International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 613621
Section 4.6.3.3.3. Icosahedral phases ^{a}Laboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, WolfgangPauliStrasse 10, CH8093 Zurich, Switzerland 
A structure that is quasiperiodic in three dimensions and exhibits icosahedral diffraction symmetry is called an icosahedral phase. Its holohedral Laue symmetry group is . All reciprocalspace vectors can be represented on a basis , , where , and , the angle between two neighbouring fivefold axes (Fig. 4.6.3.28). This can be rewritten as where are Cartesian basis vectors. Thus, from the number of independent reciprocalbasis vectors needed to index the Bragg reflections with integer numbers, the dimension of the embedding space has to be six. The vector components refer to a Cartesian coordinate system (V basis) in the physical (parallel) space.

Perspective (a) parallel and (b) perpendicularspace views of the reciprocal basis of the 3D Penrose tiling. The six rationally independent vectors point to the edges of an icosahedron. 
The set of all diffraction vectors remains invariant under the action of the symmetry operators of the icosahedral point group . The symmetryadapted matrix representations for the pointgroup generators, one fivefold rotation α, a threefold rotation β and the inversion operation γ, can be written in the form
Blockdiagonalization of these reducible symmetry matrices decomposes them into nonequivalent irreducible representations. These can be assigned to the two orthogonal subspaces forming the 6D embedding space , the 3D parallel (physical) subspace and the perpendicular 3D subspace . Thus, using , we obtainwhere , , , . The column vectors of the matrix W give the parallel (above the partition line) and perpendicularspace components (below the partition line) of a reciprocal basis in V. Thus, W can be rewritten using the physicalspace reciprocal basis defined above and an arbitrary constant c, yielding the reciprocal basis , in the 6D embedding space (D space) The symmetry matrices can each be decomposed into two matrices. The first one, , acts on the parallelspace component, the second one, , on the perpendicularspace component. In the case of , the coupling factor between a rotation in parallel and perpendicular space is 2. Thus a rotation in physical space is related to a rotation in perpendicular space (Figs. 4.6.3.28 and 4.6.3.29).

Schematic representation of a rotation in 6D space. The point P is rotated to P′. The component rotations in parallel and perpendicular space are illustrated. 
With the condition , the basis in direct 6D space is obtained: The metric tensors G, are of the type with for the reciprocal space and , for the direct space. For we obtain hypercubic direct and reciprocal 6D lattices.
The lattice parameters in reciprocal and direct space are and with , respectively. The volume of the 6D unit cell can be calculated from the metric tensor G. For it is simply
The best known example of a 3D quasiperiodic structure is the canonical 3D Penrose tiling (see Janssen, 1986). It can be constructed from two unit tiles: a prolate and an oblate rhombohedron with equal edge lengths (Fig. 4.6.3.30). Each face of the rhombohedra is a rhomb with acute angles . Their volumes are , and their frequencies ::1. The resulting point density (number of vertices per unit volume) is . Ten prolate and ten oblate rhombohedra can be packed to form a rhombic triacontahedron. The icosahedral symmetry of this zonohedron is broken by the many possible decompositions into the rhombohedra. Removing one zone of the triacontahedron gives a rhombicosahedron consisting of five prolate and five oblate rhombohedra. Again, the singular fivefold axis of the rhombicosahedron is broken by the decomposition into rhombohedra. Removing one zone again gives a rhombic dodecahedron consisting of two prolate and two oblate rhombohedra. Removing the last remaining zone leads finally to a single prolate rhombohedron. Using these zonohedra as elementary clusters, a matching rule can be derived for the 3D construction of the 3D Penrose tiling (Levine & Steinhardt, 1986; Socolar & Steinhardt, 1986).

The two unit tiles of the 3D Penrose tiling: a prolate and an oblate rhombohedron with equal edge lengths . 
The 3D Penrose tiling can be embedded in the 6D space as shown above. The 6D hypercubic lattice is decorated on the lattice nodes with 3D triacontahedra obtained from the projection of a 6D unit cell upon the perpendicular space (Fig. 4.6.3.31). Thus the edge length of the rhombs covering the triacontahedron is equivalent to the length of the perpendicularspace component of the vectors spanning the 6D hypercubic lattice .

Atomic surface of the 3D Penrose tiling in the 6D hypercubic description. The projection of the 6D hypercubic unit cell upon gives a rhombic triacontahedron. 
There are several indexing schemes in use. The generic one uses a set of six rationally independent reciprocalbasis vectors pointing to the corners of an icosahedron, , , , , with , the angle between two neighbouring fivefold axes (setting 1) (Fig. 4.6.3.28). In this case, the physicalspace basis corresponds to a simple projection of the 6D reciprocal basis . Sometimes, the same set of six reciprocalbasis vectors is referred to a differently oriented Cartesian reference system (C basis, with basis vectors along the twofold axes) (Bancel et al., 1985). The reciprocal basis is
An alternate way of indexing is based on a 3D set of cubic reciprocalbasis vectors (setting 2) (Fig. 4.6.3.32): The Cartesian C basis is related to the V basis by a rotation around , yielding , followed by a rotation around : Thus, indexing the diffraction pattern of an icosahedral phase with integer indices, one obtains for setting 1 . These indices transform into the second setting to with the fractional cubic indices , , . The transformation matrix is
The diffraction symmetry of icosahedral phases can be described by the Laue group . The set of all vectors H forms a Fourier module of rank 6 in physical space. Consequently, it can be considered as a projection from a 6D reciprocal lattice, . The parallel and perpendicular reciprocalspace sections of the 3D Penrose tiling decorated with equal point scatterers on its vertices are shown in Figs. 4.6.3.33 and 4.6.3.34. The diffraction pattern in perpendicular space is the Fourier transform of the triacontahedron. All Bragg reflections within are depicted. Without intensitytruncation limit, the diffraction pattern would be densely filled with discrete Bragg reflections.
The 6D icosahedral space groups that are relevant to the description of icosahedral phases (six symmorphous and five nonsymmorphous groups) are listed in Table 4.6.3.2. These space groups are a subset of all 6D icosahedral space groups fulfilling the condition that the 6D point groups they are associated with are isomorphous to the 3D point groups and 235 describing the diffraction symmetry. From 826 6D (analogues to) Bravais groups (Levitov & Rhyner, 1988), only three fulfil the condition that the projection of the 6D hypercubic lattice upon the 3D physical space is compatible with the icosahedral point groups : the primitive hypercubic Bravais lattice P, the bodycentred Bravais lattice I with translation 1/2(111111), and the facecentred Bravais lattice F with translations further cyclic permutations. Hence, the I lattice is twofold primitive (i.e. it contains two vertices per unit cell) and the F lattice is 16fold primitive. The orientation of the symmetry elements in the 6D space is defined by the isomorphism of the 3D and 6D point groups. The action of the fivefold rotation, however, is different in the subspaces and : a rotation of in is correlated with a rotation of in . The reflection and inversion operations are equivalent in both subspaces.

The structure factor of the icosahedral phase corresponds to the Fourier transform of the 6D unit cell, with 6D diffraction vectors , parallelspace atomic scattering factor , temperature factor and perpendicularspace geometric form factor . is equivalent to the conventional Debye–Waller factor and describes random fluctuations in perpendicular space. These fluctuations cause characteristic jumps of vertices (phason flips) in the physical space. Even random phason flips map the vertices onto positions that can still be described by physicalspace vectors of the type . Consequently, the set of all possible vectors forms a module. The shape of the atomic surfaces corresponds to a selection rule for the positions actually occupied. The geometric form factor is equivalent to the Fourier transform of the atomic surface, i.e. the 3D perpendicularspace component of the 6D hyperatoms.
For the example of the canonical 3D Penrose tiling, corresponds to the Fourier transform of a triacontahedron: where is the volume of the 6D unit cell projected upon and is the volume of the triacontahedron. and are equal in the present case and amount to the volumes of ten prolate and ten oblate rhombohedra: . Evaluating the integral by decomposing the triacontahedron into trigonal pyramids, each one directed from the centre of the triacontahedron to three of its corners given by the vectors , one obtains with running over all sitesymmetry operations R of the icosahedral group, , , , , and the volume of the parallelepiped defined by the vectors (Yamamoto, 1992b).
The radial structurefactor distributions of the 3D Penrose tiling decorated with point scatterers are plotted in Fig. 4.6.3.35 as a function of parallel and perpendicular space. The distribution of as a function of their frequencies clearly resembles a centric distribution, as can be expected from the centrosymmetric unit cell. The shape of the distribution function depends on the radius of the limiting sphere in reciprocal space. The number of weak reflections increases as the power 6, that of strong reflections only as the power 3 (strong reflections always have small components).
The weighted reciprocal space of the 3D Penrose tiling contains an infinite number of Bragg reflections within a limited region of the physical space. Contrary to the diffraction pattern of a periodic structure consisting of point atoms on the lattice nodes, the Bragg reflections show intensities depending on the perpendicularspace components of their diffraction vectors.
4.6.3.3.3.5. Relationships between structure factors at symmetryrelated points of the Fourier image
The weighted 3D reciprocal space exhibits the icosahedral point symmetry . It is invariant under the action of the scaling matrix : The scaling transformation leaves a primitive 6D reciprocal lattice invariant as can easily be seen from its application on the indices: The matrix leaves invariant, for any with all even or all odd, corresponding to a 6D facecentred hypercubic lattice. In a second case the sum is even, corresponding to a 6D bodycentred hypercubic lattice. Blockdiagonalization of the matrix S decomposes it into two irreducible representations. With we obtain the scaling properties in the two 3D subspaces: scaling by a factor τ in parallel space corresponds to a scaling by a factor in perpendicular space. For the intensities of the scaled reflections analogous relationships are valid, as discussed for decagonal phases (Figs. 4.6.3.36 and 4.6.3.37, Section 4.6.3.3.2.5).
References
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