_{1}

^{*}

Thirty seven years after the discovery of quasicrystals, their diffraction is completely described by harmonization between the sine wave probe with hierarchic translational symmetry in a structure that is often called quasiperiodic. The diffraction occurs in geometric series that is a special case of the Fibonacci sequence. Its members are irrational. When substitution is made for the golden section
*τ* by the semi-integral value 1.5, a coherent set of rational numbers maps the sequence. Then the square of corresponding ratios is a metric that harmonizes the sine wave probe with the hierarchic structure, and the quasi-Bragg angle adjusts accordingly. From this fact follows a consistent description of structure, diffraction and measurement.

“Physical” theories degenerate easily to common myth when the basic norms of physical practice are ignored. These include not only verification by exclusion of falsifiable hypotheses, but also rigorous implementation of the formal and informal logic that has been endorsed by scientists for over two millennia [

In crystals, the order is positive integral: n = 0 , 1 , 2 , 3 ⋯ ; whereas in the quasicrystal; the order is represented in powers of the golden section τ^{m}, where τ = (1 + 5^{1/2})/2 and m is positive or negative, m = − ∞ , − 1 , 0 , 1 , 2 , 3 ⋯ corresponding to n = 0 , 1 / τ , 1 , τ , τ 2 , τ 3 ⋯ . The quasi-Bragg law is a new law in physics: τ m λ = 2 d ′ sin ( θ ′ ) , where the apostrophes indicate compromise superpositions of many Bragg values. The geometric series is a special case of the Fibonacci sequence: in the former case, the ratio between successive terms is constant; in the latter case the ratio oscillates about τ. Such oscillations are not observed in the quasicrystal diffraction pattern.

The discovery of Shechtman et al. [_{hkl} for an indexed beam (hkl) is therefore neither unique in the quasicrystal nor periodic, so the wonder that has to be explained is how the diffraction due to such a structure can be sharp. In the following discussion, we show precisely how that occurs and how the model is verified by measurement. Meanwhile neither n nor d obey Bragg’s law.

Since d is not unique, the Bragg angle is not defined and does not obey Bragg’s law either. However, we will show how the quasi-Bragg angle θ ′ is calculated, and it is certainly not the Bragg angle. There is no Bragg diffraction.

Given the composition Al_{6}Mn, we know that Al has the atomic number 13 and Mn 25. The scattering power for electrons used in transmission electron microscopy is four times greater for the transition metal. In phase-contrast optimum defocus [

The unit cell is edge sharing. This results in the 13 atom unit cell having the stoichiometry Al_{6}Mn (

Knowing the structure, we can simulate the diffraction pattern; but it is necessary first to correct indexation.

The stereogram of principal axes of the icosahedral structure is 3-dimensional, in geometric series, simple, and complete [^{1}. All of the beams in the original data [

The sites of atoms and cell centers in icosahedral clusters are known [^{2p}:

Unitcell ( r u ) : Mn ( 0 , 0 , 0 ) Al 1 / 2 ( ± τ , 0 , ± 1 ) , 1 / 2 ( 0 , ± 1 , ± τ ) , 1 / 2 ( ± 1 , ± τ , 0 ) (1)

and

Cellorclustercenters ( 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:

^{1}Crystallographers know that the hexagonal close packed structure is sometimes indexed with four digits; sometimes with three. Equally they know that the structure is 3-dimensional R^{3}—in this it is like the quasicrystal.

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. Initially 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 quasicrystal; 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 Ncluster 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 c l u s t e r = N c c ⋅ N u , the QSF for the cluster may be calculated:

F h k l c l u s t e r = ∑ i = 1 12 cos ( 2 π ⋅ c s ( h h k l ¯ ⋅ r c c ¯ ) ) ⋅ F h k l c e l l (5)

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

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 the figure is for the simple geometric series, but all beams in the original data [_{s} = 1, are zero. In the quasicrystal there is no Bragg diffraction: all beams peak at the quasi-Bragg condition c_{s} = 0.894. As we shall see, this value is the result of harmonization of the incident, sine wave probe with the aperiodic, hierarchic structure.

In crystals, Bloch waves [_{s} = 0.894), for comparison with the blue, anharmonic, (100), pseudo-Bragg condition, c_{s} = 1.

The quasi-Bloch wave harmonizes with the hierarchic structure; the pseudo-Bragg wave does not diffract.

The golden triad contains three orthogonal golden rectangles. There are three principal planes, one for each dimension of the golden triad (

For each order of cluster, golden triads mark principal planes that locate subcluster centers at corners. These centers operate as principal scatterers for the hierarchic structure. The diffraction occurs by reflection between hierarchic

centers of unit cells, clusters and superclusters. Each reflection is weighted by parallel reflections from the bodies of respective sub-clusters or sub-superclusters. That is why, in

The principal planes determine the coherence factor and metric (_{s} ~ 0.894, as in

Geometric series | Irrational | Rational | Ratio | r^{2} | c_{s} | QSF | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

a | b | value | approx. | r | estimates | supercluster | ||||||

a+b*1.5 | rat/irr | 1 − 2*(1 − r) | (order) | |||||||||

0 | = | 0 | = | 0 | 0 | 0 | 0 | 0 | ||||

1 | = | 1 | = | 1 | 1 | 1 | 1 | 1 | ||||

t | = | t | = | 1.61803 | 1.5 | 0.927 | 0.859 | 0.854 | ||||

1 | + | t | = | t^{2} | = | 2.61803 | 2.5 | 0.955 | 0.912 | 0.91 | ||

1 | + | 2 | t | = | t^{3} | = | 4.23607 | 4 | 0.944 | 0.891 | 0.888 | |

2 | + | 3 | t | = | t^{4} | = | 6.8541 | 6.5 | 0.948 | 0.899 | 0.896 | |

3 | + | 5 | t | = | t^{5} | = | 11.0902 | 10.5 | 0.947 | 0.897 | 0.894 | |

5 | + | 8 | t | = | t^{6} | = | 17.9443 | 17 | 0.947 | 0.897 | 0.894 | |

8 | + | 13 | t | = | t^{7} | = | 29.0344 | 27.5 | 0.947 | 0.897 | 0.894 | |

13 | + | 21 | t | = | t^{8} | = | 46.9787 | 44.5 | 0.947 | 0.897 | 0.894 | sc(2) |

21 | + | 34 | t | = | t^{9} | = | 76.0132 | 72 | 0.947 | 0.897 | 0.894 | |

34 | + | 55 | t | = | t^{10} | = | 122.992 | 116.5 | 0.947 | 0.897 | 0.894 | sc(3) |

55 | + | 89 | t | = | t^{11} | = | 199.005 | 188.5 | 0.947 | 0.897 | 0.894 | |

89 | + | 144 | t | = | t^{12} | = | 321.997 | 305 | 0.947 | 0.897 | 0.894 | |

144 | + | 233 | t | = | t^{13} | = | 521.002 | 493.5 | 0.947 | 0.897 | 0.894 | |

233 | + | 377 | t | = | t^{14} | = | 842.999 | 798.5 | 0.947 | 0.897 | 0.894 | |

377 | + | 610 | t | = | t^{15} | = | 1364 | 1292 | 0.947 | 0.897 | 0.894 | |

610 | + | 987 | t | = | t^{16} | = | 2207 | 2090.5 | 0.947 | 0.897 | 0.894 | sc(6) |

r*r^{+}: ( ∵ Intensities α QSF^{2}) here calculated for principal planes only; #: values simulared QSFs including ALL planess, and these match estimates; and elementary estimates: c_{s} = 1/(1 + (t − 1.5)) and c_{s} = 1/(1−(t^{2} − 2.5)).

diffraction occurs by the QSF selection of d ′ = d ⋅ c s _{ }and consequently of θ ′ = q / c s under the quasi-Bragg law. Consistent with the measured value for the lattice parameter to be discussed below, the metric c_{s} is the ratio of corresponding terms: (coherent value)^{2}/(real value)^{2}. The square on the ratio is due to wave mechanics, where the intensities of the beams are—for centrosymmetric structures—squares of corresponding amplitudes.

All atoms scatter. Whereas the uniqueness of d and its periodicity forces Bragg diffraction in crystals to reflect as biplanes; quasiperiodicity forces multiplanar reflections in hierarchic quasicrystals (

On the well-known model, the path difference between two reflections from adjacent Bragg planes is equal to the wavelength of the light, with c_{s} = 1. QSFs imitate the corresponding interference of the quasi-Bragg rays from multiple planes at the quasi-Bragg condition (i.e. when c_{s} = 0.894). Notice that the filled green quasi-Bragg angle is larger than the corresponding filled red Bragg angle.

In Bragg diffraction, when d contracts θ dilates; in quasi-Bragg diffraction, when c_{s} (and d’) contract, θ ′ dilates. The dilatation is enforced by a constructive interference requirement for harmonic reflections. Actually, the better model for quasicrystal diffraction is shown in

The correlating roles of quasi-structure factors and quasi-Bragg law are summarized in

From these measurements the reciprocal lattice can be derived [

Comparison of Bragg parameters in crystals, with quasi-Bragg parameters in quasicrystals | ||
---|---|---|

Bragg | Quasi-Bragg | Comment |

n = 2 d sin ( q ) / λ | τ m = 2 d ′ sin ( θ ′ ) / λ | Harmonic laws Give us θ ′ = θ / c s |

F h k l = ∑ f i cos ( 2 π h h k l ⋅ r i ) | F ′ h k l = ∑ f i cos ( 2 π c s ( h h k l ⋅ r i ) ) including iteration | Structure factors Give c_{s}, a and d ′ = d ⋅ c s |

d = a / h | d ′ = a c s h | θ ′ = θ / c s |

a = 0.205 τ c s nm [ | Measured lattice parameter a ≈ Diameter of Al | |

Measured and verified |

n: Bragg order; m: Quasi-Bragg order; d: Bragg interplanar spacing; θ: Bragg angle; λ: wavelength; τ: golden section; prime: quasi-Bragg compromise; F: Structure factors; f_{i}: atomic scattering factor for atom i; c_{s}: metric; r_{i}: atom position; h_{hkl}: plane normal for indices h, k, l; a: lattice parameter (cubic) ~ Al diameter; reciprocal: lattice vector a*= 2π/a.

icosahedral point group symmetry. The dual lattice is also consistent with the analysis. Moreover, the hierarchic translational symmetry is the obvious reason for the “long range order” [

Moreover, the QSF simulations, when combined with harmonic analysis, demonstrate that the diffraction occurs on a scale that is a contraction of the irrational hierarchic scale of the structure. In consequence, all structural measurements that are derived from the diffraction pattern, are subject to the measured metric.

In particular, the fact that the lattice parameter was previously tentatively measured by wrongly assuming Bragg’s law [

The analytic metric completes the union of structure with diffraction. The hierarchic structure transforms the plane incident wave into geometric space. It is time to append the context. In comparatively recent times Senechal wrote for the American Mathematical Society a paper titled, “What is a quasicrystal?” The paper began, “The short answer is no one is sure” [_{h}’” ( [_{h}.” He chose an untested law that never applies. It is not normal to comment on necessities of physical practice, nor on formal and informal logic; but 37 years are a long delay.

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

Bourdillon, A.J. (2019) Wave Harmonization in Hierarchic Quasicrystals by the Analytic Metric. Journal of Modern Physics, 10, 1364-1373. https://doi.org/10.4236/jmp.2019.1011090