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Phonon Dispersion Relations

The phonon dispersion relations are defined as the k wave vector dependence of the frequencies ω(k,j) of the normal modes for all branches j and selected directions in the crystal. The number of phonon branches, j=1,2,... 3r, is equal to the number of degree of freedom in the primitive unit cell. Each point on the phonon dispersion curve ω(k,j) gives frequency of a phonon. It can be visualized as a dynamical wave of length λ = 1/|k|, propagating along k direction. In this wave the atoms vibrate with frequency ω(k,j) and displace from the equilibrium positions as indicated by the polarization vectors E(k,j).

Typical maximal phonon frequencies range is from 10 to 30 THz, where 1 THz = 1012 1/sec. Other units are also used: 1 THz = 4.1357 meV = 33.356 cm-1. The amplitudes of vibrations are of the order of 0.03 - 0.08 Å. Conventionally, the phonon dispersion relations are drawn along crystal high-symmetry axis, but other directions might also be shown.