Measuring Lattices

TL;DR

This paper reviews methods for measuring differences between crystallographic lattices using reduced unit cells in specialized mathematical spaces, addressing the complexity of calculating meaningful lattice distances.

Contribution

It presents a simplified, best-practice approach for lattice distance measurement using Delaunay-reduced unit cells in S6 and C3 spaces, improving upon more complex methods.

Findings

Uses Delaunay-reduced unit cells for accurate lattice comparison

Provides a simplified process for lattice distance calculation

Addresses computational complexity in crystallography measurements

Abstract

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. We present a review of the current best practice using Delaunay-reduced unit cells in the six-dimensional real space of Selling scalar cells S6 and the equivalent three-dimensional complex space C3. 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 G6. 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 to boundary-interrupted paths, looking for a path of minimal length.