Debye model and dispersion relation

[c299792458]

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...

 

[c299792458]

Please, somebody?

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?