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 following Bragg'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. 53 1951-1953] . 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/$au$|.... [ D.Levine and P.Steinhardt (1986) "Quasicrystals I. Definition and structure" Phys. Rev. 34 596-616] , where the golden ratio $au$=(1+5^1/2)/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 in Fibonacci 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"₆"Mn" [http://www.UHRL.net] ] . 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$).

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