Approximation to the Energy-Transfer Cross-Section

This final expression is extremely useful since it allows us to determine the differential energy-transfer cross-section if the angular differential cross-section is known, or if the center-of-mass scattering angle and impact parameter are known. 

The total cross-section for a scattering process is determined by setting the probability functions described by (4.11) and (4.12b) equal to unity. This leads to 

Approximate values of the energy-transfer differential cross-section can be obtained using a power-law approximation to the potential (Winterbon et al. 1970 Nastasi et al. 1996). The power-law energy-transfer differential cross-section has the form 

If %(r) is taken as the Thomas-Fermi screening function, Winterbon et al. (1970) have shown 

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IV. Spatially dependent velocity distributions. When the spectrum is independent of position, the central problem is the determination of the energy-transfer cross sections. The calculation of the spectrum once these cross sections are known is a straightforward procedure. The cross-section aspect of the problem is both more difficult from a physics point of view, and more time consuming from the point of view of machine computation. This situation is reversed when we come to consider the spatial dependence of the slow neutron spectrum. The cross sections needed are the same ones that already have been computed for the infinite medium spectrum problem. The transport equation must now be solved in at least two variables, and in a form for which the existing approximate techniques are not very well adapted. The focus of the problem therefore shifts to the development of appropriate techniques for solution of the transport equation when the energy and position variables are coupled in such a way that neutrons can both gain and lose energy in a collision. 

In order to develop more efficient methods for determining position dependence of the spectrum, there are two complementary directions in which work can proceed. The first is (Item No. 2 above) the consideration of simplified models in which analytic solutions can be obtained. The most suitable model is the heavy gas model [8]. This model has been used for studies of the spatial dependence of the spectrum by Kottwitz [27], and by Kazarnovsky, et al [28]. These studies of simplified models can be useful in two ways. They serve as points of normalization for numerical techniques, and indicate possible directions for approximate methods. A possibly profitable direction would be to investigate the approximate reduction of detailed models for the energy-transfer cross sections to a form similar to the heavy-... 

The first (and still the foremost) quantum theory of stopping, attributed to Bethe [19,20], considers the observables energy and momentum transfers as fundamental in the interaction of fast charged particles with atomic electrons. Taking the simplest case of a heavy, fast, yet nonrelativistic incident projectile, the excitation cross-section is developed in the first Born approximation that is, the incident particle is represented as a plane wave and the scattered particle as a slightly perturbed wave. Representing the Coulombic interaction as a Fourier integral over momentum transfer, Bethe derives the differential Born cross-section for excitation to the nth quantum state of the atom as follows. 

The quenching cross section has its maximum when the Ea vector is approximately parallel to the CM system, that is, for the o) configuration of the atom. This is relatively independent from the energy transfer, subject, however, to experimental errors as a result of uncertainties in the CM definition. 

Abstract. Cross sections for electron transfer in collisions of atomic hydrogen with fully stripped carbon ions are studied for impact energies from 0.1 to 500 keV/u. A semi-classical close-coupling approach is used within the impact parameter approximation. To solve the time-dependent Schrodinger equation the electronic wave function is expanded on a two-center atomic state basis set. The projectile states are modified by translational factors to take into account the relative motion of the two centers. For the processes C6++H(1.s) —> C5+ (nlm) + H+, we present shell-selective electron transfer cross sections, based on computations performed with an expansion spanning all states ofC5+( =l-6) shells and the H(ls) state. 

In the avoided crossing region the experimental RKR potential of the state and the "essentially experimental" potential of the state are used to determine the crossing distance R(- and the coupling matrix element T -. in the two state approximation. These quantities are relevant to the evaluation of the total charge transfer cross section at high energy (e.g. in the Landau-Zener model). 

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