Scattering cross-section orientation-averaged

One of the effective ways to exclude luminescence influence on the Raman is UV Raman approach (Asher 2002). The total vibratiOTial Raman scattering cross section Omn (in cm /molecule sr) for a vibrational transition m-n for an isolated molecule averaged over all its orientations is (Steinfeld and Wormhoudt 1998) ... 

Orientation-Averaged Extinction and Scattering Cross-Sections... 

The orientation-averaged scattering cross-section then becomes... 

Thus, the orientation-averaged scattering cross-section for macroscopically isotropic and mirror-symmetric media is proportional to the sum of the squares of the absolute values of the transition matrix in the particle coordinate system. The same result holds true for an ensemble of randomly oriented particles illuminated by a linearly polarized plane wave. 

Despite the derivation of simple analytical formulas, the above analysis shows that the orientation-averaged extinction and scattering cross-sections for macroscopically isotropic and mirror-symmetric media do not depend on the polarization state of the incident wave. The orientation-averaged extinction and scattering cross-sections are invariant with respect to rotations and translations of the coordinate system and using these properties, Mishchenko et al. [169] have derived several invariants of the transition matrix. 

Integrating over cp, we find that the orientation-averaged scattering cross-section and the orientation-averaged mean direction of propagation of the scattered field are given by... 

We have discussed intrinsically anisotropic particles—ones with anisotropy originating in their optical constants rather than their shape—in previous chapters. In Section 5.6 we gave the solution to the problem of scattering by an anisotropic sphere in the Rayleigh approximation. From the results of that section and Section 5.5 it follows that the average cross section (C) (scattering or absorption) of a collection of randomly oriented, sufficiently small, anisotropic spheres is... 

However in most processes of practical relevance such as electron-molecule collisions in industrial plasmas and upper atmosphere, orientations of the molecules seem to be not fixed. On another hand typical interaction times for 1-30 eV of collision energy is 10 14-10 15 s. Timescale for the rotations of the polyatomic molecules at room temperature is 10 12 s and longer. This comparison allows us to assume that scattered electron responds adiabatically to the rotations of the molecules and validates the fixed-nuclei approximation19,20 implicitly assumed in equations (14) and (15). Nevertheless orientation of the molecule with respect to the incoming electron is random and therefore cross sections must be averaged over all the orientations of the molecule. This is carried out by the following technique. Inelastic differential cross sections are obtained from (11) as... 

The heart of the apparatus is the scattering region where a secondary beam crosses the path of the NO molecules in a magnetic field of variable direction. The collision partners possess a well-defined direction of average relative velocity so that the magnetic field can be made parallel and perpendicular to this direction. What is observed is the difference in attenuation of the primary beam for the two orientations of the magnetic field or (translated into cross sections) the anisotropy defined by... 

Radiation scattering by an assembly of centres is also characterized by another effect related to time fluctuations concerning either the sample or the incident radiation. In fact, in the course of time, the total spin state of the neutron-nuclei system may change, and the same remark applies to the orientations of the anisotropic polarizable elements. Thus, the cross-section of the assembly of scattering centres is averaged over a period of time. Two contributions appear. The first one, which is called coherent, reveals interference effects between scattered rays. The second one, which is called incoherent, is the sum of the cross-section of the various centres (considered as isolated). 

A sufficient number of relative initial orientations must be considered in order to achieve a convenient grid in the space of Euler angles so that an accurate average can be taken to yield a cross section for a particular process /, e.g., direct scattering or electron charge exchange... 

In 5.1, we describe how the lOSA is used to calculate photodissociation cross sections. Particular care has to be taken in averaging over the orientation-angle dependent scattering wavefunction using the bending wavefunction of the ground state. To do this we apply, to the reaction problem, a procedure developed by Segev and Shapiro for the vibrational predissociation of Van der Waals complexes [ 51]. Then in 5.2, we present our calculations of the photoabsorption spectrum for the H2O photodissociation (VII). 

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