<i>Harmony</i> Is Cause—Not Consequence—Of the <i>Quantum</i>

Real Quanta and Continuous Reduction

“…Applying this value, the calculations provided a range of intensities that matched very well the wide range of experimental values observed in the diffraction patterns. Not surprisingly, the structure factor intensities were all close to zero at Bragg scattering angles, but consistent with experimental line intensities at scattering angles that were larger than defined by Bragg's law 3 . The divergence from the Bragg condition depends on the coherence factor, c s = θ/θ': the Quasi-Bragg law was therefore deduced:…”

“…The real probe bridges a short-cut that is taken by the method of mathematical probability amplitudes. The quantum finds new expression in the peculiar diffraction that we observe in quasicrystals [3].…”

“…However, these periodic Bloch waves are incommensurate with the hierarchic quasi-lattice that is geometric and irrational. However if their scale is multiplied by the metric function (Equation ( 8)) [3] [9];…”

“…The fractional increase in ray paths causes an 11.18… per cent increase in scattering angle. The dual harmony enables the periodic probe to scatter coherently from the hierarchic lattice onto a geometric reciprocal lattice with a peculiar and precise quasi-lattice constant a′ [3]. It is obvious that the dual harmony forces the quantization of the momentum that is evident in the diffraction pattern.…”