Minimum-Strain Symmetrization of Bravais Lattices

TL;DR

This paper introduces a scale- and rotation-invariant method to measure how much a lattice deviates from ideal Bravais symmetry by calculating the minimum strain needed for symmetrization, enabling a detailed lattice classification.

Contribution

It proposes a novel strain-based metric for Bravais lattice classification that overcomes scale-dependence issues of traditional methods, providing a more intuitive and physically meaningful analysis.

Findings

The method quantifies lattice symmetry deviation using minimum strain.

It creates a 14-dimensional map of Bravais lattice space.

Software implementation is publicly available.

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, fulfilment 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 fulfil 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.