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Higher-Dimensional Approach

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Book cover Crystallography of Quasicrystals

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 126))

Abstract

The nD approach elegantly restores hidden symmetries and correlations of quasiperiodic structures. Since it is based on reciprocal space information, it is directly accessible from experimental diffraction data. nD crystallography is an extension of the well-developed 3D crystallography and many well-established powerful 3D methods can be adapted for nD structure analysis. The nD approach is also a prerequisite for understanding phason modes and the structural relationships between quasicrystals and their approximants. In this chapter, the nD embedding will be quantitatively and detailed discussed of 1D, 2D, and 3D quasiperiodic tilings, which have been presented in Chap. 1, on tilings and coverings. In all cases, the direct and reciprocal space symmetry as well as the periodic average structures are treated as well.

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References

  1. P.A. Bancel, P.A. Heiney, P.W. Stephens, A.I. Goldman, P.M. Horn, Structure of Rapidly Quenched Al-Mn. Phys. Rev. Lett. 54, 2422–2425 (1985)

    Article  ADS  Google Scholar 

  2. S. Bhagavantam, T. Venkatarayudu, Theory of Groups and its Application to Physical Problems. Academic Press: New York London (1969)

    Google Scholar 

  3. A. Cervellino, W. Steurer, General periodic average structures of decagonal quasicrystals. Acta Crystallogr A 58, 180–184 (2002)

    Article  MathSciNet  Google Scholar 

  4. A. Cervellino, T. Haibach, W. Steurer, Derivation of the Proper Basis of Quasicrystals. Phys. Rev. B 57, 11223–11231 (1998)

    Article  ADS  Google Scholar 

  5. N.G. de Bruijn, Algebraic theory of Penrose’s nonperiodic tilings of the plane, I, II. Proc. Kon. Nederl. Akad. Wetenschap. A 84, 39–52, 53–66 (1981)

    Google Scholar 

  6. P.M. de Wolff, The Pseudo-symmetry of Modulated Crystal Structures. Acta Crystallogr. A 30, 777–785 (1974)

    Article  ADS  Google Scholar 

  7. K. Edagawa, K. Suzuki, M. Ichihara, S. Takeuchi, T. Shibuya, High-order Periodic Approximants of Decagonal Quasicrystal in Al70Ni15Co15. Phil. Mag. B 64, 629–638 (1991)

    Google Scholar 

  8. V. Elser, Indexing Problems in Quasicrystal Diffraction. Phys. Rev. B 32, 4892–4898 (1985)

    Article  ADS  Google Scholar 

  9. A.I. Goldman, R.F. Kelton, Quasicrystals and Crystalline Approximants. Rev. Mod.Phys. 65, 213–230 (1993)

    Article  ADS  Google Scholar 

  10. D. Gratias, A. Katz, M. Quiquandon, Geometry of Approximant Structures in Quasicrystals. J. Phys.: Condens. Matter 7, 9101–9125 (1995)

    Article  ADS  Google Scholar 

  11. C.L. Henley, Sphere Packings and Local Environments in Penrose Tilings. Phys. Rev. B 34, 797–816 (1986)

    Article  ADS  Google Scholar 

  12. C. Hermann, Kristallographie in Räumen beliebiger Dimensionszahl. I. Die Symmetrieoperationen. Acta Crystallogr. 2, 139–145 (1949)

    MathSciNet  Google Scholar 

  13. C.S. Herz, Fourier Transforms Related to Convex Sets. Ann. Math. 75, 81–92 (1962)

    Article  MathSciNet  Google Scholar 

  14. H. Hiller, The Crystallographic Restriction in Higher Dimensions. Acta Crystallogr. A 41, 541–544 (1985)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. A. Janner, Decagrammal Symmetry of Decagonal Al78Mn22 Quasicrystal. Acta Crystallogr. A 48, 884–901 (1992)

    Article  Google Scholar 

  16. T. Janssen, Crystallography of Quasi-Crystals. Acta Crystallogr. A 42, 261–271(1986)

    Article  Google Scholar 

  17. T. Janssen, Aperiodic Crystals: a Contradictio in Terminis? Phys. Rep. 168, 55–113 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  18. T. Janssen, The Symmetry Operations for N-dimensional Periodic and Quasi-periodic Structures. Z. Kristallogr. 198, 17–32 (1992)

    Article  Google Scholar 

  19. T. Janssen, Crystallographic Scale Transformations. Phil. Mag. B 66, 125–134 (1992)

    Google Scholar 

  20. T. Janssen, Tensors in quasiperiodic structures. In: International Tables for Crystallography, vol. D) Kluwer Academic Publisher, Dordrecht, pp. 243–264, (2003)

    Google Scholar 

  21. T. Janssen, A. Janner, A, Looijenga-Vos, P.M. Wolff, de: Incommensurate and Commensurate Modulated Crystal Structures. In: International Tables for Crystallography, vol. C, Kluwer Academic Publisher, Dordrecht, pp. 797–844 (1992)

    Google Scholar 

  22. M.V. Jaric, Diffraction from Quasi-Crystals - Geometric Structure Factor. Phys. Rev. B 34, 4685–4698 (1986)

    Article  ADS  Google Scholar 

  23. M.A. Kaliteevski, V.V. Nikolaev, R.A. Abram, S. Brand, Bandgap structure of optical Fibonacci lattices after light diffraction. Opt. Spectr. 91, 109–118 (2001)

    Article  ADS  Google Scholar 

  24. M. Kalning, S. Kek, H.G. Krane, V. Dorna, W. Press, W. Steurer, Phason-strain Analysis of the Twinned Approximant to Decagonal Quasicrystal Al70Co15Ni15: Evidence for a One-dimensional Quasicrystal. Phys. Rev. B 55, 187–192 (1997)

    Article  ADS  Google Scholar 

  25. E. Koch, Twinning. (International Tables for Crystallography, vol. C, Kluwer Academic Publisher, Dordrecht, pp. 10–14 (1992)

    Google Scholar 

  26. P. Kramer, M. Schlottmann, Dualization of Voronoi Domains and Klotz Construction - a General-Method for the Generation of Proper Space Fillings. J. Phys. A: Math. Gen. 22, L1097–L1102 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  27. D. Levine, P.J. Steinhardt, Quasicrystals. I. Definition and Structure. Phys. Rev. B 34, 596–616 (1986)

    Google Scholar 

  28. L.S. Levitov, J. Rhyner, Crystallography of Quasicrystals; Application to Icosahedral Symmetry. J. Phys. France 49, 1835–1849 (1988)

    Article  MathSciNet  Google Scholar 

  29. J.M. Luck, C. Godrèche, A. Janner, T. Janssen, The Nature of the Atomic Surfaces of Quasiperiodic Self-similar Structures. J. Phys. A: Math. Gen. 26, 1951–1999 (1993)

    Article  MATH  ADS  Google Scholar 

  30. Z. Masakova, J. Patera, J. Zich, Classification of Voronoi and Delone tiles of quasicrystals: III. Decagonal acceptance window of any size. J. Phys A - Math. Gen. 38, 1947–1960 (2005)

    MATH  MathSciNet  ADS  Google Scholar 

  31. K. Niizeki, A Classification of the Space Groups of Approximant Lattices to a Decagonal Quasilattice. J. Phys. A: Math. Gen. 24, 3641–3654 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  32. K. Niizeki, The Space Groups of Orthorhombic Approximants to the Icosahedral Quasilattice. J. Phys. A: Math. Gen. 25, 1843–1854 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  33. A. Pavlovitch, M. Kléman, Generalized 2D Penrose Tilings: Structural Properties. J. Phys. A: Math. Gen. 20, 687–702 (1987)

    Article  MATH  ADS  Google Scholar 

  34. R. Penrose, The Rôle of Aesthetics in Pure and Applied Mathematical Research. Bull. Inst. Math, Appl. 10, 266–271 (1974)

    Google Scholar 

  35. M. Quiquandon, D. Gratias, Unique six-dimensional structural model for Al-Pd-Mn and Al-Cu-Fe icosahedral phases. Phys. Rev. B 74, - art. no. 214205 (2006)

    Google Scholar 

  36. D.A. Rabson, N.D. Mermin, D.S. Rokhsar, D.C. Wright, The Space Groups of Axial Crystals and Quasicrystals. Rev. Mod. Phys. 63, 699–733 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  37. P. Repetowicz, J. Wolny, Diffraction pattern calculations for a certain class of N-fold quasilattices. Journal of Physics a-Mathematical and General 31, 6873–6886 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  38. D.S. Rokshar, N.D. Mermin, D.C. Wright, The Two-dimensional Quasicrystallographic Space Groups less than 23–fold. Acta Crystallogr. A 44, 197–211 (1988)

    Article  Google Scholar 

  39. J.E.S. Socolar, Simple Octagonal and Dodecagonal Quasicrystals. Phys. Rev. B 39, 10519–10551 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  40. J.E.S. Socolar, P.J. Steinhardt, Quasicrystals. II., Unit Cell Configurations. Phys. Rev. B 34, 617–647 (1986)

    Article  ADS  Google Scholar 

  41. B. Souvignier, Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6. Acta Crystallogr. A 59, 2003

    Google Scholar 

  42. W. Steurer, Experimental aspects of the structure analysis of aperiodic materials. In: Axel, F., Gratias, D. (eds.): Beyond Quasicrystals. Les Edition de Physique, Les Ulis, Springer, Berlin (1995)

    Google Scholar 

  43. W. Steurer, The Structure of Quasicrystals. Physical Metallurgy, vol. 1, Elsevier Science North Holland, Amsterdam, pp. 371–411 (1996)

    Google Scholar 

  44. W. Steurer, Twenty years of structure research on quasicrystals. Part 1. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals. Z. Kristall. 219, 391–446 (2004)

    Article  MathSciNet  Google Scholar 

  45. W. Steurer, T. Haibach, The Periodic Average Structure of Particular Quasicrystals. Acta Crystallogr. A 55, 48–57 (1999)

    Article  Google Scholar 

  46. W. Steurer, T. Haibach, Reciprocal Space Images of Aperiodic Crystals. International Tables for Crystallography, vol. B, Kluwer Academic Publishers: Dordrecht, pp. 486–518, (2001)

    Google Scholar 

  47. D. Sutter-Widmer, S. Deloudi, W. Steurer, Prediction of Bragg-scattering-induced band gaps in phononic quasicrystals. Phys. Rev. B 75, art. no. 094304 (2007)

    Google Scholar 

  48. S. van Smaalen, Incommensurate Crystallography. International Series of Monographs in Physics. Oxford University Press: Oxford, UK (2007)

    Google Scholar 

  49. R. Wang, Y. Wenge, C. Hu, D. Ding, Point and Space Groups and Elastic Behaviours of One-dimensional Quasicrystals. J. Phys.: Condens. Matter 9, 2411–2422 (1997)

    Article  ADS  Google Scholar 

  50. B.T.M. Willis, A.W. Pryor, Thermal vibrations in Crystallography. Cambridge University Press, Cambridge (1975)

    Google Scholar 

  51. A. Yamamoto, Ideal Structure of Icosahedral Al-Cu-Li Quasicrystals. Phys. Rev. B 45, 5217–5227 (1992)

    Article  ADS  Google Scholar 

  52. H. Zhang, K.H. Kuo, Giant Al-M (M = Transitional Metal) Crystals as Penrose-tiling Approximants of theDecagonal Quasicrystal. Phys. Rev. B 42, 8907–8914 (1990)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. ETH Zürich, Department of Materials, Laboratory of Crystallography, Wolfgang-Pauli-Str. 10, 8093, Zürich, Switzerland

    Professor Dr. Walter Steurer & Dr. Sofia Deloudi

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  1. Professor Dr. Walter Steurer
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  2. Dr. Sofia Deloudi
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Correspondence to Walter Steurer .

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Steurer, W., Deloudi, S. (2009). Higher-Dimensional Approach. In: Crystallography of Quasicrystals. Springer Series in Materials Science, vol 126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01899-2_3

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  • DOI: https://doi.org/10.1007/978-3-642-01899-2_3

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