Symmetry and lattices of single-wall nanotubes

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Abstract
The full Euclidean symmetry groups for all the single-wall carbon nanotubes are non-Abelian non-symorphic line groups, enlarging the groups reported in the literature. For the chiral tubes (n1,n2) (n1>n2>0) the groups are Lqp22 = TrqDn, where n is the greatest common divisor of n1 and n2, q = 2(n21+n1n2+n22)/n, while the parameters r and p are expressed in the closed forms as functions of n1 and n2. The number is three if n1-n2 is a multiple of 3n and one otherwise; it divides the tubes into two bijective classes. The line group uniquely determines the tube, unless q = 2n (then r = 1), when both the zig-zag (n,0) ( = 1) and the armchair (n,n) ( = 3) tubes are obtained, with the line group L(2n)n/mcm = Tn2nDnh having additional mirror planes. Some physical consequences are discussed: metallic tubes, quantum numbers and related selection rules, electronic and phonon bands, and their degeneracy, and applications to tensor properties.