3.5 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 $\omega =vk$ (Debye model).