If we fill a cold box with fermions, they will all go into different low-energy states. In fact, if the temperature is low enough, they will go into the lowest energy states.
If we fill up all the states up to some energy, that energy is called the Fermi energy.
All the states with energies lower than
are filled, and all the states with energies
(Non zero temperature will put some particles in excited states, but, the idea of the
Fermi energy is still valid.)
The number of states within the radius is
We can now relate the Fermi energy to the number of particles in the box.
We can also integrate to get the total energy of all the fermions.
The step in which and is related to is often useful.
Jim Branson 2013-04-22