Magnetization and susceptibility of non-interacting dipoles
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By evaluating the canonical partition function, derive the magnetization and the susceptibility of a system of N non-interacting magnetic dipoles in the presense of an external magnetic field in the high temperature limit.
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Solution Summary
We derive the partition function for a system of N non-interacting magnetic dipoles. From this we extract a formula for the magnetization. We then consider the high temperature expansion by expanding in the parameter mu H /kT <<1. The formula for the susceptibility in the high temperature limit is then derived.
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The partition function of the N noninteracting dipoles (Z_N)factorizes into N single dipole partition functions (Z_1):
Z_N = (Z_1)^N (1)
Evaluate Z_1 by integrating over mu:
Z_1 = Integral[d^2 mu Exp[beta (mu dot H)]] (2)
where beta = 1/(k T) and d^2 mu denotes integration over all directions of mu.
In Eq. (2) I've neglected the correct normalization. In classical statistical mechanics you can't get the normalization of the partition fuction right (it involves planck's constant). Usually (and in this case) the correct normalizaton isn't important. But if you are rigorous you should multiply the integral in (2) by 1/(2 pi g mu). To see this, consider Z_1 at infinite temperature. Then Z_1 is just the total number of states. The total number of states for a particle with spin total spin s is 2 s+1. The g factor ...
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