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 LaMnO₃ crystals are calculated using a non-spherical atomic scattering model of the Mn³⁺ ion. Several low-order electron structure factors showed pronounced change with orbital ordering, in which the e¹_g electron orders in the 3d(3z² - r²) orbital leaving the 3d(x² - y²) 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 e¹_g electron of the Mn³⁺ ion in an LaMnO₃ crystal. A comparison of methods for structure-factor measurement is given, including Bragg X-ray and γ-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.