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. 607613
Section 4.6.3.3.2. Decagonal phases ^{a}Laboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, WolfgangPauliStrasse 10, CH8093 Zurich, Switzerland 
A structure quasiperiodic in two dimensions, periodic in the third dimension and with decagonal diffraction symmetry is called a decagonal phase. Its holohedral Laue symmetry group is . All reciprocalspace vectors can be represented on a basis (V basis) and (Fig. 4.6.3.12) as . The vector components refer to a Cartesian coordinate system in physical (parallel) space. Thus, from the number of independent reciprocalbasis vectors necessary to index the Bragg reflections with integer numbers, the dimension of the embedding space has to be at least five. This can also be shown in a different way (Hermann, 1949).

Reciprocal basis of the decagonal phase in the 5D description projected upon (above left) and (above right). Below, a perspective physicalspace view is shown. 
The set of all vectors H remains invariant under the action of the symmetry operators of the point group . The symmetryadapted matrix representations for the pointgroup generators, the tenfold rotation , the reflection plane (normal of the reflection plane along the vectors – with modulo 5) and the inversion operation may be written in the form
By blockdiagonalization, these reducible symmetry matrices can be decomposed into nonequivalent irreducible representations. These can be assigned to the two orthogonal subspaces forming the 5D embedding space , the 3D parallel (physical) subspace and the perpendicular 2D 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 space. Thus, W can be rewritten using the physicalspace reciprocal basis defined above as yielding the reciprocal basis , in the 5D embedding space (D space): The symmetry matrices can each be decomposed into a matrix and a matrix. 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 3. Thus, a rotation in physical space is related to a rotation in perpendicular space (Fig. 4.6.3.12).
With the condition , a basis in direct 5D space is obtained: The metric tensors G, are of the type with for the reciprocal space and for the direct space. Thus, for the lattice parameters in reciprocal space we obtain , ; ; , ; , , and for those in direct space , ; ; , ; . The volume of the 5D unit cell can be calculated from the metric tensor G:
Since decagonal phases are only quasiperiodic in two dimensions, it is sufficient to demonstrate their characteristics on a 2D example, the canonical Penrose tiling (Penrose, 1974). It can be constructed from two unit tiles: a skinny (acute angle ) and a fat (acute angle ) rhomb with equal edge lengths and areas , (Fig. 4.6.3.13). The areas and frequencies of these two unit tiles in the Penrose tiling are both in a ratio 1 to τ. By replacing each skinny and fat rhomb according to the inflation rule, a τinflated tiling is obtained. Inflation (deflation) means that the number of tiles is inflated (deflated), their edge lengths are decreased (increased) by a factor τ. By infinite repetition of this inflation operation, an infinite Penrose tiling is generated. Consequently, this substitution operation leaves the tiling invariant.
From Fig. 4.6.3.13 it can be seen that the sets of vertices of the deflated tilings are subsets of the set of vertices of the original tiling. The τdeflated tiling is dual to the original tiling; a further deflation by a factor τ gives the original tiling again. However, the edge lengths of the tiles are increased by a factor , and the tiling is rotated around . Only the fourth deflation of the original tiling yields the original tiling in its original orientation but with all lengths multiplied by a factor .
Contrary to the reciprocalspace scaling behaviour of , the set of vertices of the Penrose tiling is not invariant by scaling the length scale simply by a factor τ using the scaling matrix S: The square of S, however, maps all vertices of the Penrose tiling upon other ones: corresponds to a hyperbolic rotation with in superspace (Janner, 1992). However, only operations of the type , , scale the Penrose tiling in a way which is equivalent to the (4nth) substitutional operations discussed above. The rotoscaling operation , also a symmetry operation of the Penrose tiling, leaves a pentagram invariant as demonstrated in Fig. 4.6.3.14 (Janner, 1992). Blockdiagonalization of the scaling matrix S decomposes it into two nonequivalent irreducible representations which give the scaling properties in the two orthogonal subspaces of the 4D embedding space, , the 2D parallel (physical) subspace and the perpendicular 2D subspace . Thus, using , we obtain where
The 2D Penrose tiling can also be embedded canonically in the 5D space. Canonically means that the 5D lattice is hypercubic and that the projection of one unit cell upon the 3D perpendicular space , giving a rhombicosahedron, defines the atomic surface. However, the parallelspace image , , with , of the 5D basis is not linearly independent. Consequently, the atomic surface consists of only a subset of the points contained in the rhombicosahedron: five equidistant pentagons (one with diameter zero) resulting as sections of the rhombicosahedron with five equidistant parallel planes (Fig. 4.6.3.15). The linear dependence of the 5D basis allows the embedding in the 4D space. The resulting hyperrhombohedral hyperlattice is spanned by the basis , , discussed above. The atomic surfaces occupy the positions , , on the body diagonal of the 4D unit cell. Neighbouring pentagons are in an anti position to each other (Fig. 4.6.3.16). Thus the 4D unit cell is decorated centrosymmetrically. The edge length of a Penrose rhomb is related to the length of physicalspace basis vectors by , with the smallest distance . The point density (number of vertices per unit area) of a Penrose tiling with Penrose rhombs of edge length can be calculated from the ratio of the relative number of unit tiles in the tiling to their area: This is equivalent to the calculation from the 4D description, where and are the area of the atomic surface and the volume of the 4D unit cell, respectively. The pentagon radii are and for the atomic surfaces in with and . A detailed discussion of the properties of Penrose tiling is given in the papers of Penrose (1974, 1979), Jaric (1986) and Pavlovitch & Kleman (1987).
The indexing of the submodule of the diffraction pattern of a decagonal phase is not unique. Since corresponds to a module of rank 4 with decagonal point symmetry, it is invariant under scaling by : . Nevertheless, an optimum basis (low indices are assigned to strong reflections) can be derived: not the metrics, as for regular periodic crystals, but the intensity distribution characterizes the best choice of indexing.
A correct set of reciprocalbasis vectors can be identified experimentally in the following way:
The diffraction symmetry of decagonal phases can be described by the Laue groups or . The set of all vectors H forms a Fourier module of rank 5 in physical space which can be decomposed into two submodules . corresponds to a module of rank 4 in a 2D subspace, corresponds to a module of rank 1 in a 1D subspace. Consequently, the first submodule can be considered as a projection from a 4D reciprocal lattice, , while the second submodule is of the form of a regular 1D reciprocal lattice, . The diffraction pattern of the Penrose tiling decorated with equal point scatterers on its vertices is shown in Fig. 4.6.3.17. All Bragg reflections within are depicted. Without intensitytruncation limit, the diffraction pattern would be densely filled with discrete Bragg reflections. To illustrate their spatial and intensity distribution, an enlarged section of Fig. 4.6.3.17 is shown in Fig. 4.6.3.18. This picture shows all Bragg reflections within . The projected 4D reciprocallattice unit cell is drawn and several reflections are indexed. All reflections are arranged along lines in five symmetryequivalent orientations. The perpendicularspace diffraction patterns (Figs. 4.6.3.19 and 4.6.3.20) show a characteristic starlike distribution of the Bragg reflections. This is a consequence of the pentagonal shape of the atomic surfaces: the Fourier transform of a pentagon has a starlike distribution of strong Fourier coefficients.

Schematic diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_{r} = 4.04 Å). All reflections are shown within and . 

The perpendicularspace diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_{r} = 4.04 Å). All reflections are shown within and . 
The 5D decagonal space groups that may be of relevance for the description of decagonal phases are listed in Table 4.6.3.1. These space groups are a subset of all 5D decagonal space groups fulfilling the condition that the 5D point groups they are associated with are isomorphous to the 3D point groups describing the diffraction symmetry. Their structures are comparable to 3D hexagonal groups. Hence, only primitive lattices exist. The orientation of the symmetry elements in the 5D space is defined by the isomorphism of the 3D and 5D point groups. However, the action of the tenfold rotation 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 for the decagonal phase corresponds to the Fourier transform of the 5D unit cell, with 5D diffraction vectors , N hyperatoms, parallelspace atomic scattering factor , temperature factor and perpendicularspace geometric form factor . is equivalent to the conventional Debye–Waller factor and describes random fluctuations along the perpendicularspace coordinate. These fluctuations cause characteristic jumps of vertices in physical space (phason flips). Even random phason flips map the vertices onto positions which 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 2D perpendicularspace component of the 5D hyperatoms.
For example, the canonical Penrose tiling corresponds to the Fourier transform of pentagonal atomic surfaces: where is the area of the 5D unit cell projected upon and is the area of the kth atomic surface. The area can be calculated using the formula Evaluating the integral by decomposing the pentagons into triangles, one obtains with running over the five triangles, where the radii of the pentagons are , , and the vectors
As shown by Ishihara & Yamamoto (1988), the Penrose tiling can be considered to be a superstructure of a pentagonal tiling with only one type of pentagonal atomic surface in the nD unit cell. Thus, for the Penrose tiling, three special reflection classes can be distinguished: for and the class of strong main reflections is obtained, and for the classes of weaker first and secondorder satellite reflections are obtained (see Fig. 4.6.3.18).
This section deals with the reciprocalspace characteristics of the 2D quasiperiodic component of the 3D structure, namely the Fourier module . The radial structurefactor distributions of the Penrose tiling decorated with point scatterers are plotted in Figs. 4.6.3.21 and 4.6.3.22 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 4D subunit cell. The shape of the distribution function depends on the radius of the limiting sphere in reciprocal space. The number of weak reflections increases to the power of four, that of strong reflections only quadratically (strong reflections always have small components). The radial distribution of the structurefactor amplitudes as a function of perpendicular space clearly shows three branches, corresponding to the reflection classes with , and (Fig. 4.6.3.23).
The weighted reciprocal space of the 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 (Figs. 4.6.3.19, 4.6.3.20 and 4.6.3.22).
4.6.3.3.2.5. Relationships between structure factors at symmetryrelated points of the Fourier image
Scaling the Penrose tiling by a factor by employing the matrix scales at the same time its reciprocal space by a factor : Since this operation increases the lengths of the diffraction vectors by the factor τ in parallel space and decreases them by the factor in perpendicular space, the following distribution of structurefactor magnitudes (for point atoms at rest) is obtained: The scaling operations , , the rotoscaling operations (Fig. 4.6.3.14) and the tenfold rotation , where connect all structure factors with diffraction vectors pointing to the nodes of an infinite series of pentagrams. The structure factors with positive signs are predominantly on the vertices of the pentagram while the ones with negative signs are arranged on circles around the vertices (Figs. 4.6.3.24 to 4.6.3.27).
References
Hermann, C. (1949). Kristallographie in Räumen beliebiger Dimensionszahl. I. Die Symmetrieoperationen. Acta Cryst. 2, 139–145.Ishihara, K. N. & Yamamoto, A. (1988). Penrose patterns and related structures. I. Superstructure and generalized Penrose patterns. Acta Cryst. A44, 508–516.
Janner, A. (1992). Decagrammal symmetry of decagonal Al_{78}Mn_{22} quasicrystal. Acta Cryst. A48, 884–901.
Jaric, M. V. (1986). Diffraction from quasicrystals: geometric structure factor. Phys. Rev. B, 34, 4685–4698.
Pavlovitch, A. & Kleman, M. (1987). Generalized 2D Penrose tilings: structural properties. J. Phys. A Math. Gen. 20, 687–702.
Penrose, R. (1974). The role of aesthetics in pure and applied mathematical research. Bull. Math. Appl. 10, 266–271.
Penrose, R. (1979). Pentaplexity. A class of nonperiodic tilings of the plane. Math. Intell. 2, 32–37.
Rabson, D. A., Mermin, N. D., Rokhsar, D. S. & Wright, D. C. (1991). The space groups of axial crystals and quasicrystals. Rev. Mod. Phys. 63, 699–733.