An algorithm for the arithmetic classification of multilattices

TL;DR

This paper presents an algorithm for classifying multilattices in any dimension, enabling explicit point location, equivalence checking, and applications to complex crystal structures and high-dimensional symmetry analysis.

Contribution

It introduces a novel algorithm based on integer matrix theory and Smith normal form for multilattice classification and symmetry determination.

Findings

Algorithm explicitly locates points in multilattices.

It determines arithmetic equivalence of multilattices.

Software implementation for classifying crystalline and high-dimensional structures.

Abstract

A procedure for the construction and the classification of multilattices in arbitrary dimension is proposed. The algorithm allows to determine explicitly the location of the points of a multilattice given its space group, and to determine whether two multilattices are arithmetically equivalent. The algorithm is based on ideas from integer matrix theory, in particular the reduction to the Smith normal form. Among the applications of this procedure is a software package that allows the classification of complex crystalline structures and the determination of their space groups. Also, it can be used to determine the symmetry of regular systems of points in high dimension, with applications to the study of quasicrystals and sets of points with noncrystallographic symmetry in low dimension, such as viral capsid structures.