To elucidate charge and orbital order below the Verwey transition temperature
TV∼125 K, a thin layer of magnetite partially detwined by growth on the
stepped MgO(001) substrate has been studied by means of soft x-ray diffraction
at the Fe L2,3 resonance. The azimuth angle, incident photon polarization, and
energy dependence of the (0012)c and (001)c reflection intensities have been
measured, and analyzed using a configuration-interaction FeO6 cluster model.
The azimuth dependence of the (0012)c reflection intensities directly
represents the space-group symmetry of the orbital order in the initial state
rather than indirectly through the intermediate-state level shifts caused by
the order-induced lattice distortions. From the analysis of the (0012)c
reflection intensities, the orbital order in the t2g orbitals of B sites below
TV is proved to have a large monoclinic deformation with the value of
Re[Fxy]/Re[Fyz]∼2. This finding contradicts the majority of theories on the
Verwey transition so far proposed. We show that the experimentally observed
resonance spectra cannot be explained by orbital and charge orders obtained
with recent LDA+U and GGA+U band structure calculations but by a complex-
number orbital order with excellent agreement
