Complete Solution for Quasicrystals

Abstract

A set of discoveries are described that complete the structural model and diffraction theory for quasicrystals. The irrational diffraction indices critically oppose Bragg diffraction. We analyze them as partly rational; while the irrational part determines the metric that is necessary for measurement. The measurement is verified by consistency with the measured lattice parameter, now corrected with the metric and index. There is translational symmetry and it is hierarchic, as is demonstrated by phase-contrast, optimum-defocus imaging. In Bragg’s law, orders are integral, periodic and harmonic; we demonstrate harmonic quasi-Bloch waves despite the diffraction in irrational, geometric series. The harmonicity is both local and long range. A breakthrough in understanding came from a modified structure factor that features independence from scattering angle. Diffraction is found to occur at a given “quasi-Bragg condition” that depends on the special metric. This is now analyzed and measured and verified: the metric function is derived from the irrational part of the index in three dimensions. The inverse of the function is exactly equal to the metric that was first discovered independently by means of “quasi-structure factors”. These are consistent with all structural measurements, including diffraction by the quasicrystal, and with the measured lattice parameter.