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As far as I understand, in a primitive picture insulators are localized systems, and metals are itinerant electron systems.

What I do not get: I have, e.g., 3d itinerant magnets like:

Ni ([Ar] 4s2 3d8 or [Ar] 4s1 3d9 ), 
Fe ([Ar] 4s2 3d6), and 
Co ([Ar] 4s2 3d7).

Why does it make sense to speak about 3d electron shells, if the electrons are not localized to the atomic nucleus but they are itinerant, i.e., there are distributed over the whole crystal (it can be described by the probability function of the Bloch waves).

Confinement to atomic nucleus does not make sense to me, since (weak correlated) metals behave as a free electron gas.

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That's a good question. It turns out that you can make Bloch waves out of linear combinations of atomic orbitals, generally one per site, and that choosing different atomic orbitals for those linear combinations will produce Bloch waves of different characters. These manifest themselves as the different bands of the material - including core, valence and conduction bands, as well as a bunch of higher conduction bands that can be occupied at high energy.

This then lets you talk about an electron wave that's delocalized over the entire crystal but which still retains (say) a clear 3d character.

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  • $\begingroup$ Okay, I like your answer. What I still do not get, is spin-orbit interaction for itinerant systems. Normally, in the rest frame of the electron, the atomic nucleus orbits around the electron producing a magnetic field to which couples the spin of the electron. However, for itinerant systems it is not true that the nucleus orbits the electron. But the spin-orbit equation $H_{LS}\propto \vec L \cdot \vec S \frac 1{r}\frac{dU}{dr}$ holds true, anyway. So what does $r$ mean for itinerant systems? Maybe, I should open a new question. $\endgroup$
    – cerv21
    Nov 27, 2017 at 14:17
  • $\begingroup$ @opens Yeah, that's a new question, you should post it separately. If this post is resolved, you can mark the answer as accepted using the checkmark on the left. $\endgroup$
    – Emilio Pisanty
    Nov 27, 2017 at 16:47
  • $\begingroup$ Here I posted my new question: physics.stackexchange.com/questions/371163/… . I would really appreciate it, if you could comment/answer my new question, if you have any idea :) $\endgroup$
    – cerv21
    Nov 29, 2017 at 18:16

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