Glass Physics and Chemistry

, Volume 38, Issue 1, pp 41–48 | Cite as

Space group of heterogeneous cylinder packing with six 〈110〉 directions

Article
  • 39 Downloads

Abstract

The space group of a periodic cylinder packing structure is reported. Congruent cylinders of infinite length are packed parallel to the six cubic 〈110〉 directions. The structure belongs not to the cubic system but to the tetragonal system. Although the parallel cylinders form the same rhombic lattice along each of the six directions, the packing structure as a whole is heterogeneous: cylinders along two of the directions occupy one Wyckoff position, and cylinders along the remaining four directions occupy the other Wyckoff position. Their combined structure belongs to the space group I4122, which provides two enantiomorphic structures.

Keywords

cylinder packing space group heterogeneous structure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Holden, A., Shapes, Space, and Symmetry, New York: Columbia University Press, 1971.Google Scholar
  2. 2.
    Bezdek, A. and Kuperberg, W., Maximum Density Space Packing with Congruent Circular Cylinders of Infinite Length, Mathematika, 1990, vol. 37, pp. 74–80.CrossRefGoogle Scholar
  3. 3.
    Andersson, S. and O’Keeffe, M., Body-Centered Cubic Cylinder Packing and the Garnet Structure, Nature (London), 1977, vol. 267, pp. 605–606.CrossRefGoogle Scholar
  4. 4.
    O’Keeffe, M. and Andersson, S., Rod Packings and Crystal Chemistry, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr., 1977, vol. 33, pp. 914–923.CrossRefGoogle Scholar
  5. 5.
    O’Keeffe, M. and Hyde, B.G., Crystal Structures: I. Patterns and Symmetry, Washington D.C.: The Mineralogical Society of America, 1996.Google Scholar
  6. 6.
    O’Keeffe, M., Cubic Cylinder Packings, Acta Crystallogr., Sect. A: Found. Crystallogr., 1992, vol. 48, pp. 879–884.CrossRefGoogle Scholar
  7. 7.
    Lidin, S., Jacob, M., and Andersson, S., A Mathematical Analysis of Some Rod Packings, J. Solid State Chem., 1995, vol. 114, pp. 36–41.CrossRefGoogle Scholar
  8. 8.
    Sutorik, A.C., Patschke, R., Schindler, J.L., Kannewurf, C.R., and Kanatzidis, M.G., Valence Fluctuations and Metallic Behavior in K6Cu12U2S15, a New Quaternary Sulfide with a Unique Three-Dimensional Cubic Framework, Chem. Eur. J., 2000, vol. 6, pp. 1601–1607.CrossRefGoogle Scholar
  9. 9.
    O’Keeffe, M., Plévert, J., Teshima, Y., Watanabe, Y., and Ogawa, T., The Invariant Cubic Rod (Cylinder) Packings: Symmetries and Coordinates, Acta Crystallogr., Sect. A: Found. Crystallogr., 2001, vol. 57, pp. 110–111.CrossRefGoogle Scholar
  10. 10.
    O’Keeffe, M., Plévert, J., and Ogawa, T., Homogeneous Cubic Cylinder Packings Revisited, Acta Crystallogr., Sect. A: Found. Crystallogr., 1992, vol. 48, pp. 879–884.CrossRefGoogle Scholar
  11. 11.
    Rosen, B.W. and Shu, L.S., On Some Symmetry Conditions for Three-Dimensional Fibrous Composites, J. Compos. Mater., 1971, vol. 5, pp. 279–282.CrossRefGoogle Scholar
  12. 12.
    Hatta, H., Elastic Moduli and Thermal Expansion Coefficients of Three-Dimensional Fabric Composites, Nippon Fukugo Zairyo Gakkaishi (J. Jpn. Soc. Compos. Mater.), 1988, vol. 14, pp. 73–80 [in Japanese].Google Scholar
  13. 13.
    Hijikata, A. and Fukuta, K., Structure of Three-Dimensional Multi-Axis Composites, Nippon Fukugo Zairyo Gakkaishi (J. Jpn. Soc. Compos. Mater.), 1992, vol. 18, pp. 231–238 [in Japanese].Google Scholar
  14. 14.
    Parkhouse, J.G. and Kelly, A., The Regular Packing of Fibres in Three Dimensions, Proc. R. Soc. London, Ser. A, 1998, vol. 454, pp. 1889–1909.CrossRefGoogle Scholar
  15. 15.
    Ogawa, T., Teshima, Y., and Watanabe, Y., Geometry and Crystallography of Rod System, Nippon Fukugo Zairyo Gakkaishi (J. Jpn. Soc. Compos. Mater.), 1995, vol. 21, pp. 165–173 [in Japanese].Google Scholar
  16. 16.
    Teshima, Y., A Study for Possible Structures of Congruent Cylinders, Bussei Kenkyu, 1995, vol. 65, pp. 405–439 [in Japanese].Google Scholar
  17. 17.
    Ogawa, T., Teshima, Y., and Watanabe, Y., Geometry and Crystallography of Self-Supporting Rod Structures, in Katachi U Symmetry, Tokyo: Springer, 1996, pp. 239–246.CrossRefGoogle Scholar
  18. 18.
    Watanabe, Y., Teshima, Y., and Ogawa, T., Geometry of Rod Packings, Symmetry Cult. Sci., 1996, vol. 7, pp. 81–84.Google Scholar
  19. 19.
    Watanabe, Y., Geometry of Rod Packings, Dissertation, Tsukuba: University of Tsukuba Press, 1999 [in Japanese].Google Scholar
  20. 20.
    Teshima, Y., Watanabe, Y., and Ogawa, T., A New Structure of Cylinder Packing, Lect. Notes Comput. Sci., 2001, vol. 2098, pp. 351–361.CrossRefGoogle Scholar
  21. 21.
    Teshima, Y., Matsumoto, T., Watanabe, Y., Ogawa, T., and Kase, K., A New Structure of Cylinder Packing: Its Structure and Space Group, in Proceedings of the ISM Symposium “Statistics and Discrete Geometry: Application to Crystallography and Chemistry,” Tokyo, Japan, Tokyo, March 25–26, 2002, Tokyo, 2002.Google Scholar
  22. 22.
    Teshima, Y. and Matsumoto, T., Periodic Packing of Cylinders with Six Directions: Geometry, in Proceedings of the Topical Meeting of the European Ceramic Society “Geometry, Information, and Theoretical Crystallography of the Nanoworld,” St. Petersburg, Russia, July 30–August 3, 2007, St. Petersburg, 2007.Google Scholar
  23. 23.
    Teshima, Y., Matsumoto, T., Watanabe, Y., and Ogawa, T., Heterogeneous Cylinder Packing: Space Group on Periodic Structures with 〈110〉 Six Directions, in Abstracts of Papers of the XXI Congress and Assembly of the International Union of Crystallography, Osaka, Japan, August 23–31, 2008, Osaka, 2008; Acta Crystallogr., Sect. A: Found. Crystallogr., 2008, vol. 64, pp. C150–C150.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.National Institute of Advanced Industrial Science and Technology (AIST)TsukubaJapan
  2. 2.Kanazawa University (Emeritus Professor)KanazawaJapan

Personalised recommendations