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Crystal flex bases and the RUM spectrum

Part of: Discrete geometry Phase transformations in solids

Published online by Cambridge University Press:  24 November 2021

G. Badri ,
D. Kitson Open the ORCID record for D. Kitson [Opens in a new window]  and
S. C. Power Open the ORCID record for S. C. Power [Opens in a new window]
G. Badri
Affiliation:
Department of Mathematical Sciences, Umm Al-Qura University, Mecca Saudi Arabia (gmbadri@uqu.edu.sa)
D. Kitson
Affiliation:
Department of Mathematics and Computer Studies, Mary Immaculate College, Thurles, Co. Tipperary, Ireland (derek.kitson@mic.ul.ie)
S. C. Power
Affiliation:
Department of Mathematics and Statistics, Lancaster University, LancasterLA1 4YF, UK (s.power@lancaster.ac.uk)
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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 ${{\mathcal {C}}}$. The existence of a crystal flex basis for ${{\mathcal {C}}}$ is shown to be closely related to the spectral analysis of the rigid unit mode (RUM) spectrum of ${{\mathcal {C}}}$ and an associated 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.

Keywords

rigid unit modesperiodic frameworksflexibility

MSC classification

Primary: 52C25: Rigidity and flexibility of structures 74N05: Crystals
Secondary: 74N10: Displacive transformations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 4 , November 2021 , pp. 735 - 761
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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