# Debye model and dispersion relation

## Homework Statement

I have seen case studies of the 3D Debye model where the vibrational modes of a solid is taken to be harmonic with dispersion relation $\omega = c_sk$. It is said that for temperatures much less than the Debye temperature, the heat capacity at constant volume $$C_V\sim T^3$$.

Now I want to show that for bosons with dispersion relation $\omega\sim A\sqrt k$ has heat capacity $$C_V\sim T^4$$ for $T\ll T_{Debye}$.

In the case studies I have read, I can't find where the dispersion relation comes into play. I have no idea how to see this. Please help!

## Homework Equations

Debye Temperature is given by $$T_{Debye}k_B=\hbar \omega_{max}$$

## The Attempt at a Solution

Generally, I know I need to get the density of modes -- I have a suspicion that here is where the dispersion relation kicks in, but I don't know how.

After that, I should find the ultraviolet cutoff frequency $\omega_{max}$.

Then I should find the energy $$E=\int_0^\omega{max}d\omega {E(\omega)g(\omega)\over \exp(\beta(E-\mu))-1}$$ But what form does $E$ in the integrand take? I know that for photons with dispersion relation $$\omega = c_s k$$ we have $$E=\hbar \omega$$.

After that, it's just a matter of taking limits and $$C_V=\left({\partial E\over \partial T}\right)_V$$ (should) give the required result...

OK, so I know that the density of state depends on the dispersion relation. What are the general definitions of $$E, p$$ in terms of $$\omega, k$$? E.g. for the first case $$E=\hbar \omega$$ and $$p=\hbar k$$. So the question is: what are the respective values for $$E,p$$ in general?