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Generally at low temperatures, solids have $C_{v}$ $\approx$ $aT + bT^{3}$. But graphite has $C_{v}$ $\propto$ $T^{2}$. Why is it so that graphite does not show $T^{3}$ variation of specific heat at low temperature?

I don't want a qualitative explanation, can someone explain me by deriving the $C_{v}$ formula using Debye theory?

I tried doing it and each time I end up in showing a $T^{3}$ variation of $C_{v}$ at low temperature limit. What am I missing about graphite here? I know it has something to do with the 2-D nature of graphite structure of Carbon atoms. But I don't know how I implement this fact in the integration for $C_{v}$ calculation?

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the density of states per frecuency lowers( g(w) ) due to the fact that a bidimensional cristal has less K, then fewer states to excitate the system, then less energy increase for the same temperature increase

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