Hierarchic icosahedra
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Hierarchic icosahedra represent original theory in classical crystallography[i], opposed to more common, and unmeasured, ‘Quasi’ mathematical theories. The rigorous explanation is needed since without diffraction theory, quasi is speculative. The ‘long range order’ is evident in the diffraction. Equally obvious is the ‘translational symmetry’ in geometric series, because it is hierarchic.
Natural diffraction with irrational metric: Ever since the discovery of X-rays, diffraction in crystals has been fundamental in studies of peri-odic structures at atomic scales. There, it depended on Bragg’s law for rays of wavelength λ=2(d/n)sin(θ), where d is the interplanar spacing for a particular reflection; n the order; and θ the Bragg angle. In case of high energy electron scattering, at small θ, and cubic crystals—including icosahedral crystals since they contain the octahedral subgroup—Bragg’s law can be written λ≃ aΘ/(n(h2+k2+l2)1/2). Where a is the lattice parameter; the scattering angle Θ≃2sin(θ); and h, k, l are diffraction indices. The order n is positive integral, so that incident rays are periodically and harmonically scattered. Harmony is generic in quantum wave physics and is essential in Bragg orders.
‘Quasicrystals’ diffract differently. For the greatest symmetry and simplest representation, restrict consideration to icosahedral ‘quasicrystals’. The diffraction pattern is not in periodic order nΘ; but in geometric radial series aτm where τ=(1+51/2)/2 is the golden section, i.e. irrational. The ratio d/n is therefore not harmonic, and the most critical problem is how the diffraction occurs [i].
Principal axes for the icosahedral structure are easily indexed in three dimensions using powers of τ, and corresponding plane normals are simply derived [i]. Dimensions should not be multiplied without necessity. The structure is equally unproblematic: evident in electron micrographs [e.g. Quasicrystal ref.19] are many icosahedral superclusters, each constructed from ~ 1000 atoms arranged in measured circles of five tenfold inner circles, i.e. before the prior polishing of sections. The structures are hierarchically arranged, each is four icosahedral cells deep [iv]. Higher orders naturally exist by extension, but specimen roughness on a very thin section hide their 2D arrangement. Calculation of the diffraction is enabled by knowledge of the structure. It is logarithmically periodic with period τ2. Even in defective samples, the ideal hierarchies correctly model the long-range-ordered diffraction.
Moreover the micrograph shows multiple spacings d, so that, along with the geometric n, it falsifies Bragg’s law in ‘quasicrystals’. Since Bragg’s law cannot apply, we do not know a priori the relations between n, d, and Θ. However, by calculating structure factors we avoid Θ, making it possible to link n to d directly. The Quasi-structure factor that we have to use differs in two ways from structure factors in crystals [iii] and is written Fphkl=Σiall atoms cos(2πcs(τ2phhkl∙ri)).Fp-1hkl.
The general features are retained: the scattering from each atom site ri is projected onto a plane normal hhkl and the wave amplitudes are summed. However, the quasicrystal has multiple spacings d along with sharp diffraction, so the coherence factor cs is critically needed. This will turn out to be a ‘breathing strain’ that switches the quasi-Bragg diffraction on and off suddenly, like the rocking curve in crystals. Secondly, because the unit cell does not periodically repeat, the summation is made over all the atoms in the hierarchy; not just the unit cell. To compute this, the iterative procedure is needed, as indicated in the formula along with the stretching factor τ2p. The coherence factor cs is found by scanning to find its maximum. The result is twofold: there is no Bragg diffraction, i.e. where cs=1; secondly, diffraction only occurs when the metric, cs=0.894 [iii]. Calculated quasi structure factors map experimental diffraction patterns about all major axes.
But what is the metric? It is linked to the irrational indexation that is peculiar to hierarchic icosahedra. Notice that τm = Fm(1,τ) = Fm+1(0,1) + Fm(0,1)τ where the arguments represent the first two terms of the Fibonacci series Fm , and m represents the order of a corresponding member. It follows τm is separable into natural and irrational parts. Further separation yields the metric function [iii]: 1 + (τm – Fm+4/2)/ Fm+1 = 1/cs . This analytic metric function yields, as its exact inverse, the numerical coherence factor cs that was independently evaluated. Its value is common to all of the diffracted beams. When the indexation is artificially rationalized by making τ->3/2, the coherence is found to have a Bragg-like value cs->1. This confirms the fact that the metric is due to the irrational part of the index: it harmonizes the scatter from the incoming sine wave, through the hierarchic icosahedra, to the uniquely-characteristic, geometric-series, diffraction. This is illustrated in the Quasi-Bloch wave. The feature is unique to the hierarchic crystals.
The original paper by Shechtman et al. claimed “Long range order with no translational symmetry”. This was surprising at the time, but is not in fact true since the translational symmetry is in fact strictly hierarchic. The diffraction is indeed a result of long-range order; its symmetry is illustrated by quasi-Bloch waves. Experimentally, these simulate lattice images in the two-beam condition. They are the consequence of interference between an incident beam with its quasi-Bragg reflection. In the figure, calculated blue waves are not observed: they are pseudo-Bragg, Bloch waves that are commensurate on the unit cell but not at higher orders; the red waves are approximately commensurate on all orders because in them the blue wave has been stretched by the metric function. This harmonizes the irrational part of the index. Red waves are invariant on all translations aτm. Notice that the harmony is both long range and short range at all geometric series’ intercepts, and that the number of cycles between intercepts is the denominator in the metric function. The lattice parameter, a, is measured, analyzed, verified, and complete. So is the structure and diffraction [cf. v].
The consistent argument shows that whereas Bragg diffraction is coherent scattering from Bragg planes in crystals; hierarchic diffraction is coherent scattering from cluster centers. These are located on ‘principal planes’, in geometric series, within the hierarchic icosahedral structure [iii].
Postscript: Cyclopedists and Philosophes of the 17thC made sense of their world. After 40 years, ‘quasi’ becomes it.
References
[i] Bourdillon, A.J., (1987) Fine Line Structure Convergent Beam Electron Diffraction in Icosahedral Al6Mn, Phil. Mag. Lett. 55 2l-26
[ii] Bourdillon, A.J. (2013) Icosahedral stereographic projections in three dimensions for use in dark field TEM Micron, 51, 21-25. https://doi.org/10.1016/j.micron.2013.06.004
[iii] Bourdillon, A.J., (2020) Complete solution for quasicrystals, https://www.youtube.com/watch?v=OFcSDKCecDA
[iv] Bourdillon, A.J., (2011) Logarithmically Periodic Solids, Nova Science ISBN 978-1-61122-977-6 Search this book on 
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[v] Senechal, M., (2006_) What is a quasicrystal?, Notices of the American Mathematical Society, 53, 886-887
crystallography
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