Chaos in coplanar classical collisions with particles interacting throughr−2forces

Abstract

The scattering among three particles interacting through 1/r 2 forces, with opposite charges and widely different masses, is studied in a coplanar geometry. The present work shows that at low impact velocities the output of the collision presents typical characteristics of chaos. The details of the process are investigated. ͓S1063-651X͑99͒07701-6͔PACS number͑s͒: 05.45. Ϫa, 34.70.ϩe Starting from the works of Gaspard and Rice ͓1͔, chaotic scattering was found to be ubiquitous. A large amount of work has been done on potential scattering ͑i.e., scattering of an elementary particle on fixed potential centers͒ ͓2-5͔, but also the scattering among at least three interacting bodies has received great attention ͑see the work of Petit and Hénon on the scattering between a planet and two satellites ͓6͔ or the system helium nucleus plus two electrons ͓7͔͒. A monographic issue about this subject has been published in ͓8͔.In all these studies the projectile and a target preserve their internal structure during the collision, but one can analyze also reactive collisions. Kovács and Wiesenfeld ͓9͔ studied, in a collinear geometry, the scattering between an atom and a diatomic molecule: AϩBC↔ABC↔ABϩC. More recently, we studied the chaotic behavior in the reaction between a hydrogen atom ͑proton plus electron͒ and a projectile proton interacting through Coulomb forces ͓10͔. We analyzed the full three-dimensional problem at very low impact energies. We found that the transition from regular to chaotic scattering appears when impact velocity v p is reduced to a value below about 1/10 of the classical electron velocity v e .In this paper we present another investigation on the same subject: We think that it is worth consideration since collisions with rearrangement between electrically charged particles are one current topic in atomic physics, from both the theoretical and the experimental points of view ͓11͔. The phase space of a system of three particles moving in three dimensions is too large to be easily handled; limiting to two dimensions will allow us to do a more detailed and accurate study. We will address the following topics. ͑a͒ Do features of chaotic scattering appear in two-dimensional collisions? ͑b͒ If so, do they appear in the form of a sharp order-chaos transition or as a smooth transition? ͑c͒ Is it possible to detect some traces of irregular behavior also in the heavy particles motion, instead of only when looking at the lightest one? ͑d͒ Finally, an investigation of the dynamics of the electron during the scattering is done. It gives insight into how the discontinuities on the output function appear.The final state of the projectile may be defined through a set of parameters ͕A f ͖ ͑e.g., the angle of deflection and final velocity͒, which are functions of the input quantities ͕A i ͖ ͑e.g., the impact parameter and impact velocity͒. When the set of values ͕A f ͖ depends sensitively by ͕A i ͖, i.e., a finite variation of ͕A f ͖ is caused by an infinitesimal variation of ͕A i ͖, the system is chaotic.Usually, on...

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