Vacancies structure factor

It was Ziman [77] who has noted that there is little hope, at least at present, to develop an experimental technique permitting the direct measurement of these correlation functions. The only exception are the joint densities x / (r> ) information about which could be learned from the diffraction structural factors of inhomogeneous systems. On the other hand, optical spectroscopy allows estimation of concentrations of such aggregate defects in alkali halide crystals as Fn (n = 1,2,3,4) centres, i.e., n nearest anion vacancies trapped n electrons [80]. That is, we can find x mK m = 1 to 4, but at small r only. Along with the difficulties known in interpretating structure factors of binary equilibrium systems (gases or liquids), obvious specific complications arise for a system of recombining particles in condensed media which, in its turn, are characterized by their own structure factors. 

Before discussing screening, let us examine the effect of lattice imperfections on the electronic diffraction. The simplest defect is a vacancy, a missing ion at the site To. Any such change in atomic arrangement requires us to reevaluate the structure factors, which requires application of Eq. (16-5), We rewrite the sum with a prime to indicate the omission of the term i = 0, and write A — 1 to indicate the decrease in the number of ions by one ... 

We may substitute the structure factor for the vacancy from Eq. (16-20) and write fl/A/,1 as the atomic volume. The result may also be more meaningful if we postulate some number of vacancies and add the scattering rates for each, to obtain... 

The scattering due to vacancies is not in itself of much importance, but exactly the same approach is applicable to a very wide variety of problems. For example, the scattering due to a substitutional impurity gives the same result, except that w, in Eq. (16-22) is replaced by the difference w, — w g between the form factors for the host and the impurity. Similarly, one can allow atoms to be displaced from their equilibrium positions -near an impurity or elsewhere- and recalculate the structure factor to obtain the scattering rate. (A number of such examples are discussed by Harrison, 1966a, Chapter 4.)... 

If we assume as we did before that the cell contains n vacancies, the probability of an atom s presence on a site is (N - n)/N and the diffracted intensity of each peak will be decreased by a factor corresponding to this term. Furthermore, secondary peaks associated with the atomic displacements appear. The structure factor and hence the diffracted intensity are modulated (see Figure 5.3a). 

Another feature observed in these simulations of deep quenches with many vacancies was a segregation of vacancies, if one chooses a model with attractive interactions between monomers (eAA = eBb = — e, eAB = 0). This phase separation between polymers and solvent shows up via the growth of the structure factor Sp(q, t) measuring the polymer density fluctuations, Fig. 37,... 

Fig.44. Collective structure factor S(x,e) dotted vs x = qRg(e,N) for f = 1/2, N = 20 and various choices of the energy kBTe between monomers of different kinds, allowing for a volume fraction , = 0.2 of vacancies on the simple cubic lattice. Curves are a fit to Eq. (187), treating both % and Sg in Eqs. (187) — (189) as adjustable parameters, while the actual gyration radius is used for the normalization of the abscissa. Perpendicular straightline shows the value x = 1.945of Leibler s theory [43]. The symbols denote the choices eN = 0,1,2,3,4 and6 (from bottom to top). From Fried and Binder [325],...

In order to determine the structural factors, concerning host clays, which improve on the catalytic efficiency of pillared clay by the fixation of cations, the following were chosen for comparison with TSM montmorillonite (as smectite, having less of a layer charge than TSM, but an octahedral vacancy like TSM) and taeniolite (as mica, having the same layer charge as TSM, but no octahedral vacancy, unlike TSM). Table 14-2 shows the catalytic activities for cumene... 

The progress of X-ray and especially neutron diffraction techniques now allows one to obtain information on structural vacancies and antisite defects (in the percent concentration range) from the precise determination of structure factors (see for example Kogachi et ai., 1992) neutron measurements are possible up to very high temperatures. 

Polymers are a little more complicated. The drop in modulus (like the increase in creep rate) is caused by the increased ease with which molecules can slip past each other. In metals, which have a crystal structure, this reflects the increasing number of vacancies and the increased rate at which atoms jump into them. In polymers, which are amorphous, it reflects the increase in free volume which gives an increase in the rate of reptation. Then the shift factor is given, not by eqn. (23.11) but by... 

In the case of interstitial diffusion in which we have only a few diffusing interstitial atoms and many available empty interstitial sites, random-walk equations would be accurate, and a correlation factor of 1.0 would be expected. This will be so whether the interstitial is a native atom or a tracer atom. When tracer diffusion by a colinear intersticialcy mechanism is considered, this will not be true and the situation is analogous to that of vacancy diffusion. Consider a tracer atom in an interstitial position (Fig. 5.18a). An initial jump can be in any random direction in the structure. Suppose that the jump shown in Figure 5.18b occurs, leading to the situation in Figure 5.18c. The most likely next jump of the tracer, which must be back to an interstitial site, will be a return jump (Fig. 5.18c/). Once again the diffusion of the interstitial is different from that of a completely random walk, and once again a correlation factor, / is needed to compare the two situations. 

The diffusion of an impurity atom in a crystal, say K in NaCl, involves other considerations that influence diffusion. In such cases, the probability that the impurity will exchange with the vacancy will depend on factors such as the relative sizes of the impurity compared to the host atoms. In the case of ionic movement, the charge on the diffusing species will also play a part. These factors can also be included in a random-walk analysis by including jump probabilities of the host and impurity atoms and vacancies, all of which are likely to vary from one impurity to another and from one crystal structure to another. All of these alterations can be... 

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