Quasicrystals fivefold diffraction pattern

The otiier type of noncrystalline solid was discovered in the 1980s in certain rapidly cooled alloy systems. D Shechtman and coworkers [15] observed electron diffraction patterns with sharp spots with fivefold rotational synnnetry, a syimnetry that had been, until that time, assumed to be impossible. It is easy to show that it is impossible to fill two- or tliree-dimensional space with identical objects that have rotational symmetries of orders other than two, tliree, four or six, and it had been assumed that the long-range periodicity necessary to produce a diffraction pattern with sharp spots could only exist in materials made by the stacking of identical unit cells. The materials that produced these diffraction patterns, but clearly could not be crystals, became known as quasicrystals. 

Figure Bl.8.6. An electron diffraction pattern looking down the fivefold synnnetry axis of a quasicrystal. Because Friedel s law introduces a centre of synnnetry, the synnnetry of the pattern is tenfold. (Courtesy of L Bendersky.)...

Electron diffraction pattern of an AIMn quasicrystal along the fivefold axis (left) and a computed Fourier pattern of a three-dimensional Penrose tiling (right). From C. Janot, Quasicrystals, A Primer, 2nd ed. (London Oxford Univ. Press, 1994), p. 3, figure 1.24. 

Quasicrystals are solid materials exhibiting diffraction patterns with apparently sharp spots containing symmetry axes such as fivefold or eightfold axes, which are incompatible with the three-dimensional periodicity associated with crystal lattices. Many such materials are aluminum alloys, which exhibit diffraction patterns with fivefold symmetry axes such materials are called icosahedral quasicrystals. " Such quasicrystals " may be defined to have delta functions in their Fourier transforms, but their local point symmetries are incompatible with the periodic order of traditional crystallography. Structures with fivefold symmetry exhibit quasiperiodicity in two dimensions and periodicity in the third. Quasicrystals are thus seen to exhibit a lower order than in true crystals but a higher order than truly amorphous materials. 

Quasicrystals represent the third type of aperiodic materials. Quasiperiodicity may occur in one, two, or three dimensions of physical space and is associated with special irrational numbers such as the golden mean r = (1 -h /5)/2, and = 2 -F V3. The most remarkable feature of quasicrystals is the appearance of noncrystal-lographic point group symmetries in their diffraction patterns, such as 8/mmm, lO/mmm, l2/mmm, and 2lm35. The golden mean is related to fivefold symmetry via the relation r = 2 cos( r/5) r can be considered as the most irrational number, since it is the irrational number that has the worst approximation by a truncated continued fraction,... 

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