The Reciprocal Lattice in Hierarchic Quasicrystals

Abstract

Initially, all that was known about diffraction in quasicrystals was its point group symmetry; nothing was known about the mechanism. The structure was more evident, and was called quasiperiodic. From mapping the Mn atoms by phase-contrast, optimum-defocus, electron microscopy, the progress towards identifying unit cell, cluster, supercluster and extensive hierarchic structure is evident. The structure is ordered and uniquely icosahedral. From the known structure, we could calculate structure factors. They were all zero. The quasi structure factor is an iterative procedure on the hierarchic structure that correctly calculates diffraction beam intensities in 3-dimensional space. By a creative device, the diffraction is demonstrated to occur off the Bragg condition; the quasi-Bragg condition implies a metric that enables definition and measurement of the lattice constant. The reciprocal lattice is the 3-dimensional diffraction pattern. Typically, it builds on Euclidean axes with coordinates in geometric series, but it also transforms to Cartesian coordinates.