Solve Problem: Partition Function for Magnetic Moment

[Joe Cool]

Hi,
maybe someone can help me with this problem?

Homework Statement

A system consist of N Atoms that have a magnetic moment m. The Hamiltonian in the presence of a magnetic field H is
$$\mathcal{H}(p,q) - mH \sum_{i=1}^N cos(\alpha_{i})$$
where ##\alpha_i## is the angle between the magnetic field and the atom i.

Show that the induced magnetisationt M is:
$$M=Nm\coth(\theta-\frac 1 \theta), \theta=\frac {mH}{ k_BT}$$

Homework Equations

Magnetisation ##M=-\frac {\partial F} {\partial H}##
Free energy ##F=-k_B\ln(Z)##

The Attempt at a Solution

##Z=Z_{mech}* Z_{magn}##
I don't know how to calculate the magnetic partition function.

 

[Charles Link]

This problem I think is problem (7.14) in Reif's Fundamentals of Statistical and Thermal Physics. Reif gives a hint for the probability being around the angle ## \alpha_i ## (he calls it ## \theta ## ) : In the absence of a magnetic field, the probability that the magnetic moment is between ## \theta ## and ## \theta + d \theta ## is proportional to the differential solid angle ## d \Omega=2 \pi sin(\theta) d \theta ## covered by this ## d \theta ##, and in the presence of a magnetic field this will be weighted by the factor ## e^{-E/(kT)} ##, where ## E ## is the magnetic energy for the angle ## \theta ##.

 

[Joe Cool]

Thanks a lot, now I get it :-)