Diffraction pattern of an icosahedral quasicrystal
Inspired by the growing numbers and varieties of quasiperiodic crystals, the International Union of Crystallography has redefined the term crystal to mean ``any solid having an essentially discrete diffraction diagram'', thereby shifting the essential attribute of crystallinity from position space to Fourier space. Within the family of crystals one distinguishes between periodic crystals, which are periodic on the atomic scale, and aperiodic crystals which are not. This broader definition reflects our current understanding that microscopic periodicity is a sufficient but not a necessary condition for crystallinity. In our research we deal with questions concerning the symmetry of crystals. We classify crystals according to their symmetry using an approach in Fourier space which treats periodic and quasiperiodic crystals on an equal footing.
Each Bragg peak in the discrete diffraction pattern determines a wave vector at which the density has a nonvanishing coefficient in its Fourier expansion. The (reciprocal) lattice is defined as the set of all integral linear combinations of the wave vectors determined by the diffraction pettern. As so defined, it includes wave vectors at which the peaks are too weak to be detected. As the resolution is improved more peaks may appear at larger wave vectors and in the quasiperiodic case between already existing peaks. This is because quasiperiodic lattices are `dense' in the mathematical sense -- there is no requirement of minimal distance between wave vectors. The absence of such a requirement is what allows lattices to have symmetries which are forbidden in periodic crystals (n-fold rotations with n=5, and n > 6).
The minimum number of vectors needed to generate the lattice is called the rank of the lattice. A crystal is periodic if and only if the rank of its lattice is equal to the physical dimension. Only then is the lattice a conventional `reciprocal lattice' related in the familiar way to a lattice of real-space translations under which the periodic crystal is invariant.
The point group of a periodic crystal is traditionally defined as the set of rotations and reflections which leave the density itself invariant to within a translation. Consider for example a 4-fold rotation of Escher's drawing above. The densities of aperiodic crystals, however, in general possess no such symmetries.
Penrose tilings, as the ones shown here, are often used as real space realizations of structures with 5-fold and 10-fold rotational symmetry. They produce diffraction patterns like the one at the top of this document. The red tiling is a 10-fold rotated version of the blue one. A careful examination shows that any region in the rotated aperiodic crystal can be found in the unrotated crystal, but the larger the region the further away you have to look in order to find it. The two crystals contain the same statistical distribution of substructures on all scales and are said to be indistinguishable . The point group of a crystal is therefore redefined as the set of rotations and reflections that take the density into an indistinguishable one.
Color Symmetry
To make things even more interesting we add colors (or other attributes) to the crystals. The point group is then generalized to be the set of rotations and reflections that take the density into an indistinguishable one up to a global permutation of the colors. In the periodic example of Escher's fish apply a 4-fold rotation in combination with the permutation (red, blue, yellow, green) to get back the original figure, a 2-fold rotation with the permutation (red, yellow) (blue, green), etc.
In the quasiperiodic tiling below on the left, a 10-fold rotation and a horizontal mirror reflection must be accompanied by the exchange of red and orange to produce a tiling indistinguishable from the original one, whereas the vertical mirror does not require the exchane of colors. In the tiling on the right, both horizontal and vertical mirrors require the exchange of colors but the 10-fold rotation does not. The two tilings have different color point groups.
