Beyond the Icosahedral Glass Model, the Interpenetrating Pseudo — Icosahedral Clusters (IPIC) Model

Abstract

We analyze the diffraction properties of infinite “formal” quasiperiodic, resp. periodic, packings of atoms in 3 dimensional Euclidian space that are considered as quasiperiodic, resp. periodic, distributions of identically oriented pseudo — icosahedral clusters which all have a finite spatial extension and interpenetrate. Restoring the icosahedral local order in the formalism leads to count atomic sites several times, when taken into account from the center of one cluster or another. Scattering functions are treated in perturbation over site multiplicity and icosahedral symmetry. The periodic case corresponds to approximant crystals, while the quasiperiodic one to icosahedral quasicrystals: the rank of the Z — module generated by the centers of clusters is then 3, resp. m, with m>3. A periodization of the quasicrystal can be performed in Rm. Pseudo — inflation rules arise from the geometry of an average cluster. This formalism is new and provides a “continuous link” between classical crystallography and quasi-crystallography.