Ambiguities in Powder Indexing: Conjunction of a Ternary and Binary Lattice Metric Singularity in the Cubic System

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d 1. a binary singularity involving a monoclinic and a rhombohedral lattice in a subcell-supercell relationship and 2. a second type of singularity—a ternary singularity–in which two of the three lattices are in a derivative composite relationship. In this work, we describe a ternary lattice metric singularity involving a cubic P, a tetragonal P, and an orthorhombic C lattice. Furthermore, there is a binary singularity, involving a hexagonal P and orthorhombic P lattice, which is characterized by a set of unique d-spacings very close to that of the ternary singularity. The existence of such singularities is more common than once thought and requires a paradigm shift in experimental practice. In addition singularities provide opportunities in material design as they point to highly specialized lattices that may be associated with unusual physical properties.