Detection of electromagnetic multipoles by x-ray spectroscopies

Following a brief discussion on electromagnetic multipole expansion, we provide a link between several multipole moments and the measurable quantities of some x-ray spectroscopies, like resonant x-ray scattering or dichroism in absorption. A general classification is given, accompanied by some specific examples and a geometrical interpretation of the axial toroidal (nonmagnetic) quadrupole, related to the x-ray natural circular dichroism. Finally, in order to provide a physical example for the detection of magnetic parity-odd multipoles, like the toroidal moment, we perform a numerical simulation for x-ray nonreciprocal directional dichroism, at the Fe K edge, in the polar ferrimagnet GaFeO3. Our calculations describe quite well the reported experimental profile, with the correct order of magnitude for dichroism/absorption intensity ratio and a signal which is limited to pre-edge.