Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry
Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry
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.
Majumder, Surya
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Willowson, Daniel
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Elkin, Yury
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Anosova, Olga
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Cooper, Andrew I.
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Day, Graeme M.
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Kurlin, Vitaliy A.
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Majumder, Surya
ae68d4ce-d628-4084-b883-9ed2e7337f9d
Willowson, Daniel
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Elkin, Yury
372c67c1-5f7d-4ef2-98ce-b9ada9b57cad
Anosova, Olga
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Cooper, Andrew I.
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Day, Graeme M.
e3be79ba-ad12-4461-b735-74d5c4355636
Kurlin, Vitaliy A.
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Majumder, Surya, Willowson, Daniel, Elkin, Yury, Anosova, Olga, Cooper, Andrew I., Day, Graeme M. and Kurlin, Vitaliy A.
(2026)
Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry.
IUCrJ.
(In Press)
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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Accepted/In Press date: 16 March 2026
Identifiers
Local EPrints ID: 510949
URI: http://eprints.soton.ac.uk/id/eprint/510949
ISSN: 2052-2525
PURE UUID: 06331b50-3c9b-4299-9679-a13b824cc757
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Date deposited: 27 Apr 2026 16:50
Last modified: 28 Apr 2026 01:52
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Contributors
Author:
Surya Majumder
Author:
Daniel Willowson
Author:
Yury Elkin
Author:
Olga Anosova
Author:
Andrew I. Cooper
Author:
Vitaliy A. Kurlin
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