The Choice of Vector Basis for Ammann Tiling in a Context of the Average Unit Cell

Abstract

For the construction of the average unit cell (AUC) within the statistical approach, the use of a so-called reference lattice is needed. The choice of the lattice constants is in general arbitrary. However, it is convenient to bind them with the reciprocal space vectors k and q (main and modulation vector, q=k/τ) which we use for indexing the diffraction pattern, λ _k =2π/k, λ _q =2π/q. AUC is a distribution of projections of atomic positions in real space on the reference lattice. With the choice of lattice as above, the shape of the AUC is related to the shape of the atomic surface (AS), used in the higher-dimensional approach. In this paper, the discussion on the choice of the set of wave vectors k and q is provided in terms of different geometrical bases used for a construction of Ammann–Kramer–Neri tiling (simply called Ammann tiling—AT, a model for icosahedral quasicrystal) and relation of the AUC and AS shapes. The dependence of the AUC shape on the choice of wave vectors is also demonstrated. Additionally, it is proved that the diffraction pattern does not depend on the basis chosen.