Structure factor of substitutional sequences

Abstract

We study the structure factor for a large class of sequences of two elements a and b such that longer sequences are generated from shorter ones by a simple substitution rule a → σ 1 ( a, b ) and b → σ 2 ( a, b ), where the σ 's are some sequences of a 's and b '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 frequency ω at which the structure factors of these sequences have peaks that scale with L , the size of the system like L α(ω) . For a given sequence a simple one- or two-dimensional dynamical iterative map of the variable ω 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 of ω at which there are peaks of the structure factor all of which share the same value of α . Furthermore, only those values of ω which are in the basin of attraction of the origin can have α ( ω )=2. All other peaks will grow less rapidly with L . We show how to construct many sequences which, like the Thue-Morse sequence, have no L 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.