Triakontadipole and high-order dysprosium multipoles in the antiferromagnetic phase of DyB${}_2$C${}_2$

Resonant soft x-ray Bragg diffraction at the Dy M${}_{4,5}$ edges has been used to study Dy multipoles in the combined magnetic and orbitally ordered phase of DyB${}_2$C${}_2$. The analysis incorporates both the intra-atomic magnetic and quadrupolar interactions between the 3d core and 4f valence shells. Additionally, we introduce to the formalism the interference of magnetic and nonmagnetic oscillators. This allows a determination of the higher order multipole moments of rank 1 (dipole) to 6 (hexacontatetrapole). The strength of the Dy 4f multipole moments have been estimated at being between 7 and 78 % of the quadrupolar moment.