M43.00012. The analytic metric for geometric series diffraction from icosahedral quasicrystals

Presented by: Antony Bourdillon


Sharp diffraction patterns show long range order {1] in i-Al6Mn; translational symmetry is seenby phase-contrast, optimum-defocus imagingto be hierarchic. The icosahedral diffraction pattern was novel in two ways: that it included five-fold symmetric axes; having moreover diffraction orders in geometric series, imprecisely called Fibonacci. The orders differ from Bragg diffraction which is linear. How does a plane wave X-ray, or electron beam, scatter off the hierarchic structure? The diffraction angles due to scattering from the quasiperiodic solid diverge from Bragg conditions: quasi-Bragg angles are derived from quasi-structure factors with a metric that is derived numerically and analytically. The metric is a conversion factor, 1/(tau-0.5), where tau is the golden section: the metric harmonizes the incident sine wave with hierarchic scattering (all atoms scatter) while the response occurs in geometric series. The hierarchic structure is represented by triads of golden rectangles, each having principal planes, separated by spaces in geometric series, like the diffraction pattern [2]. [1] Shechtman, D, et al. (1984) Phys. Rev. Lett. 53, 1951, http://dx.doi.org/10.1103/PhysRevLett.53.1951 [2] A.J.Bourdillon, A.J., J. Mod. Phys. 10 [6] 624 (2019), DOI: 10.4263/jmp.2019.106044


  • Antony Bourdillon


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