Understanding extended homometry based on complementary crystallographic orbit sets.

Extended homometry is a phenomenon in which distinct structures have the same X-ray diffraction (XRD) intensities, which may lead to incorrect results of structural analysis based on XRD methods. It is proposed and proved herein that half of a crystallographic orbit has the same powder X-ray diffraction intensity as its complementary set; three more theorems are deduced. These results are conducive to understanding the formation of extended homometric structures. Also analyzed are some reported or potential homometric or weakly homometric structures in the Inorganic Crystal Structure Database to confirm the theorems. This work presents a quick approach to analyze and construct extended homometric structures based on crystallographic orbits.