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Canadian Journal of Physics

Feshbach resonances in positron–hydrogen collisions

T. T. Gien

Published Jul 1, 1996• 10 citations
Publisher

Abstract

Sequences of Feshbach resonances, predicted by theory to exist below the n = 2 H threshold in positron–hydrogen scattering, that have long evaded all previous numerical searches, were finally located in our close-coupling calculation employing various coupling schemes. The energy ratios and width ratios of the successive resonances of these sequences were, furthermore, found to agree very well with the values predicted by theory.

DOI
10.1139/p96-050

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Sequences of Feshbach resonances, predicted by theory to exist below the n = 2 H threshold in positron–hydrogen scattering, that have long evaded all previous numerical searches, were finally located in our close-coupling calculation employing various coupling schemes. The energy ratios and width ratios of the successive resonances of these sequences were, furthermore, found to agree very well with the values predicted by theory.
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.
Our unexpounded writing in the first paragraph of page L24 of the letter may give a false impression to some about what was really meant. We would like, therefore, to clarify it. We simply intended to stress that the deficiency of the polarizability of H(1s) and of Ps(1s) can be seen more easily at low energies in higher-partial-wave phase shifts of e + -H scattering and in higher-partial-wave elastic cross sections of Ps(1s)-p scattering, and that for the purpose of determining the sequences of S, P, D Feshbach resonances (within the assumption of a degenerate threshold limit at -0.25 Ryd, i.e. when the threshold splitting due to fine-structure and QED corrections is ignored), the use of the E6S coupling scheme is appropriate since this coupling scheme contains the 2s and 2p states of H essential for the degenerate dipole interaction. It is therefore quite obvious that for the calculations in a very narrow range of energy of less than 1 × 10 -2 Ryd beneath the degenerate threshold (resonance region), the use of the E6PS coupling scheme may not be able to yield reliable results for S, P, D scattering, since the scheme cannot handle the resonances properly, due to the absence of the 2s and 2p H states essential for the degenerate dipole interaction. This is, however, of little concern, since the range of energies that we have considered for our various accurate calculations (such as E6PS, E8PS, E6S, ME6S, E8S) is further below the degenerate threshold. Rigorously speaking, even if the 2s and 2p H states are included in the coupling scheme, care concerning the nondegeneracy of threshold might still have to be taken for calculations in the closest vicinity of the threshold.
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 ${e}^{+}+{\mathrm{H}}^{\ensuremath{-}}$ system. Here we investigate the analogous Coulomb states of ${\mathrm{Ps}}^{\ensuremath{-}}+{\mathrm{H}}^{+}$, 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 have calculated SSS-wave resonances in the positron-lithium system using the stabilization method in the framework of hyperspherical coordinates. Resonances below the Li (n=4)(n=4)(n=4) thresholds have been investigated. We have located the previously known SSS-wave resonances by Roy and Ho [J. Phys. B 35, 2149 (2002)] and observed relatively good agreement for the resonance positions. In addition, one Feshbach resonance associated with the Ps (n=2)(n=2)(n=2) threshold is also identified.