Phonon dispersion in lattices

Summary

Phonon dispersion is the relation between wave vector and frequency for small vibrations of a crystal. The crystal is modelled as a lattice of atoms joined by elastic forces. For each wave vector the possible frequencies are the square roots of the eigenvalues of a Hermitian matrix, the dynamical matrix. As the wave vector varies, they form several branches. At long wavelengths the lowest branches are linear in the wave vector and give the elastic sound waves of continuum elastodynamics. At short wavelengths the discreteness of the lattice shows, and the waves become dispersive.