Symmetry of Single-walled Carbon Nanotubes. Outline Part I (November 29) Symmetry operations Line groups Part II (December 6) Irreducible representations.

Symmetry of Single-walled Carbon Nanotubes

Outline

Part I (November 29) Symmetry operations Line groups

Part II (December 6) Irreducible representations Symmetry-based quantum numbers Phonon symmetries

Construction of nanotubes

a1 , a2 primitive lattice vectors of graphene

Chiral vector: c = n1 a1 + n2 a2

n1 , n2 integers: chiral numbers

Mirror lines: "zig-zag line” through the midpoint of bonds"armchair line” through the atoms

Sixfold symmetry: 0 < 60°

Construction of nanotubes

a1 , a2 primitive lattice vectors of graphene

Chiral vector: c = n1 a1 + n2 a2

n1 , n2 integers: chiral numbers

Mirror lines: "zig-zag line” through the midpoint of bonds"armchair line” through the atoms

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°n1 n2 0

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 (a1,a2) is rational. For the hexagonal lattice cos (a1,a2) = ½.)

Period:

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 = n1 a1 + n2 a2 + n3 a3 3T(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 1T(3) (1 generator in 3d space)

Point symmetries in line groups Cn

Rotations about the principal axisLet n be the greatest common divisor of the chiral numbers n1 and n2 .

The number of lattice points (open circles) along the chiral vector is n + 1.

Therefore there is a Cn rotation (2/n angle) about the principal axis of the line group.

Mirror planes and twofold rotationsMirror planes only in achiral nanotubes

Twofold rotations in all nanotubes

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 axistranslation along the line axis

General form of screw operations

q — number of carbon atoms in the unit celln — greatest common divisor of the chiral numbers n1 and n2

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.)

Glide planes

Only in chiral nanotubes

Combination of reflexion to a plane and a translation

Line groups and point groups of carbon nanotubesChiral nanotubs:

Lqp22

Achiral nanotubes:

L2nn /mcm

Construction of point group PG of a line group G :

(R | t ) (R | 0 ) (This is not the group of point symmetries of the nanotube!)

Chiral nanotubs:

q22 (Dq in Schönfliess notation)

Achiral nanotubes:

2n /mmm (D2nh in Schönfliess notation)

Site symmetry of carbon atomsChiral nanotubs:

1 (C1) only identity operation leaves the carbon atom invariant

Achiral nanotubes:

m (C1h) there is a mirror plane through each carbon atom