Chiral angle/vector

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. 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. 3. The multilayered structure considered. The arrows show the bias current. In the case of positive (negative) chirality the magnetization vector M of the layer F3 makes an angle 3a (—a) with the z- axis, i.e. in the case of positive chirality the vector M rotates in one direction if we go over from one F layer to another whereas it oscillates in space in the case of negative chirality.

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. 

Generation of the chiral (8,4) carbon nanotube by rolling a graphite sheet along the vector C = naj + ma2, and definition of the chiral angle 9. The reference unit vectors aj and a2 are shown, and the broken lines indicate the directions for generating achiral zigzag and armchair nanotubes. 

The tubes with m = n are called armchair and those with m = 0 arc referred to as zigzag. All others are chiral with the chiral angle 6 defined as that between the vectors C and ai 0 can be calculated from the equation... 

It is important to define the chiral vector of the nanotube Ch, whieh is given by Ch = nai + m 32 where ai and 32 are unit vectors in the two-dimensional hexagonal lattice, and n and m are integers as shown in Figure 2 [12]. The diameter, dt, is the length of the ehiral veetor divided by %. Another important parameter is the chiral angle 0, whieh is the angle between Ch and ai. 

Figure 2. Schematic diagram of the hexagonal sheet of graphite. Carbon atoms are at the vertices. The parameters that define the nanotube structure when the sheet is rolled (chiral angle, chiral vector, basis vectors ai and a2) are indicated in the figure. From reference 12.

Gao etal. have developed a quantitative structure determination technique of SWNT using NED. This, coupled with improved electron diffraction pattern quahty using NED, allows a determination of both the diameter and chiral angle, and thus the chhal vector (n, m), from individual SWNTs. The CNT they studied were grown by chemical vapor deposition (CVD). TEM observation was carried out in a JEOL201 OF TEM with a high voltage of 200 keV. 

Figure 24 Structure of a carbon nanotube, (a) the unraveled (4,2) tube, where the indices (m,n) refer to the vector (in terms of a and 32) which spans the circumference of the tube. The vector (4,-5) indicates the size of the unit cell. The angle q is the chiral angle of the tube, (b) Positions of the (m,n) vectors on a graphite sheet. Metallic chiralities are indicted with a full circles and semiconducting tubes are indicated with open circles.

From Eq. (2) it follows that 9 = 30° for the (n, n) armchair nanotube and that the n, 0) zigzag nanotube would have 9 = 60°. From Fig. 2A it follows that if we limit 9 to between 0° and 30°, then by symmetry 0=0° for a zigzag nanotube. Both armchair and zigzag nanotubes have a mirror plane and thus are considered as chiral. Differences in the nanotube diameter and chiral angle 9 give rise to differences in the properties of the various CNTs. The symmetry vector R = of the symmetry group for the... 

Figure 13.1 Construction of the (5, 3) SWNT from a graphene sheet. First step the chiral vector Ch is obtained by starting from (0,0) and moving 5 units in the ai direction and then 3 units in the U2 direction. Second step the (dashed) sheet determined by the length of the chiral vector Ch is rolled up along the chiral vector, so the origin (0,0) coincides with the end of Ch, in this example (5, 3). For this nanotube, the chiral angle formed between C and the a direction is 21.8 degrees (calculated from Equation 13.2). The inset illustrates the rolled-up structure of a typical SWNT, this is a (14,0) nanotube with zigzag structure. In this case, the chiral angle is 0.

Theoretical calculations have predicted that the electronic properties of SWNTs depend on the tube diameter d and on the helicity of the hexagonal carbon ring alignment on the nanotube surface, defined by a chiral angle 0, which in mm depends on the n and m integers, which denote the number of unit vectors noj and ma2 in the hexagonal lattice of the graphite ... 

Angle between unit vector a and the chiral vector is defined as chiral angle. In Figure 4.7, chiral angle is shown by 6. Tubes with 0 = 0 have chiral... 

The way in which these graphene sheets are rolled determines the atomic structure of the CNT, which is described in terms of chirality (helicity) of the tube, defined by the chiral vector Ch and chiral angle 6, Fig. 5.3a. The numbers (n, m) are integers and ai and a2 are the unit vectors of the hexagonal lattice of the graphene sheet. 

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