Symmetry and lattices of single-wall nanotubes

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 (n(1), n(2)) (n(1) > n(2) > 0) the groups are Lq(p)22 = (TqDn)-D-r, where n is the greatest common divisor of n(1) and n(2), q = 2(n(1)(2) + n(1)n(2) + n(2)(2))/nR, while the parameters r and p are expressed in the closed forms as functions of n(1) and n(2). The number R is three if n(1) - n(2) 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) (R = 1) and the armchair (n, n) (R = 3) tubes are obtained, with the line group L(2n)(n)/ mcm = (T2nDnh)-D-n 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.