Crystal flex bases and the RUM spectrum

TL;DR

This paper develops a theory of free spanning sets and bases for the first order flex spaces of infinite crystallographic frameworks, linking their existence to the structure of the RUM spectrum and providing explicit computations for key frameworks.

Contribution

It introduces a novel theoretical framework for free bases in flex spaces and computes these bases for important crystal structures, connecting them to the RUM spectrum.

Findings

Explicit free bases computed for honeycomb, octahedron, and kagome frameworks.

Existence of crystal flex bases is linked to the linear structure of the RUM spectrum.

Provides a new perspective on the geometric flex spectrum in crystallography.

Abstract

A theory of free spanning sets, free bases and their space group symmetric variants is developed for the first order flex spaces of infinite bar-joint frameworks. Such spanning sets and bases are computed for a range of fundamental crystallographic bar-joint frameworks, including the honeycomb (graphene) framework, the octahedron (perovskite) framework and the 2D and 3D kagome frameworks. It is also shown that the existence of crystal flex bases is closely related to linear structure in the rigid unit mode (RUM) spectrum and a more general geometric flex spectrum.