Spins in an external magnetic field

[Jolb]

This is from Kardar's Statistical Physics of Particles, p.123, question 8.

Homework Statement

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 BM_z, with a magnetization
$$M_z=\mu\sum_{i=1}^{N}m_i$$. For each spin, m_i 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 C_B and C_M are heat capacities at constant 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:
$$G=G=-k_B T\mathrm{ln}Z$$(Kardar 4.88)
$$M = -\frac{\partial G}{\partial B}$$ (Kardar 4.97)
$$H = -\frac{\partial \mathrm{ln}Z}{\partial \beta }$$ where β=k_BT (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:
$$Z=\left (\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}} \right )^N$$
$$G = -Nk_B T\mathrm{ln}\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}}$$

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.

 

[mark726]

Dear forum post author,

Thank you for your question. I have taken a look at your attempt and I believe I have found the issue. In your equations for Z and G, you are using the partition function and the Gibbs free energy for a single spin, rather than for N spins. This is why you are getting exponentials that do not cancel out.

To fix this, you need to use the partition function for N spins and the Gibbs free energy for N spins. These can be found by raising the partition function and Gibbs free energy for a single spin to the power of N. This will give you the correct expressions for CB and CM.

I hope this helps. Let me know if you have any further questions.