= Solution
The <nearly-free electron model> expands the <Bloch state> in <plane waves>,
$$
\phi_{\mathbf k+\mathbf G}(\mathbf r)=V^{-1/2}e^{i(\mathbf k+\mathbf G)\cdot\mathbf r},\qquad \psi_{b\mathbf k}=\sum_{\mathbf G}c_{b\mathbf k}(\mathbf G)\phi_{\mathbf k+\mathbf G}.
$$
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.
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