The order-disorder evolution in quasicrystals through phason flips

The problem of incorporating phason flips in the structural investigation of aperiodic systems is still an open question and a challenge in crystallography. Phasons are understood both as atomic fluctuation and positional disorder of the quasicrystalline lattice. Popular correction to diffraction peaks' intensities takes the form of generalized Debye-Waller factor, assuming Gaussian distribution of fluctuations in the perpendicular space of higher-dimensional periodic lattice. Although proven to work in case of random tiling types of structures recent evidence indicates improper handling of peaks with high perpendicular space scattering vector whenever structure is far from random tiling regime. We introduce the concept of a series expansion of the characteristic function of the statistical distribution to properly correct the peaks' intensities with respect to phasonic fluctuations. Calculations are performed upon Penrose tiling. Such approximation of the structure factor works correctly even in cases for which the Debye-Waller correction fails. Even more we investigate transition to random tiling through phason flips by means of the statistical approach which results in interesting scaling properties (ordered → disordered → random → amorphous structure).