Universal magneto-orbital ordering in the divalent $A$-site quadruple perovskite manganites $A$Mn$_7$O$_{12}$ ($A$ = Ca, Sr, Cd, and Pb)

Through analysis of variable temperature neutron powder diffraction data, we present solutions for the magnetic structures of SrMn$_7$O$_{12}$, CdMn$_7$O$_{12}$, and PbMn$_7$O$_{12}$ in all long-range ordered phases. The three compounds were found to have magnetic structures analogous to that reported for CaMn$_7$O$_{12}$. They all feature a higher temperature lock-in phase with emph{commensurate} magneto-orbital coupling, and a delocked, multi-textbf{k} magnetic ground state where emph{incommensurate} magneto-orbital coupling gives rise to a constant-moment magnetic helix with modulated spin helicity. CdMn$_7$O$_{12}$ represents a special case in which the orbital modulation is commensurate with the crystal lattice and involves stacking of fully and partially polarized orbital states. Our results provide a robust confirmation of the phenomenological model for magneto-orbital coupling previously presented for CaMn$_7$O$_{12}$. Furthermore, we show that the model is universal to the $A^{2+}$ quadruple perovskite manganites synthesised to date, and that it is tunable by selection of the $A$-site ionic radius.