Abstract
Diffraction in quasicrystals is in logarithmic order and icosahedral point group symmetry. Neither of these features are allowed in Bragg diffraction, so a special theory is required. The present work displays exact agreement between the analytic metric with a numeric description of diffraction in quasicrystals that is based on quasi-structure factors. So far, we treated the hierarchic structure as ideal; now, we detail the theory by including two significant features: firstly, the steady state wave function of the incident radiation demonstrates how harmonics, in metrical space and time, enable coherent interaction between the periodic wave packet and hierarchic quasicrystal; secondly, mapping of the hierarchic structure for any influence of defects will allow estimation of possible error margins in the analysis. The hierarchic structure has the required logarithmic periodicity: superclusters, containing about 103 atoms, convincingly map phase contrast images; while higher orders leave space for subsidiary speculation. The diffraction is completely explained for the first time.
Highlights
Diffraction in quasicrystals is in logarithmic order and icosahedral point group symmetry
Whatever may be the structural details of quasicrystals, whether systematic or accidental; the ideal hierarchic model has provided complete understanding of diffraction in geometric series with irrational indices
Quasicrystals have demonstrated that quantum physics and Bragg diffraction are not beautiful mathematics but empirical physics: the proof of the numeric metric by analytic separation of the irrational residue is a benefit of observation over speculative expectation
Summary
Digitization and harmony are discovered in the scattered wave, so we need to review generally the nature of radiant scatterers in the broader scope of physics. The brackets symbolize in turn particulate conservation laws and response that is wave-like, resonant and harmonic The former is real; the latter imaginary. You can think of these as lattice images observed in thin foils in the two-beam condition The waves, as they proceed through the QC, oscillate (by the pendellösung effect) between the two beams (in crystals: [10]; cf in QCs: [11] [2]). The equations 2 separate the propagation direction from the transverse direction, and this has many consequences including: solutions for negative mass [12]5, phase velocity [6], uncertainty, Newton’s second law, electron spin, magnetic radius and fine structure constant [13]6, reduction of the wave packet [14] etc. The formalism enables our understanding of the fundamental interaction required in the coherent diffraction
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