Momentum dependent light scattering in insulating cuprates

We investigate the problem of inelastic x-ray scattering in the spin$-\frac{1}{2}$ Heisenberg model on the square lattice. We first derive a momentum dependent scattering operator for the $A_{1g}$ and $B_{1g}$ polarization geometries. On the basis of a spin-wave analysis, including magnon-magnon interactions and exact-diagonalizations, we determine the qualitative shape of the spectra. We argue that our results may be relevant to interpret inelastic x-ray scattering experiments in the antiferromagnetic state of copper oxide materials.