What periodicities can be found in diffraction patterns of quasicrystals?

Acta Crystallogr A Found Adv. 2014 Mar;70(Pt 2):181-5. doi: 10.1107/S2053273313034384. Epub 2014 Feb 20.


The structure of quasicrystals is aperiodic. Their diffraction patterns, however, can be considered periodic. They are composed solely of series of peaks which exhibit a fully periodic arrangement in reciprocal space. Furthermore, the peak intensities in each series define the so-called `envelope function'. A Fourier transform of the envelope function gives an average unit cell, whose definition is based on the statistical distribution of atomic coordinates in physical space. If such a distribution is lifted to higher-dimensional space, it becomes the so-called atomic surface - the most fundamental feature of higher-dimensional analysis.

Keywords: Fibonacci sequence; Penrose tiling; average unit cell; diffraction pattern; higher-dimensional analysis; quasicrystals.