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Journal of Physics B Atomic Molecular and Optical Physics

Accurate algebraic coupled-state calculation of positron-hydrogen scattering at low energies

T. T. Gien

Published Jan 1, 1999• 1 citation

Abstract

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.

DOI: 10.1088/0953-4075/32/10/501

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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.

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.

The positions of these resonances were determined within an implicit assumption of a degenerate threshold limit at -0.25 Ryd for their sequences, i.e. when the various well known threshold-splitting effects are ignored. The positions of these resonances should, therefore, be understood as to be referring to this degenerate threshold limit. For example, the 1 P o (3) and 3 D e (1) resonances, within the degenerate-threshold assumption, are respectively at the positions of 2.378 × 10 -6 Ryd and 3.943 × 10 -6 Ryd from the threshold limit (as point of reference); the threshold limit will change when the threshold-splitting effects are considered.

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.

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.

We have continued our investigation into the elastic scattering of positronium atoms by hydrogen atoms, extending the work to include the spin-one (triplet) state of the two electrons. Once again we have extracted the

S

-wave phase shifts from normalizable trial functions by examining the spatial behavior of the center-of-mass wave functions. An effective-range expansion linear in the positronium energy fits the results very well and leads to a scattering length ${a}^{(\ensuremath{-})}\ensuremath{\simeq}2.36{a}_{0}$, compared with the static result ${a}^{(\ensuremath{-})}=2.476{a}_{0}$ of Hara and Fraser. The

S

-wave total and ortho-para conversion cross sections are calculated by making use of both

^{1}S

and

^{3}S

results. In the

^{3}S

case, we see no indication of any resonances below the excitation threshold at 5.1 eV.