The structure of β-Mn crystallizes in space group P4132. The pseudo 8-fold nature of the 41 axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a five-dimensional super space group containing mutually perpendicular 8-fold axes cannot generate P4132 on projection to 3-d space and how this may instead be accomplished from a six-dimensional model. A procedure for generating the actual structure of β-Mn lifted to six-dimensional space is given.
References
[1]
Wang, N.; Chen, H.; Kuo, K.H. Two dimensional quasicrystal with eightfold rotational symmetry. Phys. Rev. Lett. 1987, 59, 1010–1013, doi:10.1103/PhysRevLett.59.1010.
[2]
Wang, N.; Fung, K.K.; Kou, K.H. Symmetry study of the Mn-Si-Al octagonal quasicrystal by convergent beam electron diffraction. Appl. Phys. Lett. 1988, 52, 2120–2121, doi:10.1063/1.99754.
[3]
Jiang, J.-C.; Hovm？ller, S.; Zou, X.-D. A three dimensional model of eight-fold quasicrystals obtained by high-resolution electron mmicroscopy. Phil. Mag. Lett. 1995, 71, 123–129, doi:10.1080/09500839508241004.
[4]
Elenius, M.; Zetterling, H.M.; Dzugutov, M.; Fredrickson, D.C.; Lidin, S. Structural model for octagonal quasicrystals derived from octagonal symmetry elements arising in β-Mn crystallization of a simple monoatomic liquid. Phys. Rev. B 2009, 79, 144201:1–144201:10.
[5]
Nyman, H.; Carroll, C.E.; Hyde, B.G. Rectilinear rods of face-sharing tetrahedra and the structure on β-Mn. Z. Krist. 1991, 196, 39–46, doi:10.1524/zkri.1991.196.1-4.39.
[6]
Lidin, S.; Andersson, S. Regular polyhedral helices. Z. Anorg. Allg. Chem. 1996, 622, 164–166, doi:10.1002/zaac.19966220123.
[7]
Berger, R.; Lee, S.; Johnson, J.; Nebgen, B.; So, C.-Y. Laves phases, γ-brass, and 2 × 2 × 2 superstructures: A new class of quasicrystal approximants and the suggestion of a new quasicrystal. Chem. Eur. J. 2008, 14, 6627–6639, doi:10.1002/chem.200800336.
[8]
Deonarine, S.; Birman, J.L. Character table of the hypercubic pointgroup in six dimensions. Acta Crystallogr. A 1989, 45, 879–882, doi:10.1107/S0108767389007348.
