Superposition, defects

It remains to comment on the fact that, contrary to expectation, the integral relation for G involves only g<2) and no higher correlation functions. This arises because in writing Eq. (172) the implicit assumption is made that the probability of finding a second partner, say / , to a given defect, a, is the same as it would be if the defect a did not already have a partner / . In fact g<3,( a/ S ) has been replaced by g(2,( a/J ). This is even stronger than the Kirkwood superposition approximation33... 

The formula for specialized distribution functions makes no such assumptions and hence involves g<3). It also involves g(4), g(5),.. . since correction is made for the possibility of a defect having two, three,. . . other partners simultaneously. Using Eq. (171) and the superposition approximation one finds that for the sodium chloride type lattice... 

This breakdown of the linear relationship between the absorption coefficient a and the product of densities, Q1Q2, indicates that the observed absorption is not a binary process. Specifically, for the case at hand, one can no longer assume that the measured absorption consists of an incoherent superposition of the pair contributions. Rather, the correlations of the dipoles that are induced in subsequent binary collisions lead to a partially destructive interference, an absorption defect that occurs if the product of the time T12 between Ne-Xe collisions, and microwave frequency, /, approaches unity [404], We note that for the spectra shown above, Figs. 3.1 and 3.2, the product fx 2 is substantially greater than unity at all frequencies where experimental data are shown and, consequently, incoherent superpositions of the waves arising from different induced dipoles occur. The intercollisional absorption defect is limited to low frequencies (Lewis 1980). 

Figure 3.4 is a schematic illustration of the fact that dipoles induced in successive collisions tend to be more or less antiparallel. This anticorrelation of dipoles induced in subsequent collisions leads to the absorption defect and causes the breakdown of the pair behavior illustrated in Fig. 3.3, if the product fx 2 is of the order of unity or less. If, on the other hand, fxn 1, superposition occurs with widely varying, random phase differences which render an interference effect inefficient. 

Figure 5-5. a) Point defect potential in an ionic crystal superposition of the periodic lattice potential and the individual defect potential valley, b) Change of potential with time after a defect... 

In this Chapter the kinetics of the Frenkel defect accumulation under permanent particle source (irradiation) is discussed with special emphasis on many-particle effects. Defect accumulation is restricted by their diffusion and annihilation, A + B — 0, if the relative distance between dissimilar particles is less than some critical distance 7 0. The formalism of many-point particle densities based on Kirkwood s superposition approximation, other analytical approaches and finally, computer simulations are analyzed in detail. Pattern formation and particle self-organization, as well as the dependence of the saturation concentration after a prolonged irradiation upon spatial dimension (d= 1,2,3), defect mobility and the initial correlation within geminate pairs are analyzed. Special attention is paid to the conditions of aggregate formation caused by the elastic attraction of particles (defects). 

As in previous Chapters, for practical use this infinite set (7.1.1) has to be decoupled by the Kirkwood - or any other superposition approximation, which permits to reduce a problem to the study of closed set of densities pm,m with indices (m + mr) 2. As earlier, this results in several equations for macroscopic concentrations and three joint correlation functions, for similar, X (r,t),X-s r,t), and dissimilar defects Y(r,t). However, unlike the kinetics of the concentration decay discussed in previous Chapters, for processes with particle sources direct use of Kirkwood s superposition approximation gives good results for small dimesionless concentration parameters Uy t) = nu(t)vo < 1 only (vq is d-dimensional sphere s volume, r0 is its radius). The accumulation kinetics predicted has a very simple form [30, 31]... 

The direct consequence of this statement for Kirkwood s superposition approximation is as follows. Substitution of equation (2.3.62) into p2,i yields correct order of its magnitude, a 1, provided f] — rfl < ro, r% — r[ > ro (i.e., there is a single A in the recombination sphere around B), since two-point density p p(f, r[ t) oc and pi,i (r 2i t) oc (<7o)° (i.e., is limited as well as another density />2,o> which does not fall into category of virtual configurations). On the other hand, for coordinates satisfying ri - f[ < ro, r 2 — rj < ro (i.e., defect B has in its recombination sphere two defects A) substitution of equation (2.3.62) results in p p oc instead of the correct M Due to this the superposition approximation neglects in the limit (To oo a number of terms in equations which finally leads to a considerable error in the accumulation kinetics. 

There is no limitation for the use of Kirkwood s superposition approximation (2.3.62) in the integral terms of equation (7.1.27) since there is a single defect in the recombination sphere of dissimilar defect. For example, a product a( r2 — f[ )piti(r2 f( t) in a line with definition (7.1.3) could be replaced by <7i,i(r2 rj <). Therefore, we have... 

Equation (3) is the most widely used in analyzing experimental curves, since its form is intuitively clear the rate of the defect accumulation is determined by the fraction of free volume of the crystal not occupied by previously created defects, without taking account of the overlap of the annihilation volumes of similar defects. Evidently it is applicable only in the initial stage of accumulation kinetics at relatively low concentrations of defects, nvo superposition approximation corresponds to the first two terms of expansion (2) in powers of nvo-... 

In our opinion, this book demonstrates clearly that the formalism of many-point particle densities based on the Kirkwood superposition approximation for decoupling the three-particle correlation functions is able to treat adequately all possible cases and reaction regimes studied in the book (including immobile/mobile reactants, correlated/random initial particle distributions, concentration decay/accumulation under permanent source, etc.). Results of most of analytical theories are checked by extensive computer simulations. (It should be reminded that many-particle effects under study were observed for the first time namely in computer simulations [22, 23].) Only few experimental evidences exist now for many-particle effects in bimolecular reactions, the two reliable examples are accumulation kinetics of immobile radiation defects at low temperatures in ionic solids (see [24] for experiments and [25] for their theoretical interpretation) and pseudo-first order reversible diffusion-controlled recombination of protons with excited dye molecules [26]. This is one of main reasons why we did not consider in detail some of very refined theories for the kinetics asymptotics as well as peculiarities of reactions on fractal structures ([27-29] and references therein). 

Fig. 18. One-dimensional model for visualisation of the surface lattice by a blunt tip based on the Moire mechanism, a The signal, which is measured by the detector during lateral sliding of the two lattices over each other, represents a superposition of the two wave vectors k and 1 b The regular pattern vanishes when the correlation between the lattices is disturb ed by the irregular surface structure c The regular pattern retains though a single atom defect emerges in the surface lattice...

However, this assumption is not necessarily justified. Even for a well-faceted nanoparticle there are a number of nonequivalent adsorption sites. For example, in addition to the low-index facets, the palladium nanoparticle exhibits edges and interface sites as well as defects (steps, kinks) that are not present on a Pd(l 1 1) or Pd(lOO) surface. The overall catalytic performance will depend on the contributions of the various sites, and the activities of these sites may differ strongly from each other. Of course, one can argue that stepped/kinked high-index single-crystal surfaces (Fig. 2) would be better models (64,65), but this approach still does not mimic the complex situation on a metal nanoparticle. For example, the diffusion-coupled interplay of molecules adsorbed on different facets of a nanoparticle (66) or the size-dependent electronic structure of a metal nanoparticle cannot be represented by a single crystal with dimensions of centimeters (67). It is also shown below that some properties are merely determined by the finite size or volume of nanoparticles (68). Consequently, the properties of a metal nanoparticle are not simply a superposition of the properties of its individual surface facets. 

It gives slightly higher saturation concentration Uo — n2K, 0.69 (again the same for all dimensions d) than the superposition approximation does, but it is still essentially underestimated. For the first time the function 5 t) was successfully calculated in [31], as defined by the correlation function of similar defects, X r,t). However, the only linear corrections in the correlation functions were taken into account. The saturation concentration Uq = 1.08 for d = 3 agrees with computer simulations Uq = 1.01 0.10 [36]. However, the saturation predicted for the low dimensions, e.g., d = 1, C/q = 1-36 is much lower than computer simulations (see, e.g., [15, 35]). 

Fig. 5. Total density maps (above) and difference density maps (total minus ionic superposition below) for Fs, Fs, and Fs defects at the MgO (100) surface (from left to right). In the plots below continuous lines correspond to charge accumulation, dashed lines to charge depletion with respect to the superposition of the ionic densities. Reproduced from ref. [55]. Copyright 1997 American Institute of Physics.

Figure 10 A quadruple SiO chain defect in a matrix of double (Si04) chains of the amphibole type is shown (a) as a drawing of packing polyhedra and (b) occurring naturally in a crystal fragment of nephrite jade. In (b) the lower inset shows a superposition of the drawing in (a) at a reduced scale and the upper inset shows the computed image of such a defect (Reproduced by permission from Contrib. Mineral. Petrol., 1978, 66, 1)...

The absence of any preferred zone combination in the electron diffraction patterns suggests that the side faces of the nanorods are either not well-developed or consist of two forms, e.g. 100 and 110 with approximately equivalent surface area. HRTEM images of individual gold nanorods (Figure 9.5) show stripe patterns characteristic of the superposition of two diffraction patterns, i.e. a twinned defect structure, consistent with the SAED data. 

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