# Modified Bragg diffraction in quasicrystals

It is commonly assumed that diffraction in quasicrystals, whether of electrons, of x-rays or of neutrons, is Bragg diffraction. Though there are similarities, notably scattering from atomic planes, there are also contrasting differences [

*A.J. Bourdillon (1987) "Fine line structure in convergent-beam electron diffraction of icosahedral" "Al"*] :_{6}"Mn", Phil. Mag. Lett.**55**, 21-26**Bragg diffraction**Bragg diffraction was defined for crystals. It is used to determine their structures. Crystals are periodic under translation and contain orientational symmetries consistent with the fourteen

Bravais lattices . Interplanar spacings are regular, e.g. |"d"|"d"|"d"|... The diffraction results from constructive interference due to ordered planes of atoms followingBragg's law , n$lambda$=2d sin$heta$ , for scattering with wavelength $lambda$, order n and at Bragg angle $heta$ .**Quasicrystal diffraction**Quasicrystals display strict orientational symmetries without long range translational order. The symmetries are inconsistent with the Bravais lattices, and may contain five–fold rotations [*D. Schectman, I. Blech, D.Gratias and J.W.Cahn (1984) "Metallic phase with long-range orientational order and no translational symmetry" Phys. Rev. Lett.*] . In the narrow sense, and in the short range, typical patterns can be explained by alternating periodicities of the type |d|d/$au$|d|d/$au$|d|d/$au$|.... [**53**1951-1953*D.Levine and P.Steinhardt (1986) "Quasicrystals I. Definition and structure" Phys. Rev.*] , where the golden ratio $au$=(1+5**34**596-616^{1/2})/2 . Then 1/$au$+1=$au$; 1+$au$=$au$^{2}; ..$au$^{"m"-1}+$au$^{"m"}=$au$^{"m"+1}.., "m" being positive or negative integral. Typically, but not always, quasicrystal diffraction patterns display scattering angles inFibonacci sequences 1, $au$, $au$^{2}, $au$^{3}....**Diffraction order "n"**Such spacings are inconsistent with Bragg’s law unless the order "n" is restricted to values of 0 or 1 [

*A.J. Bourdillon (2007) "Structure of" "Al"*] . The reason given, consistent with constructive interference within the quasicrystals, requires scattered wave amplitudes between adjacent planes having phase relations of the type exp(2$pi$i d/($lambda$$au$)) . exp(2$pi$i d/$lambda$) = exp(2$pi$i d$au$/$lambda$)._{6}"Mn" [*http://www.UHRL.net*]**Double diffraction**With this restriction in n, double diffraction along one dimension is neither simulated nor observed; but double diffraction is observed in the second dimension of the electron diffraction pattern

^{ [1] , [4] }. The last observation is abnormal, and is consistent with the last equation.**Fibonacci sequences and linear patterns**Consequently, in some diffraction patterns, as in the two-fold pattern from "Al"

_{6}"Mn", both Fibonacci sequences and linear sequences are evident and superposed^{ [4] }. Composite indexations, based on the unit cube in reciprocal space, allow remarkable agreement between calculated structure factors with observed diffraction beam intensities.**Planar alignment**Which diffraction sequence is selected depends on the alignment of Bragg planes in the direction of the scattering vector

^{ [4] }. Misalignment results in incoherent scattering in the quasicrystals.**Compromise Spacing effect**The Compromise Spacing Effect

^{ [4] }, 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.**References**

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