Skip to main content
SpringerLink
Log in

Generation of Quasiperiodic Order by Maximal Cluster Covering

  • Chapter
  • First Online:
Coverings of Discrete Quasiperiodic Sets

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 180))

Abstract

In quasicrystals, certain structural motifs occur very frequently, and sometimes even cover the entire structure. This property is particularly visible in high-resolution electron micrographs (HREMs) of decagonal quasicrystals. In this chapter, these important structural motifs will be called clusters. On the basis of the observation that at least some quasicrystals can be regarded as being covered by a single kind of cluster, Burkov [1] was one of the first to propose a structure model which was explicitly given as a covering with overlapping copies of a single cluster.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Burkov: Phys. Rev. Lett. 67, 614 (1991)

    Article  ADS  Google Scholar 

  2. H.-C. Jeong, P. J. Steinhardt: Phys. Rev. Lett. 73, 1943 (1994)

    Article  ADS  Google Scholar 

  3. C. L. Henley, “Matching rules for quasiperiodic tilings”. In: Quasicrystals: The State of the Art, ed. by D. P. DiVincenzo, P. J. Steinhardt (World Scientific, Singapore 1991) pp. 185-212

    Google Scholar 

  4. L. S. Levitov: Commun. Math. Phys. 119, 627 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  5. N. G. de Bruijn: Ned. Akad. Wetensch. Proc. Ser. A 43, 39, 53 (1981)

    MATH  Google Scholar 

  6. J. E. S. Socolar: Phys. Rev. B 39, 10519 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  7. F. Gähler: J. Non-Cryst. Solids 153&154, 160 (1993)

    Article  Google Scholar 

  8. R. Klitzing, M. Baake, M. Schlottmann: Int. J. Mod. Phys. B 7, 1455 (1993)

    ADS  MathSciNet  Google Scholar 

  9. Y. Yan, S. J. Pennycook: Phys. Rev. Lett. 86, 1542 (2001)

    Article  ADS  Google Scholar 

  10. F. Gähler: Phys. Rev. Lett. 74, 334 (1995)

    Article  ADS  Google Scholar 

  11. P. Gummelt: “Construction of Penrose tilings by a single aperiodic protoset”. In: Proceedings of the 5th International Conference on Quasicrystals, Avignon 1995, ed. by Janot, R. Mosseri (World Scientific, Singapore 1995) pp. 84–87

    Google Scholar 

  12. P. Gummelt: Geometriae Dedicata 62, 1 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. P. J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, A. P. Tsai: Nature 396, 55 (1998)

    Article  ADS  Google Scholar 

  14. R. Wittmann: Z. Kristallogr. 214 (1999) 501

    Article  Google Scholar 

  15. E. Abe, T. J. Sato, A. P. Tsai: Phys. Rev. Lett. 82, 5269 (1999)

    Article  ADS  Google Scholar 

  16. E. Cockayne, Mater. Sci. Eng. A 294–296, 224 (2000)

    Google Scholar 

  17. E. Abe, Saitoh, H. Takakura, A. P. Tsai, P. J. Steinhardt, H.-C. Jeong: Phys. Rev. Lett. 84, 4609 (2000)

    Article  ADS  Google Scholar 

  18. H.-C. Jeong, P. J. Steinhardt: Phys. Rev. B 55, 3520 (1997)

    Article  ADS  Google Scholar 

  19. M. Baake, M. Schlottmann, P. D. Jarvis: J. Phys. A: Math. Gen. 24, 4637 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. J. E. S. Socolar: Commun. Math. Phys. 129, 599 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. A. Katz: “Matching rules and quasiperiodicity: the octagonal tilings”. In: Beyond Quasicrystals, ed. by F. Axel, D. Gratias (Les Editions de Physique/Springer, Les Ulis/Heidelberg 1995) pp. 141–189

    Google Scholar 

  22. F. Gähler, H.-C. Jeong: J. Phys. A: Math. Gen. 28, 1807 (1995)

    Article  ADS  Google Scholar 

  23. F. Gähler: “Cluster interactions for quasiperiodic tilings”. In: Proceedings of the 6th International Conference on Quasicrystals, Tokyo 1997, ed. by S. Takeuchi, T. Fujiwara (World Scientific, Singapore 1998) pp. 95–98

    Google Scholar 

  24. F. Gähler, R. Lück, S. I. Ben-Abraham, P. Gummelt: Ferroelectrics 250, 335 (2001)

    Article  Google Scholar 

  25. C. L. Henley, “Random tiling models”. In: Quasicrystals: The State of the Art, ed. by D. P. DiVincenzo, P.J. Steinhardt (World Scientific, Singapore 1991) pp. 429–524

    Google Scholar 

  26. P. Gummelt, C. Bandt: Mater. Sci. Eng. A 294–296, 250 (2000)

    Google Scholar 

  27. F. Gähler, M. Reichert: J. Alloys Compd. 342 (2002)

    Google Scholar 

  28. F. P. M. Beenker, TH Report 82-WSK-04 (Technische Hogeschool, Eindhoven 1982)

    Google Scholar 

  29. R. Ammann, B. Grünbaum, G. C. Shephard: Discrete Comput. Geom. 8, 1 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  30. F. Gähler: “Crystallography of dodecagonal quasicrystals”. In: Quasicrystalline Materials, ed. by C. Janot, J. M. Dubois (World Scientific, Singapore 1988) pp. 272–284

    Google Scholar 

  31. Niizeki, H. Mitani: J. Phys. A: Math. Gen. 20, L405 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  32. M. Baake, P. Kramer, M. Schlottmann, D. Zeidler: Int. J. Mod. Phys. B 4, 2217 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. F. Gähler: “Cluster coverings: a powerful ordering principle for quasicrystals”. In: Quasicrystals: Current Topics, ed. by E. Belin-Ferré, Berger, M. Qui-quandon, A. Sadoc (World Scientific, Singapore 2000) pp. 118–127

    Google Scholar 

  34. F. Gähler: Mater. Sci. Eng. A 294–296, 199 (2000)

    Google Scholar 

  35. F. Gähler, M. Baake, M. Schlottmann: Phys. Rev. B 50, 12458 (1994)

    Article  ADS  Google Scholar 

  36. M. E. J. Newman, G. T. Barkema: Monte Carlo Methods in Statistical Physics (Oxford University Press, New York 1999)

    MATH  Google Scholar 

  37. B. Grünbaum, G. C. Shephard: Tilings and Patterns (Freeman, New York 1987)

    MATH  Google Scholar 

  38. P. Gummelt: Aperiodische Überdeckungen mit einem Clustertyp, (Shaker, Aachen 1999)

    Google Scholar 

  39. L.-H. Tang, M. V. Jarić: Phys. Rev. B 41, 4524 (1990)

    Article  ADS  Google Scholar 

  40. J. Lee: Phys. Rev. Lett. 71, 211 (1993)

    Article  ADS  Google Scholar 

  41. F. Gähler: “Thermodynamics of random tiling quasicrystals”. In: Proceedings of the 5th International Conference on Quasicrystals, Avignon 1995, ed. by C. Janot, R. Mosseri (World Scientific, Singapore 1995) pp. 236–239

    Google Scholar 

  42. J. Roth, L. Henley: Phil. Mag. A 75, 861 (1997)

    Article  ADS  Google Scholar 

  43. F. Gähler, J. Rhyner: J. Phys. A: Math. Gen. 19, 267 (1986)

    Article  ADS  MATH  Google Scholar 

  44. F. Gähler, P. Stampfli: Int. J. Mod. Phys. B 7, 1333 (1993)

    Article  MATH  ADS  Google Scholar 

  45. J. C. Jiang, S. Hovmöller, X. D. Zou: Phil. Mag. Lett. 71, 123 (1995)

    Article  ADS  Google Scholar 

  46. S. I. Ben-Abraham, F. Gähler: Phys. Rev. B 60, 860 (1999)

    Article  ADS  Google Scholar 

  47. F. Gähler: unpublished

    Google Scholar 

  48. S. I. Ben-Abraham, P. Gummelt, R. Lück, F. Gähler: Ferroelectrics 250, 313 (2001)

    Article  Google Scholar 

  49. H.-U. Nissen: “A two-dimensional quasiperiodic tiling by two pentagons”. In: Quasicrystals, Networks, and Molecules of Fivefold Symmetry, ed. by I. Hargittai (VCH, New York 1990) pp. 181–199

    Google Scholar 

  50. P. Kramer: J. Phys. A: Math. Gen. 32, 5781 (1999)

    Article  MATH  ADS  Google Scholar 

  51. P. Kramer: Mater. Sci. Eng. A 294–296, 401 (2000)

    Google Scholar 

  52. P. Kramer: J. Phys. A: Math. Gen. 33, 7885 (2000)

    Article  MATH  ADS  Google Scholar 

  53. P. Kramer: J. Phys. A: Math. Gen. 34, 1885 (2001)

    Article  MATH  ADS  Google Scholar 

  54. P. Kramer, M. Schlottmann: J. Phys. A: Math. Gen. 22, L1097 (1989)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Authors
  1. Franz Gähler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petra Gummelt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shelomo I. Ben-Abraham
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076, Tübingen, Germany

    Peter Kramer  & Zorka Papadopolos  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gähler, F., Gummelt, P., Ben-Abraham, S.I. (2002). Generation of Quasiperiodic Order by Maximal Cluster Covering. In: Kramer, P., Papadopolos, Z. (eds) Coverings of Discrete Quasiperiodic Sets. Springer Tracts in Modern Physics, vol 180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45805-0_3

Download citation

  • DOI: https://doi.org/10.1007/3-540-45805-0_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43241-8

  • Online ISBN: 978-3-540-45805-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation