Continuous Maps of 2D Lattices

17^th August 2023, 14:00
Matthew Bright
MIF

Abstract

Given the massive growth in the number of periodic structures deposited in research databases, and the ability of Crystal Structure Prediction to generate hundreds of thousands of structures, many of which will be near-identical, it is becoming important to have a classification approach for structures that is finer-grained than the finite number of discrete symmetry groups into which they fall, and which allows for fast numerical comparison. We have approached the problem from the ‘bottom up’, by making a rigorous analysis of the very simplest non-trivial periodic structure - the two dimensional lattice - in a way that we hope to extend to more complex structures. The result is the Root Invariant. from which we derive the scale-agnostic Projected Invariant, both with orientation aware versions that distinguish lattices related by a reflection) - an easily computable, complete isometry invariant which can be modified to differentiate lattices related by a reflection and from which a lattice can be uniquely reconstructed up to isometry (or rigid motion, or similarity). In this talk we will give a brief overview of the theory behind the invariant and use it to display a map of 2D lattices derived from a large database of crystal structures, showing that modulo physical constraints real world geometric structures continuously occupy the space of all possible lattices.