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

A six-dimensional cluster model of icosahedral Al-Pd-Mn (i-Al-Pd-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 β-Al-Pd-Mn-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-Al-Pd-Mn (α-and β-Al-Pd-Mn-Si) by introducing appropriate cubic phason strains and taking a three-dimensional cut.