Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures

Abstract

The authors propose to use multifractal analysis in reciprocal space as a tool to characterise, in a statistical and global sense, the nature of the Fourier transform of geometrical models for atomic structures. This approach is especially adequate for shedding some new light on a class of structures introduced recently, which exhibit 'singular scattering'. Using the language of measure theory, the Fourier intensity of these models is presumably singular continuous, and therefore represents an intermediate type of order, between periodic or quasiperiodic structures, characterised by Bragg peaks (atomic Fourier transform), and amorphous structures, which exhibit diffuse scattering (absolutely continuous Fourier transform). This general approach is illustrated in several examples of self-similar one-dimensional sequences and structures, generated by substitutions. A special emphasis is put on the relationship between the nature of the Fourier intensity of these models and the f( alpha ) spectrum obtained by multifractal analysis in reciprocal space.