Article Sidebar

Submitted
Apr 27, 2017
Published
Jun 1, 2014
Download
PDF
Statistic
Read Counter : 325 Download : 233

Main Article Content

Abstract

We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine extended Coxeter groups W(A3) and W(B3)=Aut(A3). It is naturally expected that these rank-3 Coxeter-Weyl groups define the point tetrahedral symmetry and the octahedral symmetry of the cubic lattices which have extensive applications in material science. The imaginary quaternionic units are used to represent the root systems of the rank-3 Coxeter-Dynkin diagrams which correspond to the generating vectors of the lattices of interest. The group elements are written explicitly in terms of pairs of quaternions which constitute the binary octahedral group. The constructions of the vertices of the Wigner-Seitz cells have been presented in terms of quaternionic imaginary units. This is a new approach which may link the lattice dynamics with quaternion physics. Orthogonal projections of the lattices onto the Coxeter plane represent the square and honeycomb lattices. 

 

 

Keywords

Coxeter groups Lattices Quaternions.

Article Details

References

  1. COXETER, H.S.M. and MOSER, W.O.J. Generators and Relations for Discrete Groups Springer-Verlag, 1965.
  2. WEINBERG, S. A Model of Leptons. Phys. Rev. Lett., 1967, 19: 1264-1266.
  3. SALAM, A. Elementary Particle Theory: Relativistic Groups and Analyticity. Eighth Nobel Symposium, Stockholm (ed. N. Svartholm), 1968, 367-377.
  4. FRITZSCH, H., GELL-MANN, M. and LEUTWYLER, H. Advantages of the Colour Octet Gluon Picture. Phys. Lett.B, 1973, 47: 365-368.
  5. GEORGI, H. and GLASHOW, S. Unity of all elementary-particle forces. Phys. Rev. Lett., 1974, 32: 438-441.
  6. FRITZSCH, H. and MINKOWSKI, P. Unified interactions of leptons and hadrons. Ann. Phys. 1975, 93: 193-266.
  7. GÜRSEY, F., RAMOND, P. and SIKIVIE, P. A universal gauge theory model based on E6. Phys. Lett.B, 1976, 60: 177-186.
  8. KOCA, M., KOC, R. and AL-AJMI, M. Polyhedra obtained from Coxeter groups and quaternions. J. Math. Phys., 2007, 48: 113514-113527.
  9. KOCA, M., KOCA, N.O. and KOC, R. Catalan Solids Derived From 3D-Root System. J. Math. Phys., 2010, 51: 043501-043513.
  10. COTTON, F.S., WILKINSON, G., MURILLO, C.A. and BOCHMANN, M. Advanced Inorganic Chemistry (6th Edition) Wiley-Inter science, New York, 1999.
  11. CASPAR, D.L.D. and KLUG, A. Physical Principles in Construction of Regular Viruses. Cold Spring Harbor Symp. Quant. Biol., 1962, 27: 1-24.
  12. TWAROCK, R. A Mathematical Physicist’s Approach to the Structure and Assembly of Viruses. Phil. Trans. R. Soc.A, 2006, 364: 3357-3374.
  13. KATZ, A. and DUNEAU, M. Quasiperiodic patterns and icosahedral symmetry. J. Phys. France, 1986, 47: 181-196.
  14. ELSER, V. The diffraction pattern of projected structures. Acta Cryst. A, 1986, 42: 36-43.
  15. BAAKE, M., JOSEPH, D., KRAMER, P. and SCHLOTTMANN, M. Root lattices and quasicrystals. J. Phys. Math.and Gen. A, 1990, 23: L1037-L1041.
  16. BAAKE, M., KRAMER, P., SCHLOTTMANN, M. and ZEIDLER, D. Planar patterns with fivefold symmetry as sections of periodic structures in 4-space. Int. J. Mod. Phys. B, 1990, 4: 2217-2268.
  17. CONWAY, J.H. and SLOANE, N.J.A. Sphere Packings, Lattices and Groups Springer-Verlag, New York, 1988.
  18. KOCA, M., KOCA, N.O. and KOC, R. 12-fold Symmetric Quasicrystallography from Higher Dimensional Lattices, 2013, arXiv: 1306.3316v2.
  19. GEIM, A.K. and NOVOSOLEV, K.S. The rise of graphene. Nature Materials 2007, 6 (3): 183-191.
  20. HAMILTON, S.W. R. Lectures on Quaternions London : Whittaker, 1853.
  21. KOCA, M., KOC, R., AL-BARWANI, M. Noncrystallographic Coxeter group H4 in E8. J. Phys. A: Math. Gen. A, 2001, 34: 11201-11213.
  22. CONWAY, J.H. and SMITH, D.A. On Quaternions and Octonions: Their Geometry, Arithmetics, and Symmetry A.K. Peters, Ltd. Natick, M.A., 2003.
  23. KOCA, M., KOCA, N.O. and AL-SHUEILI, M. Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions. SQU journal for Science, 2011, 16: 82-101.
  24. ASHCROFT, N.W. and MERMIN, N.D. Solid State Physics, W.B. Saunders Company, 1976.