Cyclic boundary

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

For the extension to two dimensions we consider a square lattice with nearest-neighbor interactions on a strip with sites in one direction and M sites in the second so that, with cyclic boundary conditions in the second dimension as well, we get a toroidal lattice with of microstates. The occupation numbers at site i in the 1-D case now become a set = ( ,i, /25 5 /m) of occupation numbers of M sites along the second dimension, and the transfer matrix elements are generalized to... 

The most popular method is molecular dynamics. A suitable geometry is shown in Fig. 17.7. A certain number of water molecules are enclosed in a cubic or rectangular box. Two opposite sides of the box, at x = 0 and L, represent two metal surfaces (electrodes). Cyclic boundary conditions are imposed in the y and z directions, that is, a particle that leaves the box at y = L enters again at y = 0, and similarly for the z direction. One starts with a suitable configuration, and... 

The values of 6k are found by means of the cyclic boundary condition (Ziman 1965)... 

Equation (1.47) can be generalized to the 2D case of an (n x m) lattice. The expression then becomes, again only valid for near-neighbour interactions but without cyclic boundary conditions ... 

Even though these methods have shown some success, they require that the box containing the trial structure have cyclic boundary conditions to keep the calculations to a manageable size. This imposes an artificial translational symmetry on the structure. If the results are to converge to the observed structure, the box should either have the size and shape of the observed unit cell or else it should be sufficiently large that a small crystal can spontaneously form within it. 

Fig. 6.30. Results of computer simulations for the time-development of critical exponent a (average for each time decade) observed in the reaction A + B — 0, d = 1 and nA(0) = riB(O) = 0.4. Lattice contains 105 sites, results are for cyclic boundary conditions and 10 runs averaged. Full lines are for the excluded volume case, whereas in dashed line any number of particles could occupy a given site.

To overcome the limitations imposed by the small number of particles and the small size of the box , cyclic boundary conditions are employed. The idea of cyclic boundary conditions is rather obvious in a onedimensional system as the one-dimensional box is simply imagined to be bent back on itself. As one is interested in the interaction between pairs of particles, it is necessary to avoid multiple interactions and the convention is adopted that a given particle is assumed to interact with the nearest version of another particle. To extend this idea to a three-dimensional system, one imagines a structure made from 3x3x3 boxes. The central box is taken as the starting point and particles in this box are taken to interact with either particles in this box or particles in one of the 26 other boxes, according to rules which are a straightforward extension of the one-dimensional cyclic process. 

The constructed system of equations is a closed one. It is solved with the preset initial conditions 6j (r — 0), 0 jg(, t — 0), 6i (2, t = 0). The system of equations makes it possible to describe arbitrary distributions of particles on a surface and their evolution in time. The only shortcoming is the large dimension. The minimal fragment of a lattice on which a process with cyclic boundary conditions should be described is 4 x 4. It is, therefore, natural to raise the question of approximating the description of particle distribution to lower the dimension of the system of equations. In this connection, it is reasonable to consider simpler point-like models. 

It is necessary to state now that the rigorous fulfillment of the Bloch theorem needs an infinity lattice. In order to calculate the number of states in a finite crystal, a mathematical requirement named the Bom-Karman cyclic boundary condition is introduced. That is, if we consider that a crystal with dimensions Nxa, N2b, /V3c is cyclic in three dimensions, then [5]... 

Theorem — Let G be a bipartite regular polymer graph G with cyclic boundary conditions and with a Kekule structure K, which has edge set (K). Then the difference 8(G,e,K) between the orders of e n (K) and of es+n (K) for translationally equivalent boundary sets e varies with a period of no more than 2. The period can only be 2 if the primitive translation interchanges starred unstarred sites. 

For translationally symmetric bipartite systems with NA = NB = N and cyclic boundary conditions, either bn is zero or bn = — 6n-i, although actual values of bn for a given strip depend to a certain extend on the unit cell selected. For instance, bn is even for ladders with an even number of legs, while bn is odd for ladders with an odd number of legs. In particular,... 

Now we consider an Nx AT-site square lattice with cyclic boundary conditions. We replace each site of the lattice by a square (Fig.7) with spins s = 1/2 at its corners, making the total number of spins equal to 41V2. To avoid misunderstanding, however, from now on we continue to refer to these squares as sites. The wave... 

Figure 4.11 A percolation fractal embedded on a 2-dimensional square lattice of size 50 x 50. Cyclic boundary conditions were used. We observe, especially on the boundaries, that there are some small isolated clusters, but these are not isolated since they are actually part of the largest cluster because of the cyclic boundary conditions. Exits (release sites) are marked in dark gray, while all lighter grey areas are blocked areas. Reprinted from [87] with permission from American Institute of Physics.

Let s return to our chain of equally spaced H atoms. It turns out to be computationally convenient to think of that chain as an imperceptible bent segment of large ring (this is called applying cyclic boundary conditions). 

There are a number of different theoretical approaches to the calculation of the band structures of polymers. These are extensions of the methods employed to calculate the electronic states of molecules, which obtain molecular orbitals from a linear combination of atomic orbitals. In this case the states in question are those of an infinitely long molecule, which is approximated by a finite length system with cyclic boundary conditions, i.e. the right-hand end of the chain is, in effect, joined to the left-hand end of the chain. This is the method used for band... 

A most recently developed description of the problem interprets the amplitude mode model in terms of a molecular concept of chains with cyclic boundary conditions (Zerbi et al., 1989). An effective conjugation coordinate Qja with a corresponding force constant fja is defined. The vibrational frequencies are calculated as a function of , which plays the same role as A in the amplitude mode model. Accordingly, the Raman intensity is obtained from the respective ja components of individual modes. This new concept, like the amplitude mode model, properly describes the relative intensities of different modes, but the correct line shapes and line intensities for excitation with different laser lines can not be obtained, since transition matrix elements are not evaluated explicitly. 

When the physical geometry of the problem under consideration or the expected flow pattern has a cyclically repeating nature, cyclic or periodic boundary conditions can be used to reduce the size of the solution domain. Two types of cyclic boundary condition can be distinguished. The first is for rotationally periodic flow processes, where all the variables at corresponding periodic locations on the cyclic planes are the same. The second is for translationally periodic flow processes, where all the variables, except pressure, at corresponding periodic locations on the cyclic planes are the same. Examples of these two types are shown in Fig. 2.7. Such cyclic planes are in fact part of the solution domain (by the nature of their definitions) and no additional boundary conditions are required at these planes, except the one-to-one correspondence between the two cyclic planes. 

Using Bloch s theorem and cyclic boundary conditions [8] the atomic displacements can be expressed as a product of a maximum amplitude, ( /), and a phase factor, the wave vector k. Solutions to Eq. (4.43) are ... 

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