The nature of the atomic surfaces of quasiperiodic self-similar structures

Abstract

Quasiperiodic self-similar chains generated by substitutions (i.e. deterministic concatenation rules) and their diffraction spectra are analysed in a systematic fashion, from the viewpoint of the superspace formalism. A substitution acting on n objects generates quasiperiodic chains if, and only if, the associated substitution matrix fulfils two arithmetic conditions (Pisot property and unit determinant). The structures thus obtained can be alternatively built as sections of periodic patterns in an n-dimensional superspace, which are regular repetitions of an atomic surface. The authors derive a general algorithm to construct this atomic surface. It is a compact set of the (n-1)-dimensional internal space, which is a unit cell for a lattice of translations. The atomic surface is nevertheless not necessarily connected, and its boundary is generically an anisotropic self-similar fractal. The dimension dB of this boundary is shown to govern the anomalously slow fall-off of the intensities of Bragg diffractions, and therefore to influence physical properties.