International Tables for Crystallography (2010). Vol. B. ch. 4.6, pp. 590-624
https://doi.org/10.1107/97809553602060000778 |
Chapter 4.6. Reciprocal-space images of aperiodic crystals
Contents
- 4.6. Reciprocal-space images of aperiodic crystals (pp. 590-624) | html | pdf | chapter contents |
- 4.6.1. Introduction (pp. 590-591) | html | pdf |
- 4.6.2. The n-dimensional description of aperiodic crystals (pp. 591-598) | html | pdf |
- 4.6.2.1. Basic concepts (p. 591) | html | pdf |
- 4.6.2.2. 1D incommensurately modulated structures (pp. 591-593) | html | pdf |
- 4.6.2.3. 1D composite structures (pp. 593-594) | html | pdf |
- 4.6.2.4. 1D quasiperiodic structures (pp. 594-597) | html | pdf |
- 4.6.2.5. 1D structures with fractal atomic surfaces (pp. 597-598) | html | pdf |
- 4.6.3. Reciprocal-space images (pp. 598-621) | html | pdf |
- 4.6.3.1. Incommensurately modulated structures (IMSs) (pp. 598-602) | html | pdf |
- 4.6.3.2. Composite structures (CSs) (pp. 602-603) | html | pdf |
- 4.6.3.3. Quasiperiodic structures (QSs) (pp. 603-621) | html | pdf |
- 4.6.3.3.1. 3D structures with 1D quasiperiodic order (pp. 603-607) | html | pdf |
- 4.6.3.3.1.1. Indexing (pp. 603-604) | html | pdf |
- 4.6.3.3.1.2. Diffraction symmetry (p. 604) | html | pdf |
- 4.6.3.3.1.3. Structure factor (pp. 604-605) | html | pdf |
- 4.6.3.3.1.4. Intensity statistics (pp. 605-606) | html | pdf |
- 4.6.3.3.1.5. Relationships between structure factors at symmetry-related points of the Fourier image (pp. 606-607) | html | pdf |
- 4.6.3.3.2. Decagonal phases (pp. 607-613) | html | pdf |
- 4.6.3.3.2.1. Indexing (p. 610) | html | pdf |
- 4.6.3.3.2.2. Diffraction symmetry (pp. 610-611) | html | pdf |
- 4.6.3.3.2.3. Structure factor (pp. 611-612) | html | pdf |
- 4.6.3.3.2.4. Intensity statistics (pp. 612-613) | html | pdf |
- 4.6.3.3.2.5. Relationships between structure factors at symmetry-related points of the Fourier image (p. 613) | html | pdf |
- 4.6.3.3.3. Icosahedral phases (pp. 613-621) | html | pdf |
- 4.6.3.3.3.1. Indexing (pp. 616-618) | html | pdf |
- 4.6.3.3.3.2. Diffraction symmetry (p. 618) | html | pdf |
- 4.6.3.3.3.3. Structure factor (p. 619) | html | pdf |
- 4.6.3.3.3.4. Intensity statistics (p. 619) | html | pdf |
- 4.6.3.3.3.5. Relationships between structure factors at symmetry-related points of the Fourier image (pp. 619-621) | html | pdf |
- 4.6.3.3.1. 3D structures with 1D quasiperiodic order (pp. 603-607) | html | pdf |
- 4.6.4. Experimental aspects of the reciprocal-space analysis of aperiodic crystals (pp. 621-623) | html | pdf |
- References | html | pdf |
- Figures
- Fig. 4.6.2.1. The combination of a basic structure , with period a, and a sinusoidal modulation function , with amplitude A, period λ and , gives a modulated structure (MS) (p. 591) | html | pdf |
- Fig. 4.6.2.2. 2D embedding of the sinusoidally modulated structure illustrated in Fig. 4.6.2.1 (p. 592) | html | pdf |
- Fig. 4.6.2.3. Schematic representation of the 2D reciprocal-space embedding of the 1D sinusoidally modulated structure depicted in Figs. 4.6.2.1 and 4.6.2.2 (p. 593) | html | pdf |
- Fig. 4.6.2.4. 2D embedding of a 1D composite structure without mutual interaction of the subsystems (p. 593) | html | pdf |
- Fig. 4.6.2.5. 2D embedding of a 1D composite structure with mutual interaction of the subsystems causing modulations (p. 594) | html | pdf |
- Fig. 4.6.2.6. Schematic representation of the reciprocal space of the embedded 1D composite structure depicted in Fig. 4.6.2.4 (p. 594) | html | pdf |
- Fig. 4.6.2.7. Schematic representation of the reciprocal space of the embedded 1D composite structure depicted in Fig. 4.6.2.5 (p. 594) | html | pdf |
- Fig. 4.6.2.8. 2D embedding of the Fibonacci chain (p. 596) | html | pdf |
- Fig. 4.6.2.9. Schematic representation of the reciprocal space of the embedded Fibonacci chain depicted in Fig. 4.6.2.8 (p. 596) | html | pdf |
- Fig. 4.6.2.10. Reciprocal space of the embedded Fibonacci chain as a modulated structure (p. 597) | html | pdf |
- Fig. 4.6.2.11. 2D direct-space embedding of the Fibonacci chain as a modulated structure (p. 598) | html | pdf |
- Fig. 4.6.2.12. (a) Three steps in the development of the fractal atomic surface of the squared Fibonacci sequence starting from an initiator and a generator (p. 598) | html | pdf |
- Fig. 4.6.3.1. Schematic diffraction patterns for IMSs with (a) 1D, (b) 2D and (c) 3D modulation (p. 599) | html | pdf |
- Fig. 4.6.3.2. The relationships between the coordinates and the modulation function in a special section of the space (p. 601) | html | pdf |
- Fig. 4.6.3.3. Schematic diffraction patterns for 3D IMSs with (a) 1D harmonic and (b) rectangular density modulation (p. 602) | html | pdf |
- Fig. 4.6.3.4. The relative structure-factor magnitudes of mth-order satellite reflections for a harmonic displacive modulation are proportional to the values of the mth-order Bessel function (p. 602) | html | pdf |
- Fig. 4.6.3.5. The structure factors (below) and their magnitudes (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component of the diffraction vector (p. 604) | html | pdf |
- Fig. 4.6.3.6. The structure factors (below) and their magnitudes (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component of the diffraction vector (p. 604) | html | pdf |
- Fig. 4.6.3.7. The structure factors of the Fibonacci chain decorated with aluminium atoms (Uoverall = 0.005 Å2) as a function of the parallel (above) and the perpendicular (below) component of the diffraction vector (p. 605) | html | pdf |
- Fig. 4.6.3.8. The structure factors (below) and their magnitudes (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component of the diffraction vector (p. 606) | html | pdf |
- Fig. 4.6.3.9. The structure factors (below) and their magnitudes (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component of the diffraction vector (p. 606) | html | pdf |
- Fig. 4.6.3.10. Part … LSLLSLSL … of a Fibonacci sequence before and after scaling by the factor τ (p. 607) | html | pdf |
- Fig. 4.6.3.11. Scaling operations of the Fibonacci sequence (p. 607) | html | pdf |
- Fig. 4.6.3.12. Reciprocal basis of the decagonal phase in the 5D description projected upon (above left) and (above right) (p. 608) | html | pdf |
- Fig. 4.6.3.13. A section of a Penrose tiling (thin lines) superposed by its τ-deflated tiling (above, thick lines) and by its -deflated tiling (below, thick lines) (p. 609) | html | pdf |
- Fig. 4.6.3.14. Scaling symmetry of a pentagram superposed on the Penrose tiling (p. 609) | html | pdf |
- Fig. 4.6.3.15. Atomic surface of the Penrose tiling in the 5D hypercubic description (p. 610) | html | pdf |
- Fig. 4.6.3.16. Projection of the 4D hyper-rhombohedral unit cell of the Penrose tiling in the 4D description upon the perpendicular space (p. 610) | html | pdf |
- Fig. 4.6.3.17. Schematic diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs ar = 4.04 Å) (p. 611) | html | pdf |
- Fig. 4.6.3.18. Enlarged section of Fig. 4.6.3.17 (p. 611) | html | pdf |
- Fig. 4.6.3.19. The perpendicular-space diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs ar = 4.04 Å) (p. 611) | html | pdf |
- Fig. 4.6.3.20. Enlarged section of Fig. 4.6.3.19 showing the projected 4D reciprocal-lattice unit cell (p. 611) | html | pdf |
- Fig. 4.6.3.21. Radial distribution function of the structure factors of the Penrose tiling (edge length of the Penrose unit rhombs ar = 4.04 Å) decorated with point atoms as a function of (p. 612) | html | pdf |
- Fig. 4.6.3.22. Radial distribution function of the structure factors of the Penrose tiling (edge length of the Penrose unit rhombs ar = 4.04 Å) decorated with point atoms as a function of (p. 613) | html | pdf |
- Fig. 4.6.3.23. Radial distribution function of the structure-factor magnitudes of the Penrose tiling (edge length of the Penrose unit rhombs ar = 4.04 Å) decorated with point atoms as a function of (p. 613) | html | pdf |
- Fig. 4.6.3.24. Pentagrammal relationships between scaling symmetry-related positive structure factors of the Penrose tiling (edge length ar = 4.04 Å) in parallel space (p. 614) | html | pdf |
- Fig. 4.6.3.25. Pentagrammal relationships between scaling symmetry-related negative structure factors of the Penrose tiling (edge length ar = 4.04 Å) in parallel space (p. 614) | html | pdf |
- Fig. 4.6.3.26. Pentagrammal relationships between scaling symmetry-related structure factors of the Penrose tiling (edge length ar = 4.04 Å) in parallel space (p. 614) | html | pdf |
- Fig. 4.6.3.27. Pentagrammal relationships between scaling symmetry-related structure factors of the Penrose tiling (edge length ar = 4.04 Å) in perpendicular space (p. 614) | html | pdf |
- Fig. 4.6.3.28. Perspective (a) parallel- and (b) perpendicular-space views of the reciprocal basis of the 3D Penrose tiling (p. 615) | html | pdf |
- Fig. 4.6.3.29. Schematic representation of a rotation in 6D space (p. 615) | html | pdf |
- Fig. 4.6.3.30. The two unit tiles of the 3D Penrose tiling: a prolate and an oblate rhombohedron with equal edge lengths (p. 616) | html | pdf |
- Fig. 4.6.3.31. Atomic surface of the 3D Penrose tiling in the 6D hypercubic description (p. 616) | html | pdf |
- Fig. 4.6.3.32. Perspective parallel-space view of the two alternative reciprocal bases of the 3D Penrose tiling (p. 616) | html | pdf |
- Fig. 4.6.3.33. Physical-space diffraction patterns of the 3D Penrose tiling decorated with point atoms (p. 617) | html | pdf |
- Fig. 4.6.3.34. Perpendicular-space diffraction patterns of the 3D Penrose tiling decorated with point atoms (p. 618) | html | pdf |
- Fig. 4.6.3.35. Radial distribution function of the structure factors of the 3D Penrose tiling (p. 619) | html | pdf |
- Fig. 4.6.3.36. Parallel-space distribution of (a) positive and (b) negative structure factors of the 3D Penrose tiling of the 6D P lattice type decorated with point atoms (p. 620) | html | pdf |
- Fig. 4.6.3.37. Perpendicular-space distribution of (a) positive and (b) negative structure factors of the 3D Penrose tiling of the 6D P lattice type decorated with point atoms (p. 620) | html | pdf |
- Fig. 4.6.4.1. Simulated diffraction patterns of (a) the single-crystal approximant of decagonal Al–Co–Ni, (b) the fivefold twinned approximant, and (c) the decagonal phase itself (p. 621) | html | pdf |
- Fig. 4.6.4.2. Zero-layer X-ray diffraction patterns of decaprismatic Al73.5C21.7Ni4.8 crystals (p. 622) | html | pdf |
- Tables
- Table 4.6.2.1. Expansion of the Fibonacci sequence by repeated action of the substitution rule σ: , (p. 595) | html | pdf |
- Table 4.6.3.1. 3D point groups of order k describing the diffraction symmetry and corresponding 5D decagonal space groups with reflection conditions (see Rabson et al., 1991) (p. 612) | html | pdf |
- Table 4.6.3.2. 3D point groups of order k describing the diffraction symmetry and corresponding 6D decagonal space groups with reflection conditions (see Levitov & Rhyner, 1988; Rokhsar et al., 1988) (p. 618) | html | pdf |
- Table 4.6.4.1. Intensity statistics of the Fibonacci chain for a total of 161 322 reflections with and (p. 620) | html | pdf |