= Solution
A periodic lattice <Hamiltonian> commutes with lattice translations. The <Bloch theorem> therefore classifies its states by characters of the translation group:
$$
\psi_{n\mathbf k}(\mathbf x)=e^{i\mathbf k\cdot\mathbf x}u_{n\mathbf k}(\mathbf x),
$$
where $u$ is lattice periodic. For each <crystal momentum> $\mathbf k$, solving on a primitive cell with the corresponding quasiperiodic boundary condition gives discrete <eigenvalues> indexed by $n$. Their continuous dependence on $\mathbf k$ produces the <energy bands> $E_n(\mathbf k)$ as $\mathbf k$ ranges through the <Brillouin zone>. Bragg mixing opens gaps at suitable degeneracies, although bands can overlap in three dimensions and gaps are not guaranteed between every pair.
The quantum numbers are \b[band index, wavevector modulo reciprocal lattice vectors, and <spin> where <spin> is conserved]. In a finite crystal the allowed wavevectors are discrete; in the thermodynamic limit they become dense and the band description emerges. Spin-orbit coupling can replace a separately conserved <spin> label by an appropriate internal band label.
Dilute randomly placed impurities break exact translation symmetry, so <crystal momentum> is no longer an exact quantum number. Most extended states retain approximately the host band structure but acquire impurity scattering and disorder broadening. An impurity potential can also produce <impurity levels in a crystal>, often in a band gap: levels associated with a <donor impurity> lie near the conduction-band edge and can supply electrons, while those associated with an <acceptor impurity> lie near the valence-band edge and can create holes. The impurity's species and potential determine whether such levels exist and how deep they are; not every impurity necessarily makes a gap state. At low density these levels are nearly isolated, while overlap at increasing density can broaden them into an impurity band. Added or removed carriers shift the <chemical potential> as well.
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