High-order Ho multipoles in HoB ₂C ₂ observed with soft resonant x-ray diffraction

Soft resonant x-ray Bragg diffraction (SRXD) at the HoM _4,5 edges has been used to study Ho 4f multipoles in the combined magnetic and orbitally ordered phase of HoB ₂C ₂. A full description of the energy dependence for both and incident x-rays at two different azimuthal angles, as well as the ratio _I/ _I as a function of azimuthal angle for a selection of energies, allows a determination of the higher order multipole moments of rank 1 (dipole) to 6 (hexacontatetrapole). The Ho 4f multipole moments have been estimated, indicating a dominant hexadecapole (rank 4) order with an almost negligible influence from either the dipole or the octupole magnetic terms. The analysis incorporates both the intra-atomic magnetic and quadrupolar interactions between the 3d core and 4f valence shells as well as the interference of contributions to the scattering that behave differently under time reversal. Comparison of SRXD, neutron diffraction and non-resonant x-ray diffraction shows that the magnetic and quadrupolar order parameters are distinct. The component of the magnetic order exhibits a Brillouin type increase below the orbital ordering temperature T _Q, while the quadrupolar order increases more sharply. We conclude that the quadrupolar interaction is strong, but quadrupolar order only occurs when the magnetic order gives rise to a quasi-doublet ground state, which results in a lock-in of the orbitals at T _Q.