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Soft resonant x-ray Bragg diffraction (SRXD) at the Ho M$_{4,5}$ edges hasbeen used to study Ho $4f$ multipoles in the combined magnetic and orbitallyordered phase of HoB$_2$C$_2$. A full description of the energy dependence forboth $\sigma$ and $\pi$ incident x-rays at two different azimuthal angles, aswell as the ratio $I_\sigma/I_\pi$ as a function of azimuthal angle for aselection of energies, allows a determination of the higher order multipolemoments of rank 1 (dipole) to 6 (hexacontatetrapole). The Ho 4f multipolemoments have been estimated, indicating a dominant hexadecapole (rank 4) orderwith an almost negligible influence from either the dipole or the octupolemagnetic terms. The analysis incorporates both the intra-atomic magnetic andquadrupolar interactions between the 3d core and 4f valence shells as well asthe interference of contributions to the scattering that behave differentlyunder time reversal. Comparison of SRXD, neutron diffraction and non resonantx-ray diffraction shows that the magnetic and quadrupolar order parameter aredistinct. The $(0 0 1/2)$ component of the magnetic order exhibits a Brillouintype increase below the orbital ordering temperature T$_Q$, while thequadrupolar order increases more sharply. We conclude the quadrupolarinteraction is strong, but quadrupolar order only occurs when the magneticorder gives rise to a quasi doublet ground state, which results in a lock-in ofthe orbitals at T$_Q$.

High-order Ho multipoles in HoB2C2 observed with soft resonant x-ray diffraction