  Through a combination of neutron diffraction and Landau theory we describe
the spin ordering in the ground state of the quadruple perovskite manganite
CaMn7O12 - a magnetic multiferroic supporting an incommensurate orbital density
wave that onsets above the magnetic ordering temperature, TN1 = 90 K. The
multi-k magnetic structure in the ground state was found to be a
nearly-constant-moment helix with modulated spin helicity, which oscillates in
phase with the orbital occupancies on the Mn3+ sites via trilinear
magneto-orbital coupling. Our phenomenological model also shows that, above TN2
= 48 K, the primary magnetic order parameter is locked into the orbital wave by
an admixture of helical and collinear spin density wave structures.
Furthermore, our model naturally explains the lack of a sharp dielectric
anomaly at TN1 and the unusual temperature dependence of the electrical
polarisation.
