Scattering representation

Figure 3.6-3 Momentum conservation for stimulated anti-Stokes Raman scattering (Representation of Eq. 3.6-5).

For 3-D representations, similar functional parameter values can be shown for a 3-D feature, such as surfaces, volumes and cut planes. To represent the magnitude of the functional parameter values, arrows can also be used to give a less scattered representation. Arrow plots are usually used to represent the flow directions and vortexes formed and they are particularly suitable for representing flow directions within the solution domain as these arrows can give clear directional information. 

Most recently, Freed (84) has developed a generalized theory in terms of a multiple scattering representation of scattering of fluid waves by the polymer molecules and derived the precise result ... 

K. F. Freed and A. Perieo. Consideration on the multiple scattering representation of the concentration dependence of the viscoelastic properties of polymer systems. Macromolecules, 14 (1981), 1290-1298. 

Figure 18 shows porosity at the top of the reservoir obtained for the two voxel models. The second voxel model (Figure 18 (b)) does not only show a less scattered representation of the porosity but also that high porosity values reproduce better the sand lobes. 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

For themial unimolecular reactions with bimolecular collisional activation steps and for bimolecular reactions, more specifically one takes the limit of tire time evolution operator for - co and t —> + co to describe isolated binary collision events. The corresponding matrix representation of f)is called the scattering matrix or S-matrix with matrix elements... 

Figure Bl.2.2. Schematic representation of the polarizability of a diatomic molecule as a fimction of vibrational coordinate. Because the polarizability changes during vibration, Raman scatter will occur in addition to Rayleigh scattering.

Solution of the scattering amplitude may then be detemrined from the asymptotic fomi of 4 (r jdirectly or from the integral representation... 

As well as obtaining tlie scattering amplitude from the above asymptotic boundary conditions, can also be obtained from the integral representation for the scattering amplitude is... 

For electronic transitions in electron-atom and heavy-particle collisions at high unpact energies, the major contribution to inelastic cross sections arises from scattering in the forward direction. The trajectories implicit in the action phases and set of coupled equations can be taken as rectilinear. The integral representation... 

Bacic Z, Kress J D, Parker G A and Pack R T 1990 Quantum reactive scattering in 3 dimensions using hyperspherical (APH) coordinates. 4. discrete variable representation (DVR) basis functions and the analysis of accurate results for F + Hg d. Chem. Phys. 92 2344... 

Colbert D T and Miller W H 1992 A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method J. Chem. Phys. 96 1982... 

Besides these main categories, a large number of hybrid visualization techniques also exist, which arc combinations of the methods described. Well-known hybrid approaches arc the 2D or 3D glyph displays. These techniques combine the multidimensional representation capabilities of icon-based methods with the easy and intuitive representations of scatter-plot displays, Therefore these techniques can also be frequently found within chemical data analysis applications. 

Figure 10.6 Two-dimensional representation of i and i (broken lines) and their resultant ifotai (solid line) for scattering by a molecule situated at the origin and illuminated by unpolarized light along the x axis. The intensity in any direction is proportional to the length of the radius vector at that angle. (Reprinted from Ref, 2, p. 168.)...

Schematic representation for the ray paths of both unscattered and scattered electrons beneath the sample.

Rgure 3 Experimental and calculated results (a) for epitaxial Cu on Ni (001). The solid lines represent experimental data at the Cu coverage indicated and the dashed lines represent single-scattering cluster calculations assuming a plane wave final state for the Cu IMM Auger electron A schematic representation lb) of the Ni (010) plane with 1-5 monolayers of Cu on top. The arrows indicate directions in which forward scattering events should produce diffraction peaks in (a). 

The above discussion, thus, has made somewhat more precise the sense of the limiting procedure involved in writing down the representation (10-216) for the scattering amplitude. This representation also makes clear that a knowledge of Green s function... 

An alternative approach to the representation of results for solid-liquid flow is to use the two-layer model which will be described in the following section. It will be seen that the coefficient of friction between the particles and the wall of the pipe is an important parameter in the model. It is suggested that its complete absence in equation 5.24 may be an important reason for the extent of the scatter. Unfortunately, it is a quantity which has been measured in only a very few investigations. It is interesting to note that the form of equation 5.19 was obtained by NEWITT et alP2) using a force balance similar to that... 

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