Def'n: magnetic ordering wave vectors

[iibewegung]

Can someone give a clear definition of a ferromagnetic ordering wave vector
and an antiferromagnetic ordering wave vector?

I see the terms being used all over the literature (to calculate order parameters etc.)
but never truly caught onto the definitions.

Any help greatly appreciated-

 

[kanato]

A ferromagnetic ordering wave vector is q = 0. For anything else, you have oscillation of magnetic moments in space, so there will be no net magnetic polarization in a domain.

An antiferromagnetic ordering wave vector is usually a zone boundary vector, like $$q = (\pi/a,\pi/a)$$ in a 2D square lattice. That is a wavevector where the spin on every site is surrounded by opposite spins in its nearest neighbors. But if you had stripes of ferromagnetic spins that alternate (like in the FeAs compounds) then you might have a wavevector like $$q = (\pi/a,0,0)$$.

Depending on the author, sometimes people will also call wave vectors inside the first Brillouin zone as antiferromagnetic. It's more proper to call these spin spiral wave vectors, because they imply that the spins rotate through 360 degrees over a much longer wavelength than two unit cells which is the usual case for antiferromagnetism.

As a visualization tool, you could consider the spin as $$\langle \vec{S} \rangle = (\cos \vec{q}\cdot\vec{r}, \sin \vec{q}\cdot\vec{r})$$ and draw a picture of your lattice putting the spins at each site to get an idea of what a particular q vector looks like. This may or may not be the spin configuration for a particular system of interest, but it's illustrative.