Fermi energy

The Fermi energy is the energy of the highest occupied state in a system of fermions at absolute zero. Picture stacking electrons one by one into the available states, always choosing the lowest empty one, until they are all placed; the energy of the very last electron added is the Fermi energy. Below it, essentially every state is filled; above it, at zero temperature, every state is empty. It marks the sharp surface of the filled-up sea of fermions.

This single number sets the scale for an enormous range of behaviour in metals and dense matter. In a typical metal the Fermi energy corresponds to a temperature of tens of thousands of degrees, far hotter than room temperature, which is why the electron gas behaves as if it were extremely cold even when the metal is warm. Only the small minority of electrons within a thermal whisker of the Fermi energy can absorb modest energy or take part in conduction; the rest are locked in beneath, with no empty states nearby to move into.

Because so much physics hinges on what happens right at this surface, the Fermi energy explains puzzles that classical pictures got badly wrong. It accounts for why a metal's electrons contribute so little to its heat capacity, why electrical conduction works the way it does, and how the energy of a cold fermion gas feeds directly into the degeneracy pressure that supports white dwarf stars. It is the natural reference energy for any crowd of fermions.