Atomic clusters and atomic surfaces in icosahedral quasicrystals

This paper presents the basic tools commonly used to describe the atomic structures of quasicrystals with a specific focus on the icosahedral phases. After a brief recall of the main properties of quasiperiodic objects, two simple physical rules are discussed that lead one to eventually obtain a surprisingly small number of atomic structures as ideal quasiperiodic models for real quasicrystals. This is due to the fact that the atomic surfaces (ASs) used to describe all known icosahedral phases are located on high-symmetry special points in six-dimensional space. The first rule is maximizing the density using simple polyhedral ASs that leads to two possible sets of ASs according to the value of the six-dimensional lattice parameter A between 0.63 and 0.79 nm. The second rule is maximizing the number of complete orbits of high symmetry to construct as large as possible atomic clusters similar to those observed in complex intermetallic structures and approximant phases. The practical use of these two rules together is demonstrated on two typical examples of icosahedral phases, i-AlMnSi and i-CdRE (RE = Gd, Ho, Tm).