Impact parameter

The main point of this argument is to show that if particles with velocities v and v collide in the right geometric configuration with impact parameter b, such a collision will result in one of the particles having the velocity of interest, v, after the collision. These kinds of collisions which produce particles with velocity v. 

Thus die increase of particles in our region due to restituting collisions with an impact parameter between b and b + Ab and azimuthal angle between e and e + de (see figure A3.1.7 can be obtained by adjusting the expression for the decrease of particles due to a small collision cylinder ... 

The primes and subscripts on the /s refer to their velocity arguments, and the primed velocities in the gain tenn should be regarded as fiinctions of the imprimed quantities according to (A3.1.36). It is often convenient to rewrite the integral over the impact parameter and the azimuthal angle as an integral over the unit vector kas... 

There are significant differences between tliese two types of reactions as far as how they are treated experimentally and theoretically. Photodissociation typically involves excitation to an excited electronic state, whereas bimolecular reactions often occur on the ground-state potential energy surface for a reaction. In addition, the initial conditions are very different. In bimolecular collisions one has no control over the reactant orbital angular momentum (impact parameter), whereas m photodissociation one can start with cold molecules with total angular momentum 0. Nonetheless, many theoretical constructs and experimental methods can be applied to both types of reactions, and from the point of view of this chapter their similarities are more important than their differences. 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

The physical situation of interest m a scattering problem is pictured in figure A3.11.3. We assume that the initial particle velocity v is comcident with the z axis and that the particle starts at z = -co, witli x = b = impact parameter, andy = 0. In this case, L = pvh. Subsequently, the particle moves in the v, z plane in a trajectory that might be as pictured in figure A3.11.4 (liere shown for a hard sphere potential). There is a point of closest approach, i.e., r = (iimer turning point for r motions) where... 

In an ensemble of collisions, the impact parameters are distributed randomly on a disc with a probability distribution P(b) that is defined by P(b) db = 2nb db. The cross section da is then defined by... 

Figure A3.11.6. Dependence of scattering angle r on impact parameter for a 6-12 potential.

Figure Bl.23.2. (a) Shadow cone of a stationary Pt atom in a 4 keV Ne ion beam, appearing with the overlapping of ion trajectories as a fiinction of the impact parameter. The initial position of the target atom that recoils in the collision is indicated by a solid circle, (b) Plot of the nonnalized ion flux distribution density across the shadow cone in (a). The flux density changes from 0 inside the shadow cone, to much greater than l in the focusing region, converging to 1 away from the shadow cone edge, (c) Blocking cones...

A = /W//Wp, P is impact parameter and Tq is the distance of closest approach (apsis) of the collision pair. The transformations from the CM coordinates (scattering angle y) to the laboratory coordinates with the scattering angle 0 for the primary particle and (]) for the recoiled surface atoms Is given by... 

Classical ion trajectory computer simulations based on the BCA are a series of evaluations of two-body collisions. The parameters involved in each collision are tire type of atoms of the projectile and the target atom, the kinetic energy of the projectile and the impact parameter. The general procedure for implementation of such computer simulations is as follows. All of the parameters involved in tlie calculation are defined the surface structure in tenns of the types of the constituent atoms, their positions in the surface and their themial vibration amplitude the projectile in tenns of the type of ion to be used, the incident beam direction and the initial kinetic energy the detector in tenns of the position, size and detection efficiency the type of potential fiinctions for possible collision pairs. 

Taking advantage of the synnnetry of the crystal structure, one can list the positions of surface atoms within a certain distance from the projectile. The atoms are sorted in ascending order of the scalar product of the interatomic vector from the atom to the projectile with the unit velocity vector of the projectile. If the collision partner has larger impact parameter than a predefined maximum impact parameter discarded. If a... 

When an ion beam is incident on an atomically flat surface at grazing angles, each surface atom is shadowed by its neighbouring atom such that only forwardscattering (FS) is possible these are large impact parameter (p) collisions. 

The semiclassical version is obtained by the substitution mvb = ( + so that = ( + i) in tenns of the impact parameter b. Regarding as a continuous variable,... 

The transition matrix T(b)f is therefore the probability of scattering particles with impact parameter b. B2.2.6.4 DIFFERENTIAL CROSS SECTIONS... 

The Landau-Zener transition probability is derived from an approximation to the frill two-state impact-parameter treatment of the collision. The single passage probability for a transition between the diabatic surfaces H, (/ ) and R AR) which cross at is the Landau-Zener transition probability... 

Impact parameter. The distance between two particles at their closest approach, had they continued in this original direction at their original speeds. 

Interaction distance. The greatest distance between two particles at which it is discernible that they will not pass at the impact parameter. 

The impact parameter, b, is defined to be the perpendicular distance between the initial relative path (along g) and the line parallel to g through the force center (b would be the distance of closest approach of the particles, if there were no interaction) the initial angular momentum is just pbg. Conservation of angular momentum 6 is thus ... 

This determines the distance of closest approach in terms of the initial relative velocity, the impact parameter, and the dynamical quantities (masses and force law constants). The equation for the orbit of the relative motion is found from the first of Eqs. (1-8) and (1-9), using the identity (fj6) = (drjdd), as follows ... 

As charge-dipole interaction between the electron and the atom is small, the perturbation theory expansion may be used to estimate f. The odd terms of this expansion disappear after averaging over impact parameters due to isotropy of collisions. In the second order approximation only those elements of P that are bilinear in V are non-zero. Straightforward calculation showed [176] that all components of the Stark structure are broadened but only those for which m = 0 interfere with each other ... 

To obtain a more detailed idea of the impact operator, it is customary to employ a semiclassical calculation , assuming that the orbital angular momentum of colliding particles may be considered unchanged despite transitions between rotational states. In such a case scattering occurs in a collision plane determined by impact parameter b and initial velocity v. As a result... 

Employing this simplification (known as the external field approximation - EFA) in Eq. (4.58) and replacing summation over l with integration over the impact parameters, we obtain... 

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