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Spins in an external magnetic field 
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#1
Mar312, 03:25 AM

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 noninteracting 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_{z}, with a magnetization [tex]M_z=\mu\sum_{i=1}^{N}m_i[/tex]. For each spin, m_{i} takes only the 2s+1 values s, s+1, ..., s1, s. [Parts ac omitted.] d) Show that [tex]C_B C_M = c \frac{B^2}{T^2}[/tex], where C_{B} and C_{M} 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 ac 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_{B}T (Kardar 4.89) [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{1e^{\frac{(2s+1)B\mu}{k_B T}}}{1e^{\frac{B\mu}{k_B T}}} \right )^N[/tex] [tex]G = Nk_B T\mathrm{ln}\frac{1e^{\frac{(2s+1)B\mu}{k_B T}}}{1e^{\frac{B\mu}{k_B T}}}[/tex] 3. 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 ac, which are all in "my equations." Thanks. 


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