Continuous chiral distances for two‐dimensional lattices

Abstract

Chirality was traditionally considered a binary property of periodic lattices and crystals. However, the classes of two‐dimensional lattices modulo rigid motion form a continuous space, which was recently parametrized by three geographic‐style coordinates. The four non‐oblique Bravais classes of two‐dimensional lattices form low‐dimensional singular subspaces in the full continuous space. Now, the deviations of a lattice from its higher symmetry neighbors can be continuously quantified by real‐valued distances satisfying metric axioms. This article analyzes these and newer G‐chiral distances for millions of two‐dimensional lattices that are extracted from thousands of available two‐dimensional materials and real crystal structures in the Cambridge Structural Database.

A continuous map of 2.6+ million 2D lattices from the Cambridge Structural Database

“…Similar invariant parameterizations and continuous maps are being developed for 3D lattices [3][4]. The space of arbitrary periodic point sets is also parameterized by complete isometry invariants [5].…”