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Through analysis of variable temperature neutron powder diffraction data, wepresent 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. Thethree compounds were found to have magnetic structures analogous to thatreported for CaMn$_7$O$_{12}$. They all feature a higher temperature lock-inphase 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 withmodulated spin helicity. CdMn$_7$O$_{12}$ represents a special case in whichthe orbital modulation is commensurate with the crystal lattice and involvesstacking of fully and partially polarized orbital states. Our results provide arobust confirmation of the phenomenological model for magneto-orbital couplingpreviously presented for CaMn$_7$O$_{12}$. Furthermore, we show that the modelis universal to the $A^{2+}$ quadruple perovskite manganites synthesised todate, and that it is tunable by selection of the $A$-site ionic radius.

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