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Physical Review A

Positronium-hydrogen scattering using the stochastic variational method

Igor A. Ivanov, J. Mitroy, Kálmán Varga

Published Feb 5, 2002• 31 citations
Publisher

Abstract

The stochastic variational method (SVM) is used in conjunction with stabilization ideas in order to compute low-energy phase shifts. The method is tested by applying it to a simple model problem, and to s-wave positron-hydrogen scattering. The SVM is then applied to the calculation of the s- and p-wave phase shifts for Ps-H scattering. The scattering lengths obtained were ${4.34a}_{0}$ for the electronic-spin-singlet state and ${2.22a}_{0}$ for the triplet state. The present scattering lengths are probably accurate to $\ifmmode\pm\else\textpm\fi{}3%$ and are the most accurate that have so far been computed for the Ps-H systems.

DOI
10.1103/physreva.65.032703

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Physical Review A•Feb 27, 2002•52 citations
The stochastic variational method (SVM) is used in conjunction with stabilization ideas in order to compute low-energy phase shifts. The method is tested by applying it to a simple model problem, and to s-wave positron-hydrogen scattering. The SVM is then applied to the calculation of the s- and p-wave phase shifts for Ps-H scattering. The scattering lengths obtained were 4.34a0{4.34a}_{0}4.34a0​ for the electronic-spin-singlet state and 2.22a0{2.22a}_{0}2.22a0​ for the triplet state. The present scattering lengths are probably accurate to \ifmmode±\else\textpm\fi3\ifmmode\pm\else\textpm\fi{}3%\ifmmode±\else\textpm\fi3 and are the most accurate that have so far been computed for the Ps-H systems.
The stochastic variational method is used in conjunction with stabilization ideas to compute the low-energy s-wave phase shifts and scattering lengths for Ps-Ps scattering. The scattering lengths obtained were 8.44a0{8.44a}_{0}8.44a0​ for the singlet-singlet and 3.00a0{3.00a}_{0}3.00a0​ for the triplet-triplet states (the spin state refers to the coupling of the two electrons and the coupling of the two positrons in the system). The positive scattering length in the triplet-triplet channel means a stable Bose-Einstein condensate of triplet Ps atoms is physically possible. The large value of the ${}^{3}{\mathrm{Ps}}^{3}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{Ps}}}^{1}{\mathrm{Ps}}^{1}\mathrm{Ps}$ cross section, $7.41\ensuremath{\pi}{a}_{0}^{2}$ suggests the use of spin-polarized positrons as advanced by Platzmann and Mills [Phys. Rev. B 49 454 (1994)].
The stochastic variational method is used in conjunction with stabilization ideas to compute the low energy phase shifts and scattering lengths for positronium-atom scattering. Results are obtained for the Ps-H, Ps- Li+, Ps-He, and Ps-Ps systems. The Ps-H scattering lengths are probably accurate to better than 5% and are the most accurate so far computed. The results for Ps- Li+ and Ps-Ps scattering represent the first published scattering lengths for these systems. The positive scattering length for completely spin-aligned 3Ps-3Ps scattering, namely 2.95a(0), is particularly significant since it demonstrates the feasibility of forming a stable Bose-Einstein condensate of 3Ps atoms.
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.
We investigate the four-body Coulomb process of low-energy elastic positronium-hydrogen (Ps-H) scattering below the Ps(n=2n=2n=2) excitation threshold using scattering wave functions that include Hylleraas-type correlation terms. Using the complex Kohn variational method, we compute phase shifts through the 1,3H^{1,3}H1,3H wave and obtain highly accurate 1,3S^{1,3}S1,3S- and 1,3P^{1,3}P1,3P-wave phase shifts. The complex Kohn variational results compare well to a number of other calculations for this system. We present elastic differential, elastic integrated, and momentum transfer cross sections, and for the singlet, resonances through the 1F^{1}F1F wave. The differential cross section exhibits interesting features, including a change from slightly backward peaked to forward peaked scattering as the energy of the incident positronium increases and rich structure due to multiple resonances near the Ps(n=2n=2n=2) threshold. We also give a detailed analysis of the scattering lengths and effective ranges using multiple effective range theories.
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.