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
Publication details
- DOI
- 10.24200/squjs.vol19iss1pp95-104
- Journal
- Sultan Qaboos University Journal for Science, 19(1), 95
- Publisher
- Sultan Qaboos University
- Open access
- Gold open access
- License
- CC BY 4.0
Cite this article
APA 7
Koca, N. Ö., Koca, M., & Al-Sawafi, M. (2014). FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and quaternions. Sultan Qaboos University Journal for Science, 19(1), 95. https://doi.org/10.24200/squjs.vol19iss1pp95-104
MLA 9
Koca, Nazife Özdeş, et al. "FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and quaternions." Sultan Qaboos University Journal for Science, vol. 19, no. 1, 2014, pp. 95. https://doi.org/10.24200/squjs.vol19iss1pp95-104.
Chicago (author–date)
Koca, Nazife Özdeş, Mehmet Koca, and M.M.S. Al-Sawafi. 2014. "FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and quaternions." Sultan Qaboos University Journal for Science 19 (1): 95. https://doi.org/10.24200/squjs.vol19iss1pp95-104.
Harvard
Koca, N. Ö., Koca, M. and Al-Sawafi, M. (2014) 'FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and quaternions', Sultan Qaboos University Journal for Science, 19(1), pp. 95. doi:10.24200/squjs.vol19iss1pp95-104.
Vancouver
Koca NÖ, Koca M, Al-Sawafi M. FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and quaternions. Sultan Qaboos University Journal for Science. 2014;19(1):95. doi:10.24200/squjs.vol19iss1pp95-104
IEEE
N. Ö. Koca, M. Koca, and M. Al-Sawafi, "FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and quaternions," Sultan Qaboos University Journal for Science, vol. 19, no. 1, pp. 95, 2014, doi: 10.24200/squjs.vol19iss1pp95-104.