Event Details

Speaker: Vitaliy Kurlin (Liverpool) Title. The Crystal Isometry Principle

Abstract. For hundreds of years, crystals were studied almost exclusively by discrete tools such as symmetry groups. The classification of 230 space groups into 230 types was a great achievement at the end of the 19th century. In 2024, the Cambridge Structural Database (CSD) and other major datasets contain all together more than 2 million experimental structures, while simulated crystals emerge even in greater numbers. This scale requires a much finer (stronger) classification of all known periodic crystals into more than 230 classes. There is no practical sense to distinguish crystals that can be ideally matched by rigid motion. But we need to distinguish crystals that cannot be ideally matched (are not rigidly equivalent). Indeed, if we call ‘the same’ any crystals whose all atoms can be matched up to a small perturbation, sufficiently many perturbations can geometrically deform any crystal to any other (of the same composition if we keep atomic types).

This continuum fallacy (a version of the sorites paradox) is resolved by the following new definition: A crystal structure is an equivalence class of all periodic crystals that can be rigidly matched by rigid motion in 3-dimensional space. Such a rigid class contains infinitely many crystals represented by infinitely many different CIFs, all encoding ‘the same’ periodic arrangement of atoms. Any slight perturbation of a single atom produces a crystal in a different rigid class, so there are infinitely many rigid classes, some of which can be very close due to noise, while others are very distant from each other. A classification under isometry is only slightly weaker than under rigid motion because mirror images can be distinguished by a sign of orientation.

We developed a generically complete invariant descriptor that is preserved under isometry and continuously changes under any perturbation. This Pointwise Distance Distribution PDD(S;k) is a matrix whose each row for any atom in an asymmetric unit of a periodic crystal S contains distances from the fixed atom to its k-nearest neighbors in increasing order. Within 1 hour on a modest desktop computer, PDD(100;k) distinguished all real periodic crystals in the CSD and justified the Crystal Isometry Principle saying that all these periodic crystals live at unique locations in a common continuous space.