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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 ...
A detailed solution is given.