The full symmetry and irreducible representations of nanotori

Abstract

The full symmetry groups of carbon nanotori are investigated. It is shown that that the symmetry group of a chiral (n1, n2) nanotorus is isomorphic to D(2mq/n), where m and q are the number of lattice points on the torus circumference vector and the number of graphene hexagons in the nanotorus unit cell, respectively, and n = gcd(n1, n2). It is also shown that the symmetry group of zigzag and armchair (achiral) nanotori is D4m x Z2, where D2k and Zk are the dihedral group of order 2k and the cyclic group of order k, respectively. The irreducible representations and characters of these groups are discussed.

Symmetry group of two special types of carbon nanotori

“…As we noticed above, the meridian and longitude curves of the torus have the same length relative to the distance function on the hypergraph. This means that these models do not fit in the examples discussed in Arezoomand & Taeri (2009) and Yavari & Ashrafi (2009). If there exists a nanotorus with the above structure, then the bond between the atoms must be curled.…”

“…Carbon nanotori are of course just arrangements of carbon atoms on a torus (Dienes & Thomas, 2011;Liu et al, 2002). Their group of symmetry was studied in Arezoomand & Taeri (2009) and Yavari & Ashrafi (2009). However, in these papers it was assumed that the meridian of the torus was much smaller than the longitude of the torus.…”

Symmetry groups associated with tilings on a flat torus

“…These motions are not distance preserving and hence are not elements of G. Nevertheless, we can regard them as four-dimensional symmetries in G Ã acting on H Ã with analogous effects as threedimensional non-rigid motions acting on the nanotorus. In related literature (Bovin et al, 2001;Arezoomand & Taeri, 2009;Zhao et al, 2012), these are also considered as symmetry elements.…”

“…In theory, a carbon nanotorus is a nanosized material obtained when a hexagonal monolayer of carbon atoms (graphene) is rolled into a cylindrical tube along v 1 (the transverse vector) and then joined end to end along v 2 (the longitudinal vector) to form a toroidal nanostructure (Arezoomand & Taeri, 2009). In this work, a geometric model of a carbon nanotorus is obtained as follows.…”

“…It is analogous to the approach used by De Las Peñ as et al (2014) to determine the line-group type of the symmetry group of a single-wall nanotube. Another advantage of the approach is that it allows the relevant non-rigid motions of a nanotorus considered in the literature (Bovin et al, 2001;Arezoomand & Taeri, 2009;Zhao et al, 2012) to be derived and characterized as four-dimensional symmetries. This offers a rigorous treatment of these motions as Euclidean symmetries.…”