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Physical Review A

Feshbach resonances in positron-hydrogen scattering

Richard J. Drachman

Published Jul 1, 1975• 22 citations

Abstract

Seiler, Oberoi, and Callaway first located resonances in the positron-hydrogen system below the $n=2$ 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 $1s$, $2s$, and $2p$ eigenstates of hydrogen and the $\stackrel{-}{1s}$ 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 $\stackrel{-}{1s}$ channel. It is concluded that any resonances which may occur in this system must lie closer to the $n=2$ threshold than do the corresponding electron-hydrogen resonances.

DOI: 10.1103/physreva.12.340

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Europhysics Letters (EPL)•Oct 1, 2002•14 citations

Seiler, Oberoi, and Callaway first located resonances in the positron-hydrogen system below the

n=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

1s

,

2s

, and

2p

eigenstates of hydrogen and the

\stackrel{-}{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

\stackrel{-}{1s}

channel. It is concluded that any resonances which may occur in this system must lie closer to the

n=2

threshold than do the corresponding electron-hydrogen resonances.

We reexamine the problem of calculating the positions and widths of the lowest-lying resonances in the

\mathrm{Ps}+\mathrm{H}

scattering system that consists of two electrons, one positron and one proton. The first of these resonances, for

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=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=1

and find an unexpected explanation for the shift.

The close-coupling equations for the positron-hydrogen system restricted to s-wave scattering from the l=0 levels of hydrogen and positronium are written as a set of coupled momentum space Lippmann-Schwinger equations. The matrix element for positronium formation is evaluated by integrating over the multipole projected components of the momentum space wave functions. Cross sections generated in the coupled static approximation provide a verification of the method. With a larger basis, including the 1s and 2s levels of hydrogen and the 1s and 2s levels of positronium, the broad s-wave resonance at 2.6 Ryd shifts position to 3.17 Ryd. An additional resonance is observed in the cross section at an energy of 1.10 Ryd.

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.

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.