The importance of definitions in crystallography

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 3D 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, comparison of millions of known structures in materials databases requires continuous distance metrics.

Continuous Invariant-Based Maps of the Cambridge Structural Database

“…Indeed, different rigid structures (even different rigid conformations of the same molecule such as a protein) can have different chemical properties and hence are important to distinguish. Definition 2 was also proposed in the paper 12 discussing past ambiguities and was motivated by Carolyn Brock's call 14 for "a real-space definition mentioning periodicity". We propose to separate the concepts of a periodic crystal (an object fixed in space and represented by a CIF) and a crystal structure (a class of all rigidly equivalent crystals).…”

“…In 2024, massive data produced by cheap artificial tools 11 should be validated by rigorous methods, 12 such as biometric passports for humans and DNA-style codes for crystals.…”

“…Definition 2 was also proposed in the paper 12 discussing past ambiguities and was motivated by Carolyn Brock’s call 14 for “a real-space definition mentioning periodicity”. We propose to separate the concepts of a periodic crystal (an object fixed in space and represented by a CIF) and a crystal structure (a class of all rigidly equivalent crystals).…”

“…All-vs-all comparisons will be discussed in another work but anyone right now can check that the four CIFs with IDs 4135ff7bc7, 6370e8cf.86, c6afea2d8e, and e1ea534c2c are identical texts. The GNoME contains 43 such triples, 1089 pairs, and many more thousands of numerical duplicates 12 that differ only by chemistry, not by geometry.…”