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

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Discussion Overview

The discussion centers around the equation for total free energy in a system subjected to an external magnetic field, as presented in a specific paper. Participants are exploring the derivation and components of the equation, particularly the terms related to magnetization and susceptibility.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant presents an equation for free energy involving magnetization terms and magnetic exchange coupling, expressing confusion about its derivation and the absence of certain constants.
  • Another participant requests a link to the paper for further context and understanding.
  • A third participant agrees on the need for more details about the model and references their own experience with statistical mechanics, noting the complexity of deriving such results.
  • A fourth participant provides the link to the paper, indicating its relevance to the discussion.

Areas of Agreement / Disagreement

Participants generally agree on the need for more information regarding the paper and the equation's derivation. However, there is no consensus on the understanding of the equation or its components, as confusion and uncertainty remain evident.

Contextual Notes

Participants note that free energy is distinct from internal energy, which may influence the understanding of the equation. There is also mention of the complexity involved in deriving results related to free energy in external fields.

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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