_{1}

^{*}

Initially, all that was known about diffraction in quasicrystals was its point group symmetry; nothing was known about the mechanism. The structure was more evident, and was called quasiperiodic. From mapping the
*Mn* atoms by phase-contrast, optimum-defocus, electron microscopy, the progress towards identifying unit cell, cluster, supercluster and extensive hierarchic structure is evident. The structure is ordered and uniquely icosahedral. From the known structure, we could calculate structure factors. They were all zero. The quasi structure factor is an iterative procedure on the hierarchic structure that correctly calculates diffraction beam intensities in 3-dimensional space. By a creative device, the diffraction is demonstrated to occur off the Bragg condition; the quasi-Bragg condition implies a metric that enables definition and measurement of the lattice constant. The reciprocal lattice is the 3-dimensional diffraction pattern. Typically, it builds on Euclidean axes with coordinates in geometric series, but it also transforms to Cartesian coordinates.

Rapidly cooled Al_{6}Mn has a phase, supposedly with “Long range orientational order and no translational symmetry” [

The foremost beginning lies in the formula: Mn has an atomic number that is almost double that of Al. Consequently imaging, in phase-contrast, optimum defocus, transmission, electron microscopy [_{12}Mn unit cells. The resulting stoichiometry is Al_{6}Mn, as in the melt before crystallization and as in elemental analysis. Being edge sharing the structure is not space filling. (By contrast, all crystals are both face sharing and space filling.) In icosahedral i-Al_{6}Mn, the relative atomic sizes are perfect for extremely dense packing in icosahedral coordination [^{1}. The unit cells cluster into icosahedral structures in at least four tiers of icosahedral hierarchy, seen in the image. There are good reasons for projecting the hierarchy to extend infinitely.

The stereogram of the principal axes and of principal reflecting planes in the icosahedral structure is, in its simplest form, 3-dimensional; indexed in geometric series; and complete [^{2}, as outlined below.

Because Bragg diffraction is well understood in its wide application to crystals, it is a mildly useful foil for understanding diffraction in quasicrystals, but only if differences are emphasized. Bragg’s law describes the linear series of diffraction orders, n = 0, 1, 2, 3 …, that result from periodically spaced reflecting planes of atoms. For the simple case of high energy electron diffraction from a cubic crystal, we can write approximately:

n ≈ d ⋅ Θ λ = a ⋅ Θ h k l λ ( h 2 + k 2 + l 2 ) 1 / 2 (1)

where the scattering angle, Θ ≈ 2sin(θ) for the Bragg complement to the angle of incidence θ; while d is the periodic interplanar spacing; and λ << d is the wavelength of the incident beam. For each indexed scattering angle Θ_{hkl} in a cubic crystal with lattice parameter a, there corresponds an interplanar spacing d_{hkl} = a/(h^{2} + k^{2} + l^{2})^{1/2}, h, k, and l positive integral.

By contrast, the quasicrystal (QC) does not have regular, periodic, interplanar spacings, and the orders in the diffraction pattern are in geometric series τ^{m}. Moreover, the relations between scattering angle Θ', d' and λ were a priori unknown. Supposing modified relations, the following solution is consistent with the diffraction pattern:

τ m ≈ d ′ Θ ′ λ , m positive or negative integral , (2)

with details to be determined by simulation and experimental consistency. Every atom scatters. We have to suppose that in any given QC orientation, either most atoms scatter randomly but that atoms in adjacent planes may filter coherent scattering for the appropriate d' spacing; or that all atoms in the geometric structure coherently scatter the incident periodic wave. The former supposition is falsified because known interplanar spacings do not match Equation (1); the following evaluation of quasi scattering factors is a numerical solution that describes the coherent sharp diffraction that follows Equation (2). Complete analysis is work in progress.

Meanwhile, consistency is required between structure, scattering, and measurement. In particular, and consequent on optimum defocus imaging, it is clear that the unit cell has dimensions a(1 × τ), where a is the quasilattice parameter that requires measurement, and the bracket gives the dimensions of the golden rectangle [

In crystals, Equation (1) provides the principal condition required for diffraction, and it is harmonic in order n. However, the equation is not a sufficient condition. Many indexed beams that are allowed by the equation are in fact forbidden by symmetric details within the periodic unit cells [_{6}Mn, we can proceed to compare diffracted beam intensities by simulations of Quasi Structure Factors. We need to simulate, not only the supposition of Equation (2), but also to measure the lattice parameter a, along with beam intensities and other features of the diffraction pattern. The fact that the known structure—at least when considered as ideal and defect free—is uniquely icosahedral, is confirmation of its correspondence with the point group symmetry of the diffraction pattern. The quasi structure factors are calculated from the known ideal structure.

_{6}Mn imaged in phase-contrast, optimum-defocus, electron microscopy [

The Mn atom is the first of four tiers of icosahedral structure (red circles). When combined with all of the information given by the point group symmetry of the diffraction pattern [

Whereas, in crystals, the Bragg condition (Equation (1)) determines harmonic factors n that relate λ to Θ; corresponding structure factors sum atomic scattering amplitudes that determine intensities and forbidden lines. Likewise in quasicrystals, the dual importance of the quasi-Bragg condition and quasi-structure factors applies, but especially so because line intensities are many and varied. The general application of the formulae has been previously described [

Unitcell ( r u ) : M n : ( 0 , 0 , 0 ) A l : 1 / 2 ( ± τ , 0 , ± ; 1 ) , 1 / 2 ( 0 , ± 1 , ± τ ) , and 1 / 2 ( ± 1 , ± τ , 0 ) Cellor Clustercenters ( r c c ) : 1 / 2 ( ± τ 2 , 0 , ± τ ) , 1 / 2 ( 0 , ± τ , ± τ 2 ) , 1 / 2 ( ± τ , ± τ 2 , 0 ) . (3)

In crystals (that obey Equation (1)), the structure factor formula projects each atomic site at vector r_{i} in a unit cell onto a selected plane normal having integral indices hkl (Equation (1) with c_{s} = 1). By summing the projected cosines on non-equivalent atoms, an amplitude is obtained that corresponds to the intensity of the (hkl) diffracted beam after a crystal is oriented to the Bragg condition. For example, the closest crystalline approximant to i-Al_{6}Mn is second phase, face centered cubic (fcc) Al_{x}Mn, x >> 6 [_{hkl} have values equal to either 0 or 4. The atomic scattering factor is f_{i} and, in the crystal, the metric c_{s} = 1:

F h k l = ∑ i = 1 N f i cos ( 2 π ⋅ c s ( h h k l ¯ ⋅ r i ¯ ) ) (4)

By contrast, in icosahedral quasicrystals (that obey Equation (2)), quasi-structure factors (QSFs) are more complex and more varied: because it is not structurally periodic, the summation in Equation (4) is extended to clusters and superclusters indefinitely. Write the vector from the origin to each atom in a cluster r c l ¯ as the sum of a unit cell vector r u ¯ used previously, with a vector to the cell centers in the cluster r c c ¯ : r c l ¯ = r c c ¯ + r u ¯ . Then since

∑ i N cluster exp ( h h k l ¯ ⋅ r c l ¯ ) = ∑ i 12 exp ( h h k l ¯ ⋅ r c c ¯ ) × ∑ i 13 exp ( h h k l ¯ ⋅ r u ¯ ) (5)

with corresponding summations over cell centers and unit cell sites, N_{cluster} = N_{cc}∙N_{u}, the QSF for the cluster may be calculated:

F h k l cluster = ∑ i = 1 12 cos ( 2 π ⋅ c s ( h h k l ¯ ⋅ r c c ¯ ) ) ⋅ F h k l cell (6)

and the calculation iterates on superclusters orders 1, 2, 3 ... p, by inclusion of the stretching factor τ^{2}^{p}:

F h k l p = ∑ i = 1 12 cos ( 2 π ⋅ c s ( h h k l ¯ ⋅ τ 2 p r c c ¯ ) ) × F h k l p − 1 (7)

F h k l cluster may also be written F h k l 0 . The iteration is important when the calculation is performed over large clusters, for then truncation errors due to the many additions tend to randomize answers. Though large clusters yield precise conclusions [_{s}. The negative derivative that will be calculated is used to refine former results.

^{2}00), (τ^{3}00) and (τ^{4}00), all plotted against varying values for the metric c_{s}. Each plot contains a single peak centered at c_{s} = 0.894; no other significant structure is found between 1.2 > c_{s} > 0.8. The plots confirm the geometric series diffraction that is observed in diffraction patterns, and that was discovered in the stereograms [

illustrative: all simulated peaks from the quasicrystal occur with the same metric (i.e. the value of c_{s} when the QSF peaks). The plots also emphasize the critical and creative importance of the metric for the confirmation of Equation (2).

The significance of _{s} = 1). Moreover, in the simple case of fcc Al for example, the number of values calculated is limited, typically 4 or 2 or zero, there being only four non-equivalent atoms in the unit cell. By contrast, not only do quasicrystal QSFs peak only at a precise off-Bragg condition (c_{s} = 0.894; Θ' = Θ/c_{s}; d' = d∙c_{s}) in Equation (2) and

The peaks describe a new physical effect in quasicrystals. In crystals, Bragg’s law (as in Equation (1)) supposed a simple model, with elementary mathematics and clear experimental evidence. By contrast, quasicrystals are described by

Bragg condition | Quasi-Bragg condition | Comment |
---|---|---|

nλ ≈ dΘ | τ^{m}λ ≈ d'Θ' | Including metric, ('), n, m integral m = −infinity, −1, 0 1, 2, 3… or h' = 0, τ^{−1}, 1,τ, τ^{−1}, τ^{−2}, τ^{−3}… |

d = a/h | d' = a'/h' | r is fixed within known models |

Diffraction depends on harmonic r. h in SF for Equation (1) & QSF for (2) | ||

c_{s} = 1 | c_{s} < 1 | In quasi Bragg h' c_{s} < h d ′ τ 00 = 0.205 ´ c_{s} nm d ′ 100 = 0.205 ´ c_{s} ´ τ = 0.296 nm, = a' |

equally simple mathematics (Equation (2), Equations (5)-(7)), and equally clear diffraction evidence, but require a more complex model. The equations and evidence imply that the sums on the cosines in Equation (7), when each is multiplied by planar densities taken from the structure, result in specular reflection from imaginary planes that are 10% shifted from the corresponding Bragg condition. This is the most remarkable feature of the hierarchic scattering that is observed in geometric series (Equation (2)). The shift is 1 − c_{s} at the peak.

Notice that in the experimental high energy electron diffraction pattern, the intensities of the higher orders are further and normally reduced by the deviation parameter, that result from larger scattering angles [

All atom sites in Equation (7) were summed for the supercluster order 2, but with four adjustments: edge cell atoms that were counted twice were entered by halving the atomic scattering factor f_{Al}; secondly, those that were counted thrice, at intersections of three cells, were entered with one third f_{Al}. Thirdly, mobile sites, where one atom shares two sites were counted as for edge sharing; and fourthly, some details were ignored because of their small effects and uncertainty. For example, Pauling’s observation [

Given the method of indexation described, the scattering is thus seen to be coherent when the metric c_{s} = 0.894^{3}.

_{s} is found to be −0.186, i.e. negative. A similar relationship between indexation with interplanar spacing applies to quasicrystals as to crystals (namely d_{hkl} = a/(h^{2} + k^{2} + l^{2})) so the derivative dd_{hkl}/dc_{s} must be positive. This fact is used to derive the true quasicrystal lattice parameter a' in

The parameter is at first sight ambiguous because the unit cells are edge sharing. Fixed are the stretching factor τ^{2}; the icosahedron cell length τ∙a', and its side length 1∙a'; but the two most obvious subordinate cells are overlapping cubes side τ∙a', or floating cubes side a' with severe underfilling. The measured lattice parameter is reported to be a = 0.205 nm [^{−1} and h respectively, in a (h00) line. However in the quasicrystal,

the off-Bragg condition requires correction for c_{s}. Then the cell side length measures as 0.205 ´ c_{s} ´ τ = 0.296 nm (_{6}Mn, as the normal diameter of Al in the unit cell of pure Al. We therefore define the quasi-lattice parameter as the width of the icosahedral unit cell. When we progress to the reciprocal lattice, it is convenient to express the parameter in simplified icosahedral units, a' = 1. This corresponds with the fact that in hierarchic quasicrystals, diffraction measurement occurs through the compromise multi-spacing effect, or metric, c_{s}.

Quasicrystal crystallographers sometimes ask about the reciprocal lattice in quasicrystals because the reciprocals provide important understanding in measurement from crystals.

This understanding extends to most of the solid state physics of crystals, including energy band structures, conductivity etc. In early researches, alternative bases were sometimes used to index the diffraction pattern, but the Euclidean axes of the O_{h} subgroup of the icosahedral point group are now standard.

Should we use coordinates that are Cartesian, or coordinates in geometric series? The former represent the unit cell parameter a' in linear order, and so are the more convenient choice for some purposes. More generally, the geometric series expansion of the hierarchic solid relates to the true structure.

Meanwhile, the reciprocal lattice in momentum space has the same point group symmetry as the solid structure and Euclidean axes are again the best choice. The 3-dimensional lattice may be calculated from QSFs using power values for h, k and l equal to τ^{m}, m = −infinity, −1, 0 1, 2, 3… A diffraction pattern is a projection of a reciprocal lattice onto two dimensions. The reciprocal lattice may be recorded by abscissae coordinates in either linear or geometric series. The latter series has been used to construct dispersion curves ( [^{m} coordinates. By simple transformation, extended zones can be represented on alternative linear coordinates. Another example is for high energy electrons used in imaging. Dispersion is represented in momentum ordinates with scattering vector abscissae ( [

Knowing the clear icosahedral structure, corresponding progress is also expected in defect structures (e.g. [

We do not ask “What is a quasicrystal?” [

The author declares no conflicts of interest regarding the publication of this paper.

Bourdillon, A.J. (2019) The Reciprocal Lattice in Hierarchic Quasicrystals. Journal of Modern Physics, 10, 624-634. https://doi.org/10.4236/jmp.2019.106044