Violations of Dulong-Petit rule as an upper limit to heat capacity Does any known substance have a heat capacity at constant volume ($C_V$) per mole of atoms greater than $3k_B$ ~ 24.94 J/(mol K)?
In order to count, the substance must actually be made of atoms, that is, ordinary nuclei and electrons.
If so, what are the extra degrees of freedom responsible for this unusually large heat capacity?
 A: As genneth said in a comment, any metal around (or maybe somewhat above) room temperature should have a higher heat capacity than $3k_B$ per atom.
Each vibrational degree of freedom (a.k.a. phonon mode) has a heat capacity of $k_B$ as long as the temperature $T$ and vibrational frequency $\nu$ satisfy $k_BT\gg h\nu$ (if this is not satisfied, the heat capacity is less than $k_B$). There are three phonon modes per atom, so phonons give you $3k_B$, as long as the temperature is high enough. For example, in gold, all the phonon frequencies are less than 5 THz; 5 THZ corresponds to 240K; so at room temperature the phonon heat capacity is almost $3k_B$ (but a bit less).
(I chose gold as an example because its atoms are heavy so they vibrate slowly. Metals with lighter atoms have higher vibration frequencies so a higher required temperature to get the full $3k_B$.)
On top of the phonons, a metal also has heat capacity from kinetic energy of the free electrons. So altogether it can be more than $3k_B$.
For example, I looked up gold's heat capacity (.128 or .129 J/gK) and atomic mass (196.97) and got $3.03k_B$ to $3.06k_B$ per atom.
(I'm a bit surprised it's not higher, since each atom should contribute at least one free electron, and a free electron would be expected to have $1.5k_BT$ of translational kinetic energy. I guess it's too simplistic to treat the electrons like non-interacting free particles. For example, maybe there is a ceiling on electron kinetic energy because of the band structure, or because of velocity-dependent phonon scattering? I'm not sure.)
Other possible degrees of freedom that provide extra heat capacity in some solids include plasmons, magnons, excitons, polaronic excitations, and many others. :-)
