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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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