Impact parameter dependence

Wa92a Z. Wan, J. F. Christian and S. L. Anderson, Ne -fCso Collision Energy and Impact Parameter Dependence for Endohedral Complex Formation, FVagmentation, and Charge Transfer, J. Chem. Phys. 96, 3344-3347 (1992). 

In order to integrate the coupled-channel equation (8) the time as well as the impact-parameter dependence of the matrix elements (equation (9)) have to be determined. For this purpose, the matrix elements V) (R(t)) are expanded in terms of the radial (7 ) and angular (K) parts of the internuclear vector R according to... 

The coupled-channel calculations are used as benchmark results to check simple models of the impact parameter dependence of the electronic energy loss. A detailed description of such models (convolution approximation) may be found elsewhere [25,26]. Here we present only a short outline of the method. The electronic energy loss involves a sum over all final target states for each impact parameter. Usually this demands a computational effort that precludes its direct calculation in... 

The AO results may also be used for benchmark tests of simpler models. In this context we have also checked a simple non-perturbative model, the UCA. This model includes the main features of fast heavy-ion stopping, as is shown by comparison with large-scale AO results for the impact-parameter dependent electronic energy transfer. The computation of the energy loss within the UCA is much simpler and by many orders of magnitude faster than the full numerical solution of the time-dependent Schrodinger equation. 

Impact parameter dependence of the projectile energy loss... 

Knowing the impact-parameter dependence of energy loss A (b), we can determine the stopping cross-section ... 

In Fig. 5 we show the results of calculation of the impact-parameter dependence of energy loss in collision of 100 keV protons with Ar atom. The calculations were made in the linear response approach (equations (50) and (51)). To demonstrate the effect of additional approximations, we compare this result with the calculation where the dielectric function is described by equation (53) (the static electron gas) and with calculation made in the local density approach (LDA) [20]. In the latter approach the energy loss is determined according to electron density on the ion trajectory. It is seen from the figure that both these approximations can result in significant defects of description. Particularly, the fact that the energy loss is distributed within the atomic shell (in contrast with LDA) turns out to be important. 

Fig. 5. Impact parameter dependence of energy loss in collisions of 100 keV protons with argon atoms determined using equation (50) with G(i7, given by equation (52) (thick curve). The results obtained in the LDA approach and that for the static electron gas (e(k,

Fig. 6. The impact-parameter dependence of energy loss in collision of hydrogen and carbon ions with gold atom at a projectile energy E = 0.5 MeV/a.u. For comparison the first-order perturbation results (the doted curves) and that for equivalent negative projectiles (the dashed curves) are also shown.

As an application of the approach presented above we present some studies performed to calculate the energy lost by fast protons reflected by metal surfaces. We note that for the high velocities under consideration in addition to the valence band excitations, one has to consider that the projectile can also excite electrons in the inner-shells of the target atoms. This contribution to the energy loss can be obtained in terms of the impact-parameter dependent... 

Coupled channel methods for colllnear quantum reactive calculations are sufficiently well developed that calculations can be performed routinely. Unfortunately, colllnear calculations cannot provide any Insight Into the angular distribution of reaction products, because the Impact parameter dependence of reaction probabilities Is undefined. On the other hand, the best approximate 3D methods for atom-molecule reactions are computationally very Intensive, and for this reason. It Is Impractical to use most 3D approximate methods to make a systematic study of the effects of potential surfaces on resonances, and therefore the effects of surfaces on reactive angular distributions. For this reason, we have become Interested In an approximate model of reaction dynamics which was proposed many years ago by Child (24), Connor and Child (25), and Wyatt (26). They proposed the Rotating Linear Model (RLM), which Is In some sense a 3D theory of reactions, because the line upon which reaction occurs Is allowed to tumble freely In space. A full three-dimensional theory would treat motion of the six coordinates (In the center of mass) associated with the two... 

At any total scattering energy E, elements of the multichannel S matrix In the RLM are labelled by the total angular momentum Index I, and by the Initial and final vibrational quantum numbers v and v. Equations for physical observables in the BCRLM have been given previously (24-26), and we only summarize the final results here, In order to establish a common notation. The opacity function gives the Impact parameter dependence of the reaction probabilities. 

Reaction probabilities versus impact parameter b show that rearrangement occurs only at small b, <2a0, and are nearly symmetric around b = 0. This indicates that reaction is nearly the same whether orbital and rotational angular momenta are parallel or antiparallel. Angular distributions must be obtained between 0° and 360°, because once the rotation sense is fixed one must distinguish between upper and lower half-planes. Impact parameter dependencies are shown in Fig. 6. 

Paulsen et al. (1972) developed an optical model for vibrational relaxation in reactive systems. Only collinear atom-diatom collisions were considered, i.e. impact parameter dependencies were omitted. The model was applied to vibrational relaxation of electronically excited I2 in inert gases, in which case dissociation of I2 is responsible for flux loss. Olson (1972) used an absorbing-sphere model for calculating integral cross sections of ion-ion recombination processes A++B ->A + B + AE, with A or B atoms or molecules. He employed the Landau-Zener formula to obtain a critical crossing distance Rc, and assumed the opacity to be unity for distances... 

From a very general point of view every ion-atom collision system has to be treated as a correlated many-body time-dependent quantum system. To solve this from an ab initio point of view is still impossible. So, one has to rely on various approximations. Nowadays the best method which can be applied to realistic collision systems (which we discuss here) is on the level of the non-selfconsistent time-dependent Hartree-Fock-Slater or, in the relativistic case, the Dirac-Fock-Slater method. Up-to-now no correlation beyond this approximation can be taken into account in the case of 3 or more electrons. (This is in accordance with the definition of correlation given by Lowdin [1] in 1956) In addition no QED contributions, i.e. no correction to the 1/r Coulomb interaction between the electrons, ever have been taken into account, although in very heavy collision systems this effect may become important. This will be discussed in section 5. A short survey of the theory used is followed by our results on impact parameter dependent electron transfer and excitation calculations of ion-atom and ion-solid collisions as well as first results of an ab initio calculation of MO X-rays in such complicated many particle scattering systems. 

R. Cabrera-Trujillo, E. Deumens, Y. Ohrn and J.R. Sabin, Impact parameter dependence of electronic and nuclear energy loss of swift ions H+ — He and H+ — H, Nucl. Instrum. Methods, B168 (2000) 484. 

Additional theoretical refinement is needed before we can quantitatively compare BCRLM differential cross sections with 3D. The shape of the BCRLM differential cross section contains information about the impact parameter dependence of the reaction probability, and whenever the 3D angular distribution retains only this level of dynamical detail, we would expect the BCRLM differential cross section to compare nicely. Consequently, it is likely that BCRLM will fare best when compared to 3D differential cross sections from the ground state of reactants to all product rotational states, since this type of cross section retains the least amount of rotational Information. 

INC codes-e.g., ISABEL, Yarifand Fraenkel (1981), and QGSM, Toneevet al. (1990) - are relatively successful in reproducing the fast cascade component of the reaction observables for light-ion-induced reactions above 100 MeV. They also predict the qualitative result that a broad distribution of excitation energies will be produced due to impact-parameter-dependent... 

For heavy-ion collisions well above the Fermi energy, models predict a low probability for composite-nucleus formation. Instead, most of the cross section is predicted to go into reactions that can be generalized as "participant—spectator reactions (O Fig. 3.44). In the participant-spectator scenario, the participant source is defined by those nucleons that occupy the geometrical overlap volume of the target and projectile, which is impact-parameter dependent. 

ABSTRACT. Laser and molecular beam techniques allow detailed study of many dynamical properties of single reactive collisions. The chemical scope of these methods is now very wide and includes internal state preparation of reactants, change of collision energies, state detection of products, and thus determination of state-to-state reaction rates. The great impact of laser spectroscopy on knowledge in the field of structure, molecular energy transfer and the mechanism of elementary chemical reactions is illustrated by two selected examples, i.e. studies in which laser-induced fluorescence (LIF) has been used to determine the specific impact parameter dependence of the Ca + HF -> CaF(X) + H reaction and the product state distributions for the reaction of metastable Ca with SF5. 

The energy and impact parameter dependence of the angle of deflection... 

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