Past exam of the physics A course of the University of Cambridge / 2018 / Paper 2 / Section C / C10 / b / Solution

Impose periodic boundary conditions on a large cubic box. The allowed wavevectors occupy a lattice of cell volume $(2\pi {)}^3/V$ in wavevector space. Including the two spin states, the number of states with magnitude at most $k$ isUsing $k=\sqrt{2m_eϵ}/\hslash$ and differentiating gives the total, spin-summed three-dimensional free-electron density of states,It counts states per unit energy; divide by $V$ for states per unit volume per unit energy. At zero temperature, the Fermi-Dirac distribution fills all states up to the Fermi energy, givingWriting $n=N/V$ givesThe energy zero is the bottom of the free-electron band. The electron Fermi temperature is $T_F=E_F/k_B$.