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The quadruple perovskite CaMn$_7$O$_{12}$ is a topical multiferroic, in whichthe hierarchy of electronic correlations driving structural distortions,modulated magnetism, and orbital order is not well known and may vary withtemperature. x-ray resonant elastic scattering (XRES) provides amomentum-resolved tool to study these phenomena, even in very small singlecrystals, with valuable information encoded in its polarization- andenergy-dependence. We present an application of this technique toCaMn$_7$O$_{12}$. By polarization analysis, it is possible to distinguishsuperstructure reflections associated with magnetic order and orbital order.Given the high momentum resolution, we resolve a previously unknown splittingof an orbital order superstructure peak, associated with a distinct\textit{locked-in} phase at low temperatures. A second set of orbital ordersuperstructure peaks can then be interpreted as a second-harmonic orbitalsignal. Surprisingly, the intensities of the first- and second-harmonic orbitalsignal show disparate temperature and polarization dependence. This orbitalre-ordering may be driven by an exchange mechanism, that becomes dominant overthe Jahn-Teller instability at low temperature.

Magnetic and orbital correlations in multiferroic CaMn$_7$O$_{12}$ probed by x-ray resonant elastic scattering