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arXiv · 2510.13746

Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry

Surya Majumder·Daniel Widdowson·Yury Elkin·Olga Anosova·Andrew I Cooper·Graeme M Day·Vitaliy Kurlin
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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.

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Surya Majumder, Daniel Widdowson, Yury Elkin, Olga Anosova, Andrew I Cooper, Graeme M Day, Vitaliy Kurlin. 2025-10-15. Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry. https://arxiv.org/abs/2510.13746

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