Ambiguities in powder pattern indexing: A ternary lattice metric singularity

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d-spacings. The existence of such singularities, therefore, has a practical impact on the indexing of powder patterns. Lattice metric singularities often involve lattices that are in a derivative relationship one to another. A variety of types of singularities are possible depending on the number of different lattices involved (i.e., binary, ternary, quaternary), on the nature of the derivative lattice relationship (i.e., subcell/supercell, composite), on the Bravais type of each of the lattices, and on the the volume ratio(s) of primitive cells defining the lattices. In the laboratory, an encounter with a singularity can lead one into a trap; viz., the investigator using an indexing program, or by other means, may determine only one of the lattices with a high figure of merit. When this happens, it is critical to recognize that there exists more than one indexing solution. In a previous work, a binary singularity was described involving a monoclinic and a rhombohedral lattice. In the present work, we describe a second type of singularity—a ternary singularity—in which the two of the three lattices are in a derivative composite relationship.