Analysis of multiple solutions in powder pattern indexing: the common reciprocal metric tensor approach

Powder pattern indexing routines frequently yield multiple solutions, i.e. different reciprocal lattices and unit cells. Here, a method is suggested that reveals whether or not there are numerical and geometric relationships between the solutions. It is based on the detection of a reciprocal vector triplet that is common to two or more proposed reciprocal lattices. Hence, the method can be termed a common reciprocal metric tensor approach. If no such common tensor exists, the different reciprocal lattices are unrelated, but if one exists the lattices are either in a sublattice/superlattice or in a coincidence-site lattice relationship, depending on the character of the respective orientation matrix. Furthermore, the approach can also be used to generate, from a given indexing solution, further valid indexing solutions that could also be produced by indexing routines.