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Beer-monster
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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:
[tex] \mu tanh\frac{\mu B}{kT} [/tex]
Homework Equations
Magnetic work:[tex]dW=-\mu_0 M dH [/tex]
For paramagnetic material: [tex] H=\frac{B}{\mu_0}[/tex]Energy of alignment with field: [tex] \mu B [/tex]
The Attempt at a Solution
If Free Energy is minimum [tex]dF=dE-TdS=0[/tex]
Using II Law of TD
[tex] dS=\frac{dQ}{T} \rightarrow TdS=dQ [/tex]
and I Law of TD
[tex] dQ=dE+dW = dE-\mu_0 M dH [/tex]
Gives:
[tex] dF=\mu_0 M dH = MdB[/tex]
Therefore:
[tex] \frac{dF}{dM}=M [/tex]
Using
[tex] F=-kT\ln Z[/tex]
where
[tex] Z = exp\left(\frac{-\mu B}{kT}\right) + exp\left(\frac{\mu B}{kT}\right) = 2 cosh\left(\frac{\mu B}{kT}\right) [/tex]
Differientiating with respect to B and doing some algebra gives.
[tex] \mu tanh\frac{\mu B}{kT} [/tex]
QED
Anyone see any problems, I appreciate the help.
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