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Physical Review A

Semiclassical approximation of cross sections with many rainbow peaks

Jens H. Jensen

Published Feb 1, 1992• 7 citations
Publisher

Abstract

A semiclassical formula that gives scattering cross sections down to an angular resolution of order ${\mathrm{\ensuremath{\Elzxh}}}^{4/5}$ is derived. The formula describes rainbow peaks, which have widths of order ${\mathrm{\ensuremath{\Elzxh}}}^{2/3}$, but not other interference effects, which give structure on a scale of order \ensuremath{\Elzxh}. For cross sections with many rainbow peaks, it can be easier to evaluate than conventional uniform approximations. As an example, the formula is applied to a simple two-dimensional model.

DOI
10.1103/physreva.45.1307

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A semiclassical formula that gives scattering cross sections down to an angular resolution of order \ensuremath\Elzxh4/5{\mathrm{\ensuremath{\Elzxh}}}^{4/5}\ensuremath\Elzxh4/5 is derived. The formula describes rainbow peaks, which have widths of order \ensuremath\Elzxh2/3{\mathrm{\ensuremath{\Elzxh}}}^{2/3}\ensuremath\Elzxh2/3, but not other interference effects, which give structure on a scale of order \ensuremath{\Elzxh}. For cross sections with many rainbow peaks, it can be easier to evaluate than conventional uniform approximations. As an example, the formula is applied to a simple two-dimensional model.
The leading quantum corrections for the scattering of a particle in one dimension from a pulsed potential are discussed. Depending on the model parameters, the scattering may be either regular or chaotic. The quantum corrections are obtained by applying a semiclassical formalism that gives the distribution of scattered momenta down to a resolution of order ${\mathrm{\ensuremath{\Elzxh}}}^{4/5}$. At this resolution, the main quantum effect is the softening of the classical rainbow singularities into peaks with widths of order ${\mathrm{\ensuremath{\Elzxh}}}^{2/3}$. Quantum corrections for examples of both regular and chaotic scattering are compared with exact quantum results, and a criterion is given for estimating the resolution at which the corrections first become significant.
We apply the uniform Airy approximation ( Beuc and Horvatic 1992 J. Phys. B: At. Mol. Opt. Phys. 25 1497 ) to the analytical model proposed by Devdariani et al ( 2002 J. Phys. B: At. Mol. Opt. Phys. 35 2469 ). The model describes asymptotically forbidden quasimolecular optical transitions within the frame of the semiclassical approach. Our results support conclusions that the uniform Airy approximation, originally developed for the asymptotically allowed transitions, is applicable to a broad class of asymptotically forbidden but collisionally assisted optical transitions as well. In addition, we present a new transitional formula and discuss some further approximations.
An approximate scaling formula has been determined that permits the rapid estimation of cross sections for angular-momentum-changing collisions of Rydberg atoms with a variety of targets, using information about low-energy-electron scattering from the target. The formula is obtained by fitting the results of coupled-channel and Born-approximation calculations to functions of reduced parameters. Application to Na(nd)\mathrm{Na}(\mathrm{nd})Na(nd) + He, Ne, Ar, N2{\mathrm{N}}_{2}N2​, CH4{\mathrm{H}}_{4}H4​, C3{\mathrm{C}}_{3}C3​H8{\mathrm{H}}_{8}H8​; Rb(nf)\mathrm{Rb}(\mathrm{nf})Rb(nf) + He, Ar, Xe; and Xe(nf)\mathrm{Xe}(\mathrm{nf})Xe(nf) + CO2{\mathrm{O}}_{2}O2​, suggests that the accuracy is about a factor of 2.
The scattering properties of any complex scattering potential, , can be obtained from the dynamics of a particular non-unitary two-level quantum system . The application of the adiabatic approximation to yields a semiclassical treatment of the scattering problem. We examine the adiabatic series expansion for the evolution operator of and use it to obtain corrections of arbitrary order to the semiclassical formula for the transfer matrix of v. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with large derivatives.