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

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SUMMARY

The equation for total free energy in an external magnetic field H is given by F = 0.5 * χe^(-1) * Me^2 + 0.5 * χl^(-1) * Ml^2 - Jeff * Ml * Me - (Ml + Me)H. In this equation, Ml and Me represent the magnetization of the local moment and electron subsystems, respectively, while J_eff denotes the magnetic exchange coupling. The discussion highlights the confusion surrounding the derivation of this equation, particularly regarding the absence of the permeability constant μ₀ and the distinction between free energy and internal energy.

PREREQUISITES
  • Understanding of magnetic susceptibility (χl and χe)
  • Familiarity with magnetization concepts (Ml and Me)
  • Knowledge of magnetic exchange coupling (J_eff)
  • Basic principles of statistical mechanics and free energy
NEXT STEPS
  • Study the derivation of free energy in external magnetic fields using K. Huang's statistical mechanics text.
  • Explore the role of magnetic exchange coupling in magnetic systems.
  • Research the implications of magnetic susceptibility in materials science.
  • Read the paper "Quantized Anomalous Hall Effect in Magnetic Topological Insulators" by R. Yu et al. for practical applications of these concepts.
USEFUL FOR

Students and researchers in physics, particularly those focusing on magnetism, statistical mechanics, and materials science, will benefit from this discussion.

Otterhoofd
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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.
 
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Do you have a link to the paper?
 
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.
 
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!
 

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