Multiple-Scattering Treatment

The widely used multiple-scattering treatment was developed by Waterman and Truell (1961) and Twersky (1962). The treatment is based on an approximation in which the exciting field seen by a scatteier may be represented by the total field that would exist at the scatterer if the scatterer were not present. Furthermore, it assumes that scatterers are statistically independent, i.e., the probability of finding a scatterer at one point is independent of other scatterers. The treatment yields the following expression for the effective wavenumber in terms of single-particle scattering amplitudes  

If we consider scattering by a spherical particle, the scattering amplitude can be given as 

FIGURE 5-29 Experimentally obtained and theoretically predicted sound velocity vs. solids concentration for 8 pm hollow glass beads suspended in Echogel. 

The model predicts both attenuation and phase velocity in solid suspensions of uniform spherical particles. Nonuniform particles of show very little effect on wave propagation. However, the size distribution can be treated with statistical averaging. 

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E. G. McRae, Multiple-scattering treatment of low energy electron diffraction intensities, J. Chem. Phys. 45, 3258-3276 (1966). 

ELNES on Si L ionisation edge can be used to reveal the change in coordination of Si atoms due to the reductive treatment. After background subtraction and removal of multiple scattering [49], the spectra from different areas for both samples are plotted in Fig. 5. Before reduction, the Si L ELNES from as-grown silica and from the area with Pt particles exhibit the typical Si L ELNES of Si02, identical with the one measured from the Si02 substrate after the treatment. Some new features appear in the Si L ELNES spectrum obtained from particle after the reduction. This is a... 

Data analysis procedures have developed substantially over the last few years. In particular, use of least square refinement methods have been developed. Recent progress with theoretical development for the treatment of multiple scattering has resulted in Ugand group refinement such as an imidazole. We can expect further development in this area which ought to lead us to restrained least square refinement procedures for EXAFS data analysis. This type of restrained refinement is commonly used for macromolecular crystallographic structure determination where a similar problem of imderdeterminancy exists... 

According to the ground rules laid down at the beginning of this book, multiple scattering is excluded from consideration. But it is not always prudent to pretend that multiple scattering does not exist. Fortunately, it is almost trivial—the mathematical apparatus of radiative transfer theory is unnecessary—to extend our treatment of scattering and circular polarization to multiple scattering media, and in this instance it is worth the small amount of effort required to do so. 

The considerable progress made in the studies of simple bimolecular reactions (which has led to such fundamental conclusions) was achieved by a more rigorous mathematical treatment of the problem, avoiding the use of the simplest approximations which linearize the kinetic equations. We focus main attention on the many-point density formalism developed in [26, 28, 49] since in our opinion it seems at present to be the only general approach permitting treatment of all the above-mentioned problems, whereas other theoretical methods so far developed, e.g., those of secondary quantization [19, 29-32], and of multiple scattering [72, 73], as well based on... 

Quantitative treatments of partially polarized light can be found in the texts by Born and Wolf [2], and Azzam and Bashara [5]. In this monograph, the light will be assumed to be perfectly polarized. It should be noted, however, that in many experimental situations depolarization can readily occur and care must be taken to either account for it, or to minimize this possibility. The most common source of depolarization in optical rhe-ometry is multiple scattering by such systems as dense suspensions and liquid crystals. 

In any of the steps 1, 3 and 4, multiple scattering of electrons can play an important role in the techniques of interest here. On the one hand, multiple scattering provides a greater sensitivity to various aspects of structure on the other hand, it complicates the theoretical treatment, i.e. it complicates the extraction of the structural information. For the following discussion it will be useful to first describe in more detail a couple of important features of electron scattering by individual atoms in a surface. 

The spherical-wave description has its own great advantages. It is best adapted to the scattering and emission by the spherical ion cores and it does not require the presence of any structural periodicity. In particular, it is well suited to the treatment of multiple scattering between different atoms within any cluster of atoms, in particular within a periodic unit cell as in LEED. It is also convenient for the treatment of fine structure arising from back-scattering by nearby atoms, as in (S)EXAFS, NEXAFS, EAPFS, etc. (i.e. in step 3 in our four-step description). 

Dynamical Theories. These involve a full treatment of multiple scattering by a layer method. The scattering within a single atom is determined first, then the scattering within one atomic layer and finally the scattering between layers. The technique has been extended by considering first single atom-chains, which are then... 

Figure 3.48. (a) The NIR spectra of 12 kinds of EVA copolymers before Multiplicative Scatter Correction (MSC) treatment (Shimoyama et al, 1998). (b) The effect of MSC on the spectra shown in (a). Copyright 1998 John Wiley and Sons, Inc. reproduced with permission. 

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