Symmetry of Single-walled Carbon Nanotubes

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Symmetry of Single-walled Carbon Nanotubes
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Outline

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

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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 ? ? lt 60
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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
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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 ? ? ? 30n1 ? n2 ? 0

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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
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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)
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Point symmetries in line groups
Cn
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Rotations about the principal axis
Let 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.
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Mirror planes and twofold rotations
Mirror planes only in achiral nanotubes Twofold
rotations in all nanotubes
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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
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General form of screw operations

q number of carbon atoms in the unit celln
greatest common divisor of the chiral numbers n1
and n2a 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.)
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Glide planes
Only in chiral nanotubes Combination of reflexion
to a plane and a translation
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Line groups and point groups of carbon nanotubes
Chiral 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)

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Site symmetry of carbon atoms
Chiral nanotubs 1 (C1) only identity operation
leaves the carbon atom invariant Achiral
nanotubes m (C1h) there is a mirror plane
through each carbon atom