## Phonon density of states for ZnO (Zincblende)

The lattice constant of ZnO in the zincblende structure is 4.616 Å. There are four zinc atoms and four oxygen atoms in the conventional unit cell so the atomic density is $n= 8/(4.616\times 10^{-10})^3 = 8.134 \times 10^{28}$ 1/m³. Every atom has three degrees of freedom so there are $3n = 2.44\times 10^{29}$ modes/m³. The integral of the phonon density of states over all frequencies should be $3n$.