Scattering multiple

Multiple scattering is always a troublesome phenomenon for the interpretation of experimental results. However, in an important and rather simple case, the experimentalist is able to evaluate the contribution of multiple scattering and to make the correction needed to analyse the results in terms of pair correlation. 

These very peculiar conditions are as follows. First, every collision produces a strong forward scattering (this is, for instance, the case with neutron scattering by polymers) then, the length of the trajectory of an incident neutron in the sample is nearly equal to the thickness L of this sample. 

02 = 02 + 02 (by assimilating the angle to its tangent) d0 d0y = element of solid angle. 

For z 0, it is proper to introduce the angular flux / (0W 0y, z) which at point z and in the direction determined by 0, and 0y is the neutron flux per unit time, area, and solid angle. In the isotropic case, we set 

With each scattering centre, we associate an angular cross-section (0 0y). For a concentration NjV of scatterers, the angular collision rate is 

The detector is placed at distance R from the center of the illuminated volume, in the direction defined by unit vector S. The distance R is considered to be sufficiently far in comparison to the sheet thickness t or the beam cross-sectional area A. The detector window is also open sufficiently wide that the detector is able to view all of the illuminated volume Vo- 

In other words, the intensity I (q) of single scattering, when the intensity is expressed in the sense defined in Section 1.2.1, is 

we look at the flux of the scattered beam due to double scattering. The flux reaching the volume element dV2, after having been scattered once at dV, is 

As stated earlier, the scattering function i(q) for x-rays must include the polarization factor. If the incident beam is unpolarized, the function i(q) in Equation (2.29) for single scattering should include the polarization factor P (q which, as given in Equation (1.35), is 

Similarly, in the case of an unpolarized incident beam, the product of intensity functions, i(q )i(q2), in Equation (2.33) should include the polarization factor P2(q) for double scattering given by 

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P. Meakin, Multiple Scattering of Waves in Random Media and Random Rough Surfaces, The Pennsylvania State University Press, State College, PA, 1985. 

Rehr J J 1995 Multiple-scattering approach to surface EXAFS—theory versus experiment Surf. Rev. Lett 2 63-9... 

For angles greater than only multiple scattering can occur. 

Rehr J J, Albers R C and Zabinski S I 1992 High order multiple scattering calculation of x-ray absorption fine structure Phys. Rev. Lett. 69 3397-400... 

We must describe the light scattered with interference in terms of phase differences that develop as the waves pass through a molecule consisting of multiple scattering sites. 

Clouds of Nonblack Particles The correction for nonblackness of the particles is complicated by multiple scatter of the radiation reflected by each particle. The emissivity . of a cloud of gray particles of individual surface emissivity 1 can be estimated by the use of Eq. (5-151), with its exponent multiplied by 1, if the optical thickness alv)L does not exceed about 2. Modified Eq. (5-151) would predict an approach of . to 1 as L 0°, an impossibihty in a scattering system the asymptotic value of . can be read from Fig. 5-14 as /, with albedo (0 given by particle-surface refleclance 1 — 1. Particles with a perimeter lying between 0.5 and 5 times the wavelength of interest can be handledwith difficulty by use of the Mie equations (see Hottel and Sarofim, op. cit., chaps. 12 and 13). 

Surface atomic structure. The integrated intensity of several diffracted beams is measured as a fimction of electron beam energy for different angles of incidence. The measurements are fitted with a model calculation that includes multiple scattering. The atomic coordinates of the surfiice atoms are extracted. (See also the article on EXAFS.)... 

The curve labeled geometry illustrates the kinematic energy spread due to the finite acceptance angle of the detector. The multiple scattering contribution arises from the spread in ion energies introduced by secondaiy scattering events. 

It should be noted that a comprehensive ELNES study is possible only by comparing experimentally observed structures with those calculated [2.210-2.212]. This is an extra field of investigation and different procedures based on molecular orbital approaches [2.214—2.216], multiple-scattering theory [2.217, 2.218], or band structure calculations [2.219, 2.220] can be used to compute the densities of electronic states in the valence and conduction bands. 

The depth resolution of ERDA is mainly determined by the energy resolution of the detector system, the scattering geometry, and the type of projectiles and recoils. The depth resolution also depends on the depth analyzed, because of energy straggling and multiple scattering. The relative importance of different contributions to the depth resolution were studied for some specific ERDA arrangements [3.161, 3.163]. 

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