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In order to understand why this observation is surprising, let us briefly recall one of the basic results of crystallography, the so-called crystallographic lemma or crystallographic restriction. It states that, in two and three dimensions, a lattice (and by that we mean a periodic lattice obtained by translates of its unit cell) can only have rotational symmetry axes of order one, two, three, four or six, corresponding to angles of 360°, 180°, 120°, 90° and 60°, respectively, and integer multiples of these elementary rotation angles. The proof of this lemma is elementary. If a rotation is a symmetry of the lattice, it has to map the vectors that span the lattice onto integer linear combinations of themselves. This immediately restricts the set of possible angles p to those where 2cos(p) is integer, which gives precisely the list above with their multiples. As there are obvious two-dimensional examples for these angles, it is also clear that the corresponding symmetries are indeed realized.
What does this mean for the interpretation of the diffraction patterns? Clearly, there is no three-dimensional lattice showing icosahedral symmetry, and none with a symmetry axis of order five, eight, ten, or twelve. Nevertheless, the sharpness of the observed peaks shows that the atomic positions in these solids must be long-range ordered; quasicrystals are clearly distinct from amorphous solids. The only solution that remains is to give up the idea that the atoms are arranged according to a periodic lattice, that is, the arrangement of atoms in a quasicrystal is ordered, but aperiodic. For the icosahedral phase, this is true in all directions of space, there is no periodicity whatsoever. In contrast, the T phases, as mentioned above, are periodic in one direction, so they consist of a periodic sequence of slices with twelve-, ten-, or eightfold symmetry. From the same arguments it then follows that the arrangements of atoms in these planes cannot be periodic, but again there has to be long-range order to account for the sharp diffraction spots.
But how can an aperiodic structure give rise to a sharp diffraction pattern consisting of Bragg peaks? Before quasicrystals entered the scene, it had been the folklore that only periodic arrangements result in pure point diffraction patterns, and this maybe explains why part of the scientific community had difficulties with the interpretation of quasicrystals as aperiodically ordered solids. In order to resolve this apparent dilemma, one has to be careful about the precise notion of a pure point Fourier transform, and what exactly is meant by "discrete" in this context. Indeed, there are structures that have pure point Fourier transform with non-crystallographic symmetries, but for these the Bragg peaks densely distribute over the space, though they are located at positions in space that are countable in the mathematical sense - like, for instance, the rational numbers form a countable dense subset of the real numbers, and hence have zero measure. At first, one might reject these structures because a dense set of Bragg peaks does not seem to fit the experimental observations. However, if one restricts the pattern to Bragg peaks with an intensity above a certain threshold, these sets of most intensive peaks describe the experimentally observed patterns very well. This is also corroborated by the observation that, increasing the exposure time or the sensitivity, more and more diffraction spots can also be found in experiment.
Now, what do these structures look like and how can one get hold of them? The possibly easiest way of understanding what is going on requires to leave our three-dimensional space for a while and consider higher-dimensional structures. The crystallographic lemma in the form given above only applies to two and three dimensions. In four dimensions, there exist periodic lattices with a symmetry axis of order five, eight, ten, or twelve, and icosahedral symmetry can be found in six-dimensional periodic lattices. However, our quasicrystals clearly "live" in three dimensions and do not know about higher-dimensional periodic arrangements. Nevertheless, the higher-dimensional "superspace" can be of use to describe a structure in three-dimensional space. The idea is to project part of the lattice points of the higher-dimensional lattice to two or three dimensions, choosing the projection such that one preserves the rotational symmetry. In this way, one ends up with a quasiperiodic point set or a quasiperiodic tiling of two- or three-dimensional space. By a suitable choice of the subset of lattice points that are projected to the "physical" space, one can achieve that the resulting set is both uniformly discrete, that is there exists a minimal distance between points, and relatively dense, which means that it does not contain holes of arbitrary size. In mathematical terminology, a point set having these two properties is called a Delone (or Delauney) set [R:Baa, R:Moo]. In fact, the point set obtained by projection is more special, it is a so-called harmonious set or Meyer set [B:Mey, R:Moo] because the set of difference vectors between its points, which are projections of the difference vectors of the higher-dimensional lattice, is also a Delone set. The points that are projected to the "physical" space to form the tiling are usually selected by cutting out a "slice" from the higher-dimensional lattice. Therefore, this method of constructing a quasiperiodic tiling is known as the cut-and-project method.
The diffraction pattern of the projected structure is clearly pure point since it stems from a periodic higher-dimensional lattice, but becomes dense after projecting to the two- or three-dimensional space. By construction, it also shows the desired non-crystallographic symmetry, although the point set or tiling one obtains does not, in general, obey this symmetry in the strict sense, with the possible exception of a single position with global non-crystallographic symmetry. Nevertheless, the concept of symmetry can be generalized to properly deal with this situation [R:Baa]; and the symmetry of the diffraction pattern is indeed perfect, since the intensities of the peaks reflect the symmetry of the pair correlation function, also known as the Patterson function, and not the symmetry of the original tiling. This already points to a major problem for the structure analysis - from the experimentally measured intensities, only the Patterson function can be reconstructed, but not the scattering density itself. This is due to the phase information missing in the diffraction intensities. This observation also has another consequence, it implies that one has to generalize the usual notions of space groups if one intends to treat quasicrystal symmetries on an equal footing. It has been attempted to avoid this problem by restricting the discussion to Fourier space [R:Mer, RMW87, MRW87, RMW88], but it is in fact possible to extend the conventional notions in a rather natural way that is appropriate for these structures, see [R:Baa] for details. Note that there are also rather bizarre sets that have a pure point diffraction pattern, see [BGW94] for an example.
The structures obtained by projection from higher-dimensional lattices, frequently referred to as cut-and-project sets, provide one example of the construction of quasiperiodic tilings in space, the most famous among these structures being the celebrated Penrose tiling [Penr74, Gard77, Penr78]. The concept of cut-and-project sets can be re-cast into the more general setting of model sets [B:Mey, R:Mey, R:Moo], allowing for an even richer set of examples than the quasiperiodic tilings usually considered [BMS98]. A more detailed description of other ways to construct quasiperiodic tilings and of their properties is given below.
In addition to icosahedral, dodecagonal, decagonal, and octagonal phases, also some "exotic" quasicrystals have been found. On one hand, one-dimensional quasicrystals [HLZK88, ZK90, KKKD+97] were observed that are related to the Fibonacci sequence, the paradigm of a one-dimensional quasicrystal. An electron diffraction pattern of a one-dimensional modification of decagonal Al-Co-Ni [KKKD+97] is also shown in figure 1 [RBNG+98]. On the other hand, there may, in principle, exist quasicrystals with crystallographic symmetries, that is, non-periodic solids which show a crystallographic symmetry. One such example of a "cubic quasicrystal" has been discussed in the literature [DSPH+96] which, however, can be interpreted as a modulated phase [EP99]. Also the existence of further T phases with exotic symmetries such as a sevenfold rotational axis cannot be excluded on principle grounds. Corresponding tilings could, for instance, be obtained by a cut-and-project scheme from a six-dimensional lattice, but so far no such structures have been found experimentally. However, it may be interesting to note that the symmetries observed in quasicrystals are precisely those that are related to characteristic quadratic irrationalities [Levi88, BJKS90], any other rotational symmetries would necessarily involve cubic or higher-order irrationalities.
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Uwe Grimm,
last modified on
14 September 2005