The mathematics of crystal flexes and vibrations.

11^th June 2024, 14:00
Stephen Power
Lancaster University, UK

Abstract

A crystal framework is a periodic bar-joint framework in Euclidean space. A number of characterisations of their rigidity, flexibility and zero modes (rigid unit modes of oscillation) have been obtained by mathematicians in the last fifteen years or so. Also, in materials science, over a much longer period, there have been extensive studies of stability, zero modes and phase transitions. Most of these accounts assume some form of periodic boundary conditions in modelling the dynamics. I will outline some of the mathematical results, especially the connection between the zero modes, first-order flexes, the rigidity matrix and the transfer function (or symbol function). It is the transfer function that determines the RUM spectrum (of interest in materials science) as well as a more general geometric spectrum that is of significance for surface modes. I shall also discuss necessary and sufficient conditions obtained for "absolute" first order rigidity, for which no boundary conditions are assumed (joint work with E. Kastis).