Quantitative phase analysis of natural products using whole-powder-pattern decomposition

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. For example, when ʹexperimental data from zeta-LiBO2 were indexed, two solutions (a rhombohedral and a monoclinic lattice) with approximately the same figure of merit were found. These two lattices yield the same set of unique d-spacings even though they are characterized by different reduced cells with cell volumes in the ratio 2 to 1. From the indexing point of view, both answers are correct. A singularity of this type is common and not a mathematical rarity. In fact, any rhombohedral cell of this kind has a derivative monoclinic subcell, each of which gives the same set of unique calculated d-spacings. In actual cases like this, one can run into a trap. Due to experimental error and input parameters, an indexing program may determine only one of the cells with a high figure of merit. When this happens, it is critical to recognize that another solution exists, especially if one has determined the lower symmetry lattice.