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In reality, not all rotational symmetries are allowed in crystals. The only rotational symmetries possible in crystal lattices are 2, 3, 4 and 6, because it is impossible to fill space with other symmetries (e.g. 5, 7 …). Even though quasicrystals have long-range order with symmetries, their most important feature is still the non-crystallographic point group symmetries in their diffraction patterns (e.g. 8/mmm), i.e. the point group of a quasicrystal is not necessarily one of the well-known 32 point groups. The point groups of quasicrystals cannot occur in lattice periodic structures. For instance, quasicrystals, e.g. with 5-fold rotational symmetry (see Figure 3486a), can be formed. In fact, the typical diffraction patterns of quasicrystals exhibit 5-fold or 10-fold rotational symmetry.

Two-dimensional illustration of a quasi-crystal with 5-fold rotational symmetry

Figure 3486a. Two-dimensional illustration of a quasi-crystal with 5-fold rotational symmetry.

Quasicrystals [1] do not have a three-dimensional (3-D) periodic basic structure since their basic structure is incommensurate. The symmetry groups of quasicrystals are n-dimensional crystallographic point groups. Here, n is the rank of the Fourier module. In this case, they do not mix the coordinates in physical space and the coordinates in perpendicular space. Quasicrystals represent the third type of aperiodic materials and always show a non-crystallographic point-symmetry which is forbidden in periodic structures. The approach for describing the structures of quasicrystals are different than that for the structures of modulated crystals and composite crystals. One main difference is that it is not simple to distinguish main and satellites reflections for quasicrystals.

Quasiperiodicity may exist in 1-, 2-, or 3-dimensions of physical spaces and is related to special irrational numbers such as the golden mean golden mean ≈ 1.6180339887 that is related to five-fold symmetry through the relation golden mean. Here, τ is the interesting irrational number that has the worst approximation by a truncated continued fraction,

         interesting irrational number ------------------------------ [3486]

The known 2-D quasiperiodic materials present non-crystallographic diffraction symmetries of 8/mmm (octagonal), 10/mmm (decagonal), or 12/mmm (dodecagonal). The quasiperiodicity exist only in the planes stacked along a perpendicular periodic direction. In order to index the lattice points in such a plane, four basis vectors a1, a2, a3, a4 are employed; a fifth one, a5, describes the periodic direction. Therefore, a 5-D hypercrystal is used to describe the periodicity of the quasicrystals. In reality, these quasicrystals is composed of hundreds of atoms, called clusters. There are some different tilings that show such noncrystallographic symmetries in quasicrystal structures (e.g. shown in Figure 3486b).

Octagonal Quasicrystal
Octagonal Quasicrystal
Figure 3486b. 2-D quasiperiodic tilings: (a) Dodecagonal Stampfli-type with diffraction symmetry 12mm and (b) Octagonal-type with diffraction symmetry 8mm.

Note that chemical ordering normally exists in quasicrystalline phases, and thus HAADF imaging due to Z-contrast is an effective technique to analyze such structures.











[1] The Nobel Prize in Chemistry 2011 was awarded to Dan Shechtman for the discovery of quasicrystals.