Requirements for structure determination of aperiodic crystals

Using computer simulation, we compared the Patterson functions of one-dimensional (1D) randomly packed and quasiperiodic Fibonacci lattices with or without disorder, and a 2D Penrose lattice and random packing of pentagons (icosahedral glass model). Based on these comparisons, we derived some empirical guidelines for distinguishing ideal quasicrystals from aperiodic crystals with disorder using diffraction data. In contrast to periodic crystals, it is essential to include the background to obtain correct Patterson functions of the average structure since the background contains unresolved peaks. In particular, a Bragg peak scattering measurement cannot, in general, determine the structure of aperiodic crystals. Instead, a diffuse scattering measurement is required, which determines the absolute value of the diffraction background, in addition to the Bragg peaks. We further estimate that, dependent upon the disorder present, it is necessary to include up to 75% of the total diffracted intensity in any analysis.