Covering clusters in the Katz–Gratias model of icosahedral quasicrystals

A variety of clusters with icosahedral symmetry can be identified in periodic approximant phases of quasicrystals: icosahedra, Mackay clusters (MC) and Bergman clusters (BC). For i-Al–Cu–Fe and i-Al–Pd–Mn, MCs and BCs have been proposed as complementary building blocks centred on a primary structure. However, theoretical studies showed that these 2-shell or 3-shell clusters could not cover all atomic positions given by hyperspace models. On the other hand, the recent concept of a unique covering cluster was shown to apply to 2D Penrose tilings and octagonal tilings. In this paper, we consider extended MCs of 5-shells and extended BCs of 6-shells that appear naturally in the Katz–Gratias model. We discuss the variable occupation number of some of the shells. We show that a fixed extended BC of 6-shells and 105 atoms covers about 98% of atomic positions, while extended MCs of 101 atoms only fill 93% of sites. We also prove that a variable extended BC of 6-shells covers all atomic positions of the Katz–Gratias model.