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Alternative approaches to onion‐like icosahedral fullerenes
Author(s) -
Janner A.
Publication year - 2014
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/s2053273313034219
Subject(s) - icosahedral symmetry , fullerene , graphite , affine transformation , scaling , symmetry (geometry) , series (stratigraphy) , carbon fibers , physics , crystallography , computation , materials science , combinatorics , computational chemistry , mathematics , molecular physics , chemistry , pure mathematics , quantum mechanics , geometry , algorithm , paleontology , composite number , composite material , biology
The fullerenes of the C 60 series (C 60 , C 240 , C 540 , C 960 , C 1500 , C 2160 etc. ) form onion‐like shells with icosahedral I h symmetry. Up to C 2160 , their geometry has been optimized by Dunlap & Zope from computations according to the analytic density‐functional theory and shown by Wardman to obey structural constraints derived from an affine‐extended I h group. In this paper, these approaches are compared with models based on crystallographic scaling transformations. To start with, it is shown that the 56 symmetry‐inequivalent computed carbon positions, approximated by the corresponding ones in the models, are mutually related by crystallographic scalings. This result is consistent with Wardman's remark that the affine‐extension approach simultaneously models different shells of a carbon onion. From the regularities observed in the fullerene models derived from scaling, an icosahedral infinite C 60 onion molecule is defined, with shells consisting of all successive fullerenes of the C 60 series. The structural relations between the C 60 onion and graphite lead to a one‐parameter model with the same Euclidean symmetry P 6 3 mc as graphite and having a c / a = τ 2 ratio, where τ = 1.618… is the golden number. This ratio approximates (up to a 4% discrepancy) the value observed in graphite. A number of tables and figures illustrate successive steps of the present investigation.