Fermi Energy - Three-dimensional Case

Three-dimensional Case

The three-dimensional isotropic case is known as the Fermi sphere.

Let us now consider a three-dimensional cubical box that has a side length L (see infinite square well). This turns out to be a very good approximation for describing electrons in a metal. The states are now labeled by three quantum numbers n_x, n_y, and n_z. The single particle energies are

n_x, n_y, n_z are positive integers.

There are multiple states with the same energy, for example . Now let's put N non-interacting fermions of spin 1/2 into this box. To calculate the Fermi energy, we look at the case where N is large.

If we introduce a vector then each quantum state corresponds to a point in 'n-space' with energy

The number of states with energy less than E_f is equal to the number of states that lie within a sphere of radius in the region of n-space where n_x, n_y, n_z are positive. In the ground state this number equals the number of fermions in the system.

the factor of two is once again because there are two spin states, the factor of 1/8 is because only 1/8 of the sphere lies in the region where all n are positive. We find

so the Fermi energy is given by

Which results in a relationship between the Fermi energy and the number of particles per volume (when we replace L2 with V2/3):

The total energy of a Fermi sphere of fermions is given by

Therefore, the average energy of an electron is given by: