Symmetry of SWNTs Kriza Gyorgy I

8/2/2019 Symmetry of SWNTs Kriza Gyorgy I

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Symmetry of Single-walled

Carbon Nanotubes



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 , a 2 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 , a 2 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 ,a 2) = 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 subunitsof crystals)

( R | t ) , where R O(3)

t = n a 1T (3) (1 generator in 3d space)





Rotations about the principal axis

Let n be the greatest common divisorof the chiral numbers n1 and n2 .

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.



Mirror planes and twofold rotations

Mirror planes only in achiral nanotubes

Twofold rotations in all nanotubes



Screw operations

All hexagons are equivalent in thegraphene 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



General form of screw operations

q — number of carbon atoms in the unit cell

n — 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:

Lq p22

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 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

Symmetry of SWNTs Kriza Gyorgy I

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