Impact Expansions in Classical and Semiclassical Scattering

Abstract

In the energy regime appropriate to classical and semiclassical atomic scattering theory, experimental data on diGerential cross sections a(8,E) and interference patterns are conveniently analyzed through the use of reduced variables such as r =E8, p= 8 sin8o (8,E). In forward scattering, the reduced relationship is the leading term of an impact expansion of the type p(r, E) =P E "tJ"(r). The p (r) are obtained by eliminating the impact parameter b from expansions of the classical scattering functions of the type r(h, E)=P E "r"(h), introduced by Lehmann and Leibfried. Backscattering data are to be analyzed through expansions such as o(8,E)=P"(s. -8)s"o"(E), derived by eliminating b from expansions like sr -8= y(h, E) = P h'"+'e "(E).If the scattering arises from a potential V(r), the coefficients r"(b), y"(E), etc. , are expressed in the form of integrals over the potentials which lend themselves to inversion procedures similar to Firsov's by which a lower bound to the potential can be extracted from the scattering data. In addition to deriving these expansions and testing them on several realistic interatomic potentials, we describe how the reduced variables they suggest can be applied to the presentation and analysis of experimental data.' (a) C. Lehmann and G.

Energy Relaxation of Helium Atoms in Astrophysical Gases

“…Reduced coordinates provide a unique opportunity to extrapolate and incorporate missing data inside a given domain of E and θ variables. In high-energy, forward scattering, the reduced coordinates arise from the leading terms of the impact parameter expansion and thus give a good approximation for the differential cross section (Smith et al 1966). Using reduced coordinates, the differential cross sections for several energies may be scaled to some universal curve and plotted along this curve together.…”

“…A convenient way of representing differential cross sections in the high-energy, semi-classical regime is through the use of reduced "energy" coordinates τ = Eθ and ρ = θ sin θ (dσ/dΩ) (Smith et al 1966). Reduced coordinates provide a unique opportunity to extrapolate and incorporate missing data inside a given domain of E and θ variables.…”

Scattering of Particles and Radiation in Astrophysical Environments

“…Reduced coordinates provide a unique opportunity to extrapolate and incorporate missing data inside a given domain of E and θ variables. In high-energy, forward scattering, the reduced coordinates arise from the leading terms of the impact parameter expansion and thus give a good approximation for the differential cross sections in this energy regime [34]. To further investigate the extent and utility of these scaling differential cross sections, a basis of experimental atom-molecule differential cross section data was plotted together with the quantum partial wave differential cross sections to determine if the scaling procedure could be applied to more complicated atom-molecule collisions.…”

“…For atomic and molecular scattering theory in energy regimes where it is appropriate to utilize classical and semiclassical methods, it is common to display differential cross section data using reduced coordinates 47) where E, θ, and |f (E, θ)| 2 are the center of mass energy, scattering angle and differential cross section respectively [34]. When a set of differential cross sections, covering a range of energies and scattering angles, is plotted ρ vs τ for a given collision species the data tends to clump on a single line.…”

Approximate Differential Cross Sections at Large Scattering Angles for Simple Repulsive Potentials

“…A series expansion valid at large impact parameters (small scattering angles) was developed by Lehmann and Leibfried 2 and by Smith et al 3 This series converges rapidly at high energy and has pr6ved extremely useful in interpreting experiments at small angles inc.luding the "rainbow"…”