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Homework Statement
1. State a relationship between the free energy, F, and the magnetisation, M.
2. State a partition function for the case of a system of N independent spin-1/2 paramagnets in a field, and derive an expression for its susceptibility.
The Attempt at a Solution
(1) Looking in my notes, I see that [tex]dF = - MdH - SdT[/tex] (*). It follows, then, that
[tex]M = \left(-\frac{\partial F}{\partial H}\right)_{T}[/tex]
That's the question answered, but I wonder how one arrives at this expression for F.
I know that [tex]F = U - TS[/tex] and [tex]dW = - HdM[/tex], so it would seem that
[tex]dF = TdS - dW - TdS - SdT = -dW - SdT = HdM - SdT[/tex]
ie. not [tex]MdH[/tex]. Could someone show me how (*) is derived?
(2) For the whole system, I find that
[tex] Z = 2^{N}cosh^{N}(\beta\mu H) [/tex] ([tex]H[/tex] is my field, [tex]\mu[/tex] is the Bohr magneton )
I thus find the free energy to be
[tex]F = -kTN.log[2cosh(\beta\mu H)][/tex]
And thus, by the relationship stated above,
[tex]M = N\mu tanh(\frac{\mu H}{kT})[/tex]
The susceptibility, I presume, is [tex]\frac{\partial M}{\partial H}\right)[/tex]. It comes out for me as
[tex]\frac{N\mu^{2}}{kT} sech^{2}(\frac{\mu H}{kT})[/tex]
(which, I find, tends to [tex]\frac{N\mu^{2}}{kT}[/tex] in the limit of low field or high temperature.
Does that seem sensible?
Cheers!
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