- #1

Jolb

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## Homework Statement

*Curie suceptibility*: consider

*N*non-interacting quantized spins in a magnetic field [tex]\vec{B}=B\hat{z}[/tex] and at temperature

*T*. The work done by the field is given by

*BM*, with a magnetization

_{z}[tex]M_z=\mu\sum_{i=1}^{N}m_i[/tex]. For each spin,

*m*takes only the 2

_{i}*s*+1 values

*-s, -s+1, ..., s-1, s*.

[Parts a-c omitted.]

d) Show that

[tex]C_B -C_M = c \frac{B^2}{T^2}[/tex], where

*C*and

_{B}*C*are heat capacities at constant

_{M}*B*and

*M*, respectively.

## Homework Equations

I'm not totally sure the following equations are absolutely correct, since I've tried using them to no avail. Some of them are the ones I derived in parts a-c and others are ones from Kardar. So I'll separate the ones I found from the ones in Kardar, and for the ones in Kardar, I'll include where Kardar puts them if you want to check if they apply.

Kardar equations:

[tex]G=G=-k_B T\mathrm{ln}Z [/tex](Kardar 4.88)

[tex]M = -\frac{\partial G}{\partial B}[/tex] (Kardar 4.97)

[tex]H = -\frac{\partial \mathrm{ln}Z}{\partial \beta }[/tex] where

*β=k*(Kardar 4.89)

_{B}T[tex]C_M=\frac{\partial H}{\partial T}[/tex] (given in Kardar just below 4.89)

[tex]C_B=-B\frac{\partial M}{\partial T}[/tex] (given in Kardar just below 4.98)

My equations:

[tex]Z=\left (\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}} \right )^N[/tex]

[tex]G = -Nk_B T\mathrm{ln}\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}}[/tex]

## The Attempt at a Solution

I've written down C

_{B}and C

_{M}but I have exponentials hanging around which don't cancel and therefore don't leave me with B^2/T^2 proportionality. I suspect there's something wrong with the results of my a-c, which are all in "my equations."Thanks.