Fermi sphere ($|\mathbf{k}|\le k_F$)

For an isotropic three-dimensional free electron gas at zero temperature, occupied wavevectors fill a sphere of radius $k_F$. Including two spin states, $n=k_F^3/(3{\pi }^2)$. The volume in reciprocal space counts occupied states; folding an extended-zone sphere into a first Brillouin zone does not remove states outside that first zone.