International Tables for Crystallography (2006). Vol. B. ch. 4.6, pp. 486-532 | 1 | 2 |
https://doi.org/10.1107/97809553602060000568

Chapter 4.6. Reciprocal-space images of aperiodic crystals

Contents

4.6. Reciprocal-space images of aperiodic crystals (pp. 486-532) | html | pdf | chapter contents |
W. Steurer and T. Haibach
- 4.6.1. Introduction (pp. 486-487) | html | pdf |
- 4.6.2. The n-dimensional description of aperiodic crystals (pp. 487-494) | html | pdf |
  - 4.6.2.1. Basic concepts (p. 487) | html | pdf |
  - 4.6.2.2. 1D incommensurately modulated structures (pp. 487-489) | html | pdf |
  - 4.6.2.3. 1D composite structures (pp. 489-490) | html | pdf |
  - 4.6.2.4. 1D quasiperiodic structures (pp. 490-493) | html | pdf |
  - 4.6.2.5. 1D structures with fractal atomic surfaces (pp. 493-494) | html | pdf |
- 4.6.3. Reciprocal-space images (pp. 494-515) | html | pdf |
  - 4.6.3.1. Incommensurately modulated structures (IMSs) (pp. 494-497) | html | pdf |
    - 4.6.3.1.1. Indexing (p. 495) | html | pdf |
    - 4.6.3.1.2. Diffraction symmetry (pp. 495-496) | html | pdf |
    - 4.6.3.1.3. Structure factor (pp. 496-497) | html | pdf |
  - 4.6.3.2. Composite structures (CSs) (pp. 497-498) | html | pdf |
    - 4.6.3.2.1. Indexing (p. 498) | html | pdf |
    - 4.6.3.2.2. Diffraction symmetry (p. 498) | html | pdf |
    - 4.6.3.2.3. Structure factor (p. 498) | html | pdf |
  - 4.6.3.3. Quasiperiodic structures (QSs) (pp. 498-515) | html | pdf |
    - 4.6.3.3.1. 3D structures with 1D quasiperiodic order (pp. 498-502) | html | pdf |
      - 4.6.3.3.1.1. Indexing (p. 499) | html | pdf |
      - 4.6.3.3.1.2. Diffraction symmetry (p. 499) | html | pdf |
      - 4.6.3.3.1.3. Structure factor (pp. 500-501) | html | pdf |
      - 4.6.3.3.1.4. Intensity statistics (p. 501) | html | pdf |
      - 4.6.3.3.1.5. Relationships between structure factors at symmetry-related points of the Fourier image (pp. 501-502) | html | pdf |
    - 4.6.3.3.2. Decagonal phases (pp. 503-509) | html | pdf |
      - 4.6.3.3.2.1. Indexing (p. 505) | html | pdf |
      - 4.6.3.3.2.2. Diffraction symmetry (pp. 505-506) | html | pdf |
      - 4.6.3.3.2.3. Structure factor (pp. 506-507) | html | pdf |
      - 4.6.3.3.2.4. Intensity statistics (p. 507) | html | pdf |
      - 4.6.3.3.2.5. Relationships between structure factors at symmetry-related points of the Fourier image (pp. 508-509) | html | pdf |
    - 4.6.3.3.3. Icosahedral phases (pp. 509-515) | html | pdf |
      - 4.6.3.3.3.1. Indexing (pp. 511-512) | html | pdf |
      - 4.6.3.3.3.2. Diffraction symmetry (p. 512) | html | pdf |
      - 4.6.3.3.3.3. Structure factor (pp. 512-513) | html | pdf |
      - 4.6.3.3.3.4. Intensity statistics (pp. 513-514) | html | pdf |
      - 4.6.3.3.3.5. Relationships between structure factors at symmetry-related points of the Fourier image (pp. 514-515) | html | pdf |
- 4.6.4. Experimental aspects of the reciprocal-space analysis of aperiodic crystals (pp. 516-517) | html | pdf |
  - 4.6.4.1. Data-collection strategies (pp. 516-517) | html | pdf |
  - 4.6.4.2. Commensurability versus incommensurability (p. 517) | html | pdf |
  - 4.6.4.3. Twinning and nanodomain structures (p. 517) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 4.6.2.1. The combination of a basic structure $[s({\bf r})]$ , with period a, and a sinusoidal modulation function , with amplitude A, period λ and $[t = {\bf q}\cdot {\bf r}]$ , gives a modulated structure (MS) $[s_{m}({\bf r})]$ (p. 488) | html | pdf |
  - Fig. 4.6.2.2. 2D embedding of the sinusoidally modulated structure illustrated in Fig. 4.6.2.1 (p. 488) | 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. 489) | html | pdf |
  - Fig. 4.6.2.4. 2D embedding of a 1D composite structure without mutual interaction of the subsystems (p. 489) | html | pdf |
  - Fig. 4.6.2.5. 2D embedding of a 1D composite structure with mutual interaction of the subsystems causing modulations (p. 490) | 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. 490) | 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. 490) | html | pdf |
  - Fig. 4.6.2.8. 2D embedding of the Fibonacci chain (p. 491) | 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. 492) | html | pdf |
  - Fig. 4.6.2.10. Reciprocal space of the embedded Fibonacci chain as a modulated structure (p. 493) | html | pdf |
  - Fig. 4.6.2.11. 2D direct-space embedding of the Fibonacci chain as a modulated structure (p. 493) | 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. 494) | html | pdf |
  - Fig. 4.6.3.1. Schematic diffraction patterns for IMSs with (a) 1D, (b) 2D and (c) 3D modulation (p. 494) | html | pdf |
  - Fig. 4.6.3.2. The relationships between the coordinates $[x_{1k}, x_{4k}, \bar{x}_{1}, \bar{x}_{4}]$ and the modulation function $[u_{1k}]$ in a special section of the $[(3 + d)\hbox{D}]$ space (p. 496) | html | pdf |
  - Fig. 4.6.3.3. Schematic diffraction patterns for 3D IMSs with (a) 1D harmonic and (b) rectangular density modulation (p. 497) | 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 $[J_{m}(x)]$ (p. 498) | html | pdf |
  - Fig. 4.6.3.5. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component $[|{\bf H}^{\parallel}|]$ of the diffraction vector (p. 499) | html | pdf |
  - Fig. 4.6.3.6. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component $[|{\bf H}^{\perp}|]$ of the diffraction vector (p. 500) | html | pdf |
  - Fig. 4.6.3.7. The structure factors $[F({\bf H})]$ of the Fibonacci chain decorated with aluminium atoms (U_overall = 0.005 Å²) as a function of the parallel (above) and the perpendicular (below) component of the diffraction vector (p. 500) | html | pdf |
  - Fig. 4.6.3.8. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component $[|{\bf H}^{\parallel}|]$ of the diffraction vector (p. 501) | html | pdf |
  - Fig. 4.6.3.9. The structure factors $[F({\bf H})]$ (below) and their magnitudes $[|F({\bf H})|]$ (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component $[|{\bf H}^{\perp}|]$ of the diffraction vector (p. 502) | html | pdf |
  - Fig. 4.6.3.10. Part … LSLLSLSL … of a Fibonacci sequence $[s({\bf r})]$ before and after scaling by the factor τ (p. 502) | html | pdf |
  - Fig. 4.6.3.11. Scaling operations of the Fibonacci sequence (p. 502) | html | pdf |
  - Fig. 4.6.3.12. Reciprocal basis of the decagonal phase in the 5D description projected upon $[{\bf V}^{\parallel}]$ (above left) and $[{\bf V}^{\perp}]$ (above right) (p. 503) | 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 $[\tau^{2}]$ -deflated tiling (below, thick lines) (p. 504) | html | pdf |
  - Fig. 4.6.3.14. Scaling symmetry of a pentagram superposed on the Penrose tiling (p. 504) | html | pdf |
  - Fig. 4.6.3.15. Atomic surface of the Penrose tiling in the 5D hypercubic description (p. 505) | 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. 505) | html | pdf |
  - Fig. 4.6.3.17. Schematic diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) (p. 506) | html | pdf |
  - Fig. 4.6.3.18. Enlarged section of Fig. 4.6.3.17 (p. 506) | html | pdf |
  - Fig. 4.6.3.19. The perpendicular-space diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) (p. 506) | html | pdf |
  - Fig. 4.6.3.20. Enlarged section of Fig. 4.6.3.19 showing the projected 4D reciprocal-lattice unit cell (p. 506) | html | pdf |
  - Fig. 4.6.3.21. Radial distribution function of the structure factors $[F({\bf H})]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\parallel}]$ (p. 508) | html | pdf |
  - Fig. 4.6.3.22. Radial distribution function of the structure factors $[F({\bf H})]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\perp}]$ (p. 508) | html | pdf |
  - Fig. 4.6.3.23. Radial distribution function of the structure-factor magnitudes $[|F({\bf H})|]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\perp}]$ (p. 508) | html | pdf |
  - Fig. 4.6.3.24. Pentagrammal relationships between scaling symmetry-related positive structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in parallel space (p. 509) | html | pdf |
  - Fig. 4.6.3.25. Pentagrammal relationships between scaling symmetry-related negative structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in parallel space (p. 509) | html | pdf |
  - Fig. 4.6.3.26. Pentagrammal relationships between scaling symmetry-related structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in parallel space (p. 509) | html | pdf |
  - Fig. 4.6.3.27. Pentagrammal relationships between scaling symmetry-related structure factors $[F({\bf H})]$ of the Penrose tiling (edge length a_r = 4.04 Å) in perpendicular space (p. 510) | 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. 510) | html | pdf |
  - Fig. 4.6.3.29. Schematic representation of a rotation in 6D space (p. 511) | html | pdf |
  - Fig. 4.6.3.30. The two unit tiles of the 3D Penrose tiling: a prolate $[[\alpha_{p} = \hbox{arc} \cos (5^{-1/2}) \simeq 63.44^{\circ}]]$ and an oblate $[(\alpha_{o} = 180^{\circ} - \alpha_{p})]$ rhombohedron with equal edge lengths $[a_{r}]$ (p. 511) | html | pdf |
  - Fig. 4.6.3.31. Atomic surface of the 3D Penrose tiling in the 6D hypercubic description (p. 511) | html | pdf |
  - Fig. 4.6.3.32. Perspective parallel-space view of the two alternative reciprocal bases of the 3D Penrose tiling (p. 512) | html | pdf |
  - Fig. 4.6.3.33. Physical-space diffraction patterns of the 3D Penrose tiling decorated with point atoms (p. 513) | html | pdf |
  - Fig. 4.6.3.34. Perpendicular-space diffraction patterns of the 3D Penrose tiling decorated with point atoms (p. 514) | html | pdf |
  - Fig. 4.6.3.35. Radial distribution function of the structure factors $[F({\bf H})]$ of the 3D Penrose tiling (p. 515) | 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. 515) | 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. 516) | html | pdf |
  - Fig. 4.6.4.1. Simulated diffraction patterns of (a) the $[\simeq 52\;\hbox{\AA}]$ single-crystal approximant of decagonal Al–Co–Ni, (b) the fivefold twinned approximant, and (c) the decagonal phase itself (p. 517) | html | pdf |
  - Fig. 4.6.4.2. Zero-layer X-ray diffraction patterns of decaprismatic Al_73.5C_21.7Ni_4.8 crystals (p. 518) | html | pdf |
- Tables
  - Table 4.6.2.1. Expansion of the Fibonacci sequence $[B_{n} = \sigma^{n}(\hbox{L})]$ by repeated action of the substitution rule σ: $[\hbox{S} \rightarrow \hbox{L}]$ , $[\hbox{L} \rightarrow \hbox{LS}]$ (p. 491) | 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. 507) | 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. 514) | html | pdf |
  - Table 4.6.4.1. Intensity statistics of the Fibonacci chain for a total of 161 322 reflections with $[-1000 \leq h_{i} \leq 1000]$ and $[0 \leq \sin \theta/\lambda \leq 2\;\hbox{\AA}^{-1}]$ (p. 516) | html | pdf |