,n_y,n_z are positive integers. In this case, multiple states have the same energy (known asdegenerate energy levels), for example $E_{211}=E_{121}=E_{112}$ .

Thermodynamic limit

When the box containsN non-interacting fermions of spin-⁠1/2⁠, it is interesting to calculate the energy in the thermodynamic limit, whereN is so large that the quantum numbersn_x,n_y,n_z can be treated as continuous variables.

With the vector $\mathbf {n} =(n_{x},n_{y},n_{z})$ , each quantum state corresponds to a point in 'n-space' with energy $E_{\mathbf {n} }=E_{0}+{\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}|\mathbf {n} |^{2}\,$

With $|\mathbf {n} |^{2}$ denoting the square of the usual Euclidean length $|\mathbf {n} |={\sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}$ .The number of states with energy less thanE_F +E₀ is equal to the number of states that lie within a sphere of radius $|\mathbf {n} _{\mathrm {F} }|$ in the region of n-space wheren_x,n_y,n_z are positive. In the ground state this number equals the number of fermions in the system:

$N=2\times {\frac {1}{8}}\times {\frac {4}{3}}\pi n_{\mathrm {F} }^{3}$

The free fermions that occupy the lowest energy states form asphere inreciprocal space. The surface of this sphere is theFermi surface.

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 alln are positive. $n_{\mathrm {F} }=\left({\frac {3N}{\pi }}\right)^{1/3}$ TheFermi energy is given by $E_{\mathrm {F} }={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}n_{\mathrm {F} }^{2}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}\left({\frac {3N}{\pi }}\right)^{2/3}$

Which results in a relationship between the Fermi energy and thenumber of particles per volume (whenL² is replaced withV^2/3):

E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m}}\left({\frac {3\pi ^{2}N}{V}}\right)^{2/3}

This is also the energy of the highest-energy particle (the $N {\displaystyle N}$ th particle), above the zero point energy $E_{0}$ . The $N^{'} {\displaystyle N'}$ th particle has an energy of $E_{N'}=E_{0}+{\frac {\hbar ^{2}}{2m}}\left({\frac {3\pi ^{2}N'}{V}}\right)^{2/3}\,=E_{0}+E_{\mathrm {F} }{\big |}_{N'}$

The total energy of a Fermi sphere of $N {\displaystyle N}$ fermions (which occupy all $N {\displaystyle N}$ energy states within the Fermi sphere) is given by:

$E_{\rm {T}}=NE_{0}+\int _{0}^{N}E_{\mathrm {F} }{\big |}_{N'}\,dN'=\left({\frac {3}{5}}E_{\mathrm {F} }+E_{0}\right)N$

Therefore, the average energy per particle is given by: $E_{\mathrm {av} }=E_{0}+{\frac {3}{5}}E_{\mathrm {F} }$

Density of states

Density of states (DOS) of a Fermi gas in 3-dimensions

For the 3D uniform Fermi gas, with fermions of spin-⁠1/2⁠, the number of particles as a function of the energy ${\textstyle N(E)}$ is obtained by substituting the Fermi energy by a variable energy ${\textstyle (E-E_{0})}$ :

$N(E)={\frac {V}{3\pi ^{2}}}\left[{\frac {2m}{\hbar ^{2}}}(E-E_{0})\right]^{3/2},$

from which thedensity of states (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:

$g(E)={\frac {1}{V}}{\frac {\partial N(E)}{\partial E}}={\frac {1}{2\pi ^{2}}}\left({\frac {2m}{\hbar ^{2}}}\right)^{3/2}{\sqrt {E-E_{0}}}.$

This result provides an alternative way to calculate the total energy of a Fermi sphere of $N {\displaystyle N}$ fermions (which occupy all $N {\displaystyle N}$ energy states within the Fermi sphere):

${\begin{aligned}E_{T}&=\int _{0}^{N}E\mathrm {d} N(E)=EN(E){\big |}_{0}^{N}-\int _{E_{0}}^{E_{0}+E_{F}}N(E)\mathrm {d} E\\&=(E_{0}+E_{F})N-\int _{0}^{E_{F}}N(E)\mathrm {d} (E-E_{0})\\&=(E_{0}+E_{F})N-{\frac {2}{5}}E_{F}N(E_{F})=\left(E_{0}+{\frac {3}{5}}E_{\mathrm {F} }\right)N\end{aligned}}$

Thermodynamic quantities

Degeneracy pressure

Pressure vs temperature curves of classical and quantum ideal gases (Fermi gas,Bose gas) in three dimensions. Pauli repulsion in fermions (such as electrons) gives them an additional pressure over an equivalent classical gas, most significantly at low temperature.

By using thefirst law of thermodynamics, this internal energy can be expressed as a pressure, that is $P=-{\frac {\partial E_{\rm {T}}}{\partial V}}={\frac {2}{5}}{\frac {N}{V}}E_{\mathrm {F} }={\frac {(3\pi ^{2})^{2/3}\hbar ^{2}}{5m}}\left({\frac {N}{V}}\right)^{5/3},$ where this expression remains valid for temperatures much smaller than the Fermi temperature. This pressure is known as thedegeneracy pressure. In this sense, systems composed of fermions are also referred asdegenerate matter.

Standardstars 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 byelectron degeneracy pressure. Using the Fermi gas as a model, it is possible to calculate theChandrasekhar 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 orbulk modulus of the material.

Chemical potential

See also:Fermi level

Assuming that the concentration of fermions does not change with temperature, then the total chemical potentialμ (Fermi level) of the three-dimensional ideal Fermi gas is related to the zero temperature Fermi energyE_F by aSommerfeld expansion (assuming $k_{\rm {B}}T\ll E_{\mathrm {F} }$ ): $\mu (T)=E_{0}+E_{\mathrm {F} }\left[1-{\frac {\pi ^{2}}{12}}\left({\frac {k_{\rm {B}}T}{E_{\mathrm {F} }}}\right)^{2}-{\frac {\pi ^{4}}{80}}\left({\frac {k_{\rm {B}}T}{E_{\mathrm {F} }}}\right)^{4}+\cdots \right],$ whereT is thetemperature.^[6]^[7]

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

Typical values

Metals

Under thefree 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: $E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m_{e}}}\left(3\pi ^{2}\ 10^{28\ \sim \ 29}\ \mathrm {m^{-3}} \right)^{2/3}\approx 2\ \sim \ 10\ \mathrm {eV} ,$ wherem_e is theelectron rest mass.^[8] This Fermi energy corresponds to a Fermi temperature of the order of 10⁶ kelvins, much higher than the temperature of theSun's 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 toT_F).

White dwarfs

Stars known aswhite dwarfs have mass comparable to theSun, 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:

$E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m_{e}}}\left({\frac {3\pi ^{2}(10^{36})}{1\ \mathrm {m^{3}} }}\right)^{2/3}\approx 3\times 10^{5}\ \mathrm {eV} =0.3\ \mathrm {MeV}$

Nucleus

Another typical example is that of the particles in a nucleus of an atom. Theradius of the nucleus is roughly: $R=\left(1.25\times 10^{-15}\mathrm {m} \right)\times A^{1/3}$ whereA is the number ofnucleons.

The number density of nucleons in a nucleus is therefore:

$\rho ={\frac {A}{{\frac {4}{3}}\pi R^{3}}}\approx 1.2\times 10^{44}\ \mathrm {m^{-3}}$

This density must be divided by two, because the Fermi energy only applies to fermions of the same type. The presence ofneutrons does not affect the Fermi energy of theprotons in the nucleus, and vice versa.

The Fermi energy of a nucleus is approximately: $E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m_{\rm {p}}}}\left({\frac {3\pi ^{2}(6\times 10^{43})}{1\ \mathrm {m} ^{3}}}\right)^{2/3}\approx 3\times 10^{7}\ \mathrm {eV} =30\ \mathrm {MeV} ,$ wherem_p is the proton mass.

Theradius of the nucleus admits deviations around the value mentioned above, so a typical value for the Fermi energy is usually given as 38MeV.

Arbitrary-dimensional uniform gas

Density of states

Using a volume integral on ${\textstyle d}$ dimensions, the density of states is:

$g^{(d)}(E)=g_{s}\int {\frac {\mathrm {d} ^{d}\mathbf {k} }{(2\pi )^{d}}}\delta \left(E-E_{0}-{\frac {\hbar ^{2}|\mathbf {k} |^{2}}{2m}}\right)=g_{s}\ \left({\frac {m}{2\pi \hbar ^{2}}}\right)^{d/2}{\frac {(E-E_{0})^{d/2-1}}{\Gamma (d/2)}}$

The Fermi energy is obtained by looking for thenumber density of particles: $\rho ={\frac {N}{V}}=\int _{E_{0}}^{E_{0}+E_{\mathrm {F} }^{(d)}}g^{(d)}(E)\,dE$

To get: $E_{\mathrm {F} }^{(d)}={\frac {2\pi \hbar ^{2}}{m}}\left({\tfrac {1}{g_{s}}}\Gamma \left({\tfrac {d}{2}}+1\right){\frac {N}{V}}\right)^{2/d}$ where ${\textstyle V}$ is the correspondingd-dimensional volume, ${\textstyle g_{s}}$ is the dimension for the internal Hilbert space. For the case of spin-⁠1/2⁠, every energy is twice-degenerate, so in this case ${\textstyle g_{s}=2}$ .

A particular result is obtained for $d=2$ , where the density of states becomes a constant (does not depend on the energy): $g^{(2\mathrm {D} )}(E)={\frac {g_{s}}{2}}{\frac {m}{\pi \hbar ^{2}}}.$

Fermi gas in harmonic trap

Main article:Gas in a harmonic trap

Theharmonic trap potential:

$V(x,y,z)={\frac {1}{2}}m\omega _{x}^{2}x^{2}+{\frac {1}{2}}m\omega _{y}^{2}y^{2}+{\frac {1}{2}}m\omega _{z}^{2}z^{2}$

is a model system with many applications^[5] in modern physics. The density of states (or more accurately, the degree of degeneracy) for a given spin species is:

$g(E)={\frac {E^{2}}{2(\hbar \omega _{\text{ho}})^{3}}}\,,$

where $\omega _{\text{ho}}={\sqrt[{3}]{\omega _{x}\omega _{y}\omega _{z}}}$ is the harmonic oscillation frequency.

The Fermi energy for a given spin species is:

$E_{\rm {F}}=(6N)^{1/3}\hbar \omega _{\text{ho}}\,.$

Related Fermi quantities

Related to the Fermi energy, a few useful quantities also occur often in modern literature.

TheFermi temperature is defined as ${\textstyle T_{\mathrm {F} }={\frac {E_{\mathrm {F} }}{k_{\rm {B}}}}}$ , where $k_{\rm {B}}$ is theBoltzmann constant. The Fermi temperature can be thought of as the temperature at which thermal effects are comparable to quantum effects associated with Fermi statistics.^[9] The Fermi temperature for a metal is a couple of orders of magnitude above room temperature. Other quantities defined in this context areFermi momentum ${\textstyle p_{\mathrm {F} }={\sqrt {2mE_{\mathrm {F} }}}}$ , andFermi velocity^[10] ${\textstyle v_{\mathrm {F} }={\frac {p_{\mathrm {F} }}{m}}}$ , which are themomentum andgroup velocity, respectively, of afermion at theFermi surface. The Fermi momentum can also be described as $p_{\mathrm {F} }=\hbar k_{\mathrm {F} }$ , where $k_{\mathrm {F} }$ is the radius of the Fermi sphere and is called theFermi wave vector.^[11]

Note that these quantities arenot 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 athermodynamic potential. For an ensemble ofidentical fermions, the best way to derive a potential is from thegrand canonical ensemble with fixed temperature, volume andchemical potentialμ. 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 ${\mathcal {Z}}$ can be written as

{\mathcal {Z}}(T,V,\mu )=\sum _{\{q\}}e^{-\beta (E_{q}-\mu N_{q})}=\prod _{q}\sum _{n_{q}=0}^{1}e^{-\beta (\varepsilon _{q}-\mu )n_{q}}=\prod _{q}\left(1+e^{-\beta (\varepsilon _{q}-\mu )}\right),

where $\beta ^{-1}=k_{\rm {B}}T$ , $\{q\}$ indexes the ensembles of all possible microstates that give the same total energy ${\textstyle E_{q}=\sum _{q}\varepsilon _{q}n_{q}}$ and number of particles ${\textstyle N_{q}=\sum _{q}n_{q}}$ , ${\textstyle \varepsilon _{q}}$ is the single particle energy of the state ${\textstyle q}$ (it counts twice if the energy of the state is degenerate) and ${\textstyle n_{q}=0,1}$ , its occupancy. Thus thegrand potential is written as

\Omega (T,V,\mu )=-k_{\rm {B}}T\ln \left({\mathcal {Z}}\right)=-k_{\rm {B}}T\sum _{q}\ln \left(1+e^{\beta (\mu -\varepsilon _{q})}\right).

The same result can be obtained in thecanonical andmicrocanonical ensemble, as the result of every ensemble must give the same value atthermodynamic limit ${\textstyle (N/V\rightarrow \infty )}$ . Thegrand canonical ensemble is recommended here as it avoids the use ofcombinatorics andfactorials.

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: $\Omega (T,V,\mu )=-k_{\rm {B}}T\int _{-\infty }^{\infty }D(\varepsilon )\ln \left(1+e^{\beta (\mu -\varepsilon )}\right)\,d\varepsilon$ whereD(ε) 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 $N=-\left({\frac {\partial \Omega }{\partial \mu }}\right)_{T,V}=\int _{-\infty }^{\infty }D(\varepsilon ){\mathcal {f}}\left({\frac {\varepsilon -\mu }{k_{\rm {B}}T}}\right)\,\mathrm {d} \varepsilon$ where the derivative is taken at fixed temperature and volume, and it appears ${\mathcal {f}}(x)={\frac {1}{e^{x}+1}}$ also known as theFermi–Dirac distribution.

Similarly, the total internal energy is $U=\Omega -T\left({\frac {\partial \Omega }{\partial T}}\right)_{V,\mu }-\mu \left({\frac {\partial \Omega }{\partial \mu }}\right)_{T,V}=\int _{-\infty }^{\infty }D(\varepsilon ){\mathcal {f}}\!\left({\frac {\varepsilon -\mu }{k_{\rm {B}}T}}\right)\varepsilon \,d\varepsilon .$

Exact solution for power-law density-of-states

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Many systems of interest have a total density of states with the power-law form: $D(\varepsilon )=Vg(\varepsilon )={\frac {Vg_{0}}{\Gamma (\alpha )}}(\varepsilon -\varepsilon _{0})^{\alpha -1},\qquad \varepsilon \geq \varepsilon _{0}$ for some values ofg₀,α,ε₀. The results of preceding sections generalize tod dimensions, giving a power law with:

α =d/2 for non-relativistic particles in ad-dimensional box,
α =d for non-relativistic particles in ad-dimensional harmonic potential well,
α =d for hyper-relativistic particles in ad-dimensional box.

For such a power-law density of states, the grand potential integral evaluates exactly to:^[12] $\Omega (T,V,\mu )=-Vg_{0}(k_{\rm {B}}T)^{\alpha +1}F_{\alpha }\left({\frac {\mu -\varepsilon _{0}}{k_{\rm {B}}T}}\right),$ where $F_{\alpha }(x)$ is thecomplete Fermi–Dirac integral (related to thepolylogarithm). From this grand potential and its derivatives, all thermodynamic quantities of interest can be recovered.

Extensions to the model

Relativistic Fermi gas

Radius–mass relations for a model white dwarf, relativistic relation vs non-relativistic. TheChandrasekhar limit is indicated asM_Ch.

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 respectiverest mass, the equations ofspecial relativity are applicable. Where single-particle energy is given by: $E={\sqrt {(pc)^{2}+(mc^{2})^{2}}}.$

For this system, the Fermi energy is given by: $E_{\mathrm {F} }={\sqrt {(p_{\mathrm {F} }c)^{2}+(mc^{2})^{2}}}-mc^{2}\approx p_{\mathrm {F} }c,$ where the $\approx$ equality is only valid in theultrarelativistic limit, and^[13] $p_{\mathrm {F} }=\hbar \left({\frac {1}{g_{s}}}6\pi ^{2}{\frac {N}{V}}\right)^{1/3}.$

The relativistic Fermi gas model is also used for the description of massive white dwarfs which are close to theChandrasekhar limit. For the ultrarelativistic case, the degeneracy pressure is proportional to $(N/V)^{4/3}$ .

Fermi liquid

In 1956,Lev Landau developed theFermi 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 orquasiparticles, each with a differenteffective mass andmagnetic moment.

See also

References

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^Zannoni, Alberto (1999). "On the Quantization of the Monoatomic Ideal Gas".arXiv:cond-mat/9912229.An english translation of the original work of Enrico Fermi on the quantization of the monoatomic ideal gas, is given in this paper
^Schwabl, Franz (2013-03-09).Statistical Mechanics. Springer Science & Business Media.ISBN 978-3-662-04702-6.
^Regal, C. A.; Greiner, M.; Jin, D. S. (2004-01-28). "Observation of Resonance Condensation of Fermionic Atom Pairs".Physical Review Letters.92 (4): 040403.arXiv:cond-mat/0401554.Bibcode:2004PhRvL..92d0403R.doi:10.1103/PhysRevLett.92.040403.PMID 14995356.S2CID 10799388.
^^a ^b ^cGiorgini, Stefano; Pitaevskii, Lev P.; Stringari, Sandro (2008-10-02)."Theory of ultracold atomic Fermi gases".Reviews of Modern Physics.80 (4):1215–1274.arXiv:0706.3360.Bibcode:2008RvMP...80.1215G.doi:10.1103/RevModPhys.80.1215.S2CID 117755089.
^Kelly, James J. (1996)."Statistical Mechanics of Ideal Fermi Systems"(PDF).Universidad Autónoma de Madrid. Archived fromthe original(PDF) on 2018-04-12. Retrieved2018-03-15.
^"Degenerate Ideal Fermi Gases"(PDF). Archived fromthe original(PDF) on 2008-09-19. Retrieved2014-04-13.
^Nave, Rod."Fermi Energies, Fermi Temperatures, and Fermi Velocities".HyperPhysics. Retrieved2018-03-21.
^Torre, Charles (2015-04-21)."PHYS 3700: Introduction to Quantum Statistical Thermodynamics"(PDF).Utah State University. Retrieved2018-03-21.
^Nave, Rod."Fermi level and Fermi function".HyperPhysics. Retrieved2018-03-21.
^Ashcroft, Neil W.; Mermin, N. David (1976).Solid State Physics.Holt, Rinehart and Winston.ISBN 978-0-03-083993-1.
^Blundell (2006). "Chapter 30: Quantum gases and condensates".Concepts in Thermal Physics. Oxford University Press.ISBN 9780198567707.
^Greiner, Walter; Neise, Ludwig;Stöcker, Horst (1995).Thermodynamics and Statistical Mechanics. Classical Theoretical Physics. Springer, New York, NY. pp. 341–386.doi:10.1007/978-1-4612-0827-3_14.ISBN 9780387942995.

Further reading

Neil W. Ashcroft andN. David Mermin,Solid State Physics (Harcourt: Orlando, 1976)
Charles Kittel,Introduction to Solid State Physics, 1st ed. 1953 – 8th ed. 2005,ISBN 0-471-41526-X

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