Editing Fermi energy

{{Short description|Highest particle energy in a Fermi gas at absolute zero}}
The '''Fermi energy''' is a concept in [[quantum mechanics]] usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting [[fermion]]s at [[absolute zero]] [[temperature]].
In a [[Fermi gas]], the lowest occupied state is taken to have zero kinetic energy, whereas in a metal, the lowest occupied state is typically taken to mean the bottom of the [[conduction band]].

The term "Fermi energy" is often used to refer to a different yet closely related concept, the [[Fermi level|Fermi ''level'']] (also called [[electrochemical potential]]).<ref group="note">The use of the term "Fermi energy" as synonymous with [[Fermi level]] (a.k.a. [[electrochemical potential]]) is widespread in semiconductor physics. For example:  [https://books.google.com/books?id=n0rf9_2ckeYC&pg=PA49 ''Electronics (fundamentals And Applications)''] by D. Chattopadhyay, [https://books.google.com/books?id=lmg13dHPKg8C&pg=PA113 ''Semiconductor Physics and Applications''] by Balkanski and Wallis.</ref>
There are a few key differences between the Fermi level and Fermi energy, at least as they are used in this article:
* The Fermi energy is only defined at absolute zero, while the Fermi level is defined for any temperature.
* The Fermi energy is an energy ''difference'' (usually corresponding to a [[kinetic energy]]), whereas the Fermi level is a total energy level including kinetic energy and potential energy.
* The Fermi energy can only be defined for [[Fermi gas|non-interacting fermions]] (where the potential energy or band edge is a static, well defined quantity), whereas the Fermi level remains well defined even in complex interacting systems, at thermodynamic equilibrium.

Since the Fermi level in a metal at absolute zero is the energy of the highest occupied single particle state,
then the Fermi energy in a metal is the energy difference between the Fermi level and lowest occupied single-particle state, at zero-temperature.

==Context==
{{Main|Fermi gas}}
In [[quantum mechanics]], a group of particles known as [[fermion]]s (for example, [[electron]]s, [[proton]]s and [[neutron]]s) obey the [[Pauli exclusion principle]]. This states that two fermions cannot occupy the same [[quantum state]]. Since an idealized non-interacting Fermi gas can be analyzed in terms of single-particle [[stationary state]]s, we can thus say that two fermions cannot occupy the same stationary state. These stationary states will typically be distinct in energy. To find the ground state of the whole system, we start with an empty system, and add particles one at a time, consecutively filling up the unoccupied stationary states with the lowest energy. When all the particles have been put in, the '''Fermi energy''' is the kinetic energy of the highest occupied state.

As a consequence, even if we have extracted all possible energy from a Fermi gas by cooling it to near [[absolute zero]] temperature, the fermions are still moving around at a high speed. The fastest ones are moving at a velocity corresponding to a kinetic energy equal to the Fermi energy. This speed is known as the '''Fermi velocity'''. Only when the temperature exceeds the related '''Fermi temperature''', do the particles begin to move significantly faster than at absolute zero.

The Fermi energy is an important concept in the [[solid state physics]] of metals and [[superconductor]]s. It is also a very important quantity in the physics of [[Superfluid|quantum liquid]]s like low temperature [[helium]] (both normal and superfluid <sup>3</sup>He), and it is quite important to [[nuclear physics]] and to understanding the stability of [[White dwarf|white dwarf stars]] against [[gravitational collapse]].

==Formula and typical values==
The Fermi energy for a three-dimensional, non-[[Special relativity|relativistic]], ''non-interacting'' ensemble of [[identical particles|identical]] [[spin-1/2|spin-{{frac|1|2}}]] fermions is given by<ref>{{Cite book |title=[[Introduction to Solid State Physics]] |last=Kittel |first=Charles |date=1986 |publisher=Wiley |language=en |chapter=Ch. 6: Free electron gas}}</ref>
<math display="block">E_\text{F} = \frac{\hbar^2}{2m_0} \left( \frac{3 \pi^2 N}{V} \right)^{2/3},</math>
where ''N'' is the number of particles, ''m''<sub>0</sub> the [[rest mass]] of each fermion, ''V'' the volume of the system, and <math>\hbar</math> the reduced [[Planck constant]].

===Metals===
Under the [[free electron model]], the electrons in a metal can be considered to form a Fermi gas. The number density <math>N/V</math> of conduction electrons in metals ranges between approximately 10<sup>28</sup> and 10<sup>29</sup> electrons/m<sup>3</sup>, which is also the typical density of atoms in ordinary solid matter. This number density produces a Fermi energy of the order of 2 to 10&nbsp;[[electronvolts]].<ref>{{Cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/fermi.html |title=Fermi Energies, Fermi Temperatures, and Fermi Velocities |last=Nave |first=Rod |website=[[HyperPhysics]] |access-date=2018-03-21}}</ref>

===White dwarfs===
Stars known as [[white dwarfs]] 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. Their Fermi energy is about 0.3&nbsp;MeV.

===Nucleus===
Another typical example is that of the [[nucleon]]s in the nucleus of an atom. The [[Nuclear size|radius of the nucleus]] admits deviations, so a typical value for the Fermi energy is usually given as 38&nbsp;[[MeV]].

== Related quantities ==
Using this definition of above for the Fermi energy, various related quantities can be useful.

The '''Fermi temperature''' is defined as
<math display="block">T_\text{F} = \frac{E_\text{F}}{k_\text{B}},</math>
where <math>k_\text{B}</math> is the [[Boltzmann constant]], and <math>E_\text{F}</math> the Fermi energy. The Fermi temperature can be thought of as the temperature at which thermal effects are comparable to quantum effects associated with [[Fermi statistics]].<ref>{{Cite web |url=http://www.physics.usu.edu/torre/3700_Spring_2015/Lectures/08.pdf |title=PHYS 3700: Introduction to Quantum Statistical Thermodynamics |last=Torre |first=Charles |date=2015-04-21 |website=Utah State University |access-date=2018-03-21}}</ref> 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'''
<math display="block">p_\text{F} = \sqrt{2 m_0 E_\text{F}}</math>
and '''Fermi velocity'''
<math display="block">v_\text{F} = \frac{p_\text{F}}{m_0}.</math>

These quantities are respectively the [[momentum]] and [[group velocity]] of a [[fermion]] at the [[Fermi surface]].

The Fermi momentum can also be described as
<math display="block">p_\text{F} = \hbar k_\text{F},</math>
where <math>k_\text{F} = (3\pi^2 n)^{1/3}</math>, called the '''Fermi wavevector''', is the radius of the Fermi sphere.<ref>{{cite book |title=Solid State Physics |last1=Ashcroft |first1=Neil W. |last2=Mermin |first2=N. David |publisher=Holt, Rinehart and Winston |year=1976 |isbn=978-0-03-083993-1 |url-access=registration |url=https://archive.org/details/solidstatephysic00ashc}}</ref> 
<math>n</math> is the electron density. 

These quantities may ''not'' be well-defined in cases where the [[Fermi surface]] is non-spherical.

==See also==
* [[Fermi–Dirac statistics]]: the distribution of electrons over stationary states for non-interacting fermions at ''non-zero'' temperature.
* [[Fermi level]]
* [[Quasi Fermi level]]

== Notes ==
<references group="note" />

==References==
{{Reflist}}

== Further reading ==
*{{cite book |author1=Kroemer, Herbert |author2=Kittel, Charles | title=Thermal Physics (2nd ed.) | publisher=W. H. Freeman Company | year=1980 | isbn=978-0-7167-1088-2}}

{{Authority control}}

[[Category:Condensed matter physics]]
[[Category:Fermi–Dirac statistics]]