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Density of States > The Fermi Sphere


Image of Fermi sphere

We define the function N(E) to be the 'density of states' (i.e., the number of states per unit volume at energy E).

The number of states in the shell is
N(E)dE = (8π/h3)    (2mE)(2m)dE/(2E) and thus
N(E) = (4π/h3)(2m)(2m)E

This is a parabola - it is plotted in Density of States Parabola.

The volume of the positive quadrant of the Fermi sphere is just the volume of a sphere of radius p, that is:

1
 
4πp3
8 3

The volume of an incremental shell from radius p to p+dp (shown in red) is the differential of this, πp2dp/2. This shell can hold:

πp2dp
2(h/2L)3

states or

8πp2dp
(h/L)3

electrons.

In unit volume (L=1) this becomes 8π2dp/h3 per unit volume.

From E=p2/2m we find:

p2 = 2mE so p = (2mE) and
dp =
(2m) dE
2E
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