2011/02/14 by Power, S. C.
#52C25 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1102.2744
To each discrete translationally periodic bar-joint framework \C in \bRd we associate a matrix-valued function Φ_\C(z) defined on the d-torus. The rigid unit mode 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 → \rank Φ_\C(z) and also to the set of wave vectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium, which corresponds to Φ_\C(z) being square, the determinant of Φ_\C(z) gives rise to a unique multi-variable polynomial p_\C(z1,…,zd). For ideal zeolites the algebraic variety of zeros of p_\C(z) on the d-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealised 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 2 and 3 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 idealised zeolites.