Penrose patterns and related structures. II. Decagonal quasicrystals

Structural relations of the two decagonal phases in Al-Mn and Al-Fe alloys with the Penrose pattern are discussed based on structure-factor calculations and symmetry considerations. The electron diffraction patterns are explained by models with layer structures which consist of the stacking of four kinds of layers constructing the Penrose pattern. Al-Mn has six layers within a period along the tenfold axis while Al-Fe includes eight. A projection along the axis shows the Penrose pattern in both models. The symmetries of Al-Mn and Al-Fe are expressed by the five-dimensional superspace groups P10₅/mmc and P10₅mc. These give the observed systematic extinction rules. In Al-Fe, an additional extinction rule due to the symmetry element of the superspace groupoid exists.