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$$E_{\text{ex}}=-2 J_{\text{ex}} \mathbf{S}_i \mathbf{S}_j =-2 J \mathbf{S}_i \mathbf{S}_j \text{cos}\phi$$

I'm studying magnetism and often see sentences like "spin $i$ product spin $j$" things.

The image I captured is an example. It is about exchange interaction in ferromagnetism.

So, what is exactly a "spin S" in those sentences?

First, I don't think it is spin angular momentum of a electron.

Free electron moves around metal, so it couldn't be seem like "arrow attached in space"

Second, it can't be spin angular momentum of a nucleus neither.

I heard that spin of nucleus is very small, because it is consisted of protons and neutrons which have opposite spins.

Finally, is it a sum of a atom's whole ingredient?

It is large and attached in space, so it makes sense.

But can exchange interaction occurs between two entire atoms?

I thought it occurs between two electrons, of two protons, or something very simple things.

What is the answer? If you can give me some paper or article, I'll be thankful.

Sancol.
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Song
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1 Answers1

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First up, ferromagnetism comes mainly from electron spin, with a small contribution from orbital angular momentum. The nucleus has no significant effect on it, as far as I am aware, and therefore should not contribute in the exchange interaction. However, this is as far as I will go explaining magnetism to you - Any undergraduate textbook on condensed matter physics will do a much better job at it than me.

What I will try to explain to you is what spin is, or at least, clean up some misconceptions you might have about it. Spin is not angular momentum. It is simply property of particles which only becomes significant when dealing with them on the quantum scale, and happens to have a very similar mathematical description to angular momentum - classical angular momentum is described by rotations in 3D real space (the group $\text{SO}(3)$) while spin is described by the group $\text{SU}(2)$, which is, in simple terms, isomorphic to two copies of $\text{SO}(3)$ (mathematicians say that it is a double cover). The difference between spin and other fundamental qualities of particles like mass or charge is that spin is interpreted as a direction instead of a magnitude, which is why it is commonly modeled as a vector - an "arrow in space" as you put it. It does not need to be attached to anything to exist, the free electrons in the crystal lattice of a metal all have a spin. So to answer your question what the spin $\textbf{S}_j$ means in your equation: It is simply an operator measuring (or representing, depending on what you are calculating) the direction of the spin of an electron participating in the exchange.

paulina
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