Chiral vector

The circumference of any carbon nanotube is expressed in terms of the chiral vector = nai ma2 which connects two crystallographically equivalent sites on a 2D graphene sheet [see Fig. 16(a)] [162]. The construction in... 

Fig. 1. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The chiral vector OA or Cf, = nOf + tnoi defined on the honeycomb lattice by unit vectors a, and 02 and the chiral angle 6 is defined with respect to the zigzag axis. Along the zigzag axis 6 = 0°. Also shown are the lattice vector OB = T of the ID tubule unit cell, and the rotation angle 4/ and the translation r which constitute the basic symmetry operation R = (i/ r). The diagram is constructed for n,m) = (4,2).

Fig. 3. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The pairs of integers ( , ) in the figure specify chiral vectors Cy, (see Table I) for carbon nanotubes, including zigzag, armchair, and chiral tubules. Below each pair of integers (n,m) is listed the number of distinct caps that can be joined continuously to the cylindrical carbon tubule denoted by (n,wi)[6]. The circled dots denote metallic tubules and the small dots are for semiconducting tubules.

Fig. 4. The relation between the fundamental symmetry vector R = p3] -1- qa2 and the two vectors of the tubule unit cell for a carbon nanotube specified by (n,m) which, in turn, determine the chiral vector C, and the translation vector T. The projection of R on the C, and T axes, respectively, yield (or x) and t (see text). After N/d) translations, R reaches a lattice point B". The dashed vertical lines denote normals to the vector C/, at distances of L/d, IL/d, 3L/d,..., L from the origin.

Studies on the electronic structure of carbon nanotube (CNT) is of much importance toward its efficient utilisation in electronic devices. It is well known that the early prediction of its peculiar electronic structure [1-3] right after the lijima s observation of multi-walled CNT (MWCNT) [4] seems to have actually triggered the subsequent and explosive series of experimental researches of CNT. In that prediction, alternative appearance of metallic and semiconductive nature in CNT depending on the combination of diameter and pitch or, more specifically, chiral vector of CNT expressed by two kinds of non-negative integers (a, b) as described later (see Fig. 1). 

Fig. 1. Schematic representation of generation of tube (a, b). Note a>b. Hexagons with chiral vectors satisfying 2a + b = 3/V in condition 1 are shadowed.

Table 2. Calculated energy gap due to an in-plane Kekul distortion for CNTs having chiral vector L/a = (m, 2m). The critical magnetic flux (p. and the corresponding magnetic field are also shown. The coupling constant is A, = 1.62.

Fig. 16. (a) The chiral vector OA or Ch = nhi + md2 is defined on the honeycomb lattice of carbon atoms by unit vectors ai and a of a graphene layer and the chiral angle with respect to the zigzag axis (9 = 0°). Also shown are the lattice vector... 

Fig. 10.7 Chirality vector and folding scheme for semiconducting and metallic nanotube (a). Zig-zag, armchair, and chiral nanotubes by rolling-up of the graphite lattice (b) (Reprinted from Terrones 2003. With permission from Annual Reviews)... 

It is to be noted that the QSPR/QSAR analysis of nanosubstances based on elucidation of molecular structure by the molecular graph is ambiguous due to a large number of atoms involved in these molecular systems. Under such circumstances the chiral vector can be used as elucidation of structure of the carbon nanotubes (Toropov et al., 2007c). The SMILES-like representation information for nanomaterials is also able to provide reasonable good predictive models (Toropov and Leszczynski, 2006a). 

Toropov AA, Leszczynska D, Leszczynski J (2007b) Predicting water solubility and octanol water partition coefficient for carbon nanotubes based on the chiral vector Comp. Biol. Chem. 31 127-128. 

The structure of SWCNTs is characterized by the concept of chirality, which essentially describes the way the graphene layer is wrapped and is represented by a pair of indices (n, m). The integers n and m denote the number of unit vectors (a a2) along the two directions in the hexagonal crystal lattice of graphene that result in the chiral vector C (Fig. 1.1) ... 

Fig. 1.1 (a) Schematic of unrolled SWCNT showing chiral vector Cn and the effect of m and n on the electronic properties of SWCNTs. (b, c, d) The direction of the chiral vector affects the appearance of the nanotube showing (b) (4,4) armchair, (c) (6,0) zigzag and (d) (5,3) exemplary chiral shape. With kind permission from [18],... 

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

When the graphene sheet is rolled up forming a nanotube, the two ends of the chiral vector meet one another. The chiral vector thus forms the circumference of the CNTs circular cross-section. Different values of n and m give different nanotube structures with different diameters (Figure 12.9). 

We have, at low energy, half vector and half chiral vector theory SU( 2) x SU(2)p. On the physical vacuum, we have the vector gauge theory described by A1 = A2 and B3 = V x A3 + (ie/H)A1 x A2 and the theory of weak interactions with matrix elements of the form vy ( 1 y5)e and are thus half vector and chiral on the level of elements of the left- and right-handed components of doublets. We then demand that on the physical vacuum we must have a mixture of vector and chiral gauge connections, within both the electromagnetic and weak interactions, due to the breakdown of symmetry. This will mean that the gauge potential A 3 will be massive and short-ranged. 

Helical (or chiral) vector Ch defined from the director vectors (a-1) and (a2) of the graphene sheet by using a pair of integers (n, m) Ch = na-, + ma2 and chiral angle 0. Reprint from Carbon, vol. 33, No. 7, Dresselhaus M.S., Dresselhaus G., Saito R., Physics of carbon nanotubes, pages 883-891, Copyright (1995) with permission from Elsevier. 

Let us recall that nanotubes can be considered as graphene sheets rolled up in different ways. If we consider the so-called chiral vectors c = nai + na2, in which a and a2 are the basis vectors of a 2D graphite lattice, depending on the value of the integers n and m, one can define three families of tubes armchair tubes (n = m), zig-zag tubes (n or m = 0), and chiral tubes (n m 0). Band structure calculations have demonstrated that tubes are either metallic compounds, or zero-gap semiconductors, or semiconductors [6,7]. More commonly, they are divided into metallic tubes (when n-m is a multiple of 3) or semiconducting ones. 

Toropov et al. [66] constructed two models predicting water solubilities and -octanol/water partition coefficients for carbon nanotubes utilizing the chiral vector components as descriptors. They obtained very high values of correlation coefficients (R2=0.999) for both the training and the validation sets. But, since both... 

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