Einstein solid state model exercise

[besebenomo]

Homework Statement

The system is composed of a set of N non-interacting harmonic oscillators which can be described as a two-dimensional Einstein solid model in the x, y plane. Suppose that each oscillator has mass m and charge q and that there is a magnetic field HH directed along the z direction.

Relevant Equations

Compute (when the system at thermal equilibrium at temperature T):
Internal Energy
Magnetization
Specific heat capacity\bigm

I tried to solve it considering the canonical ensemble (since the system is at the equilibrium with temperature T) and started finding the partition function:

The problem is I am not sure if I have done it correctly and need help because I don't really know where to check.

 

[vanhees71]

Why are you adding the two terms in your expression for the partition sum? You just have to use the expression and do the two sums over $n_r$ and $n_l$. Obviously these two sums factorize!