Impact parameter method

C. The Quasi-Classical Approximation The Impact Parameter Method... 

Thus, in order to determine the scattering cross sections we must find the wavefunctions of the system after the scattering for a known interaction potential. This is a very complicated problem in the case of many-electron systems and can be solved only with various approximate methods. We will only briefly discuss the results obtained in the Born approximation and in the quasi-classical impact parameter method. A detailed discussion of various approximate methods can be found in special monographs (e.g. in Refs. 104 and 107) or in reviews (see Refs. 105, 108-112). 

Within the semi-classical impact parameter method [2], the relative motion of the nuclei is described as a straightline constant velocity trajectory (figure 1)... 

As mentioned above, the processes (19.24)-(19.29) have not been studied for electronically excited states of H2. An estimate of the cross-sections for processes (19.24), (19.27)-(19.29) for N > 2 can be made by the SSH model. The available and relatively simple theoretical methods (such as the Born, Born-Rudge and impact-parameter method) can be used to calculate also the cross-section for reactions (19.26), but the difficulties of determining the higher resonant states of prevent easy estimates of cross-sections for reactions (19.25) when N > 2. 

Some of the gaps in the database can be filled by routine (but tedious and time-consuming) calculations using simplified theoretical models (e.g., BGG-model for electron-impact excitation and ionization of H2(NA v) states, the more involved impact-parameter method for excitation, etc.) For some of above listed processes (such as those mentioned under (d) and (e)), however, it is necessary to develop appropriate theoretical models for description of their dynamics. 

In order to demonstrate the physical significance of asymjjtotic nonadiabatic transitions and especially the aiialj-tical theory developed an application is made to the resonant collisional excitation transfer between atoms. This presents a basic physical problem in the optical line broadening [25]. The theoretical considerations were mad( b( for< [25, 27, 28, 29, 25. 30] and their basic id( a has bec n verified experimentally [31]. These theoretical treatments assumed the impact parameter method and dealt with the time-dependent coupled differenticil equations imder the common nuclear trajectory approximation. At that time the authors could not find any analytical solutions and solved the coupled differential equations numerically. The results of calculations for the various cross sections agree well with each other and also with experiments, confirming the physical significance of the asymptotic type of transitions by the dipole-dipole interaction. 

It is emphasized that the concept defined by equation (7) introduces for the first time a dynamically curved projectile trajectory in the impact-parameter method. Thus the projectile motion is coupled to the motion of the active electron. However, since the projectile interacts with a mean electronic field, there is only approximate conservation of energy and momentum. For small projectile scattering angles this deficiency can be circumvented. In this case conservation of energy and momentum may be forced by applying the Eikonal transformation [34]. 

Virtually all non-trivial collision theories are based on the impact-parameter method and on the independent-electron model, where one active electron moves under the influence of the combined field of the nuclei and the remaining electrons frozen in their initial state. Most theories additionally rely on much more serious assumptions as, e.g., adiabatic or sudden electronic transitions, perturbative or even classical projectile/electron interactions. All these assumptions are circumvented in this work by solving the time-dependent Schrodinger equation numerically exact using the atomic-orbital coupled-channel (AO) method. This non-perturbative method provides full information of the basic single-electron mechanisms such as target excitation and ionization, electron capture and projectile excitation and ionization. Since the complex populations amplitudes are available for all important states as a function of time at any given impact parameter, practically all experimentally observable quantities may be computed. 

When colliding particles are heavy and their interactions are long range, the impact parameter method conveniently describes the problem (9, 10, 32, 57). This method is based on the concept that the motion of nucleus is described classically and that of electrons is described quantum mechanically. If the angular momentum of colliding system is larger than K, the trajectory of an incident or a scattered particle can be defined. The impact parameter method will be useful where the total scattering is determined mainly by these processes. 

In the impact parameter method, the model is such that the trajectory and velocity of the incident particle is little changed by the interaction forces between the ion and the neutral particle, and each trajectory haS a reaction probability P b, v) determined by its impact parameter b and velocity u. Thus the reaction cross-section Ur is given by... 

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