arXiv subjects

Andrew I Cooper

Publications and source records attributed to Andrew I Cooper.

5 recordsLinked to original sources

arXiv2025

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.

math.MG↗

arXiv2022

Geographic-style maps for 2-dimensional lattices

This paper develops geographic-style maps containing 2D lattices in all known crystals parameterised by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of 2D lattices is a square with identified edges or a sphere without one point. The new continuous maps show all Bravais classes as low-dimensional subspaces, visualise hundreds of thousands of real crystal lattices from the Cambridge Structural Database, and motivate the development of continuous and invariant-based crystallography.

cs.CG↗

arXiv2021

A complete and continuous map of the Lattice Isometry Space for all 3-dimensional lattices

This paper extends the recently obtained complete and continuous map of the Lattice Isometry Space (LISP) to the practical case of dimension 3. A periodic 3-dimensional lattice is an infinite set of all integer linear combinations of basis vectors in Euclidean 3-space. Motivated by crystal structures determined in a rigid form, we study lattices up to rigid motion or isometry, which is a composition of translations, rotations and reflections. The resulting space LISP consists of infinitely many isometry classes of lattices. In dimension 3, we parameterise this continuous space LISP by six coordinates and introduce new metrics satisfying the metric axioms and continuity under all perturbations. This parameterisation helps to visualise hundreds of thousands of real crystal lattices from the Cambridge Structural Database for the first time.

cs.CG↗

arXiv2021

Fast predictions of lattice energies by continuous isometry invariants of crystal structures

Crystal Structure Prediction (CSP) aims to discover solid crystalline materials by optimizing periodic arrangements of atoms, ions or molecules. CSP takes weeks of supercomputer time because of slow energy minimizations for millions of simulated crystals. The lattice energy is a key physical property, which determines thermodynamic stability of a crystal but has no simple analytic expression. Past machine learning approaches to predict the lattice energy used slow crystal descriptors depending on manually chosen parameters. The new area of Periodic Geometry offers much faster isometry invariants that are also continuous under perturbations of atoms. Our experiments on simulated crystals confirm that a small distance between the new invariants guarantees a small difference of energies. We compare several kernel methods for invariant-based predictions of energy and achieve the mean absolute error of less than 5kJ/mole or 0.05eV/atom on a dataset of 5679 crystals.

cond-mat.mtrl-sci↗

arXiv2021

The asymptotic behaviour and a near linear time algorithm for isometry invariants of periodic sets

The fundamental model of a periodic structure is a periodic point set up to rigid motion or isometry. Our recent paper in SoCG 2021 defined isometry invariants (density functions), which are complete in general position and continuous under perturbations. This work introduces much faster isometry invariants (average minimum distances), which are also continuous and distinguish some sets that have identical density functions. We explicitly describe the asymptotic behaviour of the new invariants for a wide class of sets including non-periodic. The proposed near linear time algorithm processed a dataset of hundreds of thousands of real structures in a few hours on a modest desktop.

cond-mat.mtrl-sci↗