Please check- Statistical paramagnetism.

[Beer-monster]

Homework Statement

A paramagnetic crystal of N ions of spin 1/2, at constant temperature T, is subjected to a magnetic field B. Using the fact that the Helmholtz Free Energy is minimized show that the mean magnetic moment M is:

\mu tanh\frac{\mu B}{kT}

Homework Equations

Magnetic work:dW=-\mu_0 M dH

For paramagnetic material: H=\frac{B}{\mu_0}Energy of alignment with field: \mu B

The Attempt at a Solution

If Free Energy is minimum dF=dE-TdS=0

Using II Law of TD

dS=\frac{dQ}{T} \rightarrow TdS=dQ

and I Law of TD

dQ=dE+dW = dE-\mu_0 M dH

Gives:

dF=\mu_0 M dH = MdB

Therefore:

\frac{dF}{dM}=M

Using
F=-kT\ln Z

where
Z = exp\left(\frac{-\mu B}{kT}\right) + exp\left(\frac{\mu B}{kT}\right) = 2 cosh\left(\frac{\mu B}{kT}\right)

Differientiating with respect to B and doing some algebra gives.

\mu tanh\frac{\mu B}{kT}

QED

Anyone see any problems, I appreciate the help.

 

[Beer-monster]

Please anyone? I really need this because I'm tutoring and don't want to give a solution I'm not 100% about.