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, Yashas May 31 at 16:26

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Pieter, Thomas Fritsch, glS, Phonon, Yashas
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ Under what circumstance do you recover the Dulong-Petit law in Debye's model? What material properties affect that? $\endgroup$ – jacob1729 May 28 at 19:26
  • $\begingroup$ I am not really sure $\endgroup$ – 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.


Not the answer you're looking for? Browse other questions tagged or ask your own question.