# High temperature phonon heat capacity and equipartition theorem

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?

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 $$3Nk_BT$$.