Curie's Law - Derivation With Classical Statistical Mechanics

Derivation With Classical Statistical Mechanics

An alternative treatment applies when the paramagnetons are imagined to be classical, freely-rotating magnetic moments. In this case, their position will be determined by their angles in spherical coordinates, and the energy for one of them will be:

where is the angle between the magnetic moment and the magnetic field (which we take to be pointing in the coordinate.) The corresponding partition function is

We see there is no dependence on the angle, and also we can change variables to to obtain

Now, the expected value of the component of the magnetization (the other two are seen to be null (due to integration over ), as they should) will be given by

To simplify the calculation, we see this can be written as a differentiation of :

(This approach can also be used for the model above, but the calculation was so simple this is not so helpful.)

Carrying out the derivation we find

where is the Langevin function:

This function would appear to be singular for small, but it is not, since the two singular terms cancel each other. In fact, its behavior for small arguments is, so the Curie limit also applies, but with a Curie constant three times smaller in this case. Similarly, the function saturates at for large values of its argument, and the opposite limit is likewise recovered.