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)
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.)
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)
Top Related