By Geoffrey R. Grimmett, David R. Stirzaker

"The better half quantity to likelihood and Random strategies, third variation this booklet comprises one thousand+ workouts at the topics of ordinary facets of chance and random variables, sampling, Markov chains, convergence, desk bound procedures, renewals, queues, Martingales, Diffusion, Mathematical finance and the Black-Scholes model."

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The symmetry vector R is then defined as the site vector (shown by O R in Fig. * The C h component of R, or c h . 11) az), *If p and q were to have a common divisor, rl then, R/r would have a smaller component in the direction of Ch than does R. R L IRxTI, - T Or T*R T - IChXRI L ’ where L and T are given by Eqs. 8), respectively. Here we assume that between C h and T,as in Fig. 3. R lies 42 CHAPTER 3. SINGLE-WALL CARBON NANOTUBE where ( t l q - t z p ) on the right hand side of Eq. 11 is an integer.

Chiral nanotubes exhibit a spiral symmetry whose mirror image cannot be superposed on to the original one. We call this tube a chiral nanotube, since such structures are called axially chiral in the chemical nomenclature. Axial chirality is commonly discussed in connection with optical activity. We have thus a variety of geometries in carbon nanotubes which can change diameter, chirality and cap structures. 1 and is further discussed in Sect. 2 and Sect. 6. 2. 1: Classification of carbon nanotubes.

2) is defined as the angle between the vectors Ch and a1 , with values of 0 in the range 0 5 101 5 30°, because of the hexagonal symmetry of the honeycomb lattice. The chiral angle 0 denotes the tilt angle of the hexagons with respect to the direction of the nanotube axis, and the angle 0 specifies the spiral symmetry. 4) thus relating 0 to the integers ( n ,m) defined in Eq. 1). In particular, zigzag and armchair nanotubes correspond to 0 = 0’ and 0 = 30°, respectively. 3 Translational Vector: T The translation vector T is defined t o be the unit vector of a 1D carbon nanotube.

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