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The orbital contribution to the magnetic moment of the transition metal ionin the isostructural weak ferromagnets ACO$_3$ (A=Mn,Co,Ni) and FeBO$_3$ wasinvestigated by a combination of first-principles calculations, non-resonantx-ray magnetic scattering and x-ray magnetic circular dichroism. A non-trivialevolution of the orbital moment as a function of the $3d$ orbitals filling isrevealed, with an anomalously large value found in the Co member of the family.Here, the coupling between magnetic and lattice degrees of freedom produced bythe spin-orbit interaction results in a large single-ion anisotropy and apeculiar magnetic-moment-induced electron cloud distortion, evidenced by theappearance of a subtle scattering amplitude at space group-forbiddenreflections and significant magnetostrictive effects. Our results, whichcomplement a previous investigation on the sign of the Dzyaloshinskii$-$Moriyainteraction across the series, highlight the importance of spin-orbit couplingin the physics of weak ferromagnets and prove the ability of modernfirst-principles calculations to predict the properties of materials where theDzyaloshinskii$-$Moriya interaction is a fundamental ingredient of the magneticHamiltonian.

Investigation into the role of the orbital moment in a series of isostructural weak ferromagnets