# Why does diamond's specific heat capacity deviate from the Dulong-Petit theory at room temperatures? [closed]

Why does diamond deviate from the prediction of $$c_v = 3k_B$$ at room temperature more than other solids. Why do we require Debye's theory instead?

## closed as off-topic by Pieter, Thomas Fritsch, glS, Phonon, YashasMay 31 at 16:26

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• Under what circumstance do you recover the Dulong-Petit law in Debye's model? What material properties affect that? – jacob1729 May 28 at 19:26
• I am not really sure – user208480 May 28 at 19:38

Recall that the Debye model is based on a phonon dispersion $$\omega = c_s k$$ with $$c_s$$ the speed of sound. The Debye frequency $$\omega_D$$ can only depend upon the speed of sound and the lattice spacing $$a$$ so on dimensional grounds $$\omega_D \sim c_s / a$$.
In the high temperature limit the Debye model must reproduce the Dulong-Petit law, but how high is high? Again, there are not many parameters in the problem and the only possible energy we can cook up is $$\hbar \omega_D$$. This gives a Debye temperature $$k_BT_D = \hbar \omega_D$$ above which the Dulong-Petit rule $$3R$$ holds fairly accurately and below which it is a poor approximation. But then $$T_D \sim \omega_D \sim c_s$$ so the Debye temperature is proportional to the speed of sound, which in turn measures how stiff a material is. Diamond is known to be exceptionally stiff (speed of sound is $$\sim 12,000\text{m/s}$$) and so has a high Debye temperature (around $$2000K$$, much higher than room temperature) and as such the Dulong-Petit rule is almost always invalid.