Magnetic system, partition function

[nicksauce]

Homework Statement

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.

Homework Equations

$$F = -\frac{\ln{Z}}{\beta}$$
$$U = F - T\frac{\partial F}{\partial T}$$

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.