Basic scattering theory

Some basic scattering theory is reviewed here and in Appendix C, but an excellent resource for specific scattering and diffraction techniques is Modern X-ray Physics by Nielsen and McMorrow. ... 

Although the electron scattering theory, basic principles of experimentation and interpretation are the same at any temperature, high-temperature conditions pose specific problems in the experiments and interpretation of the data . ... 

As in the case of photodissociation, the quantum theory of reactive molecular scattering was initially entirely based on time-independent scattering theory [4-7,100-123]. There were several early attempts to apply time-dependent quantum theory to reactive scattering processes [124—131]. But the modern era of the field really began with the seminal work of Kosloff et al. [37] and the subsequent application of his grid-based methods to the reactive scattering problem by Neuhauser and Baer and coworkers [45,132]. There have been many developments in the field [93,133-138], and several reviews and a book have been written on the topic [10,139-141]. My aim in the next section will be to outline the basic methods of time-dependent quantum theory used in reactive scattering calculations. While the review will cover many aspects of the theory, it will not cover all the approaches currently in use (as of 2003). 

Scattering theory is covered in detail in textbooks [82,83] hence, only a brief introduction to some of the basic principles will be given here. 

It is usual for the scattering theory to expand ip in terms of eigenfunctions of T0, and then to solve the time-dependent equations for the corresponding probability amplitudes. An alternative approach would be to use an adiabatic basic set k obtained by diagonalization of at each configuration of the nuclei ... 

The scope of this chapter will be on these recent developments of the last three years, 1971 to 1973. We restrict ourselves to the interaction of neutral partners, with no restriction to the energy or state of the species. The article starts with a brief review of basic scattering laws and experimental techniques used in the experiments (Section II). Then the theory of elastic scattering is reviewed, the different cross section features are summarized and their sensitivity to the potential is discussed. Perturbations of these results caused from anisotropic potentials, inelasticities or potential crossings are described (Section 111). The next Section IV contains a survey of methods of how to determine the potential from the data. The emphasis will be on the inversion procedures. Finally, the results are summarized in Section V where, besides the atom-atom interactions in the ground states, new results for metastable species and molecules are included. 

In principle, the correspondence between the two theories is not complete, because scattering theory is the more general formulation. For our purposes, however, the fact that the applications to atomic physics obtained by both methods are quite consistent with each other is an important and useful conclusion. The same result and connections have been obtained independently by Komninos and Nicolaides [378]. Both [373] and [378] noted that the derivation of MQDT from Wigner s scattering theory establishes its basic structure and theorems without special assumptions about the asymptotic forms of wavefunctions. The approach of Komninos and Nicolaides [378] is designed for applications involving Hartree-Fock and multiconfigurational Hartree-Fock bases. In the present exposition, we follow the approach and notation of Lane [379] and others [380, 381], who exploit the analytic K-matrix formalism and include photon widths explicitly when interferences occur. 

While the basic formalism of quantum scattering theory can be found in a variety of general physics textbooks [i, 2, 3, 4, 5, 6 and 7] and textbooks that are concerned with scattering theory in a broad sense [8, 9,10 and... 

Basic quantum mechanics textbooks that include one or more chapters on scattering theory as applied to physics ... 

The most popular effective medium theories are the Maxwell Garnett theory [18], which was derived from the classical scattering theory, and the Bruggeman theory [19]. With these theories, an effective dielectric function is calculated from the dielectric functions of both basic materials by using the volume filling factor. At some extensions of these theories, a unique particle shape for all particles is assumed. There is also an other concept based on borders for the effective dielectric functions. The borders are valid for a special nanostructure. Between these borders, the effective dielectric function varies depending on the nanostructure of the material. The Bergman theory includes a spectral density function g(x) that is used as fit function and correlates with the nanostructure of the material [20]. 

This expression is the basic equation in scattering theory for two reasons. First, all scattering phenomena (light, x-ray, and neutron) can be interpreted in terms of this equation. These techniques differ mainly in the structural entities that contribute to the Kj term. For light, as pointed out previously, the refractive index or polarizability is the principal contributor. For x-rays, the electron density is important, and, for neutrons, the nature of the scattering nucleus [2]. Secondly, equation... 

The basic question posed in reactive scattering theory today is simple. Given a potential energy surface for the interacting molecules, what will be the outcome of a collision between them. The outcome is described in various terms such as a reaction probability, distribution of final states, differential cross section, rate constant etc. All of these quantities depend on the initial state of the colliders. 

R. J. Garbacz, Basic Relations Between Antennas and Scattering Parameters, from short course. Antenna and Scattering Theory Recent Advances, Ohio State University, Columbus, OH, 1966. 

The dynamical behavior of strongly non-adiabatic multimode systems exhibits complex features that can perhaps be better understood by trying first to underline the aspects that they share with simple one-dimensional models basically derived from scattering theory. To this purpose we introduce a minimal model for the Cl and then discuss the limit of validity of an interpretation scheme grounded on the Landau-Zener formula as well as of a simple one-dimensional time-dependent model obtained by treating quantum-mechanically only the coupling mode. 

Other Effects. Some of the other approximations in the usually applied theory will briefly be mentioned here, i.e. neglect of exchange scattering and of atom distortion and polarization, and the use of non-relativistic scattering theory. The basic theory necessary to handle these efiFects is given by Mott and Massey " especially in the Chapters 9 and IS—18. Bonham"" has shown that the change correction probably is unimportant at the commonly applied electron wavelengths. 

---