Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Chapter 4.6. Reciprocal-space images of aperiodic crystals

W. Steurera* and T. Haibacha

aLaboratory of Crystallography, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland
Correspondence e-mail:


Axel, F. & Gratias, D. (1995). Editors. Beyond quasicrystals. Les Ulis: Les Editions de Physique and Berlin: Springer-Verlag.
Bancel, P. A., Heiney, P. A., Stephens, P. W., Goldman, A. I. & Horn, P. M. (1985). Structure of rapidly quenched Al–Mn. Phys. Rev. Lett. 54, 2422–2425.
Böhm, H. (1977). Eine erweiterte Theorie der Satellitenreflexe und die Bestimmung der modulierten Struktur des Natriumnitrits. Habilitation thesis, University of Munster.
Cummins, H. Z. (1990). Experimental studies of structurally incommensurate crystal phases. Phys. Rep. 185, 211–409.
Dehlinger, U. (1927). Über die Verbreiterung der Debyelinien bei kaltbearbeiteten Metallen. Z. Kristallogr. 65, 615–631.
Dräger, J. & Mermin, N. D. (1996). Superspace groups without the embedding: the link between superspace and Fourier-space crystallography. Phys. Rev. Lett. 76, 1489–1492.
Estermann, M., Haibach, T. & Steurer, W. (1994). Quasicrystal versus twinned approximant: a quantitative analysis with decagonal Al70Co15Ni15. Philos. Mag. Lett. 70, 379–384.
Goldman, A. I. & Kelton, K. F. (1993). Quasicrystals and crystalline approximants. Rev. Mod. Phys. 65, 213–230.
Gouyet, J. F. (1996). Physics and fractal structures. Paris: Masson and Berlin: Springer-Verlag.
Hausdorff, F. (1919). Dimension und äusseres Mass. Math. Ann. 79, 157–179.
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 Al78Mn22 quasicrystal. Acta Cryst. A48, 884–901.
Janner, A. & Janssen, T. (1979). Superspace groups. Physica A, 99, 47–76.
Janner, A. & Janssen, T. (1980a). Symmetry of incommensurate crystal phases. I. Commensurate basic structures. Acta Cryst. A36, 399–408.
Janner, A. & Janssen, T. (1980b). Symmetry of incommensurate crystal phases. II. Incommensurate basic structures. Acta Cryst. A36, 408–415.
Janner, A., Janssen, T. & de Wolff, P. M. (1983a). Bravais classes for incommensurate crystal phases. Acta Cryst. A39, 658–666.
Janner, A., Janssen, T. & de Wolff, P. M. (1983b). Determination of the Bravais class for a number of incommensurate crystals. Acta Cryst. A39, 671–678.
Janot, Chr. (1994). Quasicrystals. A primer. Oxford: Clarendon Press.
Janssen, T. (1986). Crystallography of quasicrystals. Acta Cryst. A42, 261–271.
Janssen, T. (1988). Aperiodic crystals: a contradictio in terminis? Phys. Rep. 168, 55–113.
Janssen, T. (1995). From quasiperiodic to more complex systems. In Beyond quasicrystals, edited by F. Axel & D. Gratias, pp. 75–140. Les Ulis: Les Editions de Physique and Berlin: Springer-Verlag.
Janssen, T., Janner, A., Looijenga-Vos, A. & de Wolff, P. M. (2004). Incommensurate and commensurate modulated crystal structures. In International tables for crystallography, Vol. C, edited by E. Prince, ch. 9.8. Dordrecht: Kluwer Academic Publishers.
Jaric, M. V. (1986). Diffraction from quasicrystals: geometric structure factor. Phys. Rev. B, 34, 4685–4698.
Kalning, M., Kek, S., Burandt, B., Press, W. & Steurer, W. (1994). Examination of a multiple twinned periodic approximant of the decagonal phase Al70Co15Ni15. J. Phys. Condens. Matter, 6, 6177–6187.
Kato, K. (1990). Strukturverfeinerung des Kompositkristalls im mehrdimensionalen Raum. Acta Cryst. B46, 39–44.
Kelton, K. F. (1995). Quasicrystals and related structures. In Intermetallic compounds. Principles and practice, Vol. 1, edited by J. H. Westbrook & R. L. Fleischer, pp. 453–491. Chichester: John Wiley & Sons.
Korekawa, M. (1967). Theorie der Satellitenreflexe. Habilitation thesis, University of Munich.
Lancon, F., Billard, L., Burkov, S. & DeBoissieu, M. (1994). On choosing a proper basis for determining structures of quasicrystals. J. Phys. I France, 4, 283–301.
Levine, D. & Steinhardt, P. J. (1986). Quasicrystals. I. Definition and structure. Phys. Rev. B, 34, 596–616.
Levitov, L. S. & Rhyner, J. (1988). Crystallography of quasicrystals; application to icosahedral symmetry. J. Phys. France, 49, 1835–1849.
Luck, J. M., Godréche, C., Janner, A. & Janssen, T. (1993). The nature of the atomic surfaces of quasiperiodic self-similar structures. J. Phys. A Math. Gen. 26, 1951–1999.
Paciorek, W. A. & Chapuis, G. (1994). Generalized Bessel functions in incommensurate structure analysis. Acta Cryst. A50, 194–203.
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 non-periodic tilings of the plane. Math. Intell. 2, 32–37.
Petricek, V., Maly, K. & Cisarova, I. (1991). The computing system `JANA'. In Methods of structural analysis of modulated structures and quasicrystals, edited by J. M. Pérez-Mato, F. J. Zuniga & G. Madriaga, pp. 262–267. Singapore: World Scientific.
Petricek, V., Maly, K., Coppens, P., Bu, X., Cisarova, I. & Frost-Jensen, A. (1991). The description and analysis of composite crystals. Acta Cryst. A47, 210–216.
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.
Rokhsar, D. S., Wright, D. C. & Mermin, N. D. (1988). Scale equivalence of quasicrystallographic space groups. Phys. Rev. B, 37, 8145–8149.
Senechal, M. (1995). Quasicrystals and geometry. Cambridge University Press.
Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953.
Smaalen, S. van (1991). Symmetry of composite crystals. Phys. Rev. B, 43, 11330–11341.
Smaalen, S. van (1992). Superspace description of incommensurate intergrowth compounds and the application to inorganic misfit layer compounds. Mater. Sci. Forum, 100 & 101, 173–222.
Smaalen, S. van (1995). Incommensurate crystal structures. Crystallogr. Rev. 4, 79–202.
Socolar, J. E. S. & Steinhardt, P. J. (1986). Quasicrystals. II. Unit-cell configurations. Phys. Rev. B, 34, 617–647.
Steurer, W. (1990). The structure of quasicrystals. Z. Kristallogr. 190, 179–234.
Steurer, W. (1995). Experimental aspects of the structure analysis of aperiodic materials. In Beyond quasicrystals, edited by F. Axel & D. Gratias, pp. 203–228. Les Ulis: Les Editions de Physique and Berlin: Springer-Verlag.
Steurer, W. (1996). The structure of quasicrystals. In Physical metallurgy, Vol. I, edited by R. W. Cahn & P. Haasen, pp. 371–411. Amsterdam: Elsevier.
Willis, B. T. M. & Pryor, A. W. (1975). Thermal vibrations in crystallography. Cambridge University Press.
Wolff, P. M. de (1974). The pseudo-symmetry of modulated crystal structures. Acta Cryst. A30, 777–785.
Wolff, P. M. de (1977). Symmetry operations for displacively modulated structures. Acta Cryst. A33, 493–497.
Wolff, P. M. de (1984). Dualistic interpretation of the symmetry of incommensurate structures. Acta Cryst. A40, 34–42.
Wolff, P. M. de, Janssen, T. & Janner, A. (1981). The superspace groups for incommensurate crystal structures with a one-dimensional modulation. Acta Cryst. A37, 625–636.
Yamamoto, A. (1982). Structure factor of modulated crystal structures. Acta Cryst. A38, 87–92.
Yamamoto, A. (1992a). Unified setting and symbols of superspace groups for composite crystals. Acta Cryst. A48, 476–483.
Yamamoto, A. (1992b). Ideal structure of icosahedral Al–Cu–Li quasicrystals. Phys. Rev. B, 45, 5217–5227.
Zobetz, E. (1993). One-dimensional quasilattices: fractally shaped atomic surfaces and homometry. Acta Cryst. A49, 667–676.