# Heat capacity due to Phonon Vibrations

Why does the speed of sound come, in the expression of Heat capacity, from phonon vibrations?

To be more precise, sound velocity appears in the Debye's expression for specific heat of a crystalline solid. The reason is that in Debye's theory the exact expression for the internal energy of the harmonic solid: $$E = \int D(\omega)\frac{\hbar \omega}{e^{\beta \hbar \omega}-1} {\text d}\omega$$ where $$\beta=1/k_BT$$ and $$D(\omega)$$ is the phonon density of states, can be approximated, at sufficiently low temperatures, by using as density of states, the density of states of the acoustic branches only. Since the dispersion on an acoustic branch is given by $$\omega({\bf k})= c_s^{(i)}\left| {\bf k} \right|,$$ where $$c_s^{(i)}$$is the sound speed in the direction identified by that acoustic branch, the resulting density of states depends on the speed of sound.