Characterization and decoration of the two-dimensional Penrose lattice

The self-similarity property of the two-dimensional Penrose lattice is utilized to characterize it in terms of the distribution of different kinds of vertices, the Voronoi cells, and their nearest neighborhoods. Striking similarities are observed between the layer structures of the crystalline δ-${\mathrm{Al}}_{11}$ ${\mathrm{Mn}}_4$, ${\mathrm{Al}}_6$Mn, ${\mathrm{Al}}_{13}$ ${\mathrm{Fe}}_4$, ${\mathrm{Pt}}_5$ ${\mathrm{P}}_2$, and ${\mathrm{Ni}}_3$ ${\mathrm{Si}}_2$ with a sublattice of the Penrose lattice. The latter can be described in terms of cells which need not have fivefold rotational symmetry. Following the atomic distributions in the crystalline ${\mathrm{Al}}_6$Mn and ${\mathrm{Al}}_{13}$ ${\mathrm{Fe}}_4$, decorations of such lattices are suggested to model the T phase of Al-Mn and other related quasicrystals. We find two types of layers with fivefold rotational symmetry in the T phase. This is in agreement with the electron diffraction from such quasicrystals.