130_notes.dvi

13.1.1 Filling the Box with Fermions

If we fill a cold box withNfermions, they will all go into different low-energy states. In fact,if the
temperature is low enough, they will go into the lowest energyNstates.

If we fill up all the states up to some energy, that energy is called theFermi energy. All the states
with energies lower thanEFare filled, and all the states with energies larger thanEF are empty.
(Non zero temperature will put some particles in excited states, but, the idea of the Fermi energy is
still valid.)

π^2 ̄h^2

2 mL^2

(n^2 x+n^2 y+n^2 z) =

π^2 ̄h^2

2 mL^2

r^2 n< EF

Since the energy goes liken^2 x+n^2 y+n^2 z, it makes sense to define a radiusrninn-space out to which
the states are filled.

Fermi Surface

Occupied States

nx

ny

nz

rn

Unoccupied States

The number of states within the radius is

N= (2)spin

1

8

4

3

πrn^3

where we have added a factor of 2 because fermions have two spin states. This is anapproximate
counting of the number of statesbased on the volume of a sphere inn-space. The factor of^18
indicates that we are just using one eighth of the sphere inn-space because all the quantum numbers
must be positive.