Symmetry of Nanostructures: Nanotubes and Nanowires.

Robert A. Evarestov

Department of Quantum Chemistry, St.Petersburg State University, 26 Universitetskiy Prospekt,

Stary Peterhof 198504, Russia

The  symmetry  groups of three-dimensional objects translationally  periodic  along a line (stereoregular  polymers, nanotubes and nanorods) are known as the commensurate Line Groups (LG). The thirteen families of  LGs  are considered.

The nanotube of the chirality (n1, n2) is  modeled  by  rolling up  the layer in a way that the chiral vector  R becomes the circumference of the tube. It is demonstrated  that  the translational symmetry of the nanotube  can exist for any arbitrary chirality  (n1,n2)  if the nanotube is obtained by  rolling up  the layer with square or hexagonal  plane lattices. The symmetry connection between   the symmetry  groups  of the layers  (layer groups ) and  the symmetry of the nanotubes  obtained by the  layer  rolling up  is analyzed.

The symmetry groups of nanoribbons and nanowires (75 Rod Groups) form a finite subset of an infinite number of line groups. The connection between two factorizations of  RGs  is considered.

The symmetry of the rutile and perovskite based  nanotubes and nanowires is analyzed.