Multiple-scattering processes

Fig. 22. The diagrams contributing to electrophoresis and the definition of lithe box. The reason why we only retain this class of terms is that we want to allow long-range propagation by the fluid— as shown in Fig. 21—which is clearly impossible if ions a and fi overlap in the box, because this would correspond to multiple scattering processes on particles a and fi.

The scattering from a distorted surface region is no longer governed by the dynamical scattering in the bulk of the crystal. In the bulk, due to multiple scattering processes, the refractive index is not quite unity and the result is that... 

Besides structural information (which is of minor interest in this case), EXAFS spectra reveal experimental scattering phases for the P-0 system and allow us to distinguish between resonances caused by single scattering processes and those caused by multiple scattering processes (so-called shape resonances). 

Several analytic methods have been proposed to solve the equation of radiative transfer in an absorbing and scattering atmosphere, but they can only be applied for the most simple cases. To obtain quantitative solutions, numerical methods are generally used, such as the Monte-Carlo method, DART method, iterative Gauss, discrete ordinate method, etc. A complete summary of these techniques is provided by Lenoble (1977), and a detailed discussion of multiple scattering processes in plane parallel atmospheres is given in the book by Liou (2002). 

The region in the continuum where multiple scattering processes are relevant is called the XANES region. 

In dealing with dilute systems and not too intense light sources one can eliminate, from the current density operator, the term with the vector potential A f). This eliminates from consideration multiple scattering processes and inelastic processes. The propagator ((Jx k ), jx(,k)))E is then approximated with its counterpart derived from the equations of motion of matter (electrons). 

The calculation of the diffracted intensities usually proceeds in two steps. The first step is the construction of the crystal potential and the calculation of the scattering amplitudes from a single atom, and the second step is the calculation of scattering processes within a single atomic layer and the calculation of scattering between different atomic layers. In the second step the multiple scattering processes are based on the condition that the scattered wave from one atom is an incident wave on all other atoms. This leads to a set of linear equations that is solved by matrix inversion. The formulation of the theory is based on the KKR (Korringa-Kohn-Rostocker) method used for band structure calculations. 

Apart from the oscillations caused by nuclear hyperfine interactions, another oscillation due to multiple-scattering process of nuclear decay exists for the SR passing through a relative thick sample containing resonance nuclei. This oscillation is called dynamic beat (DB), it has been shown theoretically [22,23] and experimentally [24]. In the case of a single resonance (absence of nuclear hyperfine interactions), DB takes the form [22]... 

Both the classical and the quantum dynamics of this system can be described as a multiple scattering process which is governed by the Perron-Frobenius matrix defined as ... 

It might be expected that the intensity of a spectral line from a sample would be directly proportional to the amount of the corresponding element in the sample. In practice the intensity can deviate considerably from the expected linear relation due to absorption in the matrix material and multiple scattering processes. However, it is possible to correct for such effects and very reliable quantitative analyses can be performed. X-ray fluorescence measurements on alloys have an elemental sensitivity of about 10 ppm (ppm parts per million, 1 10 ). The typical penetration depth of the radiation in the metal is about 1 /zm and thus, primarily, the surface is analysed. X-ray emission techniques have been discussed in [5.3,4]. 

If multiple scattering processes occur an erroneously long lifetime will be obtained, as for several of the above-mentioned methods. The opposite effect is obtained if, in addition to the fluorescence light, non-shifted stray light from the modulated light source is recorded. If the modulation is not perfectly sinusoidal, the first Fourier component can be isolated and the phase shift for this component will still yield the lifetime. 

In the second method, going beyond Bom, we examined the density of states within the coherent potential approximation (CPA) which takes into account multiple scattering processes. One might think that on this level impurity states are introduced in the gap. However, we find [16] that the existence of such localized impurity states strongly depends on the relative strength of site vs. bond impurity. Only states in the gap due to disorder can be found if the site amplitude f/s is stronger than the bond amplitude Ufc. Since CPA is an effective medium theory this result might be questionable in one dimension. 

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