Continuum theory scattering

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

In Figure 4 our calculated cross sections for the 3cTg level are compared with the results of the Stleltjes moment theory (STHT) approach (25) and of the continuum multiple scattering model (CMSM)... 

The spectra of protons of incident energy about 20 MeV scattered inelastic-ally from moderately heavy elements lose the group structure shown with lower energy protons and lighter elements and have to be interpreted by continuum theory. The experiments (Sect. 38) then lead to an estimation of relative level densities of the target nuclei. The excitation of low-lying levels of beryllium by 190 MeV electrons has also been observed (McIntyre et al. ). 

Photoinduced ET in binuclear complexes with localized electronic states provides at the moment the best test of theory predictions for the solvent dependent ET barrier. This type of reaction is also called metal-metal charge-transfer (MMCT) or intervalence transfer (IT). The application of the theory to IT energies for valence localized biruthenium complexes and the acetylene-bridged biferricenium monocation revealed its superiority to continuum theories. The plots of E vs. E p are less scattered, and the slopes of the best-fit lines are closer to unity. As a major merit, the anomalous behavior of some solvents in the continuum description - in particular HMPA and occasionally water - becomes resolved in terms of the extreme sizes, as they appear at the opposite ends of the solvent diameter scale. 

Continuum theories are not satisfactory for powder type layers, like in TLC. The scattering and absorption characteristics of a medium are reflected in only two constants, K and S, but no reference is made regarding the nature of the particles that are inside a layer. Bodo (8), Melamed (9), Johnson (10) have developed well-known discontinuum theories for the determination of absolute optical constants from the properties of individual sample particles. 

Moreover, in quantitative TLC the continuum theories for absorption and scattering of powder layers are not satisfactory, because, in addition to the influence of the particle size, the very important problem of nonuniform distribution of a sample (the absorbing molecules), throughout the layer is completely ignored. 

What for the case of small particles is called backward and forward scatter is, for the case of particles with large, flat surfaces, the sum of reflection and transmission. For infinitesimally small particles, continuum theories of diffuse reflection may be applied. As particles get larger, it becomes more likely that the terms for geometrical optics will be applied and discontinuum theories are more relevant. 

A continuum theory implicitly assumes a model for the absorption by and scatter from a particle of infinitesimal size. This model is only a reasonable approximation for samples in which the fraction of light absorbed by an individual particle is a very small fraction of the light incident upon it. The advantage of this model is that it is simple, though the mathematics that describe it are not. The discontinuum method we will describe uses mathematics no more complex than the continuum theories. However, the description of the sample is more complex. This is both the power and limitation of the discontinuum theories they can describe more complex situations, but doing so requires a more detailed description. 

As noted above, the phenomenological two-fiux theories that have been developed on the basis of the radiation transfer equation can be considered continuum theories. Continuum theories consider the absorption and scattering coefficients as properties of an irradiated isotropic layer of infinitesimal thickness. On the other hand, discontinuum theories consider layers containing a collection of particles. Consequently, the thickness of a layer is dictated by the size of the scattering and absorbing particles. Optical constants can then be determined from the scattering and absorption properties of these particles. 

This chapter has provided a brief overview of the application of optimal control theory to the control of molecular processes. It has addressed only the theoretical aspects and approaches to the topic and has not covered the many successful experimental applications [33, 37, 164-183], arising especially from the closed-loop approach of Rabitz [32]. The basic formulae have been presented and carefully derived in Section II and Appendix A, respectively. The theory required for application to photodissociation and unimolecular dissociation processes is also discussed in Section II, while the new equations needed in this connection are derived in Appendix B. An exciting related area of coherent control which has not been treated in this review is that of the control of bimolecular chemical reactions, in which both initial and final states are continuum scattering states [7, 14, 27-29, 184-188]. 

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