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## 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 [itex]\omega = c_sk[/itex]. It is said that for temperatures much less than the Debye temperature, the heat capacity at constant volume [tex]C_V\sim T^3[/tex].

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

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 [tex]T_{Debye}k_B=\hbar \omega_{max}[/tex]

## 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 [itex]\omega_{max}[/itex].

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 [tex]\omega = c_s k[/tex] we have [tex]E=\hbar \omega[/tex].

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