Magnetic system, partition function

[nicksauce]

1. The problem statement, all variables and given/known data
A certain magnetic system has N independent molecules per unit volume, each of which as 4 distinct energy levels: 0, \Delta - \mu_BB, \Delta, \Delta + \mu_BB.
i) Write down the partition function, and hence find an expression for the Hemholtz function
ii) Use this expression to find the internal energy, U, and the magnetization M.


2. Relevant equations
F = -\frac{\ln{Z}}{\beta}
U = F - T\frac{\partial F}{\partial T}


3. The attempt at a solution
So I think I found the correct equations for the partition function, the hemholtz function and the energy, but I am not quite sure on how to calculate the magnetization. Any ideas?

[Edward G]

There are other ways of writing the internal energy of a system that include magnetic energy.

[eys_physics]

The magnetization is given by (Schroeder. "Thermal Physics")
M=N\bar{\mu_{z}}
where
\bar{\mu_{z}}=\sum_{s}{\mu_{z}(s)P(s)}
where P(s) is the probability for state s.