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Physical Review A

Possible high-lying resonances in the positronium-hydrogen system

Joseph DiRienzi, Richard J. Drachman

Published Sep 11, 2007• 10 citations
Publisher

Abstract

We were previously successful in representing a series of low-lying $\mathrm{Ps}+\mathrm{H}$ scattering resonances as due to quasibound Coulomb states in the closed-channel ${e}^{+}+{\mathrm{H}}^{\ensuremath{-}}$ system. Here we investigate the analogous Coulomb states of ${\mathrm{Ps}}^{\ensuremath{-}}+{\mathrm{H}}^{+}$, which are also accessible from the $\mathrm{Ps}+\mathrm{H}$ scattering system, but at considerably higher collision energies. We locate several such states and suggest that they may be scattering resonances, but because there are many open-channel thresholds lying below they may have large widths and shifts.

DOI
10.1103/physreva.76.032705

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We were previously successful in representing a series of low-lying Ps+H\mathrm{Ps}+\mathrm{H}Ps+H scattering resonances as due to quasibound Coulomb states in the closed-channel e++H\ensuremath−{e}^{+}+{\mathrm{H}}^{\ensuremath{-}}e++H\ensuremath− system. Here we investigate the analogous Coulomb states of Ps\ensuremath−+H+{\mathrm{Ps}}^{\ensuremath{-}}+{\mathrm{H}}^{+}Ps\ensuremath−+H+, which are also accessible from the Ps+H\mathrm{Ps}+\mathrm{H}Ps+H scattering system, but at considerably higher collision energies. We locate several such states and suggest that they may be scattering resonances, but because there are many open-channel thresholds lying below they may have large widths and shifts.
We reexamine the problem of calculating the positions and widths of the lowest-lying resonances in the Ps+H\mathrm{Ps}+\mathrm{H}Ps+H scattering system that consists of two electrons, one positron and one proton. The first of these resonances, for L=0,L=0,L=0, was found by the methods of complex rotation and stabilization, and later described as a Feshbach resonance lying close to a bound state in the closed-channel ${e}^{+}+{\mathrm{H}}^{\ensuremath{-}}$ system. Recently, results for the L=1L=1L=1 and 2 scattering states were published, and it was found, surprisingly, that there is a large shift in the positions of these resonances. In this paper we repeat the analysis for L=1L=1L=1 and find an unexpected explanation for the shift.
Seiler, Oberoi, and Callaway first located resonances in the positron-hydrogen system below the n=2n=2n=2 inelastic threshold, using the algebraic close-coupling method without including the open positronium channel. Several other calculations including both open channels have failed to find any resonances. In an attempt to clarify the situation, a calculation has been performed which is an extension of the work of Seiler, Oberoi, and Callaway. A wave function including the 1s1s1s, 2s2s2s, and 2p2p2p eigenstates of hydrogen and the 1s−\stackrel{-}{1s}1s− ground state of positronium was used in a calculation that is a simplification of the usual close-coupling technique and that resembles the optical-potential method. In this approximation it is found that the lowest resonance of Seiler, Oberoi, and Callaway is destroyed by opening the 1s−\stackrel{-}{1s}1s− channel. It is concluded that any resonances which may occur in this system must lie closer to the n=2n=2n=2 threshold than do the corresponding electron-hydrogen resonances.
We investigate the four-body Coulomb process of low-energy elastic positronium-hydrogen (Ps-H) scattering below the Ps(n=2n=2n=2) excitation threshold using scattering wave functions that include Hylleraas-type correlation terms. Using the complex Kohn variational method, we compute phase shifts through the 1,3H^{1,3}H1,3H wave and obtain highly accurate 1,3S^{1,3}S1,3S- and 1,3P^{1,3}P1,3P-wave phase shifts. The complex Kohn variational results compare well to a number of other calculations for this system. We present elastic differential, elastic integrated, and momentum transfer cross sections, and for the singlet, resonances through the 1F^{1}F1F wave. The differential cross section exhibits interesting features, including a change from slightly backward peaked to forward peaked scattering as the energy of the incident positronium increases and rich structure due to multiple resonances near the Ps(n=2n=2n=2) threshold. We also give a detailed analysis of the scattering lengths and effective ranges using multiple effective range theories.
We have investigated the elastic scattering of positronium atoms by hydrogen atoms below the excitation threshold at 5.1 eV, limiting the investigation to the spin-zero (singlet) state of the two electrons. Nonresonant SSS-wave phase shifts have been obtained from normalizable trial functions by extracting the center-of-mass wave function and examining its spatial behavior. An effective-range expansion is fitted to the phase shifts and the previously obtained positronium-hydride bound-state energy to yield a scattering length $a\ensuremath{\cong}5.3{a}_{0}$ compared with Fraser's corrected exchange result, a=7.28a0a=7.28{a}_{0}a=7.28a0​ (a0{a}_{0}a0​ is the Bohr radius). The stabilization method suggests that a resonance occurs at a scattering energy of 4.5 eV, about 0.6 eV below the excitation threshold. Using the complex rotation method, we have obtained evidence confirming the existence of the resonance and an estimate of its width of about 6 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ eV.
Positronium-hydrogen (Ps-H) scattering is of interest, as it is a fundamental four-body Coulomb problem. We have investigated low-energy Ps-H scattering below the Ps(n=2) excitation threshold using the Kohn variational method and variants of the method with a trial wavefunction that includes highly correlated Hylleraas-type short-range terms. We give an elegant formalism that combines all Kohn-type variational methods into a single form. Along with this, we have also developed a general formalism for Kohn-type matrix elements that allows us to evaluate arbitrary partial waves with a single codebase. Computational strategies we have developed and use in this work will also be discussed.With these methods, we have computed phase shifts for the first six partial waves for both the singlet and triplet states. The 1S and 1P phase shifts are highly accurate results and could potentially be viewed as benchmark results. Resonance positions and widths for the 1S-, 1P-, 1D-, and 1F-waves have been calculated.We present elastic integrated, elastic differential, and momentum transfer cross sections using all six partial waves and note interesting features of each. We use multiple effective range theories, including several that explicitly take into account the long-range van der Waals interaction, to investigate scattering lengths for the 1,3S and 1,3P partial waves and effective ranges for the 1,3S-wave.