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, 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.

Two-dimensional ferroelectrics from high throughput computational screening

“…For materials that do not have an out-of-plane polar axis one needs a non-polar reference state that can be adiabatically connected with the polar states in order to calculate the spontaneous polarization. We have used the module evgraf [52] to generate the centrosymmetric structure of highest similarity to the polar structure [53] of all materials. This structure is then relaxed under the constraint of inversion symmetry and the result is taken FIG.…”

Generating random unit cells

“…Methods for Bravais lattice determination of experimental unit cells continue to be described, including by Larsen et al (2020). Andrews & Bernstein (2014) listed some of the metric methods that have been described (and gave a formal introduction of space G 6 , already discussed in 1988).…”

“…Introduction Niggli (1928) and Delaunay (1932) first described the solutions to the problem of recognizing the appropriate Bravais lattice type of a particular unit cell. Practical methods of computationally determining that relationship for a particular cell continue to be of recurring interest [see, for example, Larsen et al (2020)]. In some cases, perturbations past boundaries into regions of unreduced cells may be useful, but in many cases reduction of the perturbed cells is necessary for further analysis.…”