Skip to main content
SpringerLink
Log in

From Quasiperiodic to More Complex Systems

  • Conference paper
Beyond Quasicrystals

Part of the book series: Centre de Physique des Houches ((LHWINTER,volume 3))

Abstract

For a long time solid state systems were considered as either crystalline, i.e. lattice periodic and with space group symmetry, or amorphous. In the former case the diffraction peaks are infinitely sharp for a perfect infinite crystal, in the latter there are no sharp peaks. The presence of some disorder does not eliminate sharp Bragg peaks as long as long-range order is preserved. Moreover, the sharp Bragg peaks lie on a lattice, the reciprocal lattice, such that each peak can be labelled with three integer coefficients. It was also known that sometimes additional peaks appear, satellites. For example, a crystalline material with magnetic moments may show an ordering of the moments that can be described as a helix. If the period of this helix is not a period of the underlying lattice, the total system consisting of crystal and magnetic moments does not have lattice periodicity. This was not a problem as long as one did not want to describe the symmetries of crystal and magnetic system simultaneously. Moreover, for a long period it was not clear whether there was a common (super)lattice for crystal and magnetic moments, or not.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight 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. Aalst W. van, Hollander J. den, Peterse W.J.A.M., Wolff P.M. de, Acta Cryst. B32, 47 (1976).

    Google Scholar 

  2. Ascher E., Janner A., Helv. Phys. Acta 38, 551 (1965).

    MathSciNet  MATH  Google Scholar 

  3. Ascher E., Janner A. Commun. Math. Phys. 11, 138 (1968).

    Article  ADS  MATH  Google Scholar 

  4. Aubry S., Le Daeron P.Y. Physica D 8, 381 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Axe J.D., Iizumi M., Shirane G. Phys. Rev. B22, 3408 (1980).

    Article  Google Scholar 

  6. Axel F., Terauchi H., Phys. Rev. Lett. 66, 208 (1991).

    Article  Google Scholar 

  7. Bak P., Boehm J. von, Phys. Rev. B 21, 5297 (1980).

    Article  Google Scholar 

  8. Benoit C., J.Phys. C 20, 765 (1987).

    Article  ADS  Google Scholar 

  9. Benoit C., Poussigue G., Azougarh A. J.Phys. C.M. 2, 2519 (1990).

    Google Scholar 

  10. Bernard L., Currat R., Delamoye P., Zeyen C.M.E., Hubert S., Kouchkovsky R. de, J.Phys. C 16, 433 (1983).

    Google Scholar 

  11. Bieberbach L., Math. Ann. 70, 297 and 401 (1911).

    Google Scholar 

  12. Bienenstock A., Ewald P.P., Phys. Rev. 6, 720 (1961).

    Google Scholar 

  13. Bohr H., Acta Math. 45, 29 (1924), 46, 101 (1925), 52, 127 (1928).

    MathSciNet  MATH  Google Scholar 

  14. Brouns E., Visser J.W., Wolff P.M.de Acta Cryst. B 25, 2665 (1969).

    Google Scholar 

  15. Brown H., Math. of Comp. 23, 499 (1969).

    Article  MATH  Google Scholar 

  16. Brown H., Bülow H., Neuböser J., Wondratschek H., Zassenhaus H., “Crystallographic Groups of Four-Dimensional Space”, J. Wiley & Sons (New York, 1978 ).

    Google Scholar 

  17. Cailleau H., Moussa F., Mons J., Sol. St. Comm, 31, 521 (1979).

    Article  ADS  Google Scholar 

  18. Cailleau H., Moussa F., Zeyen C.M.E., Bouillot J., Sol. Stat. Comm. 33, 407 (1980).

    Article  ADS  Google Scholar 

  19. Cockayne E., Phys. Rev. B49, 5896 (1994).

    Google Scholar 

  20. Comes R., Shirane G., Shapiro S.M., Garito A.F., Heeger A.J., Phys. Rev. B14, 2376 (1976).

    Article  ADS  Google Scholar 

  21. Currat R., Janssen T. in “Solid State Physics 41”, (Ehrenreich H., Turnbull D. eds. ), Academic Press (New York, 1988 ), 201.

    Google Scholar 

  22. Durand D., Dénoyer F., Currat R., Lambert M., “Incommensurate Phases in Dielectrics 2” (eds. R. Blinc, A.P. Levanyuk ), Elsevier (Amsterdam, 1986 ), 101.

    Book  Google Scholar 

  23. Elliott R.J., Phys. Rev. 124, 346 (1961).

    Article  ADS  Google Scholar 

  24. Fast G., Janssen T., J. Comp. Phys. 7, 1 (1971).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Frenkel Y.I., Kontorova T., Zh. Eksp. Teor. Fiz. 8, 1340 (1938).

    MATH  Google Scholar 

  26. Friedel J., Dénoyer F., C.R. Acad. Sci. Paris Ser. II 305, 171 (1987).

    Google Scholar 

  27. Futama H., Shiozaki Y., Chiba A., Tanaka E., Mitsui T., Furuichi J., Phys. Lett. 25A, 8 (1967).

    Article  Google Scholar 

  28. Gähler F., Dissertation ETH Zürich (1988).

    Google Scholar 

  29. Godrèche C., Luck J.M., Janner A., Janssen T., J.de Physique 3, 1921 (1993).

    Article  ADS  Google Scholar 

  30. Goursat E., Ann. Sci. Ecole Norm. Super. Paris 6, 9 (1889).

    Google Scholar 

  31. Hafner J., Krajci M., Europhys. Lett. 21, 31 (1993).

    Article  ADS  Google Scholar 

  32. Halsey T.C., Jensen M.H., Kadanoff L.P., Procaccia I., Shraiman B.I., Phys. Rev. A33, 1141 (1986).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Hastings J.M., Pouget J.P., Shirane G., Heeger A.J., Miro N.D., McDiarmid A.G., Phys. Rev. Lett. 39, 1484 (1977).

    Article  ADS  Google Scholar 

  34. Haydock R., Heine V., Bullet D., Kelly M.J., in “Solid State Physics 35”, Academic Press (New York 1980 ).

    Google Scholar 

  35. Heesch H., Z. Kristallogr. 73, 325 (1930).

    MATH  Google Scholar 

  36. Heilmann I.U. Axe J.D., Hastings J.M., Shirane G., Heeger A.J., MacDiarmid A.G., Phys. Rev. B 20, 751 (1979).

    Article  ADS  Google Scholar 

  37. Hendricks S., Teller E., J. Chem. Phys. 10, 147 (1942).

    Article  ADS  Google Scholar 

  38. Hermann C., Acta Cryst. 2, 139 (1949).

    Article  MathSciNet  Google Scholar 

  39. Iizumi M., Axe J.D., Shirane G., Phys. Rev. B 15, 4392 (1977).

    Article  ADS  Google Scholar 

  40. Janner A., Janssen T. Physica 53, 1 (1971).

    Article  MathSciNet  ADS  Google Scholar 

  41. Janssen T., Ferroelectrics 66, 203 (1986).

    Article  Google Scholar 

  42. Janssen T., Physics Reports 168, 57 (1988).

    Article  ADS  Google Scholar 

  43. Janssen T., Janner A., Physica A 126, 163 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. Janssen T., Kohmoto M., Phys.Rev. B 38, 5811 (1988).

    Article  ADS  Google Scholar 

  45. Janssen T., Los J., Phase Trans. 33, 29 (1991).

    Google Scholar 

  46. Janssen T., Tjon J.A., Phys. Rev. B 24, 2245 (1981).

    Article  ADS  Google Scholar 

  47. Janssen T., Tjon J.A., Phys. Rev. B 25, 3767 (1982).

    Article  ADS  Google Scholar 

  48. Janssen T., Tjon J.A., J. Phys. C 16, 4789 (1983).

    Article  ADS  Google Scholar 

  49. Kohmoto M., Phys. Rev. A37, 1345 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  50. Kohmoto M., Kadanoff L.P., Tang Chao, Phys. Rev. Lett. 50, 1870 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  51. Lange C. de, Janssen T., J. Phys. C 14, 5269 (1981).

    Article  ADS  Google Scholar 

  52. Lange C. de, Janssen T., Phys. Rev. B 28, 195 (1983).

    Article  Google Scholar 

  53. Lange C. de, Janssen T., Physica A 127, 125 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  54. Los J., Janssen T., J.Phys. Cond.Matter 2, 9553 (1990).

    Article  ADS  Google Scholar 

  55. Los J., Janssen T., Gähler F., J. Non-Cryst. Sol., 153/4, 581 (1993).

    Google Scholar 

  56. Los J., Janssen T., Gähler F., J.Physique France 3, 107 (1993).

    Article  ADS  Google Scholar 

  57. Los J., Janssen T., Gähler F., J. de Physique 3, 1431 (1993).

    Article  ADS  Google Scholar 

  58. Los J., Janssen T., Gähler F., Int.J.Mod.Phys. B 7, 1505 (1993).

    Article  ADS  MATH  Google Scholar 

  59. Luck J.M., Godrèche C., Janner A., Janssen T., J.Phys.A 26, 1951 (1993).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  60. Mermin N.D., Rev. Mod. Phys. 64, 3 (1992).

    Article  MathSciNet  ADS  Google Scholar 

  61. Okamura K., Iwasaki H., Ogawa S., J. Phys. Soc. Japan 24, 569 (1968).

    Article  Google Scholar 

  62. Paasschens J., Janssen T., preprint (1994).

    Google Scholar 

  63. Pater C.J. de, Axe J.D., Currat R., Phys. Rev. B 19, 4684 (1979).

    Article  Google Scholar 

  64. Plesken W., Pohst M., M.th.Comp. 31, 536 and 552 (1977).

    Google Scholar 

  65. Pouget J.P., Shirane G., Hastings J.M., Heeger A.J., Miro N.D., McDiarmid A.G., Phys. Rev. B 18, 3645 (1978).

    Article  ADS  Google Scholar 

  66. Poussigue G., Benoit C., de Boissieu M., R. Currat, J.Phys. CM 6, 659 (1994).

    Google Scholar 

  67. Rabson D.A., Ho Tin-Lun, N.D.Mermin, Acta Cryst. A 45, 538 (1989).

    MathSciNet  MATH  Google Scholar 

  68. Rabson D.A., Mermin N.D., Rokhsar D.S., Wright D.C., Rev. Mod. Phys. 63, 699 (1991).

    Article  MathSciNet  ADS  Google Scholar 

  69. Sasisekharan V., Baranidharan S., Balagurusamy V.S.K., Srinivasan A., and Gopal E.S.R., Pramana 33, 405 (1989).

    Article  ADS  Google Scholar 

  70. Selke W., in “Phase Transitions and Critical Phenomena Vol.15”, C.Domb and J.L. Lebowitz, eds., Academic Press, ( New York, 1992 ), 1.

    Google Scholar 

  71. Shechtman D., Blech I., Gratias D., Cahn J.W., Phys. Rev. Letters 53, 1951 (1984).

    Article  ADS  Google Scholar 

  72. Stephens P.W., Goldman A.I., Phys. Rev. Lett. 56, 1168 (1986).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, University of Nijmegen and University of Utrecht, 6525 ED, Nijmegen, The Netherlands

    T. Janssen

Authors
  1. T. Janssen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Université Paris VII-Denis Diderot, France

    Françoise Axel

  2. CECN-CNRS, Vitry, France

    Denis Gratias

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Janssen, T. (1995). From Quasiperiodic to More Complex Systems. In: Axel, F., Gratias, D. (eds) Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03130-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-03130-8_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59251-8

  • Online ISBN: 978-3-662-03130-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation