Computing the bridge length: the key ingredient in a continuous isometry classification of periodic point sets

TL;DR

This paper introduces a practical algorithm to compute the bridge length of periodic point sets, a crucial invariant for classifying crystal structures under isometries, aiding in material design.

Contribution

The paper presents a new algorithm for efficiently calculating the bridge length of periodic point sets, enabling improved classification and inverse design of crystal structures.

Findings

Algorithm successfully tested on large crystal dataset

Enables continuous classification of periodic crystals

Facilitates inverse material design

Abstract

The fundamental model of any periodic crystal is a periodic set of points at all atomic centres. Since crystal structures are determined in a rigid form, their strongest equivalence is rigid motion (composition of translations and rotations) or isometry (also including reflections). The recent classification of periodic point sets under rigid motion used a complete invariant isoset whose size essentially depends on the bridge length, defined as the minimum `jump' that suffices to connect any points in the given set. We propose a practical algorithm to compute the bridge length of any periodic point set given by a motif of points in a periodically translated unit cell. The algorithm has been tested on a large crystal dataset and is required for an efficient continuous classification of all periodic crystals. The exact computation of the bridge length is a key step to realising the inverse design of materials from new invariant values.