Power, Stephen
(2014)
*Crystal frameworks, matrix-valued functions and rigidity operators.*
In:
Concrete operators, spectral theory, operators in harmonic analysis and approximation : 22nd International Workshop in Operator Theory and its Applications, Sevilla, July 2011.
Operator Theory: Advances and Applications
.
Springer, Basel, pp. 405-420.
ISBN 9783034806473

## Abstract

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 ℝd, 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 SiO4 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.