Graphene lattice

Fig. 2. Depiction of conformal mapping of graphene lattice to [4,3] nanotube. B denotes [4,3] lattice vector that transforms to circumference of nanotube, and H transforms into the helical operator yielding the minimum unit cell size under helical symmetry. The numerals indicate the ordering of the helical steps necessary to obtain one-dimensional translation periodicity.

The role of heteroatoms has had a very long research history, and some well-established facts have thus emerged. First, oxygen cannot be considered to be present substitutionally in the graphene lattice. Second, because of its ubiquity in surface functionalities, some of which can be deprotonated, its influence is intimately connected to that of pH, as discussed in the previous section. 

Figure 1.14 Atomic force microscopic image of highly ordered pyrolytic graphite (HOPG). The graphene lattice is indicated by the white lines. The different shading of carbon atoms results from the different situation in the atomic layer underneath ( A. Schwarz).

The graphene layer is rolled up in a way to make the ideal ends of an open tube be a zig-zagged edge (Figure 3.2a). It means that the rolling up is done in parallel to the unit vector 3i of the graphene lattice. 

The vector model has proven versatQe to this purpose. It employs the unit vectors of the graphene s two-dimensional unit ceU as a reference dimension. The vector C running in paraUel to the coiling direction is a Unear combination of integer multiples of the units vectors. It is an interconnection of two identical points on the graphene lattice (Figure 3.2). C describes a straight line that represents the uncoiled perimeter of the respective nanotube. It also defines the orientation of the nanotube as the tubular axis f is perpendicular to the tube s cross-section, which on its own part Ues in a plane defined by the perimeter (or the coUed C). 

First experiments and calculations revealed the electronic properties of carbon nanotubes to be in parts rather extraordinary. The small diameter, for instance, causes the occurrence of quantum effects. The tubes behave like a quasi-one-dimensional molecular wire, which is very useful for some electronic apphcations. However, the electronic properties of nanotubes are also related to those of the two-dimensional graphene as the first formally result from the roUing up of the latter. Changes and unexpected phenomena then arise, for example, from the curvature of the graphene lattice. 

Fig. 3.1 The action of the basic Stone-Wales rotations in both the direct and dual graphene lattice representations is shown, (a) SW g flips the arrowed central bond of the four shaded hexagons originating two 517 pairs in gray, (b) SW /y splits the two pairs along dashed direction swapping one of them with two nearby hexagons (shaded). The dashed SW /y operator represents the next available rotation...

Fig. 3.2 (a) Dual representation of the graphene lattice armchair orientated, the direct lattice being also represented. The dashed region individuate the lattice supercell along the diagonal direction, (b-c) Dual representation of the SW mechanisms (a, b) given Fig. 3.1. Hexagons, pentagons, heptagons are represented by white, shaded, black circles respectively. The dashed SW6/7 operator represents the next available rotation...

Molecular graphs made of equivalent atoms (e.g. the graphene lattice or the Cso stable fullerene molecule) all w,- have the same value w, becoming W ... 

In Fig. 3.2 the graphene lattice is represented under an up-down armchair orientation it has N starred nodes and the closed form for its Wiener index is (Cataldo et al. 2011) ... 

Table 3.1 Topological characterization of the 200 nodes of the reference dual graphene lattice when a SW defect is present. Nodes are sorted by their w,- values. Dual graph is made of 200 vertices with W = 116,015 Af = 10 total bonds in the graph B = 600 summing bn entries gives B value twice sum of >v, values is W. The most stable vertices are the two heptagons V91, Vlll on the contrary the two adjacent pentagons V90, V92 are the less stable ones...

Fig. 3.9 The dependence of the topological potential W from the number of the rotated bonds is reported during the multi-diagonal diffusion of the 517 defects in the closed lattice G. Ideal graphene starting ideal graphene lattice has Wq = 116,500, whereas configurations D (Fig. 3.7) and D2 (Fig. 3.8) have W = 112,500 and W = 112,500 respectively...

An SWNT s rolling vector, or chirality, has the most profound effect on its electronic properties. Figure 1 shows the three possible types of carbon nanotubes as defined by chirality. If a carbon nanotube is formed by connecting one carbon atom in the graphene lattice to another carbon atom that is located directly along one of the imit vectors of the surface, the result is what is known as an armchair carbon nanotube. If an SWNT is formed in such a way that one atom cormects and atom which is 30° from the zigzag direction, the result is an... 

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