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 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.
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.
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.
The total elastic scattering and ortho-para conversion cross sections of low-energy ortho-positronium with hydrogen atoms have been calculated in the static exchange approximation including the l=0, 1 and 2 partial waves with the incident positronium wavenumber k ranging from 0 to 3.0 (in atomic units) for l=0 and from 0 to 1.2 for l=1 and 2. The scattering lengths are a (+) =7.275 a 0 and a (-) =2.476 a 0 for the space-symmetric and anti-symmetric cases respectively. A resonance is found in the anti-symmetric s-wave scattering at k approximately=2.07, but its real existence is of course questionable because of neglected effects.
Calculations are reported for positronium (Ps) scattering by atomic hydrogen (H) in the energy range 0--6.5 eV in a coupled-pseudostate approximation in which excitation and ionization channels of both the Ps and the H are taken into account. The approximation contains an accurate representation of the van der Waals coefficient. Results are presented for phase shifts, scattering lengths, effective ranges, and various cross sections including partial wave, total, and ortho-para conversion cross sections. An analysis of the possible spin transitions is provided and the energy of the positronium hydride (PsH) bound state is determined. Substantial differences are found from earlier work within the frozen target approximation, now clearly confirming the importance of target excitation channels. Good agreement is obtained with recent calculations of S-wave phase shifts and scattering lengths using the stabilization method. Convergence to the exact binding energy for PsH appears to be slow. Resonances corresponding to unstable states of the positron orbiting ${\mathrm{H}}^{\ensuremath{-}}$ are seen in the electronic spin singlet partial waves. The importance of the ${\mathrm{H}}^{\ensuremath{-}}$ formation channel is discussed.
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 1S and 3S results. In the 3S case, we see no indication of any resonances below the excitation threshold at 5.1 eV.
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 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− 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.