This is from Kardar's Statistical Physics of Particles, p.123, question 8.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Curie suceptibility: considerNnon-interacting quantized spins in a magnetic field [tex]\vec{B}=B\hat{z}[/tex] and at temperatureT. The work done by the field is given byBM, with a magnetization_{z}

[tex]M_z=\mu\sum_{i=1}^{N}m_i[/tex]. For each spin,mtakes 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], whereCand_{B}Care heat capacities at constant_{M}BandM, 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:

[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] whereβ=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]

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 a-c, which are all in "my equations."

Thanks.

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# Spins in an external magnetic field

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