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!

 

[sardar]

I can provide some clarification on the equation for magnetic energy in an external field. The first two terms in the equation represent the energy contributions from the local moment and electron subsystems, respectively. The χe and χl values represent the susceptibility of each subsystem, which is a measure of how easily they can be magnetized in the presence of an external field. The Me and Ml values represent the magnetization of each subsystem.

The third term, - Jeff*Ml*Me, represents the magnetic exchange coupling between the local moment and electron subsystems. This term takes into account the interaction between the two subsystems and how it affects their magnetization.

The last term, (Ml+Me)H, represents the energy contribution from the applied external field, H. This term takes into account the energy required to align the magnetization of both subsystems with the applied field.

The presence of mu_zero is not necessary in this equation, as it is already accounted for in the definition of H as the applied field. The overall equation for magnetic energy in an external field takes into account the contributions from the different subsystems and their interactions with each other and the applied field.