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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 () is obtained by substituting the Fermi energy by a variable energy ():
A result is the Fermi–Dirac distribution of particles over these states where no two particles can occupy the same state, which has a considerable effect on the properties of the system. Fermi–Dirac statistics is most commonly applied to electrons , a type of fermion with spin 1/2 .
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 (also called electrochemical potential).
The density of states related to volume V and N countable energy levels is defined as: = = (()). Because the smallest allowed change of momentum for a particle in a box of dimension and length is () = (/), the volume-related density of states for continuous energy levels is obtained in the limit as ():= (()), Here, is the spatial dimension of the considered system and the wave vector.
Notice that in using this continuum approximation, also known as Thomas−Fermi approximation, the ability to characterize the low-energy states is lost, including the ground state where n i = 1. For most cases this will not be a problem, but when considering Bose–Einstein condensation , in which a large portion of the gas is in or near the ...
An important application of the grand canonical ensemble is in deriving exactly the statistics of a non-interacting many-body quantum gas (Fermi–Dirac statistics for fermions, Bose–Einstein statistics for bosons), however it is much more generally applicable than that. The grand canonical ensemble may also be used to describe classical ...
Fermi fled Italy in 1939 and after arriving in Chicago, he built the first nuclear reactor, which induced and controlled a nuclear chain reaction, causing uranium atoms to continually split.
Specifically for the electron degeneracy pressure, m is substituted by the electron mass m e and the Fermi momentum is obtained from the Fermi energy, so the electron degeneracy pressure is given by = / /, where ρ e is the free electron density (the number of free electrons per unit volume).