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The Journal of Chemical Physics

Semiclassical approximations for scattering by nonlocal potentials. II Simple applications to atomic collisions

R. Benny Gerber

Published Jun 1, 1973• 1 citation

Abstract

The effective-mass method for treating scattering by nonlocal potentials described in a previous publication. is applied to some problems in atomic collision theory. The applications are based on an effective equation for scattering in the elastic channel when coupling with other channels is included, giving rise to a nonlocal effective potential. We use the effective-mass theory to analyze the influence of coupling with closed channels on the elastic scattering in atom-atom collisions and point out that this may lead to a new type of rainbow effect. We show that strong interchannel coupling results in the appearance of poles and zeros in the effective mass and that the latter may lead to compound-state resonances and orbiting effects. It is observed that the occurences of Fesbach resonances in molecular encounters is always reflected in the presence of zeroes in the corresponding effective mass. We also employ the effective-mass method to deal briefly with the familiar problem of curve crossing and nonadiabatic transitions in atomic collisions.

DOI: 10.1063/1.1679082

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The effective-mass method for treating scattering by nonlocal potentials described in a previous publication. is applied to some problems in atomic collision theory. The applications are based on an effective equation for scattering in the elastic channel when coupling with other channels is included, giving rise to a nonlocal effective potential. We use the effective-mass theory to analyze the influence of coupling with closed channels on the elastic scattering in atom-atom collisions and point out that this may lead to a new type of rainbow effect. We show that strong interchannel coupling results in the appearance of poles and zeros in the effective mass and that the latter may lead to compound-state resonances and orbiting effects. It is observed that the occurences of Fesbach resonances in molecular encounters is always reflected in the presence of zeroes in the corresponding effective mass. We also employ the effective-mass method to deal briefly with the familiar problem of curve crossing and nonadiabatic transitions in atomic collisions.

Studies of cold atom collisions and few-body interactions often require the energy dependence of the scattering phase shift, which is usually expressed in terms of an effective-range expansion.We use accurate coupled-channel calculations on 6 Li, 39 K, and 133 Cs to explore the behavior of the effective range in the vicinity of both broad and narrow Feshbach resonances.We show that commonly used expressions for the effective range break down dramatically for narrow resonances and near the zero crossings of broad resonances.We present an alternative parametrization of the effective range that is accurate through both the pole and the zero crossing for both broad and narrow resonances.However, the effective-range expansion can still fail at quite low collision energies, particularly around narrow resonances.We demonstrate that an analytical form of an energy and magnetic-field-dependent phase shift, based on multichannel quantum defect theory, gives accurate results for the energy-dependent scattering length.

Muon transfer and elastic scattering in low-energy collisions of t+d\ensuremath{\mu}(1s) are studied by the hyperspherical coupled-channel method in which many closed channels are explicitly coupled to obtain converged cross sections. A hybrid procedure of the diabatic-by-sector and the traditional adiabatic-basis-expansion methods is developed for solving a larget set of coupled equations influenced by the nearly singular nature of couplings induced by numerous avoided crossings. It is shown that the adiabatic states belonging to the n=2 excited states in the separated-atom limit contribute significantly to the muon transfer. The discrepancy between the previous hyperspherical calculation of Fukuda et al. [Phys. Rev. A 41, 145 (1990)] and other existing theoretical calculations has been resolved by taking into account the couplings with these excited states.

The effect of nonadiabatic couplings on low-energy s-wave scattering of two hydrogen atoms is investigated. Coupling matrix elements are computed in a wide range of internuclear distances. The resulting scattering equations are numerically unstable and therefore are integrated only approximately. Computations are performed for H, D, and T atoms. The phase shifts in the zero velocity limit are inversely proportional to the nuclear reduced mass ${\ensuremath{\delta}}_{0}\ensuremath{\approx}0.392/\ensuremath{\mu}.$ This leads to infinite scattering lengths.

The calculations of the elastic scattering properties for collisions between two 23Na atoms at ultra-cold temperatures are reported and compared with experimental and theoretical results in the detail. We present a parametric model potential for the triplet molecular state of the 23Na molecule, and calculate the scattering length ɑ(subscript t) and the effective range r(subscript e) by using the semiclassical and Numerov methods, respectively. The shape resonances occur in the collisions at low energies, and resonance positions and widths are computed by Briet-Wigner formula. In addition, we also present other results that include differential cross sections at zero energy limit.

A general approximate method for computing the cross sections for certain inelastic collision processes is developed. It is based on the possibility of obtaining, with the use of the impact-parameter method, an exact expression for the transition amplitude in a special case. The dependence of the cross section on the parameters of coupling constant, velocity, and energy difference between states is discussed for a two-level model. Detailed calculations are made of cross section for collisions of the type $A(^{2}P_{\frac{3}{2}})+B\ensuremath{\rightleftharpoons}A(^{2}P_{\frac{1}{2}})+B$, where

A

is an excited atom of an alkali metal and

B

is an atom of a noble gas. An Appendix contains a review of the effective long-range interaction of atoms.