Symmetrical scattering

Friedel used the local perturbation f/—as a model for the scattering potential, reduced it even to one spherically symmetric scatterer and calculated the scattering by free electrons. By that, for the scatterer at the origin labeled by j = 0,... 

Hansen, N. K., Study of the Electron Density Distribution in Molecular Crystals by Analysis of X-ray Diffraction Data Using Non-Spherically Symmetric Scattering Functions, Thesis, University of Arhus, Denmark (1978). 

For indistinguishable particles, this has to by symmetrized appropriately. The 4He nucleus is a boson, and hence the total wave function does not change sign when interchanging the nuclei. One obtains for the properly symmetrized scattering amplitude... 

Figure 4.1 3 Molecular beam contour diagram showing symmetric scattering...

Reactions involving collision complexes are sometimes found. Here an intermediate is produced, which lasts long enough for it to perform several rotations, and this results in symmetrical scattering. Typical reactions are... 

We assume that the intensity of the initially scattered light from a single scattering center is the same for symmetric scattering angles 9S. This assumption does not imply that the inten-... 

Oriented polymers will no longer show a symmetric scattering pattern. For such systems a two dimensional detector is necessary. A commercial data aquisition system (Westinghouse), based on a Vidicon tube, is shown in Fig. 32. The video signal... 

As the resolution of the Bragg reflection data is improved, it becomes possible to obtain information on the more minute details of electron density in a molecule. At high enough resolution information can be obtained on the redistribution of electron density (deformation density) around atoms when they combine to form a molecule. Electrons in molecules ma -form bonds or exist as lone pairs, thereby distorting the electron density around each atom and requiring a more complicated function to describe this overall electron density than normally used, in which it is treated as if it were spherically symmetrical (deformed to an ellipsoid in order to account for anisotropic displacements). This assumption is inherent in the use of spherically-symmetrical scattering factors although the elec-... 

Thus the value of p depends on the degree of asymmetry in the tensor. For pure symmetric scattering, as in the normal Raman effect for instance, Opp = pp and Pi = 3/4 (depolarised scattering). For pure antisymmetric scatteringOpp = - pp and therefore G = 0, giving Pi = (inverse polarisation). For an asymmetric tensor lappi Iflppl and < Pi < ° , which is called anomalous polarisation. 

Active area corrections (non-specular reflectivity). The active area correction for non-specular reflectivity becomes more complicated because it relies on determining the overlap of the incident and detector footprints, which might not have a simple analytical form. In Figures 11B-D, we show representative examples. For a four-circle spectrometer in the symmetric scattering mode (i.e., a, = af), with an incident beam shape of At = 0.5 mm by Av= 0.1 mm and a square 10 mm by 10 mm sample, we show the incident and detector footprints at selected values of Qz for a non-specular CTR with Qn =1.2 A-1 and Qz varying from 1.8 A-1 to 7.2 A-1. This calculation shows that the beam footprint becomes highly skewed as Qz increases at constant Qn, and the overlap of the incident... 

The symmetry properties of resonance Raman lines can be predicted on the basis of Eq. 19.15. For totally symmetric vibrations, a and =0. Then Eq. 19.15 gives 0 sp <. Nontotally symmetric vibrations (q=0) are classified into two types those which have symmetric scattering tensors, and those which have antisymmetric scattering tensors. If the tensor is symmetric, = 0 and y, 9 0. Then Eq. 19.15 gives Py, = (depolarized). If the tensor is... 

These matrices correspond stmcturally with the Mueller matrix of so-called macroscopically isotropic and mirror-symmetric scattering media composed of randomly oriented particles with plane of symmetry and/or equal numbers of randomly oriented particles and their mirror-symmetric counterparts [46] ... 

The above theory predicts a cylindrically symmetrical scattering pattern (independent of n). This is a consequence of the random... 

Figure 1 gives the solution of (7) in case of spherically symmetric scattering, (7p = 0. For practical purposes... 

Abstract. The Boltzmann s equation is solved in the case of monoenergetic neutrons created by a plane or point source in an infinite medium which has spherically symmetric scattering. The customary solution of the diffusion equation appears to be multiplied by a constant factor which is smaller than 1. In addition to this term the total neutron density contains another term which is important in the neighborhood of the source. It goes with 1/r in the neighborhood of a point source. 

It may be worthwhile to remark that the calculation of the first two sections could be carried out also in case of a not spherically symmetric scattering. The exponent in JVi, i.e. ac, has already been calculated in A-21 for the case that the differential cross section contains, in addition to a constant term, a term proportional to the cosine of the scattering angle. 

Elementary considerations, based on the assumption of spherically symmetric scattering, give for the number C(r, E) dE of scattering collisions which the neutrons of the energy range E E + dE make in unit time and unit volume... 

When there is more than one open channel, the complex symmetric scattering matrix S has elements 5y. The diagonal S-matrix element in incoming channel i has magnitude Su < 1 and may be written in terms of a comp/ex phase shift hi E) with a positive imaginary part [37],... 

In insertion of Fig. 7.1 the dependence of overloading coefficient mean values k (k = 0.5(Kj + k )) on value D l is adduced. As one can see, a good linear correlation between the indicated parameters is observed, with the exception of only two polymers (PTFE and PSF) which have considerable, although symmetrical, scatter. This gives the possibility to express as function as follows [9] ... 

Mishchenko, M. L, and Travis, L. D. (1998) Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers, / Quant. Spectrosc. Radiat. Transfer, 60, 309-324. 

It must be pointed out here, that the scattering of light by small particles is determined by a spherical symmetric scattering function, P . Upon using verticaUy polarized incident light, i is equal to io, the intensity of light scattered at zero observation angle, that is, P = i /io = 1. 

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