Abstract
The metric, that enables measurement of structural data from diffraction in quasicrystals, is analyzed. A modified compromise spacing effect is the consequence of scattering of periodic electromagnetic or electron waves by atoms arranged on a geometric grid in an ideal hierarchic structure. This structure is infinitely extensive, uniquely aligned and uniquely icosahedral. The approximate analytic factor that converts the geometric terms base τ, into periodic terms modulo 2π, is . It matches the simulated metric cs=0.947, consistently used in second (Bragg) order, over a wide scale from atomic dimensions to sixth order superclusters.
Highlights
The quasi-Bragg law for quasicrystals was discovered, firstly by visual inspection of the quasi-crystal diffraction pattern, and secondly by three-dimensional indexation and simulation of the pattern
One method was to expand the cosine function used in calculations of quasi-structure factors into its regular series derived from complex exponentials: the members of both the diffraction pattern series and the cosine series expansion belong to the same geometric series ([1] Appendix A.1)
What is needed is the factor that converts those geometric terms in Equation (4) that result from the locations of x′, y′, and z′ on a logarithmic grid base τ, into the corresponding series that is periodic in space common to the incident and diffracted beams, i.e. modulo 2π
Summary
The quasi-Bragg law for quasicrystals was discovered, firstly by visual inspection of the quasi-crystal diffraction pattern, and secondly by three-dimensional indexation and simulation of the pattern. One method was to expand the cosine function used in calculations of quasi-structure factors into its regular series derived from complex exponentials: the members of both the diffraction pattern series and the cosine series expansion belong to the same geometric series ([1] Appendix A.1). This fact turns out to be related to a second explanation for simulated values for the metric: the mid terms of the geometric series, around g0, have almost half integral values [5] that distort the regularity observed in Bragg diffraction from crystals. The metric is the key to understanding any measurement dependent on diffraction in quasicrystals
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