Fermi momentum ($p_F$)

Fermi momentum is the magnitude of the momentum at the boundary of filled one-particle states in an isotropic zero-temperature Fermi gas. For the uniform three-dimensional two-component electron model, a short state count givesTo see the factors, impose periodic boundary conditions on a cubic box of side $L$, so $V=L^3$ and each momentum component is $p_i=2\pi \hslash j_i/L$ for an integer $j_i$. Each allowed momentum point occupies a cell of volume $(2\pi \hslash /L{)}^3$ in momentum space. Divide the sphere volume $4\pi p_F^3/3$ by this cell volume and multiply by two for the two spin states. Replacing the discrete count with a volume is the large-box approximation; surface and shell effects are neglected.^[ref]

The factor two is not automatic for every Fermi gas. If only one internal state is populated, its number density $n_{\sigma }$ instead satisfies $p_{F,\sigma }=\hslash (6{\pi }^2{n}_{\sigma }{)}^{1/3}$. For two equally populated states, $n_{\sigma }=n/2$ recovers the displayed result. A Fermi surface in a crystal can have a more complicated shape, so a single spherical Fermi momentum is a property of this isotropic model.

The group velocity of an excitation evaluated on the Fermi surface. For an isotropic dispersion relation $ϵ(p)$, $v_F=(dϵ/dp{)}_{p_F}$, where $p_F$ is the Fermi momentum.

The Fermi wavevector is the radius of the occupied sphere in wavevector coordinates for an isotropic Fermi gas. In the uniform two-component electron model, $k_F=(3{\pi }^{2}n{)}^{1/3}$ and $E_F={\hslash }^2{k}_F^2/(2m_e)$ for nonrelativistic particles. Its units are inverse length, whereas the Fermi momentum has units of momentum.

The wavelength associated with a specified Fermi wavevector. A local-density response normally requires variation on a scale large compared with $k_F^{-1}$; numerical factors such as $2\pi$ must be kept explicit when comparing a variation scale with the full wavelength.