Elastic scattering approximate methods

Approximate methods may be employed in solving tiiis set of equations for tlie (r) however, the asymptotic fonn of the solutions are obvious. For the case of elastic scattering... 

We first follow the flow chart for the simple case of elastic scattering of structureless atoms. The number of internal states, Nc, is one, quantum scattering calculations are feasible and recommended, for even the smallest modem computer. The Numerov method has often been used for such calculations (41), but the recent method based on analytic approximations by Airy functions (2) obtains the same results with many fewer evaluations of the potential function. The WKB approximation also requires a relatively small number of function evaluations, but its accuracy is limited, whereas the piecewise analytic method (2) can obtain results to any preset, desired accuracy. 

Backscattering of y rays and X rays depends on the mass per unit area and the effective average atomic number Z. The saturation value of backscattering decreases approximately with this number. For example, the composition of ores can be determined by this method. Elastic scattering of y radiation ((y,/) process) can also be applied for analytical purposes. High selectivity is obtained by resonance absorption, i.e. by application of a radionuclide that decays into a stable ground state of the element to be determined. The y rays emitted by the (y,/) process are measured. 

The use of elastic scattering in reactive systems as already described involves a number of approximations and assumptions. However, the method may represent a virtually independent procedure for evaluating Sr(E) with practically no recourse to direct measurements of the yield of reaction product. 

Elastic scattering was also the theoretical workhorse in earlier years for developing many approximate theoretical methods, and though these approximations are no longer needed for elastic scattering calculations they still serve as useful guides for dealing with more complex processes. 

For atom-diatom collisions, the prototype system has been H 4 H2, for which both a very accurate ground state surface and a three-dimensional quantum scattering study have been reported. In addition, the H + H2 system has proved to be valuable in the development and comparative testing of more approximate quantum and semiclassical methods for reaction dynamics. This system has also proved useful for computations of cross sections for nonreactive inelastic collisions as well as elastic scattering. Most of these studies, however, have been concerned with thermal or low-energy collisions. 

As one would expect, developments in the theory of such phenomena have employed chemical models chosen more for analytical simplicity than for any connection to actual chemical reactions. Due to the mechanistic complexity of even the simplest laboratory systems of interest in this study, moreover, application of even approximate methods to more realistic situations is a formidable task. At the same time a detailed microscopic approach to any of the simple chemical models, in terms of nonequilibrium statistical mechanics, for example, is also not feasible. As is well known, the method of molecular dynamics discussed in detail already had its origin in a similar situation in the study of classical fluids. Quite recently, the basic MD computer model has been modified to include inelastic or reactive scattering as well as the elastic processes of interest at equilibrium phase transitions (18), and several applications of this "reactive" molecular dynamicriRMD) method to simple chemical models involving chemical instabilities have been reported (L8j , 22J. A variation of the RMD method will be discussed here in an application to a first-order chemical phase transition with many features analogous to those of the vapor-liquid transition treated earlier. 

Ed being the energy of the fast electron. To a good approximation, the effect of inelastically scattered electrons on the elastic electron wave field may be treated via a first order perturbation method. From Equation (4) we have... 

A discussion of elastic positron-atom scattering is most conveniently introduced in the context of positron-hydrogen scattering, and we therefore describe this system in considerable detail and use it to illustrate some of the more important methods of approximation used in positron collision... 

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