Simple approximations for lattice vibrations

The optical and acoustic branch of the phonon dispersion relation are displayed in the following figure:

For further simplification we approximate both branches:

The phonons with a weak dependence on \(k\) are approximated by \(\omega = const.\) (Einstein model).
The phonons with a linear dependence for small \(k\) values are approximated by \(\omega = v k \) (Debye model).