Abstract
An ion and a polar molecule interact by an anisotropic ion-dipole potential scaling as $- \alpha \cos (\theta)/r^2$ at large distances. Due to its long-range character, it modifies the properties of angular wave functions, which are no longer given by spherical harmonics. In addition, an effective centrifugal potential in the radial equation can become attractive for low angular momenta. In this paper, we develop a general framework for an ion-dipole reactive scattering, focusing on the regime of large $\alpha$. We introduce modified spherical harmonics as solutions of the angular part of the Schr\"odinger equation and derive several useful approximations in the limit of large $\alpha$. We present a formula for the scattering amplitude expressed in terms of the modified spherical harmonics and we derive expressions for the elastic and reactive collision rates. The solutions of the radial equation are given by Bessel functions, and we analyse their behaviour in two distinct regimes corresponding, basically, to attractive and repulsive long-range centrifugal potentials. Finally, we study reactive collisions in the universal regime, where the short-range probability of loss or reaction is equal to unity.
Highlights
Hybrid systems involving cold atoms and ions are gaining increasing attention both in theory and experiment [1]
We present a formula for the scattering amplitude expressed in terms of the modified spherical harmonics and we derive expressions for the elastic and reactive collision rates
The solutions of the radial equation are given by Bessel functions, and we analyze their behavior in two distinct regimes corresponding, basically, to attractive and repulsive long-range centrifugal potentials
Summary
Hybrid systems involving cold atoms and ions are gaining increasing attention both in theory and experiment [1]. We study the scattering problem for the ion-dipole potential focusing on the regime of very large α In such a case, the wave functions at r → 0 are singular, and one needs to impose some supplemental boundary conditions, defining the short-range behavior of the wave function. This could be done, for instance, in the spirit of the quantum-defect theory (QDT) [82,83,84,85,86], where one introduces some short-range parameters, that weakly depend both on the collision energy and on the angular momentum of the relative motion [87] Such a treatment can be extended to the case of the reactive scattering, where apart from the phase parameters, one introduces amplitude of the short-range reaction processes [50,51].
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