Discrete lattice sites

Although the theory of solutions has been widely used in formulating problems of defects in solids the problems encountered differ in certain respects. The most obvious point is that defects are restricted to discrete lattice sites, whereas the ions in a solution can occupy any position in the fluid. Sometimes no allowance is made for this fact. For example, it has not been demonstrated that at very low concentrations, in the absence of ion-pair effects, the activity coefficients are identical with those of the Debye-Hiickel theory. It can be plausibly argued51 that at sufficiently low concentrations the effect of discreteness is likely to be negligible, but clearly in developing a theory for any but the lowest concentrations the effect should be investigated. A second point... 

The coarse graining procedure used earlier results in the smoothing of physical variables as one proceeds from the atomic domain to Kadanoff block structures. The denumeration of discrete lattice sites are then replaced by the continuum distance variable r, whereby physical properties such as the order parameter are sensibly expected to become functions of r. Close to criticality, the system contains islands of correlated spins of average extension A given by the correlation length f(7), within which one encounters a magnetization... 

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

Discrete Cellular State Space the discrete lattice of cells or sites upon which CA live , and their dynamics unfolds. C can be one-dimensional, two-dimensional... 

A completely discrete phase space (i.e. discrete values of lattice-site positions, particle velocities and time, so that particles move from site-to-site and collisions taken within discrete time steps). 

To discretize this theory, we attach the field to every lattice site n and make, the substitution for its derivative ... 

Cellular automata are simple mathematical idealizations of natural systems. They consist of a lattice of discrete identical sites, each site taking on a finite set of say integer values. The values of the sites evolve in discrete time steps according to deterministic rules that specijy the value of each site in terms of the values of neighboring sites. Cellular automata may thus be considered as... 

Producing a reasonably good accuracy for analytically defined surfaces, this scheme of calculation is very inaccurate when the field is specified by the discrete set of values (the lattice scalar field). The surface in this case is located between the lattice sites of different signs. The first, second, and mixed derivatives can be evaluated numerically by using some finite difference schemes, which normally results in poor accuracy for discrete lattices. In addition, the triangulation of the surface is necessary in order to compute the integral in Eq. (8) or calculate the total surface area S. That makes this method very inefficient on a lattice in comparison to the other methods. 

In LGCA models, time and space are discrete this means that the model system is defined on a lattice and the state of the automaton is only defined at regular points in time with separation St. The distance between nearest-neighbor sites in the lattice is denoted by 5/. At discrete times, particles with mass m are situated at the lattice sites with b possible velocities ch where i e 1, 2,. .., b. The set c can be chosen in many different ways, although they are restricted by the constraint that... 

Consider the approximation of four discrete molecular orientations along the axes of a square lattice ( At/4 kBT, AU4 < 0). To conveniently describe orientations, we introduce, at each lattice site, two spin variables, a m = l and s m = 1, which are related to unit vectors nm as follows (Fig. 2.18) ... 

In the structures of compounds of the type M3UF7 the seven F atoms are statistically distributed over fluorite lattice sites.153 The nine-coordinate thorium atom in (NH ThFg is surrounded by a distorted tricapped trigonal prismatic array of fluorine atoms, with the prisms sharing edges to form chains, whereas the uranium(IV) compound contains discrete dodeca-hedrally coordinated [UF8]4 ions. The protactinium(IV), neptunium(IV) and plutonium(IV) analogues are isostructural with the uranium compound.154... 

In computer simulations a discrete lattice of sites is considered, each site is occupied by not more than one particle. Particles A are localized in their sites for r seconds and then possess hops. Thus, the mean diffusion coefficient Da = a2/(2dr) could be introduced. We assume that particles B are immobile, Dq = 0, since it permits to reduce greatly simulation time. Moreover, for Da = Dq and d = 1,2 the kinetics turn out to be quite similar. A hop of particle A into the site occupied by particle B results in their instant recombination. 

In order to study theoretically defect aggregation, several methods of physical and chemical kinetics were developed in recent years. Irrespective of the particular method used, the two basic approaches - a continuous and discrete-lattice ones - are used. In the former model intrinsic defect volume is ignored and thus a number of similar defects in any volume element is unlimited. In its turn, in the latter model any lattice site could be occupied by no more than a single particle (v or i) [15]. 

In studying processes of accumulation of the Frenkel defects, one uses three different types of simple models the box, continuum, and discrete (lattice) models. In the simplest, box model, which was proposed first in [22], one studies the accumulation of complementary particles in boxes having a certain capacity, with walls impenetrable for diffusion of particles among the boxes. The continuum model treats respectively a continuous medium the intrinsic volume of similar defects at any point of the space is not bounded. In the model of a discrete medium a single cell (e.g., crystalline lattice site) cannot contain more than one defect (v or i). 

The simulation takes place on a discrete lattice with coordination number 2. Each lattice site is given a lattice vector l. The state of the site l is represented by the lattice variable 07, which may depend on the state of the catalyst site (e.g., promoted or not) and on the coverage with a particle. Let us assume we deal with the simple case in which all catalyst sites are equal... 

In the last region K = 0 for T -C c/(p2/imp) we come back to the strong pinning case, discussed in section 3.3 before, and calculate the pair correlation function exactly. Taking into account that the hi s are independent on different lattice sites, i.e., hihj oc the (discrete) phase correlation function is given... 

The situation with II-VI semiconductors such as ZnO is similar to the situation with the elemental and the III-V semiconductors in respect of the location of the impurity atoms and their influences on the electric property. It is reported in ZnO that P, As, or S atom replaces either Zn or O site, and a part of them are also located at an interstitial site, as well as at a substitutional site [2,5-7], The effect of a few kind of impurities such as group-IIIA and -VA elements on the electric property of ZnO was extensively studied, especially when the impurity atoms were located at a substitutional site. The effects of the greater part of elements in the periodic table on the electric property of ZnO are, however, not well understood yet. The purpose of the present study is to calculate energy levels of the impurity atoms from Li to Bi in the periodic table, to clarify the effect of impurity atoms on the electric property of ZnO. In the present paper, we consider double possible configuration of the impurity atoms in ZnO an atom substitutes the cation lattice site, while another atom also substitutes the anion sublattice site. The calculations of the electronic structure are performed by the discrete-variational (DV)-Xa method using the program code SCAT [8,9],... 

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