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I hope someone knows a reference or knows how to solve the following question. (I am pretty sure I won't be getting any answers, but why not try?! :-)).

Find an integral expression for the heat capacity: $$C_V=\partial_T \sum_{\vec{k},s}\frac{\omega_s(\vec{k})}{e^{\beta \hbar \omega_s(\vec{k})}-1}$$

in a general $n$ dimensions given density of states $D(\omega)$, for an isotropic crystal with an infinite volume $V\to \infty$.

Thanks.

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It looks like the integral expression cannot exist because a crystal, or any body for that matter, of infinite volume is by definition indeterminate.

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  • $\begingroup$ An integral expression can exist perhaps asymptotically, it doesn't converge so we need to take a cutoff. $\endgroup$ – MathematicalPhysicist Jan 5 '17 at 3:38

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