Crystal flex bases and the RUM spectrum

Badri, Ghada and Kitson, Derek and Power, Stephen (2021) Crystal flex bases and the RUM spectrum. Proceedings of the Edinburgh Mathematical Society, 64 (4). pp. 735-761. ISSN 0013-0915

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Abstract

A theory of infinite spanning sets and bases is developed for the first order flex space of an infinite bar-joint framework, together with space group symmetric versions for a crystallographic bar-joint framework $\C$. The existence of crystal flex basis for $\C$ is shown to be closely related to the spectral analysis of the rigid unit mode (RUM) spectrum of $\C$ and an associated \emph{geometric flex spectrum}. Additionally, infinite 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.