Symmetry and lattices of single-wall nanotubes

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 (n 1 , n 2 ) (n 1 > n 2 > 0) the groups are Lq p 22 = T rq D n , where n is the greatest common diviser of n 1 and n 2 , q = 2(n 21 +n 1 n 2 +n 22 )/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 = T n2n D nh 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.