Three Applications of 2D Lattice Invariants.

31^st March 2023, 14:00
Matthew Bright
MIF

Abstract

The first application will be a deeper investigation of our first map - that of 2.7 million lattices derived from 3D structures in the CSD. We will discuss the key structural features of the complete map, before demonstrating how invariants can be used to investigate the impact of chemical properties such as molecular weight on lattice symmetry.
The second will be an application to the geometry of 2D Materials. Since the isolation of graphene, the number of both actual and theoretical 2D materials in the literature has been rapidly growing. Materials with a strongly asymmetric lattice are of interest to chemists, since they are likely to have anisotropic properties - conductivity, for example, may be higher in one direction and lower in another.
We will discuss the available data on these materials, and demonstrate the application of chirality distances in two cases: 2DMatPedia - one of the largest publicly available databases of materials [1] - and the smaller 2D Materials Database [2]. The use of chirality distances allows us to seek out potentially highly asymmetric structures, and to investigate changes in the symmetry of potential 2D materials as layers are extracted from the bulk crystal.
Finally, we consider the more theoretical topic of ‘random lattices’. Randomly selecting from the space of all possible lattices (at a given fixed scale) is equivalent to defining a ‘uniform’ probability measure on the space of all lattices modulo isometry and basis change.
The work of Prof Jens Marklof et al. [3] on the explicit definition of this measure can be used to implement random lattice generation. An interesting question is whether the distribution of naturally occurring 2D lattices in chemistry is similar to this ‘Haar Random Lattice’ distribution - that is, does nature explore all of ‘lattice space’? We might expect this not to be the case, but our early investigations have shown surprising similarities between the two distributions.
[1] J.Zhou et al, ‘2DMatPedia, an open computational database of two-dimensional materials from top-down and bottom-up approaches’ Scientific Data 86, 669 (2019).
[2] N.Mounet et al, ‘Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds’. Nature Nanotechnology 13, 246 (2018).
[3] J.Marklof. ‘Random lattices in the wild: from Polya’s orchard to quantum oscillators’. LMS newsletter 493 43-50 (2021).