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Icosahedral symmetry breaking: C 60 to C 84 , C 108 and to related nanotubes
Author(s) -
Bodner Mark,
Bourret Emmanuel,
Patera Jiri,
Szajewska Marzena
Publication year - 2015
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.742
H-Index - 83
ISSN - 2053-2733
DOI - 10.1107/s2053273315003824
Subject(s) - icosahedral symmetry , physics , combinatorics , fullerene , order (exchange) , crystallography , symmetry group , geometry , chemistry , mathematics , quantum mechanics , finance , economics
This paper completes the series of three independent articles [Bodner et al. (2013). Acta Cryst . A 69 , 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 C 60 . The icosahedral symmetry of C 60 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 C 60 passing through the centers of two opposite edges between two hexagons on the surface of C 60 . 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.