= Solution
The <nearly-free electron model> treats a weak periodic potential as a perturbation of plane waves. Near a <Bragg plane>, states with wavevectors $\mathbf k$ and $\mathbf k-\mathbf G$ are nearly degenerate and must be mixed together. Their two-state Hamiltonian is
$$
\begin{pmatrix}\varepsilon_{\mathbf k}&V_{\mathbf G}\\V_{\mathbf G}^*&\varepsilon_{\mathbf k-\mathbf G}\end{pmatrix},\qquad E_\pm=\frac{\varepsilon_{\mathbf k}+\varepsilon_{\mathbf k-\mathbf G}}2\pm\sqrt{\left(\frac{\varepsilon_{\mathbf k}-\varepsilon_{\mathbf k-\mathbf G}}2\right)^2+|V_{\mathbf G}|^2}.
$$
At the unperturbed crossing, the <band gap> is $2|V_{\mathbf G}|$. The corresponding standing waves place different charge densities near the ions and between them, giving different potential energies. Far from a crossing, ordinary perturbation theory is adequate; near it, the degenerate two-wave calculation is essential. In three dimensions, gaps at individual zone boundaries do not guarantee a global insulating gap because bands from different directions can overlap.
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