Theory of Anomalous X-ray Scattering in Orbital Ordered Manganites

We study theoretically the anomalous X-ray scattering as a new probe to observe the orbital orderings and excitations in perovskite manganites. The scattering matrix is given by the virtual electronic excitations from Mn $1s$ level to unoccupied Mn $4p$ level. We find that orbital dependence of the Coulomb interaction between Mn $3d$ and Mn $4p$ electrons is essential to bring about the anisotropy of the scattering factor near the K edge. The calculated results in $MnO_6$ clusters explain the forbidden reflections observed in $La_{0.5}Sr_{1.5}MnO_4$ and $LaMnO_3$. A possibility of the observation of the orbital waves by the X-ray scattering is discussed.