Point-pair distance distribution

In order to validate the point-process model introduced in Section 24.3.2, we consider three different characteristics of stationary point processes the distribution function of (spherical) contact distances (0, oo) (0,1], the nearest-neighbor distance distribution function D [0, OO) (0,1], and the pair-correlation function g [0, oo) [0, oo), which can be found, for example, in Illian et al. (26). 

Thus in principle there are two pieces of information relating to each pair of atoms that can be determined from the electron diffraction data (provided we ignore more complicated matters such as anharmonicity). Refinement of stmctures involves constructing an appropriate mathematical model of the molecule, such as an initial set of Cartesian coordinates. While this is a straightforward choice, it actually requires the refinement of more variables than is strictly necessary, as there are only 3A 6 degrees of freedom for an A-atom molecule (Section 8.2.1), and this number can be further reduced if symmetry is taken into account. This point is extremely important we have already seen in the examples of P3AS and PBrF2S that different atom-pair distances can fall under the same peak in the radial distribution curve, with the result that we could have more variables to define than we have data. 

The description of the atomic distribution in noncrystalline materials employs a distribution function, (r), which corresponds to the probability of finding another atom at a distance r from the origin atom taken as the point r = 0. In a system having an average number density p = N/V, the probability of finding another atom at a distance r from an origin atom corresponds to Pq ( ). Whereas the information given by (r), which is called the pair distribution function, is only one-dimensional, it is quantitative information on the noncrystalline systems and as such is one of the most important pieces of information in the study of noncrystalline materials. The interatomic distances cannot be smaller than the atomic core diameters, so = 0. 

Derivation of the formula for the attraction field caused by an infinitely thin line with the density X is very simple, and is illustrated in Fig. 4.5b. We will consider the field at the plane y = 0. Due to the symmetry of the mass distribution, we can always find a pair of elementary masses Xdy and —Xdy, which when summed do not create the field component gy directed along the y-axis, and respectively the total field generated by all elements of the line has only the component located in the plane y — 0. Here r is the coordinate of the cylindrical system with its origin at the point 0, and the line with masses is directed along its axis. As is seen from Fig. 4.5b the component dg at the point located at the distance r from the origin 0 is... 

The Mahalanobis distance considers the distribution of the object points in the variable space (as characterized by the covariance matrix) and is independent from the scaling of the variables. The Mahalanobis distance between a pair of objects xA and xB is defined as... 

When the distance between each A reactant is very large compared with that between each pair of B reactants, at a point about midway between a pair of A reactants, the concentration of B reactants is unlikely to be significantly affected by the presence of the A reactants. Smoluchowski suggested that such B reactants are effectively an infinite distance from the A reactants under discussion. By effectively an infinite distance is meant perhaps 1000 times the molecular diameter or encounter distance R. In this region, the concentration of B reactants at any time during the reaction is very close to the initial concentration, i.e. [B](1000iZ) [B]0 for all time (t > 0). From the definition of the density distribution, eqn. (2), this boundary condition as r - °° is... 

The irradiation of all kinds of solids produces pairs of the point Frenkel defects hereafter denoted just AB-vacancies, v, and interstitial atoms, i, which usually are well correlated spatially [1-10]. In many ionic crystals these Frenkel defects are the so-called F and H centres discussed in Chapter 3. The function of the initial distribution of complementary defects - v, i pairs (called also geminate) over relative distances depends strongly not only on the particular mechanism of defect creation but also on the particular irradiation kind (e.g., X-rays or photons) [9]. Under creation of v, i pair an interstitial... 

Equation (7.1.16) is asymptotically (cto — oo) exact. It shows that the accumulation kinetics is defined by (i) a fraction of AB pairs, 1 — u>, created at relative distances r > r0, (ii) recombination of defects created inside the recombination volume of another-kind defects. The co-factor (1 - <5a - <5b ) in equation (7.1.16) gives just a fraction of free folume available for new defect creation. Two quantities 5a and <5b characterizing, in their turn, the whole volume fraction forbidden for creation of another kind defects are defined entirely by quite specific many-point densities pmfl and po,m > he., by the relative distribution of similar defects only (see equation (7.1.17)). 

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

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