Past exam of the physics A course of the University of Cambridge / 2011 / Paper 1 / Section C / C11 / a (internal energy) / Solution

For independent conduction-electron states labelled by band, wavevector and spin, combine those labels into $j$. ThenHere $g$ counts the same spin and band states as the sum, and $\mu (T)$ is fixed by the number constraint $N=\int g{f}_{\mathrm{FD}}dϵ$. The energy zero is a fixed band reference. Use the total DOS for total internal energy or the molar DOS for molar internal energy; these conventions must not be mixed.