# Spins in an external magnetic field

by Jolb
Tags: external, field, magnetic, spins
 P: 419 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.