Nearly-free-electron model

The nearly-free-electron model describes electrons in a crystal as free particles weakly perturbed by the lattice’s periodic potential. It explains how energy gaps form at Brillouin-zone boundaries.

Bloch's theorem: A theorem stating that wavefunctions in a periodic potential equal plane waves multiplied by lattice-periodic functions. It supplies the wavefunction form used when the lattice perturbs free-electron states.
Free-electron model: A model of electrons treated as independent particles moving without a crystal potential. The nearly-free model begins with these states before adding weak lattice effects.
Tight-binding model: A band-theory model that builds crystal states from localized atomic orbitals with limited overlap. It is the complementary limit, emphasizing strong localization rather than weak lattice perturbations.
Alkali metals: The chemical elements in Group 1, including lithium, sodium, and potassium. Their relatively simple valence-electron behavior makes them useful nearly-free-electron examples.
Bragg scattering: Coherent scattering from a periodic structure that reinforces waves at particular directions or wavevectors. At zone boundaries, scattering couples counter-propagating electron waves and opens gaps.
Schrödinger equation: The equation governing quantum-mechanical wavefunctions and their energies. Solving it with a periodic potential yields the model's electron states.
Pseudopotential: An effective potential that replaces the strong ionic potential experienced by valence electrons with a smoother one. Pseudopotential methods extend weak-potential ideas to practical calculations for real solids.
Copper: A metallic chemical element with atomic number 29 and a face-centered cubic crystal structure. Its conduction electrons are often approximated as nearly free, though its filled d bands complicate the picture.
Brillouin zone: A primitive cell in reciprocal space bounded by planes of Bragg reflection. Its boundaries mark where the model predicts strong wave coupling and band gaps.
Reciprocal lattice: The lattice in wavevector space associated with a crystal's real-space periodicity. Its vectors specify which plane-wave states the periodic potential can couple.

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