Symmetry and physical aspects of 'forbidden' reflections excited by a local polarization anisotropy of the X-ray susceptibility are surveyed. Such reflections are observed near absorption edges where the anisotropy is caused by distortions of the atomic electronic states owing to interaction with neighbouring atoms. As a consequence, they allow for extracting nontrivial information about the resonant atom's local environment and their physical conditions. The unusual polarization properties of the considered reflections are helpful to distinguish them from other types of 'forbidden' reflections. When such reflections are excited, it is, for example, possible to determine not only the intrinsic anisotropy of an atomic form factor but also additional anisotropy induced by thermal motion, point defects and/or incommensurate modulations. Even the local ;chirality' of atoms in centrosymmetric crystals is accessible. Unsolved key problems and possible future developments are addressed.

We observe and explain theoretically a dramatic evolution of the Dzyaloshinskii-Moriya interaction (DMI) in the series of isostructural weak ferromagnets, ${\mathrm{MnCO}}_3$, ${\mathrm{FeBO}}_3$, ${\mathrm{CoCO}}_3$, and ${\mathrm{NiCO}}_3$. The sign of the interaction is encoded in the phase of the $x$-ray magnetic diffraction amplitude, observed through interference with resonant quadrupole scattering. We find very good quantitative agreement with first-principles electronic structure calculations, reproducing both sign and magnitude through the series, and propose a simplified “toy model” to explain the change in sign with $3d$ shell filling. The model gives insight into the evolution of the DMI in Mott and charge transfer insulators.

Three experimental techniques sensitive to the sign of the Dzyaloshinskii-Moriya interaction are discussed: neutron diffraction, Mössbauer γ-ray diffraction, and resonant x-ray scattering. Classical...