Modified Bragg diffraction

Bragg diffraction is the chief method used to identify atomic structures in crystals. It is sometimes assumed that diffraction in quasicrystals, whether of electrons, x-rays or neutrons, is Bragg diffraction. Though there are some similarities - including scattering from atomic planes - there are also significant differences: In crystals diffraction patterns always form sequences in linear order; in quasicrystals the series are generally geometric.
Bragg Diffraction was defined for crystals. Crystals are periodic under translation and contain orientational symmetries consistent with the fourteen Bravais lattices. Interplanar spacings are regular. Bragg diffraction is used to determine the structure of crystals. It results from constructive interference due to ordered planes of atoms. The diffraction follows Bragg’s Law.
Quasicrystals display strict orientational symmetries without long range linear translational order. The symmetries are inconsistent with the Bravais lattices, and may contain 5-fold rotations. In the narrow sense, and in the short range, typical patterns can be explained by alternating periodicities: t:1:t:1:t where the golden ratio t=(1+root(5))/2 . Typically, but not uniquely, quasicrystal diffraction patterns display spacings in Fibonacci sequences.
Such spacings are inconsistent with Bragg’s law unless the order n is restricted to values of 0 or 1. The reason has been given that is consistent with constructive interference, within the quasicrystals, that is satisfied only by scattered wave amplitudes between adjacent planes having special phase relations.
With the restriction in n, double diffraction along one dimension is neither simulated nor observed; but double diffraction is observed in the second dimension of the diffraction pattern.
Consequently, in some diffraction patterns, as in the 2-fold pattern from Al<sub>6</sub>Mn, both Fibonacci sequences and linear sequences are evident and superposed. Composite indexations, based on the unit cube in reciprocal space, allow remarkable agreement between calculated structure factors with observed diffraction beam intensities.
Which diffraction sequence is selected depends on the alignment of Bragg planes in the direction of the scattering vector. Misalignment results in incoherent scattering in the quasicrystals .
The Compromise spacing effect , that is found both analytically and by simulation, provides a real quasilattice parameter that is larger than the corresponding Bragg interplanar spacing, d. This spatial effect is critical in fitting atoms into a theoretical structure.
A consequence of their extraordinary diffraction, the electronic band structures of quasicrystals are most conveniently represented on graphs having logarithmic scales .

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