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    Magnetization of europium.

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    A state with magnetic moment mu will have an energy of -mu B in a magnetic field of B. The magnetization of a system is the average total magnetic moment. In this case it is the average magnetic moment of an atom. We can evaluate this as follows. Suppose that state r of the system has a magnetic moment of mu_r and thus an energy of the form

    E_r = e_r - mu_r B (1)

    Then the average magnetic moment is:

    <mu> = Sum over r of P_r mu_r (2)

    P_r is the probability that the system is in state r:

    P_r = exp(-beta E_r)/Z = exp[-beta e_r + beta mu_r B]/Z (3)

    We can express ...

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