X-Ray Resonant Scattering as a Direct Probe of Orbital Ordering in Transition-Metal Oxides

X-ray resonant scattering at the $K$-edge of transition-metal oxides is shown to measure the orbital order parameter, supposed to accompany magnetic ordering in some cases. Virtual transitions to the $3d$ orbitals are quadrupolar in general. In cases with no inversion symmetry, such as ${\mathrm{V}}_2{\mathrm{O}}_3$, treated in detail here, a dipole component enhances the resonance. Hence, we argue that the detailed structure of orbital order in ${\mathrm{V}}_2{\mathrm{O}}_3$ is experimentally accessible.