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Spins in an external magnetic field

  1. Mar 3, 2012 #1
    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 [tex]\vec{B}=B\hat{z}[/tex] and at temperature T. The work done by the field is given by BMz, with a magnetization
    [tex]M_z=\mu\sum_{i=1}^{N}m_i[/tex]. For each spin, mi takes only the 2s+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], 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:
    [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=kBT (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{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 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."

  2. jcsd
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