# Phonon density of states

How can I easily calculate phonon density of states from phonon dispersion? I want to compare DOS of graphene and Si from phonon dispersion. Is there a better alternative to Debye DOS = $$\frac{w^2}{2\pi^2v^3}$$ approximation?

If the question is "Can I calculate the Density of States of phonons from the dispersion relationship" then the answer is yes.

The dispersion relationship is $$\omega = f(k)$$ where $$f$$ is some funciton, $$\omega$$ the angular frequency, and $$k$$ the momentum.

In one dimension (1D), the phonon density of states $$D^{(1D)}(\omega)$$ is defined as the number of modes per unit frequency per unit (real space) volume. The latter is just the length of a 1D system so $$L$$.
This gives: $$D^{(1D)}(\omega) = \frac{1}{L} \frac{\mathrm{d}N}{\mathrm{d}\omega} = \frac{1}{L} \frac{\mathrm{d}N}{\mathrm{d}k} \frac{\mathrm{d}k}{\mathrm{d}\omega},$$where the chain rule was used in the last step.

Now, $$\frac{\mathrm{d}k}{\mathrm{d}\omega} = 1/(\frac{\mathrm{d}\omega}{\mathrm{d}k})$$ and you can get $$\frac{\mathrm{d}\omega}{\mathrm{d}k}$$ from the density of states $$\omega = f(k)$$.
Secondly, the separation between $$k$$ points in $$\pi/L$$ (the typical boundary conditions for $$\sin(n \pi x/L)|_{x=0,a}=0$$) so $$\frac{\mathrm{d}k}{\mathrm{d}N} = \pi/L .$$

Plugging that back into the above:

$$D^{(1D)}(\omega) = \frac{1}{\pi} \frac{1}{\mathrm{d}\omega/\mathrm{d}k}.$$

Same goes for 2D and 3D.
In 3D, the procedure would give you $$D^{(3D)}(\omega) = \frac{k^2}{2\pi^2} \frac{1}{\mathrm{d}\omega/\mathrm{d}k}.$$

In the Debye model, the dispersion relation is linear so $$\omega = ck$$. If you plug that into the $$D^{(3D)}(\omega)$$ expression, you get:

$$D^{(3D)}(\omega) = \frac{k^2}{2\pi^2} \frac{1}{c} = \frac{\omega^2}{2\pi^2c^3},$$

which is the expression you quote in the question.

So, bottom line. The Debye model is one model that gives you a specific dispersion relation. But you can calculate the DOS from any generic dispersion relation.