- #1

cozycoz

## Homework Statement

A certain magnetic system contains n independent molecules per unit volume, each of which has four energy levels given by 0, ##Δ-gμ_B B##, ##\Delta##, ##\Delta +gμ_B B##. Write down the partition function, compute Helmholtz function and hence compute the magnetization ##M##. Hence show that the magnetic susceptibility ##χ## is given by [tex]χ = \lim_{B\rightarrow 0} {\frac{μ_0 M}{B}} = \frac{2 n μ_0 g^2 {μ_B}^2}{k_B T (3+e^{Δ/k_B T})} [/tex]

(Sorry I don't know how to type limit in mathematical form)

\lim_{n\rightarrow +\infty} {\frac{\sin(x)}{x}}

## Homework Equations

##F=-k_B T \ln Z##, ##m=-(\frac{∂F}{∂B})##, ##m=MV##

## The Attempt at a Solution

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If there are N=Vn molecules, the partition function is [tex] Z=(e^{0}+e^{-β(Δ-gμ_B)}+e^{-βΔ}+e^{-β(Δ+gμ_B)})^N [/tex].

Then Helmholtz free energy function F is

[tex] F=-k_B T[-βΔ+\ln (1+2\cosh {βgμ_B B})] [/tex]

Now I have to differentiate F by B, but I've noticed βΔ part would be gone because it doesn't have B at all.

My answer is

[tex] χ = \frac{2 n μ_0 g^2 {μ_B}^2}{3 k_B T}[/tex]

where βΔ part is omitted.

If you tell me what's wrong it would be lovely.