_{1}

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This structural study of quasicrystals is based on extremely dense icosahedral unit cells that are systematically and consistently measured for the first time. The structure and pattern indexation are 3-dimensional. A formula is given for scattering from atoms in hierarchic arrangement and geometric series. The Quasi-Bragg law is a new law in physics, with possible applications beyond crystallography. The structure is compared with previous, unsuccessful, and contradictory, attempts at analysis.

The structural solution for icosahedral quasicrystals has been described in several journal publications [

The term “quasicrystallography” was coined by its practitioners [

Quasicrystals were discovered in 1982 and have been long discussed. The first solid, i-Al_{6}Mn, was supposedly icosahedral in structure [

The following summary of features suffices for the solution:

1) A unique unit cell that is icosahedral and extremely dense:

• With 15 close packed planes through its center;

• With 20 close packed planes on its surface;

• And with a density increase of 17% compared with the equivalent cell in the face-centered cubic phase. This is a crystalline phase that forms the matrix in rapidly quenched Al_{6}Mn [

2) With a stoichiometric alloy ratio of 6:1. This occurs on a unit cell with 12 shared surface atoms and one completely contained central atom, in this case Mn.

3) With a significant ratio of solute metal atom diameter/solvent diameter of

4) With an icosahedral cell that is edge-sharing; not face sharing as in crystals. Notice that many glasses, such as amorphous silica, also have dense, edge-sharing cells; but the diffraction patterns are diffuse and circular because the sharing is by single, uncorrelated edges. By contrast, the quasicrystal cells share two or more adjacent edges, and these result in unique orientation with sharp diffraction patterns by a mechanism that is simulated.

5) With an agglomeration of cells that is represented in an ideal hierarchic structure that is uniquely icosahedral, uniquely aligned and infinitely extensive;

6) With a structure that is open to simulation.

1) With structural and diffraction pattern sequences that are geometric (as below);

2) With complete, simple and systematic 3-dimensional indexation of icosahedral axes and diffraction planes [

3) With 3-dimensional pattern indexation that is simple, consistent and complete;

4) With an explanation for the diffraction that is systematically, completely, and simply described by means of the Quasi-Bragg law that has been derived and is now enhanced in appendix A.1 ~ A.2.

1) With simulated quasi-structure factors that match experimental values extraordinarily well [

2) With a structure that is simulated and systematically measured without guessing [cf. [

3) With a simple demonstration of the measured metric [

1) As derived above consistently, especially in sections 2.1.1 to 2.1.3;

2) With a logarithmic electron energy band structure [

Notice that the ‘logarithmically periodic solid’ described is the only proposed solution with a single unit cell and obvious driving force, and is the only solution that has been systematically and consistently measured. The structure is illustrated in

Their two-fold patterns are inconsistent (_{6}Mn [citations in ref. 1]. Two assumptions are therefore fairly made: his data are mistakenly transcribed; and a fundamental feature of crystallography has passed unnoticed. The pity is that the experimental discovery is so novel.

This sequence has the following properties: each term is the sum of its two preceding terms

and the ratio between consecutive terms in a series,

It is interesting to note, nevertheless, that the geometric series on the ratio _{1} = a; f_{2} = b, is too loose and inaccurate a description of the diffraction patterns in quasicrystals. In this application, f_{2} is strictly constrained:

In real space, the structure is unambiguously 3-dimensional. However for reasons given in the previous section, neither the structure nor the diffraction pattern are 6-dimensional in any sense that can survive Occam’s razor: dimensions “should not be multiplied without necessity”. Unfortunately, unphysical dimensionality is the most strongly held tenet of quasicrystallography [

A reader will easily find a simple understanding for their widespread belief in 6-dimensional quasicrystals: indexation, for quasicrystallographers, is based on a double Fibonacci series construction, though the series is, in fact, simply three dimensional and geometric – as in real life. An illustration can be given for the relationship between those Fibonacci sequences and the single geometric series, base_{1}, F_{2}, one of them multiplied by _{1} and f_{2} being the first two terms in the same Fibonacci series. Furthermore, the golden section is an irrational number, which is why G is not linearly harmonic (as quoted in Section 3.6 below, but typical only in crystals).

As in the stereogram, geometric, 3-dimensional indexation is easily and completely represented, without obfuscation [

This law applies to periodic crystals, as it does to amorphous materials. In each case, the scattering is specular and is commonly represented on 2-dimensional ray diagrams. The scattering is constrained by the Ewald sphere construction, and also by absorption, crystal orientation and other features. Such features vary according to applications in X-ray optics or in electron or neutron optics etc. In particular, electron diffraction relaxes the Ewald sphere constraint [

Many consequences follow an inadequate understanding of the special diffraction; two (in imaging and in structural analysis) will be described below as illustrations. Quasicrystallographers have failed to understand or explain the special diffraction in quasicrystals that relates to the multiple interplanar spacings: multi-dimensionality multiplies only confusion; the interplanar spacings are indeed evident in phase-contrast electron microscopy e.g. [

High resolution transmission electron microscopy (HRTEM) is complicated science that is typically ambiguous and that is easily misrepresented [

How should a crystallographer interpret real structure from a diffraction pattern that he does not understand? An example is given in the following review [

“We know that for every reflection with index H and its higher harmonics with indices nH, there exists a periodic average structure (PAS) with period d_{H} which has to be used in Bragg’s equation. Then it can be applied as usual. The scaling symmetry by powers of _{H }of the respective scales PAS properly.”

The trouble is, he doesn’t know how to make the proper choice, so his simulation is without measurement, without consistency and without significance. Moreover, what mental confusion allows a “periodic average structure” in a “non-periodic material”? Furthermore, his “harmonics” are not observed. If there are harmonics, they are in geometric space, but only sound visual inspection proves it. Any interpretation that lacks the measured metric is unphysical.

Worse still, is the misallocation of effort. Fourier transform analysis hides defects where these are of the greatest significance for structure, especially in rapidly quenched material, like Shechtman’s. That is why sterogram’s such as that in

Mathematically, it is as easy to multiply cells as to multiply dimensions, but neither contributes to understanding why and how quasicrystals form. Amorphous silica can be constructed from an infinite number of unit cells. Quasicrystallographers typically contrive cells that could contribute to icosahedral structure [

Multiplicity of cells fails many requirements of acceptable theory: it does not explain stoichiometry in this type of alloy; it does not explain peculiar atomic sizes; it does not reveal the driving force for these unusual materials. Only the ideal of logarithmic periodicity satisfies these requirements. Crystallography abhors multiple unit cells!

The structure of icosahedral quasicrystals is solved. It depends on an icosahedral unit cell that is extremely dense and that is edge-sharing. By contrast, seven quasi-conventional blunders are described: Shechtman’s data are not icosahedral; the diffraction pattern is not Fibonacci; quasicrystals are not 6-dimensional; there is no Bragg diffraction; HRTEM does not image atoms; Fourier transform analysis, without the measured metric, is misleading; and crystallography abhors multiple unit cells. The Quasi-Bragg law is a new law in physics.

A.1. Fourier Summations on Geometric Series In grating optics and Fraunhofer diffraction, Fourier transforms of the grating are used to describe diffraction patterns observed in the image plane. In crystal optics, it is often possible to approximate atoms to point scatterers and so apply Fourier series to the analysis. We did this before in simulations of quasi-structure factors. The calculations were performed in 3-dimensions over a truncated and ideal hierarchic model that is, in principle, infinitely extensive. Now we show how the method can be used to understand quasi-Bragg diffraction on atoms in linear geometric series.

Start with the Quasi-Bragg law described earlier,_{hkl} for a reflection with Miller indices h, k, l, is written:

summed over atoms j, each with atomic scattering amplitude_{j}, v_{j} and w_{j}_{,}, where the superscript _{s} that has unit value in crystals, but that has a smaller measured value in quasicrystals because the multiple interplanar spacings are here non-uniform and geometric. Then the quasi-Miller indices _{n} factors are integral values that express the number of times a particular interplanar space is contained in u_{j}. The factors are inherent in the simulations cited earlier. Quasi structure factors, calculated for superclusters up to order 6, with 10^{8} atomic sites, match experimental values.

Now we introduce a further feature that is the evident result of many calculations. We have found the line width of quasi-structure factors to be dependent on sample sizes used in simulations. For example in a cluster of ~100 atoms the line width is ~5% of the quasi-Bragg angle; reducing to 0.005% in the supercluster order 6. The most probable reason for the systematic trend is the oscillatory behavior in the cosine term in Equation (A1).

We now proceed to further details in the crystallography of quasicrystals. Begin by noticing the octahedral subgroup within the icosahedral point group symmetry. This is enabling, both for the 3-dimensional indexation, while it also allows us to write for an interplanar spacing

with

In quasicrystals, linearly periodic electromagnetic waves are scattered by atoms in an approximately hierarchic structure having logarithmic periodicity, in such a way that the resulting diffraction pattern is geometric as simulated.

A.2. Geometric and Fibonacci Series Comparedwith logarithmic periodicity. Start with the simplest Fibonacci sequence based on 0 and 1 (column 1) then calculate the ratios between consecutive terms. Notice that they vary about τ (column 2); that they tend to τ at high order; and that the quasicrystal diffraction series that is observed is not Fibonacci because it lacks the property of varying ratios [