# Why does Debye model predict that, for big wavelenghts, the relationship between the frequency and the wave number of phonons is linear?

I'm studying Debye model for the specific heat capacity of solids. In class we found first of all that the number of modes per frequency is proportional to the frequency squared. We found that there is a maximum frequency, which is the Debye frequency. And then we derived the formula for the total internal energy, and studying its limit for low temperatures we found that $$c_{V}=\frac{12\pi^4R}{5}\left(\frac{T}{T_{D}}\right)^{3},$$ where $$T_{D}$$ is the Debye temperature. Of course we underlined that one of the hypothesis for this model is that the solid is considered as a continuum, so its vibrations propagate making the atoms vibrate together, therefore the wavelength has to be much longer than the distance between atoms.

After that, we analysed a one-dimensional lattice of atoms and derived the dispersion relation $$\omega^2=\frac{4b}{M}\sin^2\left(\frac{ka}{2}\right),$$ where $$b$$ is the elastic constant, $$M$$ is the mass of the atom, $$k$$ is the wave vector and a is the distance between two atoms. Of course if $$k$$ is small (meaning $$\lambda$$ is big, so we are in the hypothesis of the Debye model) this relation becomes $$\omega=\sqrt\frac{b}{M}ka.$$ We then concluded that $$\omega$$ is linear in $$k$$, as predicted by Debye model.

What I don't understand is why we can say this, where Debye model predicts that there is a linear relationship between $$\omega$$ and $$k$$.

• Errm, you already derived the correct behaviour and the correct limit, so what is it that you are actually asking? May 8, 2023 at 5:25
• The Debye model is essentially defined by the assumption that there is a single sound speed $c_{s}$ for all the modes. Then it is automatic that $\omega=c_{s}k$, because the group velocity of a propagating sound wave has to be $c_{g}=d\omega/dk$.
– Buzz
May 8, 2023 at 5:41
• You can see the Debye derivation yourself on Wikipedia en.m.wikipedia.org/wiki/Debye_model May 8, 2023 at 5:41
• @Buzz thank you, I thought that given that assumption Debye predicted that omega was linear in k, while it is actually the opposite, so I think that's what was confusing me
– Fede
May 8, 2023 at 19:56