Numbers of electrons on the Fermi surface

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