Direct Observation of Charge and Orbital Ordering in ${\mathrm{La}}_{0.5}{\mathrm{Sr}}_{1.5}{\mathrm{MnO}}_4$

Charge ordering and orbital ordering in ${\mathrm{La}}_{0.5}{\mathrm{Sr}}_{1.5}{\mathrm{MnO}}_4$ have been investigated by using synchrotron x-ray diffraction. The alternating pattern of ${\mathrm{Mn}}^{3+}/{\mathrm{Mn}}^{4+}$ in real space is observed directly by making use of their anomalous dispersion. The first clear evidence of the orbital ordering of $e_g$ electrons on ${\mathrm{Mn}}^{3+}$ is also presented based on the measurements of ATS (anisotropy of the tensor of susceptibility) reflection near the Mn $K$-absorption edge. The present observations indicate that the charge ordering and orbital ordering occur simultaneously at a transition temperature higher than that of the spin ordering.