Minimum-strain symmetrization of Bravais lattices

Abstract

Bravais lattices are the most fundamental building blocks of crystallography. They are classified into groups according to their translational, rotational, and inversion symmetries. In computational analysis of Bravais lattices, fulfillment of symmetry conditions is usually determined by analysis of the metric tensor, using either a numerical tolerance to produce a binary (i.e., yes or no) classification or a distance function which quantifies the deviation from an ideal lattice type. The metric tensor, though, is not scale invariant, which complicates the choice of threshold and the interpretation of the distance function. Here, we quantify the distance of a lattice from a target Bravais class using strain. For an arbitrary lattice, we find the minimum-strain transformation needed to fulfill the symmetry conditions of a desired Bravais lattice type; the norm of the strain tensor is used to quantify the degree of symmetry breaking. The resulting distance is invariant to scale and rotation, and is a physically intuitive quantity. By symmetrizing to all Bravais classes, each lattice can be placed in a 14-dimensional space, which we use to create a map of the space of Bravais lattices and the transformation paths between them. A software implementation is available online under a permissive license.