Classical dynamics near the triple collision in a three-body Coulomb problem

We investigate the classical motion of three charged particles with both attractive and repulsive interaction.The triple collision is a main source of chaos in such three body Coulomb problems.By employing the McGehee scaling technique, we analyse here for the first time in detail the three-body dynamics near the triple collision in 3 degrees of freedom.We reveal surprisingly simple dynamical patterns in large parts of the chaotic phase space. The underlying degree of order in the form of approximate Markov partitions may help in understanding the global structures observed in quantum spectra of two-electron atoms.Comment: 4 pages, 3 figure

Classical Mechanics of Collinear Positron-Hydrogen Scattering

We study the classical dynamics of the collinear positron-hydrogen scattering system below the three-body breakup threshold. Observing the chaotic behavior of scattering time signals, we in- troduce a code system appropriate to a coarse grained description of the dynamics. And, for the purpose of systematic analysis of the phase space structure, a surface of section is introduced being chosen to match the code system. Partition of the surface of section leads us to a surprising conjec- ture that the topological structure of the phase space of the system is invariant under exchange of the dynamical variables of proton with those of positron. It is also found that there is a finite set of forbidden patterns of symbol sequences. And the shortest periodic orbit is found to be stable, around which invariant tori form an island of stability in the chaotic sea. Finally we discuss a possible quantum manifestation of the classical phase space structure relevant to resonances in scattering cross sections.Comment: 14 pages, 9 figure

Scaling laws for the photo-ionisation cross section of two-electron atoms

The cross sections for single-electron photo-ionisation in two-electron atoms show fluctuations which decrease in amplitude when approaching the double-ionisation threshold. Based on semiclassical closed orbit theory, we show that the algebraic decay of the fluctuations can be characterised in terms of a threshold law $\sigma \propto |E|^{\mu}$ as $E \to 0_-$ with exponent $\mu$ obtained as a combination of stability exponents of the triple-collision singularity. It differs from Wannier's exponent dominating double ionisation processes. The details of the fluctuations are linked to a set of infinitely unstable classical orbits starting and ending in the non-regularisable triple collision. The findings are compared with quantum calculations for a model system, namely collinear helium.Comment: 4 pages, 1 figur

The classical dynamics of two-electron atoms near the triple collision

The classical dynamics of two electrons in the Coulomb potential of an attractive nucleus is chaotic in large parts of the high-dimensional phase space. Quantum spectra of two-electron atoms, however, exhibit structures which clearly hint at the existence of approximate symmetries in this system. In a recent paper,(Phys. Rev. Lett. 93, 054302 (2004)), we presented a study of the dynamics near the triple collision as a first step towards uncovering the hidden regularity in the classical dynamics of two electron atoms. The non-regularisable triple collision singularity is a main source of chaos in three body Coulomb problems. Here, we will give a more detailed account of our findings based on a study of the global structure of the stable and unstable manifolds of the triple collision.Comment: 21 pages, 17 figure

Total photoionization cross section of planar helium: scaling laws and collision orbits

The total photoionization cross section of planar helium has been calculated up to the single ionization threshold I22 of triple P states. The cross section shows chaotic fluctuations as the energy E approaches the double ionization threshold E=0. By analyzing the fluctuating part of the cross section, we show that its amplitude decreases as ∣E∣ mu for E-> 0- as predicted in Byunet al(2007 Phys. Rev. Lett., 98, 113001). The Fourier transform of the fluctuating part reveals peaks at the classical actions of closed triple collision orbits. Furthermore, the relative height of the peaks is consistent with the semiclassical predictions. Our findings underline that the fluctuating part of the photoionization cross section can be described by classical triple collision orbits in the semiclassical limit. These orbits all lie in the collinear eZe subspace, demonstrating that the fluctuations are dominated by the dynamics of this low dimensional phase space