Quasiperiodic lattice

A relationship actually exists between periodic and quasiperiodic patterns such that any quasilattice may be formed from a periodic lattice in some higher dimension (Cahn, 2001). The points that are projected to the physical three-dimensional space are usually selected by cutting out a slice from the higher-dimensional lattice. Therefore, this method of constmcting a quasiperiodic lattice is known as the cut-and-project method. In fact, the pattern for any three-dimensional quasilattice (e.g., icosahedral symmetry) can be obtained by a suitable projection of points from some six-dimensional periodic space lattice into a three-dimensional subspace. The idea is to project part of the lattice points of the higher-dimensional lattice to three-dimensional space, choosing the projection such that one preserves the rotational symmetry. The set of points so obtained are called a Meyer set after French mathematician Yves Meyer (b. 1939), who first studied cut-and-project sets systematically in harmonic analysis (Lalena, 2006). 

One way to solve the problem of unphysically short atomic distances is to project onto the Rpm subspace only those grid points included in a selected strip (gray area), with width of a (cos a + sin a) in the A per subspace. The slope of RPai shown in Fig. 1 is 0.618..., an irrational number related to the golden mean [( /5 + l)/2 = 1.618...]. As a result, the projected ID structure contains two segments (denoted as L and S), and their distribution follows a ID quasiperiodic Fibonacci sequence [2] (c.f. Table 1). From another viewpoint, the ID quasiperiodic structure on the par subspace can be conversely decomposed into periodic components (square lattice) in a (higher) 2D space. The same strip/projection scheme holds for icosahedral QCs, which are truly 3D objects but apparently need a more complex and abstract 6D... 

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. 

The symmetry treatment of incommensurate structures is beyond the scope of this chapter. From Equation (33) it is readily seen that for indexing, whatever the reflection of the diffraction pattern of an incommensurately modulated structure, we need to specify 3 + d integers (h, k, I, m, m2... m fl. It can be demonstrated that the observed 3D structure can be considered as a projection of a periodic structure m3 + d dimensions over the real 3D space, which is a hyper-plane not cutting the points of the 3 + d lattice except the origin. The superspace approach of de Wolff, Janssen and Janner is now well established and has become the routine way of treating the symmetry of the displacive incommensurate structures. The same approach has been extended to study general quasiperiodic structures (composite structures and quasicrystals). 

Quasiperiodic tilings in R may be obtained by projecting certmn subsets of lattices from a higher-dimensional space R" into R . This is described by a cut-and-project scheme, summarised in the following diagram ... 

On a mathematically rigorous level, it may be possible to show the equality of the critical points for SAPs and SAWs on quasiperiodic tilings by appropriately modifying the existing proofs for the hypercubic lattice [68]. Furthermore, it would be interesting to carry out an analysis to determine if random walk behaviour can be proved for dimensions greater than four [4]. 

Fig.1.3-12a-d Some 2-D quasiperiodic tilings the corresponding four basis vectors ai,. .., 04 are shown. Linear combinations of r = M,a, reach all lattice points, (a) Penrose tiling with local symmetry 5mm and diffraction symmetry... 

Kaneko, K. Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-hke structures and spatial intermittency in coupled logistic lattice. Progress of Theoretical Physics 72, 480-486 (1984)... 

Quasiperiodicity in Antiferro-Like Structures and Spatial Intermittency in Coupled Logistic Lattice—Towards a Prelude of a Field-Theory of Chaos. 

To date, aU known 2-D quasiperiodic materials exhibit noncrystallographic diffraction symmetries of 8/mmm, 10/mmm, or 12/mram. The structures of these materials are called octagonal, decagonal, and do-decagonal structures, respectively. Quasiperiodicity is present only in planes stacked along a perpendicular periodic direction. To index the lattice points in a plane, four basis vectors a, 2. 4 are needed a fifth one,... 

Scheme 2 (a) Amphiphilic dendritic building blocks that promote self-organization into (b) lattices and quasiperiodic arrays. Part (b) adapted with permission from [43]. Copyright 2011 Wiley-VCH Verlag GmbH Co. KGaA, Weinheim... 

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