Measuring lattices

Unit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three-dimensional crystallographic lattices. This is a surprisingly complex and computationally demanding problem. A review is presented of the current best practice using Delaunay-reduced unit cells in the six-dimensional real space of Selling scalar cells S-6 and the equivalent three-dimensional complex space C-3. The process is a simplified version of the process needed when working with the more complex six-dimensional real space of Niggli-reduced unit cells G(6). Obtaining a distance begins with identification of the fundamental region in the space, continues with conversion to primitive cells and reduction, analysis of distances to the boundaries of the fundamental unit, and is completed by a comparison of direct paths with boundary-interrupted paths, looking for a path of minimal length.