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.

翻译：任何周期性晶体的基本模型是所有原子中心构成的周期点集。由于晶体结构以刚性形式确定，其最强等价关系为刚体运动（平移与旋转的组合）或等距变换（亦包含反射）。近期基于刚体运动的周期点集分类采用了完全不变集（isoset），其规模本质上取决于桥长度——定义为连接给定集合中任意两点所需的最小“跨越”距离。本文提出一种实用算法，用于计算由周期性平移单胞内点阵基元所定义的任意周期点集的桥长度。该算法已在大型晶体数据集上进行测试，并为实现所有周期晶体的高效连续分类所必需。桥长度的精确计算是实现基于新不变量值的材料逆向设计的关键步骤。