Real Space Structure Factor for Different Quasicrystals

Abstract

AbstractThe statistical approach to the description of aperiodic structures using the concept of the so called Average Unit Cell (AUC) is presented. The use of this method is shown for 1D, 2D, and 3D structures. We start with the presentation of the basic ideas for 1D simple sinusoidal modulation (1q) and the Fibonacci chain (a 1D model of a quasicrystalline structure). Up to now the AUC concept has been most thoroughly studied for 2D decagonal quasicrystals. The idea of structure factor derivation for the decorated Penrose tiling, a quasilattice for the description of decagonal quasicrystals, is presented. Two examples of structure refinement of decagonal phases from the Al‐Ni‐Co system are discussed. The potential of the cluster description of decagonal structures is also presented. The AUC construction and structure factor derivation is shown for the 3D Amman–Kramer–Neri tiling, which is a model quasilattice for the description of icosahedral phases. The AUC concept can also be extended to structures with singular‐continuous Fourier spectrum. An example for the 1D Thue–Morse sequence is studied.