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I have seen the derivation of the high temperature heat capatcity sing integration of the density of states. The heat capacity is $C=3Nk_B$ with $N$ the number of atoms in the lattice. According to my lecturer 'this is in agreement with the equipartition theorem'. While I agree that the dependence on the temperature and the number of atoms is the same, I would think that the equiparition theorem would predict $C=\frac{3}{2}Nk_B$, i.e. $1/2 k_B T$ of energy per quadratic degree of freedom, and there are 3N modes: N atoms, 2 transverse and 1 lngitudinal vibrational mode. Where did the factor of 2 go?

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There are actually six quadratic degrees of freedom per atom: the three vibrational potential energies, which are quadratic in displacement, and the three kinetic energies, which are quadratic in velocity. This gives you the proper specific heat.

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Each mode is an independent harmonic oscillator. Thus, there is a quadratic kinetic energy term and a quadratic potential energy term in the Hamiltonian of each normal mode. Therefore the average total energy is $3Nk_BT$.

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