Polynomials for crystal frameworks and the rigid unit mode spectrum

Abstract

To each discrete translationally periodic bar-joint framework C in Rd, we associate a matrix-valued function ΦC(Z) defined on the d-torus. The rigid unit mode (RUM) spectrum Ω(C) of C is defined in terms of the multi-phases of phase-periodic infinitesimal flexes and is shown to correspond to the singular points of the function Z → rankΦC(Z) and also to the set of wavevectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium, which corresponds to ΦC(Z) being square, the determinant of ΦC(Z) gives rise to a unique multi-variable polynomial p(C)(Z1, . . . , Zd). For ideal zeolites, the algebraic variety of zeros of pC(Z) on the d-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealized framework rigidity and flexibility, and in particular leads to an explicit formula for the number of supercell-periodic floppy modes. In the case of certain zeolite frameworks in dimensions two and three, direct proofs are given to show the maximal floppy mode property (order N). In particular, this is the case for the cubic symmetry sodalite framework and some other idealized zeolites.