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Journal of Physics B Atomic and Molecular Physics

Low-energy ortho-positronium scattering by hydrogen atoms

S Hara, P A Fraser

Published Dec 21, 1975• 56 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/8/18/002

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arXiv:1409.3371•Sep 11, 2014•1 citation
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.
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 SSS-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 SSS-wave total and ortho-para conversion cross sections are calculated by making use of both 1S^{1}S1S and 3S^{3}S3S results. In the 3S^{3}S3S case, we see no indication of any resonances below the excitation threshold at 5.1 eV.
We have calculated cross sections for elastic and spin-flip scattering processes of electrons on positronium atoms in the para and ortho states in the energy region scrE\ensuremath{\le}0.05 eV; elastic and symmetric charge-exchange cross sections for the same system were calculated without spin interactions for the scattering energies scrE\ensuremath{\le}4 eV. This energy region has been treated for the first time. The calculations have been done by the adiabatic method, where this three-body problem reduces to the multichannel-scattering problem. Very high scattering cross sections were obtained, expecially near the threshold for spin-flip processes.
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 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.
The scattering of orthopositronium (Ps) by hydrogen atoms has been investigated in a five-state coupled-channel model allowing for Ps(1s)H(2s,2p)\mathrm{Ps}(1s)\mathrm{H}(2s,2p)Ps(1s)H(2s,2p) and Ps(2s,2p)H(1s)\mathrm{Ps}(2s,2p)\mathrm{H}(1s)Ps(2s,2p)H(1s) excitations using a recently proposed electron-exchange model potential. The higher (n> 3)(n>~3)(n> 3) excitations and ionization of the Ps atom are calculated using the first Born approximation. Calculations are reported of scattering lengths, phase shifts, elastic, Ps and H excitation, and total cross sections. Remarkable correlations are observed between the S-wave Ps-H binding energy and the singlet scattering length, effective range, and resonance energy obtained in various model calculations. These correlations suggest that if a Ps-H dynamical model yields the correct result for one of these four observables, it is expected to lead to the correct result for the other three. The present model, which is constructed so as to reproduce the Ps-H resonance at 4.01 eV, automatically yields a Ps-H bound state at $\ensuremath{-}1.05$ eV that compares well with the accurate value of $\ensuremath{-}1.067$ eV. The model leads to a singlet scattering length of 3.72a0{3.72a}_{0}3.72a0​ and effective range of 1.67a0,{1.67a}_{0},1.67a0​, whereas the correlations suggest the precise values of 3.50a0{3.50a}_{0}3.50a0​ and 1.65a0{1.65a}_{0}1.65a0​ for these observables, respectively.
The elastic collision between two ortho-positronium (e.g. S=1) atoms is studied using an ab-initio static-exchange model (SEM) in the center of mass (CM) frame considering the system as a four-body Coulomb problem where all the Coulomb interaction terms in the direct and exchange channels are treated exactly. A coupled-channel methodology in momentum space is used to solve Lippman-Schwinger equation following the integral approach. A new SEM code is developed in which the Born-Oppenheimer (BO) scattering amplitude acts as input to derive the SEM amplitude adapting the partial wave analysis. The s-, p- and d- wave elastic phase shifts and the corresponding partial cross sections for the spin alignment S=0, i.e. singlet (+) and S=2, i.e. triplet (-) states are studied. An augmented-Born approximation is used to include the contribution of higher partial waves more accurately to determine the total/integrated elastic cross section ( ), the quenching cross section ( ) and ortho to para conversion ratio ( ). The effective range theory is used to determine the scattering lengths and effective ranges in the s-wave elastic scattering. The theory includes the non-adiabatic short-range effects due to exchange.