Applications and Consequences
More generally, a Bloch-wave description applies to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. It is generally treated in the various forms of the dynamical theory of diffraction.
The plane wave vector (Bloch wave vector) k, which when multiplied by the reduced Planck's constant is the particle's crystal momentum, is unique only up to a reciprocal lattice vector, so one only needs to consider the wave vectors inside the first Brillouin zone. For a given wave vector and potential, there are a number of solutions, indexed by n, to Schrodinger's equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wave vectors, it is called a (complete) band gap. The band structure is the collection of energy eigenstates within the first Brillouin zone. All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wave functions, at least within the independent electron approximation.
A corollary of this result is that the Bloch wave vector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the group velocity of the wave is conserved. This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and that electrical resistance in a crystalline conductor only results from imperfections and finite size which break the periodicity and induce interaction with phonons.
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