Cross scattering

In order to extract microscopic quantities such as spatial probabilities and RDFs from collected differential cross scattering data, analysis is typically performed following reverse Monte Carlo (RMC) or empirical potential structure refinement (EPSR) procedures [7]. The latter procedure can be viewed as a Monte Carlo simulation of system utilising a model potential similar to a classical molecular mechanics force field. This model potential is modified in order to bring the total structure factor calculated from the model system as close as possible to the ejq)erimental data. From configurations generated with this refined potential, standard quantities (such as RDFs) may be calculated. 

McCourt F R, Beenakker J, Kohler W E and Kuscer I 1991 Nonequilibrium Phenomena In Polyatomic Gases. 2. Cross-sections, Scattering and Rarefied Gases (Oxford Clarendon)... 

The more conventional, energy domain fonnula for resonance Raman scattering is the expression by Kramers-Heisenberg-Dirac (KHD). The differential cross section for Raman scattering into a solid angle dD can be written in the fomi... 

In a third step the S-matrix is related to state-selected reaction cross sections a., in principle observable in beam scattering experiments [28, 29, 30, 31, 32, 33, 34 and 35], by the fiindamental equation of scattering theory... 

Actually equation (A3.4.72) for o. is still fomial, as practically observable cross sections, even at the highest quantum state resolution usually available in molecular scattering, correspond to certain sums and averages of... 

Flowever, in order to deliver on its promise and maximize its impact on the broader field of chemistry, the methodology of reaction dynamics must be extended toward more complex reactions involving polyatomic molecules and radicals for which even the primary products may not be known. There certainly have been examples of this notably the crossed molecular beams work by Lee [59] on the reactions of O atoms with a series of hydrocarbons. In such cases the spectroscopy of the products is often too complicated to investigate using laser-based techniques, but the recent marriage of intense syncluotron radiation light sources with state-of-the-art scattering instruments holds considerable promise for the elucidation of the bimolecular and photodissociation dynamics of these more complex species. 

The measurable quantity in a three dimensional scattering experiment is the differential cross section da (9)/dQ. This is defined as... 

This differential cross section may be integrated over scattering angles to define an integral cross section a as follows ... 

Equations A3.11.114(b) and A3.11.115(b) are in a fonn that is convenient to use for potential scattering problems. One needs only to detemiine the phase shift 5 for each i, then substitute into these equations to detemiine the cross sections. Note that in the limit of large i, finiist vanish so that the infinite sum over partial waves iwill converge. For most potentials of interest to chemical physics, the calculation of finiist be done numerically. 

The cross section a -is related to the partial wave reactive scattering matrix , tln-ough the partial wave sum... 

Experimentally, it is these invariants (equation (B 1.3.17), equation (B 1.3.18) and equation (B 1.3.19)) that can be obtained by scattering intensity measurements, though clearly not by measuring the total cross-section only. 

Measurement of the total Raman cross-section is an experimental challenge. More connnon are reports of the differential Raman cross-section, doj /dQ, which is proportional to the intensity of the scattered radiation that falls within the element of solid angle dQ when viewing along a direction that is to be specified [H]. Its value depends on the design of the Raman scattering experiment. 

We have seen that the strength of Raman scattered radiation is directly related to the Raman scattering cross-section (Oj ). The fact that this cross-section for Raman scattering is typically much weaker than that for absorption (oj limits conventional SR as a sensitive analytical tool compared to (Imear) absorption... 

The metal substrate evidently affords a huge ( 10 and even as high as 10 [84, 85]) increase in the cross-section for Raman scattering of the adsorbate. There are two broad classes of mechanisms which are said to contribute to this enhancenient [, and Ml- The first is based on electromagnetic effects and the second on cheniicaT effects. Of these two classes the fomier is better understood and, for the most part, the specific mechanisms are agreed upon the latter is more complicated and is less well understood. SERS enhancenient can take place in either physisorbed or chemisorbed situations, with the chemisorbed case typically characterized by larger Raman frequency shifts from the bulk phase. 

For a circularly polarized light experiment, one can measure the cross sections for either right (r) or left (1) polarized scattered light. Suppose that right polarized light is made incident on a Raman active sample. The general expressions for the Raman cross sections are [176]... 

Bethe provided the theoretical basis for understanding the scattering of fast electrons by atoms and molecules [3, 4]. We give below an outline of the quantum-mechanical approach to calculating the scattermg cross section. 

The cross section for scattering into the differential solid angle dD centred in the direction (9,(l)), is proportional to the square of the scattering amplitude ... 

In this expression, factors that describe the incident and scattered projectile are separated from the square modulus of an integral that describes the role of the target in detemiining the differential cross section. The temi preceding the... 

---