The chemical and geometrical genesis of quasicrystals: Relationship of icosahedral aluminum alloy quasicrystal structures to icosahedral structures in elemental boron

Abstract

Chemically based structural models of icosahedral aluminum alloy quasicrystals use lattices of polyhedra of icosahedral local symmetry which are closely related to icosahedral structures found in boron chemistry, particularly the two rhombohedral allotropes of elemental boron. The structures of both β-rhombohedral boron and cubic R-AI 5CuLi 3 can be constructed from 60-vertex truncated icosahedra (the 'C 60 polyhedron') but linked in very different ways in a three-dimensional crystalline lattice. Electron-precise models for the chemical bonding topology of both structures can be described using methods similar to those used to treat isolated globally delocalized deltahedral boranes such as B l2H 12 2-. In the structure of R-AI 5CuLi 3 the truncated icosahedra form the surfaces of 84-vertex Samson complexes retaining icosahedral local symmetry where each of the peripheral vertices is shared with an adjacent Samson complex leading to 54-atom building blocks. Simple pairwise rotation of the 30 pairs of vertices connecting pentagonal faces in the peripheral truncated icosahedron of each 54-atom building block in this lattice leads to a closely related lattice of 54-atom Mackay icosahedra. Such processes may convert crystal lattices such as R-AI 5CuLi 3 into closely related icosahedral quasicrystal, structures such as T2-Al 5CuLi 3.