Phonon Density of States

The phonon density of states g(ω) describes the number of phonon modes of a selected frequency ω(k,j) in a given frequency interval (ω - ½Δω,  ω + ½Δω), if the density of wave vectors k in the Brillouin zone is homogeneously distributed. The phonon density of states is normalized ∫₀^∞ dω g(ω) = 1. The phonon density of states g(ω) spreads from zero to the maximal phonon frequency existing in a given crystal. The acoustic phonons, behaving linearly in vicinity of k=Γ, and cause that g(ω) is proportional to ω² close to ω=0. In simple crystals the ω² dependence may cover a substantial part of frequency interval, but in complex crystals the quadratic dependence is limited to the close vicinity of the ω=0 point. Phonon dispersion relations which are flat, may lead to sharp peaks in g(ω). In simple crystals with a large mass difference of the constituents the vibrations could be separated to a low-frequency phonon band, caused mainly by oscillations of heavy atoms, and to a high-frequency band, occupied by light atoms. The bands could be separated by a frequency gap.

To describe the vibrations of a specific atom μ moving along i-direction a partial phonon density of states g_μ,i(ω) is introduced. It is normalized to ∫₀^∞ dω g_μ,i(ω) = 1/r, where r is the number of degree of freedom in the primitive unit cell.