Six-dimensional model of icosahedral Al-Pd-Mn quasicrystals

A six-dimensional cluster model of icosahedral Al-Pd-Mn $(i-\mathrm{Al}-\mathrm{Pd}-\mathrm{Mn})$ quasicrystals used in the recent structure refinement is described in detail. This model is based on five kinds of clusters located at vertices, edge centers of the inflated three-dimensional Penrose pattern (3DPP) with an edge length of about 20 Å, and two body-diagonal positions of each acute rhombohedron. Two kinds of 20 Å clusters, one of which is seen in $\beta -\mathrm{Al}-\mathrm{Pd}-\mathrm{Mn}-\mathrm{Si},$ are located at even- and odd-parity 12-fold vertices of the inflated 3DPP. This is proved to give small R factors ${(R}_w=0.055,R=0.049)$ for 493 independent reflections and leads to the 1/1 and 2/1 approximant structures of $i-\mathrm{Al}-\mathrm{Pd}-\mathrm{Mn}$ $(\alpha -$ and $\beta -\mathrm{Al}-\mathrm{Pd}-\mathrm{Mn}-\mathrm{Si})$ by introducing appropriate cubic phason strains and taking a three-dimensional cut.