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Bodner M.,11 Academy Drive | Bourret E.,University of Montreal | Patera J.,11 Academy Drive | Patera J.,University of Montreal | And 3 more authors.
Acta Crystallographica Section A: Foundations and Advances | Year: 2015

This paper completes the series of three independent articles [Bodner et al. (2013). Acta Cryst. A69, 583-591, (2014), PLOS ONE, 10.1371/journal.pone.0084079] describing the breaking of icosahedral symmetry to subgroups generated by reflections in three-dimensional Euclidean space as a mechanism of generating higher fullerenes from C60. The icosahedral symmetry of C60 can be seen as the junction of 17 orbits of a symmetric subgroup of order 4 of the icosahedral group of order 120. This subgroup is noted by A 1 × A 1, because it is isomorphic to the Weyl group of the semi-simple Lie algebra A 1 × A 1. Thirteen of the A 1 × A 1 orbits are rectangles and four are line segments. The orbits form a stack of parallel layers centered on the axis of C60 passing through the centers of two opposite edges between two hexagons on the surface of C60. These two edges are the only two line segment layers to appear on the surface shell. Among the 24 convex polytopes with shell formed by hexagons and 12 pentagons, having 84 vertices [Fowler & Manolopoulos (1992). Nature (London), 355, 428-430; Fowler & Manolopoulos (2007). An Atlas of Fullerenes. Dover Publications Inc.; Zhang et al. (1993). J. Chem. Phys. 98, 3095-3102], there are only two that can be identified with breaking of the H 3 symmetry to A 1 × A 1. The remaining ones are just convex shells formed by regular hexagons and 12 pentagons without the involvement of the icosahedral symmetry. © 2015 International Union of Crystallography.

Bodner M.,11 Academy Drive | Chadzitaskos G.,Czech Technical University | Patera J.,11 Academy Drive | Patera J.,University of Montreal | And 2 more authors.
Acta Polytechnica | Year: 2013

About 60 years ago, I. Shmushkevich presented a simple ingenious method for computing the relative probabilities of channels involving the same interacting multiplets of particles, without the need to compute the Clebsch-Gordan coefficients. The basic idea of Shmushkevich is "isotopic non-polarization" of the states before the interaction and after it. Hence his underlying Lie group was SU(2). We extend this idea to any simple Lie group. This paper determines the relative probabilities of various channels of scattering and decay processes following from the invariance of the interactions with respect to a compact simple a Lie group. Aiming at the probabilities rather than at the Clebsch-Gordan coefficients makes the task easier, and simultaneous consideration of all possible channels for given multiplets involved in the process, makes the task possible. The probability of states with multiplicities greater than 1 is averaged over. Examples with symmetry groups O(5), F(4), and E(8) are shown. © Czech Technical University in Prague, 2013.

Bodner M.,11 Academy Drive | Patera J.,11 Academy Drive | Patera J.,University of Montreal | Szajewska M.,University of Montreal | Szajewska M.,University of Bialystok
Acta Crystallographica Section A: Foundations of Crystallography | Year: 2013

The icosahedral symmetry group H 3 of order 120 and its dihedral subgroup H 2 of order 10 are used for exact geometric construction of polytopes that are known to exist in nature. The branching rule for the H 3 orbit of the fullerene C60 to the subgroup H 2 yields a union of eight orbits of H 2: four of them are regular pentagons and four are regular decagons. By inserting into the branching rule one, two, three or n additional decagonal orbits of H 2, one builds the polytopes C70, C80, C90 and nanotubes in general. A minute difference should be taken into account depending on whether an even or odd number of H 2 decagons are inserted. Vertices of all the structures are given in exact coordinates relative to a non-orthogonal basis naturally appropriate for the icosahedral group, as well as relative to an orthonormal basis. Twisted fullerenes are defined. Their surface consists of 12 regular pentagons and 20 hexagons that have three and three edges of equal length. There is an uncountable number of different twisted fullerenes, all with precise icosahedral symmetry. Two examples of the twisted C60 are described. © 2013 International Union of Crystallography Printed in Singapore - all rights reserved.

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