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There are at least three different mechanisms which can give rise to ferromagnetic order in iron.

  1. First is due to the band electrons called band magnetism or itinerant magnetism which is an exchange interaction between conduction electrons.

The page 91 of Fundamentals of Many-body physics by W. Nolting, page 251 (sidenote 1) of Oxford Solid State Basics by Steven H. Simons tells that Fe is itinerant.

  1. Second is indirect exchange i.e. exchange between unpaired d electrons and conduction electrons.

  2. The third is the direct exchange between localized magnetic moments of two neighbouring Fe ions as described by the Heisenberg model.

Which one of them is responsible for ferromagnetism in iron (and also cobalt and nickel) and why? I expect that the third effect would be least because d orbitals are inner orbitals and do not have much overlap.

I read this, this, this and the question titled "What is the difference between a localized and itinerant magnetism?". None seem to address my concern.

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    $\begingroup$ Very nice question, I thought my2cents was right until I checked many papers. It seems like Fe's ferromagnetism is due to itinerant electrons. Apparently there was a HUGE debate over the years and your option 3, which was favored in the early days, lost the battle. See for example annualreviews.org/doi/pdf/10.1146/annurev.ms.14.080184.000245. $\endgroup$ – thermomagnetic condensed boson Jun 19 '19 at 20:03
  • $\begingroup$ I think we can rule out #2 because the only conduction electrons are d-electrons themselves, making the distinction impossible. If you are thinking of s- electrons, they should be well below the fermi energy and inactive. $\endgroup$ – KF Gauss Oct 24 '19 at 18:13
  • $\begingroup$ @KFGauss The $4sp$-band is very wide and crosses the Fermi level. $\endgroup$ – Pieter Oct 24 '19 at 18:22
  • $\begingroup$ @thermomagneticcondensedboson That is a paper from 1984, which was a while ago. It is clear that localized moments persist above the Curie temperature, which is hard to understand from a band-magnetism point of view. $\endgroup$ – Pieter Oct 24 '19 at 18:30
  • $\begingroup$ @Pieter, my mistake. But regarding the 1984 paper, the actual paper takes into account the fact that the susceptibility above Tc implies moments still exist, but they are non-integer. This would disagree with a naive local moment picture suggested by option 3. $\endgroup$ – KF Gauss Oct 24 '19 at 19:39
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A lot of work has been done since the 1984 review by Moriya and Takahashi, both theory and experiments.

On the theoretical side, calculations have been refined a lot. The effect of the on-site $dd$ correlation energy $U$ was taken into account first by LSDA+U, then by LSD+DMFT (a dynamic mean field theory), for example this paper from 2001 about high temperature: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.87.067205

On the experimental side, instead of studying the spectra of the paramagnetic phase by heating up the sample, techniques were developed to just heat the electrons with ultra-fast laser pulses. For example this paper from 2017 by Eich: https://advances.sciencemag.org/content/advances/3/3/e1602094.full.pdf Figure 1B shows what would be expected in the photoemission spectrum contrasting the case of an itinerant Stoner picture and the case of a Heisenberg-like picture with local moments. The data support the latter.

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For ferromagnetic materials like iron the cause of alignment of the individual atomic magnetic moments is the direct exchange interaction, that is number 3. This is the mainstream explanation as given in the Feynman lectures ( https://en.m.wikipedia.org/wiki/Ferromagnetism).

Fe is definitely not an itinerant magnet. The electrons responsible for the magnetism are localised. The magnetic transition is an order-disorder transition. In itinerant magnets delivalised electrons are responsible and the transition is a phase transition. No magnetism persists above the transition temperature, apart of course from the lo al moments. Note that the d electrons form filled bands of one spin direction. A full band Slater determinant can be written equivalently in local and in crystal orbitals. This fact may be relevant to the debate.

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  • $\begingroup$ I agree. One can also Google for titles like "Stoner I or Hubbard U". The reason for abandoning the itinerant explanation for iron etc is that the local moments persist above the Curie temperature. $\endgroup$ – Pieter May 13 '19 at 11:45
  • $\begingroup$ @my2cats All the interactions I have written are different types of exchange interactions. Look at the book Oxford Solid State Basics by Steven H. Simon which clearly says "Most of the ferromagnets that we are familiar with, such as iron, are itinerant." Page 251. $\endgroup$ – mithusengupta123 May 14 '19 at 6:06
  • $\begingroup$ @Pieter See my comment above. I cite a counter reference. This creates confusion because it does not explain why. $\endgroup$ – mithusengupta123 May 14 '19 at 6:07
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    $\begingroup$ Hey, I have been quite intrigued by the question and after reading many papers, it seems that the conclusion is that Fe's ferromagnetism is definitely due to itinerant electrons. See for example annualreviews.org/doi/pdf/10.1146/annurev.ms.14.080184.000245. $\endgroup$ – thermomagnetic condensed boson Jun 19 '19 at 20:01
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    $\begingroup$ @my2cts do you have any source for your claims in the second paragraph? $\endgroup$ – user137661 Oct 24 '19 at 20:31
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Normally, it is 1. This is when the unpaired outer valence electrons of neighboring atoms overlap and the distribution of their electric charge in space are farther apart when they have parallel spins. This reduces the electrostatic energy so when they have parallel spin, it is more stable.

Normally this is very short ranged, called intra-atomic (electrons in the same atom) exchange or direct exchange between neighboring atoms.

But iron has a characteristic, it is able to create something called magnetic domains. Though the spins are aligned inside the domains, the domains themselves cancel out each other and the whole peace of iron will not be magnetic.

Now this is when longer ranged interactions can occur via intermediary atoms, called superexchange, or indirect exchange, that is 2 and 3.

Please see here:

https://arxiv.org/ftp/cond-mat/papers/0701/0701423.pdf

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    $\begingroup$ This answer contains some inaccuracies. It states that Fe has itinerant ferromagnetism but then describes direct exchange. The paper it referred to us about Anderson superexchange which is relevant mainly to oxides. This mechanism is different from the three listed by the OP. $\endgroup$ – my2cts May 13 '19 at 5:51

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