Lattice regular

For larger systems N> 1000 or so, depending on the potential range) anotlier teclmique becomes preferable. The cubic simulation box (extension to noncubic cases is possible) is divided into a regular lattice of n x n x n cells see figure B3.3.7. These cells are chosen so that the side of the cell = L/n is greater than the... 

However, the B.E.T. and modificated B.E.T as well as isotherm of d Arcy and Watt fit the experimental data only in some range of the relative humidities up to about 80-85%. At the same time the adsorption in the interval 90-100% is of great interest for in this interval the A— B conformational transition, which is of biological importance, takes place [17], [18]. This disagreement can be the result of the fact that the adsorbed water molecules can form a regular lattice, structure of which depends on the conformation of the NA. To take into account this fact we assume that the water binding constants depend on the conformational variables of the model, i.e ... 

Crystals are sohds. Sohds, on the other hand can be crystalhne, quasi-crystal-hne, or amorphous. Sohds differ from liquids by a shear modulus different from zero so that solids can support shearing forces. Microscopically this means that there exists some long-range orientational order in the sohd. The orientation between a pair of atoms at some point in the solid and a second (arbitrary) pair of atoms at a distant point must on average remain fixed if a shear modulus should exist. Crystals have this orientational order and in addition a translational order their atoms are arranged in regular lattices. 

The theory of crystal growth accordingly starts usually with the assumption that the atoms in the gaseous, diluted, or hquid mother phase will have a tendency to arrange themselves in a regular lattice structure. We ignore here for the moment the formation of poly crystalhne solids. In principle we should start with the quantum-mechanical basis of the formation of such lattice structures. Unfortunately, however, even with the computational effort of present computers with a performance of about 100 megaflops... 

There are cases where non-regular lattices may be of advantage [36,37]. The computational effort, however, is substantially larger, which makes the models less flexible concerning changes of boundary conditions or topological constraints. Another direction, which may be promising in the future, is the use of hybrid models, where for example local attachment kinetics are treated on a microscopic atomistic scale, while the transport properties are treated by macroscopic partial differential equations [5,6]. 

We do this as follows. The N particles are placed in a starting configuration, for example a regular lattice. Each particle is then tentatively moved at random. For each move, we calculate the change in the mutual potential energy, AU. If At/ is negative, then we allow the move. If At/ is positive, we allow the move with a probability of expi—U/kaT). 

While the locations of the spins are not random - indeed, the spins populate sites of a regular lattice - the interactions themselves are completely random. Frustration, too, has been retained. Thus, arguably, two of the three fundamental properties of real spin glass systems are satisfied. What remains to be seen, of course, is the extent to which this simplified model retains the overall physics. 

Table 7.4 compares the mean-field-theory prediction for the order of the phase transition and critical probability pc to numerical results obtained by Biduax, et.al. ([bidaux89a], [bidaux89b]) by simulating dynamics on regular lattices of dimension d = I, d = 2 and d = 4. 

A discrete regular lattice whose local neighborhood symmetry is such... 

At temperatures only slightly below the liquefaction temperatures, the liquids freeze. The solids are all simple crystals in which the atoms are close-packed in a regular lattice arrangement. The narrow temperature range over which any one of these liquids can exist suggests that the forces holding the crystal together are very much like the forces in the liquid. 

In Chapter 6 we saw that the chemistry of sodium can be understood in terms of the special stability of the inert gas electron population of neon. An electron can be pulled away from a sodium atom relatively easily to form a sodium ion, Na+. Chlorine, on the other hand, readily accepts an electron to form chloride ion, Cl-, achieving the inert gas population of argon. When sodium and chlorine react, the product, sodium chloride, is an ionic solid, made up of Na+ ions and Cl- ions packed in a regular lattice. Sodium chloride dissolves in water to give Na+(aq) and C (aq) ions. Sodium chloride is an electrolyte it forms a conducting solution in water. 

At a given ideal composition, two or more types of defects are always present in every compound. The dominant combinations of defects depend on the type of material. The most prominent examples are named after Frenkel and Schottky. Ions or atoms leave their regular lattice sites and are displaced to an interstitial site or move to the surface simultaneously with other ions or atoms, respectively, in order to balance the charge and local composition. Silver halides show dominant Frenkel disorder, whereas alkali halides show mostly Schottky defects. 

FIGURE 18.15 Plane string net of regular lattice (A) and irregular lattice indicated by a, b, c (B). 

Stability requirements for the existence of these alternative conformational states at Tg allowed us also to estimate the strength of their coupling to the regular lattice vibrations, which is determined by Tg, the material mass density, and the speed of sound. This enabled us to understand the universality of the phonon scattering at the low temperatures. 

The layout of building foundation piles always depends on the building structure, geological condition and the type of foundation piles. It is not in regular lattice pattern in most cases. Many existing designing soft wares utilize so-called G-function and they cannot support the irregular layout. The author s... 

Diffusion and migration in solid crystalline electrolytes depend on the presence of defects in the crystal lattice (Fig. 2.16). Frenkel defects originate from some ions leaving the regular lattice positions and coming to interstitial positions. In this way empty sites (holes or vacancies) are formed, somewhat analogous to the holes appearing in the band theory of electronic conductors (see Section 2.4.1). 

In random bond percolation, which is most widely used to describe gelation, monomers, occupy sites of a periodic lattice. The network formation is simulated by the formation of bonds (with a certain probability, p) between nearest neighbors of lattice sites, Fig. 7b. Since these bonds are randomly placed between the lattice nodes, intramolecular reactions are allowed. Other types of percolation are, for example, random site percolation (sites on a regular lattice are randomly occupied with a probability p) or random random percolation (also known as continuum percolation the sites do not form a periodic lattice but are distributed randomly throughout the percolation space). While the... 

A number of nodes arranged in a regular lattice each node stores a set of weights ... 

Although the SOM is a type of neural network, its structure is very different from that of the feedforward artificial neural network discussed in Chapter 2. While in a feedforward network nodes are arranged in distinct layers, a SOM is more democratic—every node occupies a site of equal importance in a regular lattice. 

In two dimensions, the nodes occupy the vertices of a regular lattice, which is usually rectangular (Figure 3.5). This layer of nodes is sometimes known as a Kohonen layer in recognition of Teuvo Kohonen s (a Finnish academician and researcher) work in developing the SOM. 

A GCS can be constructed in any number of dimensions from one upwards. The fundamental building block is a /c-dimensional simplex this is a line for k = 1, a triangle for k = 2, and a tetrahedron for k = 3 (Figure 4.2). In most applications, we would choose to work in two dimensions because this dimensionality combines computational and visual simplicity with flexibility. Whatever the number of dimensions, though, there is no requirement that the nodes should occupy the vertices of a regular lattice. 

A set of equivalent cells arranged in a regular lattice and a set of states that characterize the cells. 

The cells of the CA lie at the vertices of a regular lattice that homogenously covers a region of space. Although it is possible in principle to use an irregular lattice, we need to be able to define unambiguously the "neighborhood"... 

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

---