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Homework Help: Magnetic susceptibility integral trouble

  1. Jan 28, 2015 #1

    This is really more of an algebra question. Here is the main integral I am using for the susceptibility

    [itex] V\chi_0=\frac{d\langle \vec{m_i}\rangle}{d\vec{H_i}}=\frac{1}{Z}\int\vec{m_i}\frac{d}{d\vec{H_i}}e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m-\frac{1}{Z^2}\frac{dZ}{d\vec{H_i}}\int\vec{m_i}e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m[/itex]
    Z is the partition function, m is the magnetic moment, H is the field, beta is just a constant in these calculations.

    I have it that

    [itex] \langle\vec{m_i}\rangle=\frac{1}{\mu_0\beta}\frac{1}{Z}\frac{dZ}{d\vec{H_i}}[/itex]

    And ultimately I need to show that the susceptibility equation can be expressed as

    [itex] V\chi_0=\mu_0\beta(\langle\vec{m_i}^2\rangle-\langle\vec{m_i}\rangle^2)[/itex]
    I can get pretty close. If we look at the first term in the susceptibility equation at the top of the post


    The last line simplification was achieved using the relation given at the top of the post (second equation down).
    The second term then

    Giving a final result for the susceptibility as


    So basically I have an extra factor of [itex]\mu_0\beta[/itex] in the first term so I can't factorize it out to achieve the result I am supposed to. My apologies for the long winded nature of the post but if you could point out where I've made a mistake it would be greatly appreciated.
  2. jcsd
  3. Jan 28, 2015 #2


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    The last line line is incorrect. You should be able to interpret the expression ##\frac{1}{Z} \int\vec{m_i^2}\,e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m##.
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