For phonon gas, we have introduced two models to describe its statistical properties. Namely they are the Einstein model and the Debye model. We have learnt that for non-interacting systems, the statistical properties (i.e. the partition function) are mainly controlled by the dispersion relations and the density of states. The latter also strongly depends on the dimension of the system. (a) Discuss about the dispersion relations and the density of states of the two phonon models. Make plots to show their differences. (b) For phonons, the total number of modes are fixed. Note this does not imply the total number of phonons is fixed. Each mode is treated as a quantum harmonic oscillator, so it is a state that can be occupied by any integer number of phonons (since phonons are bosons). We are thus looking at partition functions of the "grand canonical ensemble" where number of phonons are not fixed. Write down the partition function of the Einstein model and the Debye model in three dimensions, and explain why we do not have a chemical potential in the partition function. (c) Derive the specific heat capacity for the Debye model from the partition function above. You will encounter an integration that is difficult to do, and you can leave it so. Try to do a change of variable for the integration, so the integrand is dimensionless. Explain why the integration is still dependent on temperature. (a mistake in the lecture notes was corrected). (d) Can you also derive the specific heat capacity in two-dimension using the Debye model? Note the total number of modes also depend on the spatial dimension, which will affect the Debye frequency.
5. It is stated that the chemical potential of a phonon gas is zero, and that hence it is possible to obtain the average number of phonons in a crystal with frequency ω by substituting µ = 0 into 〈N〉 = k 1 e β(ϵk−µ) − 1 Explain how it is possible that µ = 0, if the free energy of an Einstein solid or the Debye solid explicitly depends on N ?