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A set of discoveries are described that complete the structural model and diffraction theory for quasicrystals. The irrational diffraction indices critically oppose Bragg diffraction. We analyze them as partly rational; while the irrational part determines the metric that is necessary for measurement. The measurement is verified by consistency with the measured lattice parameter, now corrected with the metric and index. There is translational symmetry and it is hierarchic, as is demonstrated by phase-contrast, optimum-defocus imaging. In Bragg’s law, orders are integral, periodic and harmonic; we demonstrate harmonic quasi-Bloch waves despite the diffraction in irrational, geometric series. The harmonicity is both local and long range. A breakthrough in understanding came from a modified structure factor that features independence from scattering angle. Diffraction is found to occur at a given “quasi-Bragg condition” that depends on the special metric. This is now analyzed and measured and verified: the
* metric function* is derived from the irrational part of the index in three dimensions. The inverse of the function is exactly equal to the metric that was first discovered independently by means of “quasi-structure factors”. These are consistent with all structural measurements, including diffraction by the quasicrystal, and with the measured lattice parameter.

“A metallic phase with long range order but with no translational symmetry” [

The original solid-state phase was discovered in 1982 and published in 1984 [_{6}Mn icosahedral alloy, owing to high atomic scattering factor. Knowing all of the microscope magnification, the image pattern, and the diffraction pattern, it was obvious that the unit cell is Al_{12}Mn, with stoichiometry Al_{6}Mn because of edge sharing. These cells are hierarchically arranged, each order having 6 five-fold rotation axes. In particular, four tiers of hierarchic structure are evident in the data and this structure was shown to be infinitely extensible: it is logarithmically periodic with period τ 2 , the square of the golden section τ = ( 1 + 5 1 / 2 ) / 2 [

To calculate the diffraction pattern from the known structure, it was necessary to correctly index [

A breakthrough in understanding of the diffraction pattern followed our realization that the structure factor is independent of scattering angle and can be simply calculated from our knowledge of the structure [

The outstanding question remained, what is the metric and how does the geometric series diffraction occur? It is not consistent with Bragg diffraction; but rather diametrically opposed by geometric, aperiodic, and anharmonic orders n , and multiple spacings d at any given Bragg condition. The new quasi-Bragg law was evident: τ m λ = 2 d ’ sin ( θ ’ ) , where through multiple simulations, the role of the metric defined all of the lattice parameter a ; the order n ’ → m ; the quasi-interplanar spacing d ’ ; and quasi-Bragg angle θ ’ (the compliment to the angle of incidence). After completely understanding the metric in this unique diffraction, the varieties of data verify both the structure and the diffraction.

The sites of atoms and cell centers in icosahedral clusters are known [

Unit cell ( r u ): Mn ( 0 , 0 , 0 )

Al 1 2 ( ± τ , 0 , ± 1 ) , 1 2 ( 0 , ± 1 , ± τ ) , 1 2 ( ± 1 , ± τ , 0 ) (1)

and

Cell or cluster centers ( r c c ) 1 2 ( ± τ 2 , 0 , ± τ ) , 1 2 ( 0 , ± τ , ± τ 2 ) , 1 2 ( ± τ , ± τ 2 , 0 ) (2)

The QSF formula is adapted from classical crystallography with two differences:

F h k l = ∑ i = 1 allatoms f i cos ( 2 π ⋅ c s ( h h k l ¯ ⋅ r i ¯ ) ) (3)

firstly, because the diffraction is sharp in spite of multiple interplanar spacings d , a coherence factor c s is inserted. Its value will be derived analytically below. The factor is used as a scanned variable (

Secondly, because the unit cells are not periodic as in crystals, the summation is made over all atoms in the QC; not just the unit cell. The summation is taken in two steps: over the unit cell and cluster, and iteratively over the superclusters in hierarchic order p . 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 ¯ , 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 ¯ ) (4)

with corresponding summations over unit cell sites and cell centers, and knowing that N cluster = N c c ⋅ N u , the QSF for the cluster is 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 (5)

and repeating iteratively over superclusters by using the known stretching factor τ 2 p :

F h k l p = ∑ i = 1 allatoms cos ( 2 π ⋅ c s ( h h k l ¯ ⋅ τ 2 p r c c ¯ ) ) ⋅ F h k l p − 1 (6)

The example in

Bragg diffraction is bi-planar: the path difference between two rays reflected from neighboring Bragg planes is equal to the wavelength of light. QC diffractions multiplanar, as is observed in high resolution electron micrographs: within the “quasi-periodic solids” every atom scatters. To know how the phases of the various scattered rays add, it is necessary to calculate the QSF. The addition is iterative (Equation (6)). Subclusters locate on the corners of the golden rectangles shown in

Interplanar spacings are ordered like the diffraction pattern: 0 , 1 , τ , τ 2 , τ 3 , τ 4 , τ 5 , ⋯ It is evident that whereas Bragg diffraction occurs by coherent scattering from Bragg planes, hierarchic diffraction occurs by coherent scattering from subcluster centers. How, more precisely, this happens will be illustrated with quasi-Bloch waves. These waves differ from both the Bloch wave in crystals and from the Bragg diffracted wave beam. Bloch waves are evident as lattice images in the two-beam condition [

Meanwhile, it is obvious in

Column 7 in

Fibonacci series | Geometric series | Irrational values | Rational Approx. | Irr.-Rat. residue | Commensurate divisor | Residue/ divisor | Metric | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

τ m − 1 | c s | ||||||||||||

m | F m | F m + 1 | τ | a + 3 b / 2 | / F m + 1 | Δ = τ − 3 / 2 | 1 / ( 1 + Δ ) | ||||||

0 | τ | τ 0 | = | 1 | 1 | 0 | 0 | ||||||

1 | 0 | 1 | τ | = | τ | = | 1.618034... | 1.5 | 0.11803 | 1 | 0.118034 | 0.894427 | |

2 | 1 | + | 1 | τ | = | τ 2 | = | 2.618034… | 2.5 | 0.11803 | 1 | 0.118034 | 0.894427 |

3 | 1 | + | 2 | τ | = | τ 3 | = | 4.236068… | 4 | 0.23607 | 2 | 0.118034 | 0.894427 |

4 | 2 | + | 3 | τ | = | τ 4 | = | 6.854102 | 6.5 | 0.3541 | 3 | 0.118034 | 0.894427 |

5 | 3 | + | 5 | τ | = | τ 5 | = | 11.09017 | 10.5 | 0.59017 | 5 | 0.118034 | 0.894427 |

6 | 5 | + | 8 | τ | = | τ 6 | = | 17.944272 | 17 | 0.94427 | 8 | 0.118034 | 0.894427 |

7 | 8 | + | 13 | τ | = | τ 7 | = | 29.034443 | 27.5 | 1.53444 | 13 | 0.118034 | 0.894427 |

8 | 13 | + | 21 | τ | = | τ 8 | = | 46.978715 | 44.5 | 2.47872 | 21 | 0.1180341 | 0.894427 |

9 | 21 | + | 34 | τ | = | τ 9 | = | 76.013159 | 72 | 4.01316 | 34 | 0.1180341 | 0.894427 |

10 | 34 | + | 55 | τ | = | τ 1 0 | = | 122.99188 | 116.5 | 6.49188 | 55 | 0.1180341 | 0.894427 |

11 | 55 | + | 89 | τ | = | τ 11 | = | 199.00504 | 188.5 | 10.505 | 89 | 0.1180341 | 0.894427 |

12 | 89 | + | 144 | τ | = | τ 12 | = | 321.99691 | 305 | 16.9969 | 144 | 0.1180341 | 0.894427 |

integral or half integral. QSF calculations show it is half integral. The rational approximation to the geometric series indices is listed in col. 10. Calculation of the QSFs for this imaginary series is Bragg-like with c s = 1 . This demonstrates the fact that the metric is an expression of the irrational part of the geometric sequence in col. 7. To derive the metric, subtract the rational part from the irrational sequence (col. 7 - col. 10) to give the residue in column 11. Notice that this is a growing number down the sequence, and that it can be harmonized by the integers F m + 1 ( 0 , 1 ) in col. 12, i.e. the Fibonacci sequence F m + 1 = 0 , 1 , 1 , 2 , 3 , 5 , ⋯ , with the argument representing the first two terms. The re-normalization corresponds to the increasing number of periods between intercepts illustrated in

The result is summarized in

1 c s = 1 + τ m − F m + 4 / 2 F m + 1 = 1 0.894 (7)

To illustrate, the first six terms are shown in the figure: the metric is exactly derived from the irrational part of the index, the part that is completely absent in the Bragg formula.

The metric may be derived in several ways, one is as follows: An index τ m is separated into rational and irrational parts while τ is separated into the rational semi-integral 3/2 and an irrational residue Δ = τ − 3 / 2 (

τ m = F m + F m + 1 ⋅ τ = F m + F m + 1 + F m + 1 / 2 + F m + 1 ⋅ Δ = F m + 2 + F m + 1 / 2 + F m + 1 ⋅ Δ = F m + 4 / 2 + F m + 1 ⋅ Δ (8)

Since, by general properties of the Fibonacci sequence:

F m + 4 / 2 = F m + 3 / 2 + F m + 2 / 2 = F m + 2 / 2 + F m + 1 / 2 + F m + 2 / 2 = F m + 2 + F m + 1 / 2 (9)

Equation (8) is thereby systematically confirmed from

Δ = τ m − F m + 4 / 2 F m + 1 (10)

So that the metric function:

1 c s = 1 + Δ = 1 0.894 (11)

in complete agreement with the QSF simulations where c s = 1 / 1.1180 , and Δ = τ − 3 / 2 = 0.1180 . This irrational part (

The metric can be derived in further ways: notice, for example, that if τ is supposed to vary, then c s → 1 as τ → 0 , i.e. at the Bragg condition. Generally however, the metric commensurates and harmonizes the diffracted sine wave onto a geometric grid [

A summary of the structural result is shown in

diffraction, namely that all diatomic, icosahedral QCs have atomic diameter ratios: D solute / D solvent = ( 1 + τ 2 ) 1 / 2 − 1 , as indeed do D Mn / D Al in the figure. The fact is consistent with high local density as the structural driving force.

The true measurement of the lattice parameter, with the correction given by the metric and index under the quasi-Bragg law (

The quasicrystal has inter-related icosahedral symmetry with diffraction in geometric series: an incident X-ray or electron beam scatters off the hierarchic lattice into geometric space. The metric, that is measured and experimentally verified, is now completely understood. This is unique and novel in QCs. Quasicrystallographers have, for 38 years refused to accept this fact, though some, like Senechal have acknowledged shortcomings. For example, a sub-editor of Acta Crystallographica wrote that you don’t measure the lattice parameter, “You just have to choose ‘ d h ’”, the interplanar spacing [ [

The quasi-Bragg law is a new law in physics. Now that the structure and diffraction are clear for anybody to see and understand, we should turn our attention to outlying problems. These include quasicrystalloids in which planar five-fold, six-fold and eight-fold quasicrystal symmetries link to regular linear symmetries on planar normals. The icosahedral unit cell may share edges in various ways, and metrics are likely dependent on them individually. Another class of problems is defects, especially in rapidly quenched material. Some anomalous microstructures have been recorded, by convergent beam diffraction for example [

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

Bourdillon, A.J. (2020) Complete Solution for Quasicrystals. Journal of Modern Physics, 11, 581-592. https://doi.org/10.4236/jmp.2020.114038