Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry
Abstract
Ideal symmetry is known to break down under almost any noise. One measure of asymmetry in a periodic crystal is the relative multiplicity Z' of geometrically non-equivalent units. However, Z' discontinuously changes under almost any displacement of atoms, which can arbitrarily scale up a primitive cell. This discontinuity was recently resolved by a hierarchy of invariant descriptors that continuously change under all small perturbations. We introduce a Continuous Invariant-based Asymmetry (CIA) to quantify (in physically meaningful Angstroms) the deviation of a periodic crystal from a higher symmetry form. Our experiments on several Crystal Structure Prediction datasets show that about a half of simulated crystals have high values of CIA, while all experimental structures in these datasets have CIA=0. On another hand, many crystals with high values Z' in the Cambridge Structural Database (CSD) turned out to be close to more symmetric forms with Z'<=1 due to low values of CIAs.
Keywords
Cite
@article{arxiv.2510.13746,
title = {Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry},
author = {Surya Majumder and Daniel Widdowson and Yury Elkin and Olga Anosova and Andrew I Cooper and Graeme M Day and Vitaliy Kurlin},
journal= {arXiv preprint arXiv:2510.13746},
year = {2026}
}
Comments
The 3rd version (34 pages with appendices) included the latest computations on the Cambridge Structural Database (CSD) and pictures of the most relevant crystals. The latest version is maintained at http://kurlin.org/projects/computational-materials-science/continuous-asymmetry.pdf