Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 5 / a / Solution

The nearly-free electron model expands the Bloch state in plane waves,All basis functions have the same translation eigenvalues because $e^{i\mathbf{G}\cdot \mathbf{R}}=1$. A weak periodic potential mixes them most strongly near Bragg degeneracies and opens band gaps. Delocalized $s,p$ conduction states of simple metals, such as sodium and aluminium, are typical applications; the approximation need not describe tightly bound core states.