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Published in:
2014 | OriginalPaper | Chapter
Crystal Frameworks, Matrix-valued Functions and Rigidity Operators
Author : S. C. Power
Published in: Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation
Publisher: Springer Basel
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An introduction and survey is given of some recent work on the infinitesimal dynamics of
crystal frameworks
, that is, of translationally periodic discrete bond-node structures in ℝ
d
, for
d
= 2,3,... We discuss the rigidity matrix, a fundamental object from finite bar-joint framework theory, rigidity operators, matrix-function representations and low energy phonons. These phonons in material crystals, such as quartz and zeolites, are known as rigid unit modes, or RUMs, and are associated with the relative motions of rigid units, such as SiO
4
tetrahedra in the tetrahedral polyhedral bondnode model for quartz. We also introduce semi-infinite crystal frameworks, bi-crystal frameworks and associated multi-variable Toeplitz operators.