Numbers of electrons on the Fermi surface

[Tianwu Zang]

In free electron 3D box model, we can calculate the density of state on the Fermi surface g($$\epsilon$$f) easily, but how about the level spacing near the Fermi surface? I think this level spacing $$\Delta$$E should satisfy $$\Delta$$E=d/g($$\epsilon$$f) where d is the degree of degenerate on the Fermi surface. So how many electrons are on the Fermi surface? Are there only 6? ({kf,0,0,$$\uparrow$$};{kf,0,0,$$\downarrow$$};{0,kf,0,$$\uparrow$$};{0,kf,0,$$\downarrow$$};{0,0,kf,$$\uparrow$$};{0,0,kf,$$\downarrow$$})
In one textbook I saw level spacing near the Fermi surface is nearly $$\epsilon$$f/N, so how can d/g($$\epsilon$$f) be connected with $$\epsilon$$f/N?
Thank you so much :-)

 

[BigJDubs]

The level spacing near the Fermi surface is related to the density of states at the Fermi energy, g(εF), as well as the degree of degeneracy, d. The level spacing is given by: ΔE = (d/g(εF))*εF. The number of electrons on the Fermi surface depends on the material and its band structure. For a free electron 3D box model, the number of electrons on the Fermi surface is 6, corresponding to the 6 possible spin-orbitals you mentioned. The level spacing near the Fermi surface can also be approximated as εF/N, where N is the total number of states near the Fermi energy. This approximation is valid when the density of states is approximately constant in the vicinity of the Fermi energy.