The importance of definitions in crystallography

TL;DR

This paper emphasizes the critical role of precise definitions in crystallography, highlighting the limitations of traditional cell-based representations and advocating for continuous metrics for comparing crystal structures.

Contribution

It clarifies the importance of rigorous definitions and introduces the need for continuous distance metrics in crystal comparison methods.

Findings

Conventional cell-based representations are discontinuous under atomic displacements.

Rigid motion is the fundamental equivalence in crystal comparison.

Continuous metrics are essential for large-scale crystal database analysis.

Abstract

This paper was motivated by the articles "Same or different - that is the question" in CrystEngComm (July 2020) and "Change to the definition of a crystal" in the IUCr newsletter (June 2021). Experimental approaches to crystal comparisons require rigorously defined classifications in crystallography and beyond. Since crystal structures are determined in a rigid form, their strongest equivalence in practice is rigid motion, which is a composition of translations and rotations in 3-dimensional space. Conventional representations based on reduced cells and standardizations theoretically distinguish all periodic crystals. However, all cell-based representations are inherently discontinuous under almost any atomic displacement that can arbitrarily scale up a reduced cell. Hence comparing millions of known structures in materials databases needs continuous distance metrics.