Orbital ordering in LaMnO3: estimates of structure factors and comparison of measurement methods.

This paper shows that the phenomenon of orbital ordering should be detectable by energy-filtered quantitative convergent-beam electron diffraction (QCBED). The structure factors of LaMnO(3) crystals are calculated using a non-spherical atomic scattering model of the Mn(3+) ion. Several low-order electron structure factors showed pronounced change with orbital ordering, in which the e(g)(1) electron orders in the 3d(3z(2) - r(2)) orbital leaving the 3d(x(2) - y(2)) unoccupied. In contrast, the X-ray structure factors showed very small change. Orbital order is important in transition-metal oxides, including colossal magnetoresistive manganite oxides. The calculations show that by using QCBED it is possible to measure the subtle changes in electron structure factors due to orbital ordering of the e(g)(1) electron of the Mn(3+) ion in an LaMnO(3) crystal. A comparison of methods for structure-factor measurement is given, including Bragg X-ray and gamma-ray diffraction, X-ray Pendellösung and critical-voltage methods. New measurements by QCBED of structure factors in rutile are compared with the Bragg X-ray values. These show that QCEBD can provide an accurate extinction-free measurement of low-order structure factors, which is extremely difficult or perhaps impossible when using other methods applied to real crystals.