Symmetry analysis of magneto-optical effects: The case of x-ray diffraction and x-ray absorption at the transition metal L2,3 edge

A general symmetry analysis of the optical conductivity or scattering tensor is presented and used to rewrite the conductivity tensor as a sum of linear independent spectra multiplied by simple functions depending on the local magnetization direction. This allows one to describe the full, magnetization directional dependent, magneto-optical response of a system in arbitrary symmetry by only a few linear independent fundamental spectral functions. Using this formalism, we discuss the azimuthal dependence of the resonant x-ray diffracted intensity on magnetic Bragg reflections. We present several numerical examples at the transition metal L-2,L-3 edge. From these numerical calculations, we can conclude that for realistic parameters several fundamental spectra, not present in spherical symmetry, become important and should not be neglected. Deviations from the standard analysis in spherical symmetry become large in cases where orbital order coexists with magnetic order, even if the orbital order is at a different q vector. In the extreme case of the layered cuprates, one finds that one is not sensitive to the projection of the magnetic moments onto the plane of the ordered d(x)(-y)(2)(2) hole. Not including the correct crystal symmetry can lead to incorrectly determined magnetic orientations and structures.