Quasicrystals and the Concept of Interpenetration in $$m\dot{3}\dot{5} $$ — Approximant Crystals with Long — Range Icosahedral Atomic Clustering

Abstract

Interpenetrating Pseudo — Icosahedral atomic Clusters (IPIC) are found in various \(m\dot{3}\dot{5} \) — approximant crystals such as α — AlMnSi, R — AlLiCu, presenting one orientational family of IPIC., c — Ti2Ni, s — AlMnSi, μ, — Al4Mn, exhibiting several ones, and in many models of icosahedral or decagonal phases. We have developped a theory of G — approximant crystals based on IPIC., where G is a non — crystallographic point group (G = \(m\dot{3}\dot{5} \), Cn, D2n, with n = 5, > 6) allowing to describe what happens in reciprocal space for the intense Bragg peak positions and their intensities, which is easily generalizable to quasicrystals and provides a continuity of the perception of the symmetry group G in reciprocal space when small atomic rearrangements occur in real space during the phase transition quasicrystal approximant crystal. This theory is new and completes the edifice of the theory of crystallographic groups by considering non — crystallographic point groups.