= Solution
In the <nearly-free electron model>, a weak <periodic potential> has Fourier components $V_{\mathbf G}$ that couple <plane waves> at $\mathbf k$ and $\mathbf k-\mathbf G$. Far from a degeneracy their admixture gives small perturbative energy shifts. At a Bragg plane, however,
$$
\epsilon_{\mathbf k}=\epsilon_{\mathbf k-\mathbf G},
$$
so <degenerate perturbation theory> is required. In one dimension the first such crossing is at $k=\pm G/2$, the boundary of the first <Brillouin zone>.
Within the degenerate pair the eigenvalues are $\epsilon_{G/2}\pm|V_G|$. Thus
$$
\boxed{\text{energy gap at the Bragg crossing}=2|V_G|.}
$$
The eigenstates are standing-wave combinations of counterpropagating waves. Their charge densities sit differently relative to the ionic potential, giving different potential energies and removing the free-electron degeneracy. The resulting <avoided crossing> produces continuous allowed <energy bands> separated by a forbidden interval. A Fourier component or symmetry-allowed coupling is needed: geometry alone does not require every possible gap to be nonzero.
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