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.