Orbital ordering and resonant x-ray scattering in layered manganites

In layered manganites with orbital and charge orderings, the degeneracy of the Mn 4p orbitals as well as the 3d ones is lifted by the effects of the 4p bands and the local Coulomb interactions. We formulate the atomic scattering factor for the resonant x-ray scattering in the memory function method by taking into account these effects on an equal footing. It is shown that the polarization dependences of the scattering intensities at the orbital and charge superlattice reflections observed in LaSr₂Mn₂O₇ are caused by the local and itinerant characters of 4p electrons, respectively. We examine the type of the orbital ordered state.