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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?

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