What is the magnetic-ordering wave vector?

Question

Like ferromagnetic, antiferromagnetic, the magnetic-ordering are (0,0),(π,0), what is the definition of it? Is there a formula about it?

Answer 1

The magnetic ordering momentums (wave vectors) are the momentums at which the spin structural factor diverges (in the thermodynamic limit).

Let $\mathbf{S}(\mathbf{r})$ be the spin operator at position $\mathbf{r}$ in the material, the spin structural factor is defined as the Fourier transformation of the spin-spin correlation to the momentum space:

$$\chi(\mathbf{q})=\int\mathrm{d}^d\mathbf{r}_1\mathrm{d}^d\mathbf{r}_2\langle \mathbf{S}(\mathbf{r}_1)\cdot \mathbf{S}(\mathbf{r}_2)\rangle \mathrm{e}^{-\mathrm{i}\mathbf{q}\cdot(\mathbf{r}_1-\mathbf{r}_2)}.$$

Long-range magnetic ordering is signified (and defined) by the divergence of $\chi(\mathbf{q})$ at specific momentums $\mathbf{Q}$, s.t.

$$\chi(\mathbf{q}\to\mathbf{Q})\to\infty.$$

These momentums $\mathbf{Q}$ are called magnetic ordering momentum, and the peak of $\chi(\mathbf{q})$ around $\mathbf{q}\to\mathbf{Q}$ is called the magnetic Bragg peak. The physical significance of the magnetic ordering momentums $\mathbf{Q}$ is that they label the (periodic) patterns of magnetization in the magnet. Elastic neutron scattering (ENS) is a commonly used experimental approach to detect the magnetic ordering momentums.