Covering Clusters in Icosahedral Quasicrystals

Abstract

The structural analysis of various approximant phases of icosahedral quasicrystals shows local environments with icosahedral symmetry: icosahedra, Mackay clusters, and Bergman clusters. For the icosahedral phases i-AlCuFe and i-AlPdMn, these clusters have been proposed as complementary building blocks centered on particular nodes. However, computations have shown that these 2-shell or 3-shell clusters do not cover all atomic positions given by 6D models. On the other hand, the recent concept of a unique covering cluster has been shown to apply to 2D Penrose tilings and Ammann-Beenker tilings. In this chapter we examine the local environments in i-AlCuFe and i-AlPdMn models about Wyckoff positions of the 6D lattice. We consider extended Bergman clusters of 6 shells that appear naturally in the Katz-Gratias model. We discuss the cell decomposition of the atomic surfaces and the variable occupation number of some of the shells. We show that a fixed extended Bergman cluster of 6 shells and 106 atoms covers about 98% of atomic positions. We also prove that a variable extended Bergman cluster of 6 shells, which contains the previous fixed cluster, covers all atomic positions of the theoretical model.