1. Symmetry of Single-walled Carbon Nanotubes

2. Outline Part I (November 29) Symmetry operations Line groups Part II (December 6) Irreducible representations Symmetry-based quantum numbers Phonon symmetries

3. Construction of nanotubes a 1, a 2 primitive lattice vectors of graphene Chiral vector: c = n 1 a 1 + n 2 a 2 n 1, n 2 integers: chiral numbers Mirror lines: "zig-zag line” through the midpoint of bonds "armchair line” through the atoms Sixfold symmetry: 0   < 60°

4. Construction of nanotubes a 1, a 2 primitive lattice vectors of graphene Chiral vector: c = n 1 a 1 + n 2 a 2 n 1, n 2 integers: chiral numbers Mirror lines: "zig-zag line” through the midpoint of bonds "armchair line” through the atoms

5. Why "chiral" vector? Chiral structure: no mirror symmetry "left-handed" and "right-handed" versions If c is not along a mirror line then the structure is chiral and 60° –  pairs of chiral structures It is enough to consider 0    30° n 1  n 2  0

6. Discrete translational symmetry The line perpendicular to the chiral vector goes through a lattice point. (For a general triangular lattice, this is only true if cos (a 1,a 2 )  is rational. For the hexagonal lattice cos (a 1,a 2 )  = ½.) Period:

7. Space groups and line groups Space group describes the symmetries of a crystal. General element is an isometry: (R | t ), where R  O(3) orthogonal transformation (point symmetry: it has a fixed point) t = n 1 a 1 + n 2 a 2 + n 3 a 3  3 T(3) (superscript: 3 generators, argument: in 3d space) Line group describes the symmetries of nanotubes (or linear polymers, quasi-1d subunits of crystals) (R | t ), where R  O(3) t = n a  1 T(3) (1 generator in 3d space)

8. Point symmetries in line groups CnCn

9. Rotations about the principal axis Let n be the greatest common divisor of the chiral numbers n 1 and n 2. The number of lattice points (open circles) along the chiral vector is n + 1. Therefore there is a C n rotation (2  /n angle) about the principal axis of the line group.

10. Mirror planes and twofold rotations Mirror planes only in achiral nanotubes Twofold rotations in all nanotubes

11. Screw operations All hexagons are equivalent in the graphene plane and also in the nanotubes General lattice vector of graphene corresponds to a screw operation in the nanotube: Combination of rotation about the line axis translation along the line axis

12. General form of screw operations q — number of carbon atoms in the unit cell n — greatest common divisor of the chiral numbers n 1 and n 2 a — primitive translation in the line group (length of the unit cell) Fr(x) — fractional part of the number x  (x) — Euler function All nanotube line groups are non-symmorphic! Nanotubes are single-orbit structures! (Any atom can be obtained from any other atom by applying a symmetry operation of the line group.)

13. Glide planes Only in chiral nanotubes Combination of reflexion to a plane and a translation

14. Line groups and point groups of carbon nanotubes Chiral nanotubs: Lq p 22 Achiral nanotubes: L2n n /mcm Construction of point group P G of a line group G : (R | t )  (R | 0 ) (This is not the group of point symmetries of the nanotube!) Chiral nanotubs: q22 (D q in Schönfliess notation) Achiral nanotubes: 2n /mmm (D 2nh in Schönfliess notation)

15. Site symmetry of carbon atoms Chiral nanotubs: 1 (C 1 ) only identity operation leaves the carbon atom invariant Achiral nanotubes: m (C 1h ) there is a mirror plane through each carbon atom