I am doing the second question on 2006 paper and got stuck with showing the average phonon energy at low T is a linear function of temperature.
So the average phonon energy = total energy (i.e. the expression we differentiate to find heat capacity) divided by the total number of phonons?
total number of phonons is integral of g(omega) x bose factor
Is that right?
I'm a bit confused as I am not getting the right T dependence and also there is the zero point energy in the total energy?
I think I'm doing it wrong. Is that the right expression for the total number of phonons?