Structure factor of substitutional sequences

Abstract

We study the structure factor for a large class of sequences of two elementsa andb such that longer sequences are generated from shorter ones by a simple substitution rulea→σ ₁(a, b) andb→σ ₂(a, b), where theσ's are some sequences ofa's andb's. Such sequences include periodic and quasiperiodic systems (e.g., the Fibonacci sequence), as well as systems such as the Thue-Morse sequence, which are neither. We show that there are values of the frequencyω at which the structure factors of these sequences have peaks that scale withL, the size of the system likeL ^α(ω). For a given sequence a simple one- or two-dimensional dynamical iterative map of the variableω can easily be abstracted from the substitution algorithm. The basin of attraction of a given fixed point or limit cycle of this map is a set of values ofω at which there are peaks of the structure factor all of which share the same value ofα. Furthermore, only those values ofω which are in the basin of attraction of the origin can haveα(ω)=2. All other peaks will grow less rapidly withL. We show how to construct many sequences which, like the Thue-Morse sequence, have noL ² peaks. Other qualitative features of the structure factors are presented. Our approach unifies the treatment of a large class of apparently very diverse systems. Implications for the band structure of these systems as well as for the analysis of sequences with more than two elements are discussed.