  The orbital contribution to the magnetic moment of the transition metal ion
in the isostructural weak ferromagnets ACO$_3$ (A=Mn,Co,Ni) and FeBO$_3$ was
investigated by a combination of first-principles calculations, non-resonant
x-ray magnetic scattering and x-ray magnetic circular dichroism. A non-trivial
evolution of the orbital moment as a function of the $3d$ orbitals filling is
revealed, with an anomalously large value found in the Co member of the family.
Here, the coupling between magnetic and lattice degrees of freedom produced by
the spin-orbit interaction results in a large single-ion anisotropy and a
peculiar magnetic-moment-induced electron cloud distortion, evidenced by the
appearance of a subtle scattering amplitude at space group-forbidden
reflections and significant magnetostrictive effects. Our results, which
complement a previous investigation on the sign of the Dzyaloshinskii$-$Moriya
interaction across the series, highlight the importance of spin-orbit coupling
in the physics of weak ferromagnets and prove the ability of modern
first-principles calculations to predict the properties of materials where the
Dzyaloshinskii$-$Moriya interaction is a fundamental ingredient of the magnetic
Hamiltonian.
