The importance of definitions in crystallography

This paper proposes a rigorous definition of a periodic structure as an equivalence class of periodic sets under rigid motion, which is a composition of translations and rotations.

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.