QUASI-CRYSTALS AND THEIR APPROXIMANTS - DODECAHEDRAL LOCAL ORDERING VERSUS CANONICAL-CELL DESCRIPTION

Two models of icosahedral quasicrystals are compared and connected. These are canonical-cell ordering (CCO) over medium-length scales (about 10 angstrom and more) and dodecahedral local ordering (DLO), which describes interatomic arrangements. In the DLO model, each atom is surrounded by closest neighbours positioned at several vertices of a regular pentagon-dodecahedron; of the 20 vertices of any dodecahedron, only a few can be occupied simultaneously without conflict (eight at most). Some icosahedral quasicrystals and their crystalline approximants exhibit DLO as the main structure motif at atomic scales. DLO networks are formally described using an unconventional projection of a six-dimensional lattice. It is shown that most DLO configurations (but not all of them!) can be constructed from small atomic size canonical cells that are a factor of tau3 smaller than the original ones. Two of the small canonical cells have the forms of distorted tetrahedra. It is also shown that DLO produces naturally die two most popular decorations of the Ammann rhombohedra: the edge decoration and the vertex-face decoration. Moreover, both decorations can be identified inside die same approximant. For medium-range distances, DLO leads to CCO with special decorations of the canonical cells. Therefore, die ordering in quasicrystals and in their approximants can be constructed as a hierarchy of dodecahedral ordering (or a hierarchy of canonical cells). It is shown that within the DLO model there may be an additional ordering of closest neighbours that leads to the transition between quasicrystals with primitive and face-centred lattices. The DLO-based duality between alpha-AlMnSi and Al5Li3Cu approximants is demonstrated. Possible physical reasons for DLO are considered. Some intrinsic features and difficulties of the DLO and CCO models (frustrations, disorder etc.) are discussed.