What Is the Equation for Magnetic Energy in an External Field?

[Otterhoofd]

I was reading an article, and it said that the total free energy of the system in an external magnetic field H can be written as:
F= 0.5 * χe^(-1) * Me^2 + 0.5 * χl^(-1) * Ml^2 - Jeff*Ml*Me - (Ml+Me)H

Where Ml and Me denote the magnetization for the local moment and electron subsystem. J_eff is the magnetic exchange coupling between them. H is the applied field and χl and χe the susceptibilty of the system (local moment/electrons)

I don't understand where this expression comes from. I start out with U=-M*B, but I cannot seem to get to the right answer. The first two terms of the right hand site of the equation from the paper are mysterious to me, and I would expect a mu_zero to show up somewhere.
Can anybody please give it a try? It shouldn't be that hard I would say. Any help would be greatly appreciated! Thanks.

 

[gabbagabbahey]

Do you have a link to the paper?

 

[LawlQuals]

Agreed, could we obtain more details about what it specifically models, and a link to the paper? I just took a statistical mechanics course, and deriving results like this can be pretty trying in fact, at least to me they were. Specifically, I remember the course covering topics related to the free energy of a system subject to an external H-field. I am not competent enough in it to give you a more thorough answer though off-hand, I will have to consult K. Huang's text and my notes from the course. In the mean time, how about giving us a peak at that paper?

Recall though that the free energy dF = dU - TdS - SdT (or, however you wish to manipulate it), there are more terms included other than the internal energy U. You may have been pursuing this route, but I thought I would point out that free energy is not the same thing as internal energy (U), on the off-chance that you made this oversight.

 

[Otterhoofd]

Link to the paper:
http://www.sciencemag.org/cgi/content/abstract/329/5987/61

Quantized Anomalous Hall Effect in Magnetic Topological Insulators by R.Yu et al.

Thanks for the replies so far!