## Spins in an external magnetic field

This is from Kardar's Statistical Physics of Particles, p.123, question 8.
1. The problem statement, all variables and given/known data
Curie suceptibility: consider N non-interacting quantized spins in a magnetic field $$\vec{B}=B\hat{z}$$ and at temperature T. The work done by the field is given by BMz, with a magnetization
$$M_z=\mu\sum_{i=1}^{N}m_i$$. For each spin, mi takes only the 2s+1 values -s, -s+1, ..., s-1, s.

[Parts a-c omitted.]

d) Show that
$$C_B -C_M = c \frac{B^2}{T^2}$$, where CB and CM are heat capacities at constant B and M, respectively.

2. Relevant 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:
$$G=G=-k_B T\mathrm{ln}Z$$(Kardar 4.88)
$$M = -\frac{\partial G}{\partial B}$$ (Kardar 4.97)
$$H = -\frac{\partial \mathrm{ln}Z}{\partial \beta }$$ where β=kBT (Kardar 4.89)
$$C_M=\frac{\partial H}{\partial T}$$ (given in Kardar just below 4.89)
$$C_B=-B\frac{\partial M}{\partial T}$$ (given in Kardar just below 4.98)

My equations:
$$Z=\left (\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}} \right )^N$$
$$G = -Nk_B T\mathrm{ln}\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}}$$

3. The attempt at a solution
I've written down CB and CM 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.
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