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Physical Review Letters

Comparative Studies of Excitations and Resonances in H− , Ps− , and e++H Systems

Yan Zhou, C. D. Lin

Published Sep 18, 1995• 40 citations
Publisher

Abstract

The hyperspherical close coupling method for general Coulomb three-body systems is used to examine ${e}^{\ensuremath{-}}+\mathrm{H}$, ${e}^{+}+\mathrm{Ps}$, and ${e}^{+}+\mathrm{H}$ collisions to investigate how the excitation cross sections and resonances depend on the masses, the charges, and the quantum statistics of the system. Hyperspherical potential curves are used to help unravel the nature of Feshbach resonances, shape resonances, and overlapping resonances in these systems.

DOI
10.1103/physrevlett.75.2296

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The hyperspherical close coupling method for general Coulomb three-body systems is used to examine e\ensuremath−+H{e}^{\ensuremath{-}}+\mathrm{H}e\ensuremath−+H, e++Ps{e}^{+}+\mathrm{Ps}e++Ps, and e++H{e}^{+}+\mathrm{H}e++H collisions to investigate how the excitation cross sections and resonances depend on the masses, the charges, and the quantum statistics of the system. Hyperspherical potential curves are used to help unravel the nature of Feshbach resonances, shape resonances, and overlapping resonances in these systems.
A three-state and a six-state close-coupling approximation have been employed to investigate the e − -Ps scattering below the first excitation threshold of the Ps atom. Present s - and p -wave phase shifts and total cross-sections are compared with the corresponding predictions of Ward et al. ( J. Phys. B , 20 (1987) 127). All the Feshbach resonances below the n = 2 threshold, including 3 S e and 3 P e resonances as predicted by Ho ( Phys. Lett. A , 120 (1984) 348 and Phys. Rev. A , 60 (1999) 2834), have been obtained.
The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.
In an effort to unravel the dependence of the dynamics of three-bound-particle systems on the relative masses of the particles, an adiabatic treatment in hyperspherical coordinates is used to study the ground state and some doubly excited states of Ps('-) (positronium negative ion) and other three-particle systems, including H('-) and M('-) (muonium negative ion). Two different methods for the solution of the adiabatic eigenvalue equation are presented; one uses a "prediagonalized" basis of hyperspherical harmonics, and the other uses an asymptotic basis which consists of hydrogenic functions for the positronium. Potential curves for Ps('-) and M('-) in the ('1)S('e), ('1)P('o), ('3)P('e), ('3)P('o) and ('1)D('e) states and for H('-) in the ('1)P('o) states are presented. A method for calculating the probability density as a function of the hyperspherical coordinates (R,(alpha),(theta)(,12)) is introduced and used for the P states of Ps('-). This helps to understand the different symmetries and to distinguish among them. A remarkable linear dependence of the binding energy of the ground state and of the doubly excited states on the reduced mass of the atom, discovered in this study, is presented and discussed.
We apply the hyperspherical close-coupling method to the elastic and spin-flip processes in $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}+\mathrm{Ps}$ (or equivalently, e++Ps){e}^{+}+\mathrm{Ps})e++Ps) scattering at low collision energies. The spin-spin interactions VS{V}_{S}VS​ are included explicitly in the Hamiltonian, producing the correct hyperfine-splitting energy $\ensuremath{\Delta}\ensuremath{\epsilon}$ in the separated-atom limit. A comparison is made with a simpler method in which the effect of VS{V}_{S}VS​ is represented only by the symmetry of the wave functions with respect to the interchange of the two identical particles. Explicit inclusion of VS{V}_{S}VS​ is found to be of vital importance for the calculation of the spin-flip cross sections for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$ for both $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}{(e}^{+})+\mathrm{Ps}$ collisions. The elastic cross sections are less sensitive to VS,{V}_{S},VS​, but those for $p+p\ensuremath{\mu}$ are still affected by VS{V}_{S}VS​ for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$.
Quantum mechanical calculations of ultralow-energy collision of Coulomb three-body systems in the hyperspherical elliptic coordinates are presented. The nonadiabatic coupling between the hyperradius and hyperangular variables is treated with the slow-variable discretization method in combination with the RRR-matrix propagation technique. For scattering state calculations, the two-dimensional matching procedure using Gailitis's method [M. Gailitis, J. Phys. B 9, 843 (1976); C. Noble and R. Nesbet, Comput. Phys. Commun. 33, 399 (1984)] is implemented to determine the boundary conditions between the internal and the asymptotic wave functions. This method is proved to be very efficient and gives very accurate results. Taking advantage of this method, we accurately calculate the scattering phase shifts and the scattering lengths of Coulomb three-body systems with mass ratio varying over several orders of magnitudes. We observed jumps of the scattering length from $\ensuremath{-}\ensuremath{\infty}$ to $\ensuremath{\infty}$ at certain mass ratios and monotonic decreases between two jumps. These are closely related to the binding energy of the highest bound state through Levinson's theorem [N. Levinson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 25, 9 (1949)]. Our calculations provide a comprehensive perspective to the scattering length from the variation of mass ratio of Coulomb three-body systems.