Resonant $1\overrightarrow{s}3d$ x-ray Bragg diffraction and structure factors for transition-metal compounds

Structure factors for Bragg diffraction of x rays enhanced by electric quadrupole absorption $\left(1\mathrm{}\overrightarrow{s}3d\right)$ are calculated for several configurations of the resonant ions found to exist in transition-metal compounds. The configurations include the spontaneous order (fully compensating and noncollinear antiferromagnetism) and field-induced states of ferrous niobate, and paramagnetic iron pyrite. The calculation is couched in terms of spherical tensors, which describe the orbital magnetism of $3d$ electrons, and highlights the essential components contributing to the results, namely, use of an atomic model for the resonant contribution to the x-ray scattering length, the influence of the elements of symmetry pertinent to a resonant ion and the crystal structure, and linear and circular polarization effects. Spatial anisotropy in the $3d$ orbital moments leads to charge-forbidden reflections. Associated structure factors are calculated as functions of the canting angle of the principal axis, and the (azimuthal) angle of rotation of the crystal about the Bragg wavevector. In the model for ferrous niobate the orbital moment arises from a conventional mechanism based on the spin-orbit coupling.