# Vibrational Energy at T = 0 K using the Debye model

I mm trying to estimate the vibrational energy at $$T=0$$ for a 3-D lattice using the Debye model.

My plan is to sum over all the frequencies using:

$$U = \int_0^{\omega_D} d\omega D(\omega)\epsilon(\omega)$$

where $$D(\omega)$$ is the density of states, and $$\epsilon$$ is the energy of each phonon.

Using the density of states for a 3-D lattice: $$D(\omega)=V\omega^2/2 \pi^2v^3$$

where $$v$$ is the velocity of sound, $$V$$ the volume of the solid and $$\omega$$ the frequency, which is related to the vibration wavenumber by $$\omega = vK$$

The energy of a phonon is given by

$$(n+1/2)\hbar\omega \quad n=0,1,2..$$

But since we want to know the energy when $$T=0$$, then we can assume that every phonon will be in the basal state, i.e. $$n=0$$.

Therefore, the energy per phonon will be

$$U = \int_0^{\omega_D} d\omega D(\omega)\hbar\omega/2 = \frac{V\hbar}{4 \pi^2 v^3}\int_0^{\omega_D} \omega^3 d\omega = \frac{V\hbar}{4 \pi^2 v^3} \omega_D^4$$

Does this seem right?