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. [4] was entitled “a metallic phase with long range order an no translational symmetry”. Imaging showed multiple interplanar spacings irregularly jumbled. Bragg’s specific d_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 Al₆Mn, 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 [5] Mn is readily specified and located [3] [6] [7] . The image demonstrates a hierarchic icosahedral structure that is four tiers deep and apparently expansive indefinitely. Every atom is specified, located and measured for size.

The unit cell is edge sharing. This results in the 13 atom unit cell having the stoichiometry Al₆Mn (Figure 1). It is extremely dense owing to the precise ratio of the central Mn atom to outer Al atoms. All diatomic quasicrystals share the same ratio.

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 [8] [9] . So also are the principal diffraction planes that are normal to the axes. Following William of Ockham, dimensions should not be multiplied without necessity. Inventing dimensions does not solve physical problems¹. All of the beams in the original data [4] have been indexed on this basis ( [6] and refs.) and their quasi-structure factors (QSFs) calculated [3] [6] [7] and summarized as follows.

The sites of atoms and cell centers in icosahedral clusters are known [7] [9] [10] , including sites in higher order p of supercluster, where indices multiply by τ^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:

¹Crystallographers 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 R³—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 (Figure 2) to illustrate the variance of quasi-Bragg diffraction from Bragg diffraction in crystals.

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 [4] have been calculated to match. All structure factors, i.e. at the Bragg condition c_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 [11] [12] can be observed as lattice images observed, for example, in the two beam Bragg condition. The waves occur as interference between incident and reflected waves. Two are simulated as the red trace for the quasicrystal (Figure 3 with c_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 (Figure 4). The triad scales within the hierarchy for each order of unit cell, cluster and supercluster. The listing of interplanar spacings illustrates the origin of the geometric series in the quasicrystal diffraction pattern.

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 Figure 2, the line widths narrow with increasing power on the index, consistent with increasing numbers of scattering atoms. The diffraction consists in coherent scattering from hierarchic bodies of the same order.

The principal planes determine the coherence factor and metric (Table 1). The planes lie in geometric series. This series is irrational but a rational approximation may be constructed by the substitution of the rational fraction 3/2 for τ as shown in the table. It is obvious that atoms located on the half-integral set of “coherent values” (column 7) could diffract coherently according to Bragg’s law if such a structure could be constructed. It follows that, owing to the systemically derived value for the ratio r, atoms located on the “real” geometric-series lattice will likewise diffract coherently, but with the metric displacement c_s ~ 0.894, as in Figure 2. This fact is confirmed by the numerical QSF calculations. Coherent

r*r⁺: ( ∵ Intensities α QSF²) 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−(t² − 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 (Figure 5). Their superposition is described by the QSFs.

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 Figure 4: the diffraction consists in coherent scattering from sub-cluster centers rather than Bragg biplanes.