[Statistical Physics] Spin-1 atoms in uniform magnetic field

[timon]

Homework Statement

A crystal contains $N$ atoms which posses spin 1 and magnetic moment $\mu$. Placed in a uniform magnetic field $B$ the atoms can orient themselves in three directions: parallel, perpendicular, and antiparallel to the field. If the crystal is in thermal equilibrium at temperature $T$ find an expression for its mean magnetic moment $M$, assuming that only the interactions of the dipoles with the field $B$ need be considered. [This is a literal transcription of exercise 3.1 from the second edition of Statistical Physics by F. Mandl. ]

The Attempt at a Solution

First, I wrote down the partition function, henceforth denoted $Z$. There are two perpendicular states, which have no interaction energy. There are also two parallel states, separated by a minus sign. Therefore, if I take $x = \beta \mu B$ I get

$Z = e^0 + e^0 + e^{-\beta \mu B} + e^{\beta \mu B} = 2 + \cosh(x).$

Now I get confused. How can I calculate the mean magnetic moment if the book gives me the magnetic moment for each atom? Surely, $M = N \mu$ is a little too simple. Besides, the answer is given in the back of the book as

$M = N \mu \frac{2sinh(x)}{1+2cosh(x)}$.

 

[Nate810]

I think I should use the definition of mean energy and then the magnetization, but I'm not sure how to approach the problem from this point.