Advertisement

Model of Icosahedral Order

  • M. Widom
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 69)

Abstract

Metallic glasses, quasicrystals, and crystals may share identical local icosahedral order. This type of ordering extends to fill a three-dimensional curved space, producing an icosahedral “polytope” with perfect short and long-range icosahedral order. In this paper I demonstrate how to flatten the polytope and fill space with structures possessing the short-range order of the polytope but various types of long-range order. Both the rhombohedral packing units required to construct a three-dimensional Penrose pattern and long-range orientational order arise from rolling the polytope along special paths in three-dimensional flat space.

Keywords

Metallic Glass Flat Space Matching Rule Standard Orientation Icosahedral Cluster 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Schechtman, I. Blech, G. Gratias, and J.W. Cahn, Phys. Rev. Lett. 53, 1951 (1984)CrossRefADSGoogle Scholar
  2. 2.
    F.C. Frank, Proc. Roy. Soc. 215, 43 (1952)CrossRefADSGoogle Scholar
  3. 3.
    M. Kleman and J.F. Sadoc, J. Phys. (Paris) Lett. 40, L569 (1979);Google Scholar
  4. D.R. Nelson, Phys. Rev. B28, 5515 (1983)ADSMathSciNetGoogle Scholar
  5. 4.
    S. Sachdev and D.R. Nelson, Phys. Rev. Lett 53, 1947 (1984)CrossRefADSGoogle Scholar
  6. 5.
    H.S.M. Coxeter, “Regular Polytopes” (Dover, N.Y., 1973 )Google Scholar
  7. 6.
    J.P. Sethna, Phys. Rev. Lett. 51, 2198 (1983)CrossRefADSMathSciNetGoogle Scholar
  8. 7.
    D.R. Nelson and M. Widom, Nucl Phys. B240 [FS12], 113 (1984)CrossRefADSMathSciNetGoogle Scholar
  9. 8.
    C.L. Henley and V. Elser, Phil. Mag. B53, L59 (1986)CrossRefGoogle Scholar
  10. 9.
    M. Widom, Phys. Rev. B31, 6456 (1985)ADSGoogle Scholar
  11. 10.
    P. DuVal, “Homographics, Quaternions, and Rotations” ( Oxford, London, 1964 )Google Scholar
  12. 11.
    M. Widom, Phys. Rev. B33, July 15 (1986)Google Scholar
  13. 12.
    Actually an imperceptible adjustment must be made because the vertices below the equator of the icosahedron centered at (o,o,d) do not match precisely the locations of the vertices above the equator of the icosahedron at the origin.Google Scholar
  14. 13.
    G. Bergman, J.L.T. Waugh, and L. Pauling, Acta Cryst. 10, 254 (1957)CrossRefGoogle Scholar
  15. 14.
    D. Levine and P.J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984)CrossRefADSGoogle Scholar
  16. 15.
    V. Elser, Phys. Rev. Lett., 54, 1730 (1985);CrossRefADSGoogle Scholar
  17. See also article by P. Horn in these proceedings.Google Scholar
  18. 16.
    P. Ramachandrarao and G.V.S. Sastry, Pramana 25, L225 (1985)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • M. Widom
    • 1
  1. 1.Department of PhysicsCarnegie-Mellon UniversityPittsburghUSA

Personalised recommendations