X-ray resonant scattering studies of orbital and charge ordering in ${\mathrm{Pr}}_{1-x}{\mathrm{Ca}}_x{\mathrm{MnO}}_3$

We present the results of a systematic x-ray scattering study of the charge and orbital ordering in the manganite series ${\mathrm{Pr}}_{1-x}{\mathrm{Ca}}_x{\mathrm{MnO}}_3$ with $x=0.25,$ 0.4, and 0.5. The temperature dependence of the scattering at the charge and orbital wavevectors, and of the lattice constants, was characterized throughout the ordered phase of each sample. It was found that the charge- and orbital-order wavevectors are commensurate with the lattice, in striking contrast to the results of electron and neutron diffraction studies of samples with $x=\mathrm{0.5.}\mathrm{}$ High-momentum-transfer resolution studies of the $x=0.4\mathrm{}$ and 0.5 samples further revealed that while long-range charge order is present, long-range orbital order is never established. Above the charge and orbital ordering temperature $T_0,$ the charge-order fluctuations are more highly correlated than the orbital fluctuations. This suggests that charge order drives orbital order in these samples. In addition, a longitudinal modulation of the lattice with the same periodicity as the charge and orbital ordering was discovered in the $x=0.4\mathrm{}$ and 0.5 samples. For $x=0.25,$ only long-range orbital order was observed with no indication of charge ordering, nor of an additional lattice modulation. We also report the results of a preliminary investigation of the loss of charge and orbital ordering in the $x=0.4\mathrm{}$ sample by application of a magnetic field. Finally, the polarization and azimuthal dependence of the charge and orbital ordering in these compounds are characterized both in the resonant and nonresonant limits, and compared with the predictions of current theories. The results are qualitatively consistent with both cluster and local density approximation $+U$ calculations of the electronic structure.