,
Bose gas) in three dimensions. Pauli repulsion in fermions gives them an additional pressure over an equivalent classical gas, most significantly at low temperature. Electrons are members of a family of particles known as
fermions. Fermions, like the
proton or the
neutron, follow Pauli's principle and
Fermi–Dirac statistics. In general, for an ensemble of non-interacting fermions, also known as a
Fermi gas, each particle can be treated independently with a single-fermion energy given by the purely kinetic term, \ E = \frac{\ p^2 }{\; 2\;m \;}\ , where is the momentum of one particle and its mass. Every possible momentum state of an electron within this volume up to the Fermi momentum being occupied. The degeneracy pressure at zero temperature can be computed as P= \frac{\;\! 2 \;\!}{ 3 }\;\!\frac{~~ E_\mathsf{tot} }{ V } = \frac{\;\! 2 \;\!}{ 3 }\;\!\frac{\ p_\mathsf{F}^5 }{\ 10\;\!\pi^2\;\! m\;\! \hbar^3 }\ , where
V is the total volume of the system and
Etot is the total energy of the ensemble. Specifically for the electron degeneracy pressure, is substituted by the electron mass and the Fermi momentum is obtained from the
Fermi energy, so the electron degeneracy pressure is given by P_\mathsf{e} = \frac{\ (3\;\!\pi^2)^{\tfrac{2}{3}}\;\!\hbar^2 }{ 5\;\! m_\mathsf{e} }\ \rho_\mathsf{e}^{\tfrac{5}{3}}\ , where \ \rho_\mathsf{e}\ is the free electron
density (the number of free electrons per unit volume). For the case of a metal, one can prove that this equation remains approximately true for temperatures lower than the Fermi temperature, about
kelvins. When particle energies reach
relativistic levels, a
modified formula is required. The relativistic degeneracy pressure is proportional to \ \rho_\mathsf{e}^{\tfrac{4}{3}} ~. == Examples ==