Graphene hexagonal sheet

Geometrically, CNTs can be described in terms of a two-dimensional graphene (graphite) sheet. A chiral vector is defined on the hexagonal lattice as... 

A nanotube can be thought of as a hexagonal sheet of carbon atoms (graphene sheet), rolled up to make a cylinder and capped at the ends by a half of a buckyball, as illustrated in Figure 17.8. Tubes typically have diameters of about 1 nm. The diameter of the smallest nanotube corresponds to the diameter of the smallest buckyball (Ceo-) The length-to-diameter ratio is typically about 104. 

FIGURE 1 Schematic diagram showing that how a hexagonal sheet of graphene is rolled to form a CNT with different chirahties (A—armehair, B— zigzag, and C—ehiral). 

Figure 12.2 Schematic diagram showing how a hexagonal sheet of graphene is rolled" to form a carbon nanotube. The rolling shown in the diagram will form a (3,2) nanotube. (Reprinted with permission from Reference (15). Copyright...

These surprising results can be understood on the basis of the electronic structure of a graphene sheet which is found to be a zero gap semiconductor [177] with bonding and antibonding tt bands that are degenerate at the TsT-point (zone corner) of the hexagonal 2D Brillouin zone. The periodic boundary... 

Fig. 3. (a) Depiction of central Brillouin zone and allowed graphene sheet states for a [4,3] nanolube conformation. Note Fermi level for graphene occurs at K points at vertices of hexagonal Brillouin zone, (b) Extended Brillouin zone pie-ture of [4,3] nanotube. Note that top left hexagon is equivalent to bottom right hexagon. 

The question whieh then arises is What do we call a defect in a nanotube To answer this question, we need to define what would be a perfeet nanotube. Nanotubes are mieroerystals whose properties are mainly defined by the hexagonal network that forms the eentral cylindrical part of the tube. After all, with an aspect ratio (length over diameter) of 100 to 1000, the tip structure will be a small perturbation except near the ends. This is clear from Raman studies[4] and is also the basis for calculations on nanotube proper-ties[5-7]. So, a perfect nanotube would be a cylindrical graphene sheet composed only of hexagons having a minimum of defects at the tips to form a closed seamless structure. 

From Euler s theorem, one can derive the following simple relation between the number and type of cycles n, (where the subscript / stands for the number of sides to the ring) necessary to close the hexagonal network of a graphene sheet ... 

Next, we consider the F-point nanotube modes obtained by setting k = Q and /j. = N/2 in eqn (17). The modes correspond to 2D graphene sheet modes at the point k = (A 7r/C)Cin the hexagonal BZ. We consider how such modes transform under the symmetry operations of the groups Qj and Under the ac-... 

Single planar sheet of sp -bonded carbon atoms corresponds to one hexagonal basal plane of graphite and is termed graphene. Recently, a successful attempt to isolate such a graphene layer has been reported [3], and apparently, it becomes possible to produce... 

Each graphene sheet is composed of rings of carbon atoms arranged on a hexagonal 2D tiling. This form of carbon has n bonds in addition to the a bonds, as shown in Fig. 4.6, leading to a bond order of 1.33. 

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