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Physical Review A

Nonadiabatic treatment of hydrogen-antihydrogen collisions

Konrad Piszczatowski, Alexei Yu. Voronin, Piotr Froelich

Published Jun 5, 2014• 10 citations
Publisher

Abstract

We present a nonadiabatic treatment of the hydrogen-antihydrogen system. The technique used to describe $\mathrm{H}\text{\ensuremath{-}}\overline{\mathrm{H}}$ collisions is based on the coupled-rearrangement-channel method. Within this approach the total, nonadiabatic wave function of the system is divided into two parts: an inner and an outer one. To describe the inner part a set of square-integrable four-body functions is used. These functions are obtained by a diagonalization of the total Hamiltonian projected on a chosen ${L}^{2}$ subspace; they explicitly contain components of various arrangement channels expressed in terms of corresponding Jacobi coordinates. The outer part of the total wave function reflects its asymptotic character. Our procedure leads to a system of nonlocal integrodifferential equations that are solved iteratively and simultaneously determine the outer part of the solution and the coefficients in the four-body expansion of the inner part. To solve these equations the compact fine difference method was applied. Using this formalism we perform a one-channel calculation of the elastic scattering to obtain the $S$ matrix, the nonadiabatic scattering length, and the cross section for the low-energy elastic scattering in the $\mathrm{H}\text{\ensuremath{-}}\overline{\mathrm{H}}$ channel.

DOI
10.1103/physreva.89.062703

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We present a nonadiabatic treatment of the hydrogen-antihydrogen system. The technique used to describe H\ensuremath-H‾\mathrm{H}\text{\ensuremath{-}}\overline{\mathrm{H}}H\ensuremath-H collisions is based on the coupled-rearrangement-channel method. Within this approach the total, nonadiabatic wave function of the system is divided into two parts: an inner and an outer one. To describe the inner part a set of square-integrable four-body functions is used. These functions are obtained by a diagonalization of the total Hamiltonian projected on a chosen L2{L}^{2}L2 subspace; they explicitly contain components of various arrangement channels expressed in terms of corresponding Jacobi coordinates. The outer part of the total wave function reflects its asymptotic character. Our procedure leads to a system of nonlocal integrodifferential equations that are solved iteratively and simultaneously determine the outer part of the solution and the coefficients in the four-body expansion of the inner part. To solve these equations the compact fine difference method was applied. Using this formalism we perform a one-channel calculation of the elastic scattering to obtain the SSS matrix, the nonadiabatic scattering length, and the cross section for the low-energy elastic scattering in the H\ensuremath-H‾\mathrm{H}\text{\ensuremath{-}}\overline{\mathrm{H}}H\ensuremath-H channel.
We present a treatment of cold hydrogen-antihydrogen collisions based on the asymptotic properties of atom-antiatom interactions. We derive general formulas for the elastic and inelastic cross sections and for the scattering lengths and analyze their sensitivity to the parameters characterizing the inelasticity of the collision process. Given the inelasticity, we obtain bounds for the complex scattering length. We investigate the influence of strong nuclear forces and the isotope effects in HH‾\text{H}\overline{\text{H}}HH and DH‾\text{D}\overline{\text{H}}DH collisions and demonstrate enhancement of these effects due to the presence of the near-threshold narrow HH‾\text{H}\overline{\text{H}}HH (DH‾)(\text{D}\overline{\text{H}})(DH) states. The values of the elastic and inelastic cross sections with simultaneous account of rearrangement and strong forces are presented. General expressions for the (complex) energies of the near-threshold HH‾\text{H}\overline{\text{H}}HH states are obtained.
In view of current interest in the trapping of antihydrogen ( ) atoms at very low temperatures, we have carried out a calculation of s-wave hydrogen–antihydrogen scattering at very low energies, using the Kohn variational method, taking into account rearrangement scattering into the three channels that contain positronium in its ground state and lie closest to threshold. We find that our values for the elastic cross section are in good agreement with the values obtained by Jonsell et al (Jonsell et al 2001 Phys. Rev. A 64 052712) using a distorted wave approximation. However, our values for the total rearrangement cross section are much larger than their values. In particular, the largest such cross section is for the N = 23 s-state of protonium and positronium in its ground state, a channel that was estimated to make a negligible contribution by Jonsell et al . As a consequence of our much larger values for the total rearrangement cross section, we predict that cooling of by cold H would be considerably less efficient than was found to be the case by Jonsell et al .
The stability of antimatter in contact with matter has been investigated. The interaction between hydrogen and antihydrogen is considered as the prototype reaction. We have focused interest on the rates for proton-antiproton and/or electron-positron annihilation during hydrogen-antihydrogen collisions at low energies. In particular, we have concentrated on the calculation of the rates for the rearrangement reaction leading to formation of protonium and positronium, ending inevitably in particle-antiparticle annihilation. The cross section for the rearrangement collision has been calculated in a fully quantum-mechanical treatment. Additionally, we have calculated cross sections for direct annihilation during the collision process, which was found to be comparable to the rearrangement cross section. The elastic cross section and its low-energy limit, given by the scattering length, have been calculated, allowing, by comparison to the inelastic processes, a prediction for the efficiency of cooling antihydrogen via collisions with ultracold hydrogen atoms.
A description of the excited lepton states of the hydrogen-antihydrogen quasimolecule is presented. Potential energy curves and the leptonic part of the wave functions corresponding to a variety of such states are calculated within the Born-Oppenheimer approximation employing the Ritz variational principle. Nonadiabatic corrections to the leptonic potentials are also obtained. Basis functions are constructed as products of explicitly correlated Gaussians and spherical harmonics which describe correctly the motion of leptons with arbitrary orbital angular momentum projection onto the molecular (internuclear) axis. The hadronic part of the wave function for each leptonic level of the hydrogen-antihydrogen system is calculated by solving the Schr\"odinger equation with the obtained leptonic potentials. Corresponding solutions are generated utilizing precise B-spline representations. Employing leptonic and hadronic parts of the wave function the electron-positron and proton-antiproton annihilation rates are computed for a number of quasimolecular states. The decay rates of the hydrogen-antihydrogen system into separate positronium and protonium atoms are also estimated for the quasimolecular levels under consideration.
We calculate cross sections for helium–antihydrogen scattering for energies up to 0.01 atomic unit. Our calculation includes elastic scattering, direct antiproton–alpha particle annihilation and rearrangement into and ground-state positronium. Elastic scattering is calculated within the Born–Oppenheimer approximation using the potential calculated by Strasburger et al (2005 J. Phys. B: At. Mol. Opt. Phys. 38 3091). Matrix elements for rearrangement are calculated using the T -matrix in the distorted wave approximation, with the initial state represented by Hylleraas-type functions. The strong force, leading to direct annihilation, was included as a short-range boundary condition in terms of the strong-force scattering length.
Matter-antimatter interactions are investigated using hydrogen-antihydrogen collisions as an example. Cross sections for elastic scattering and for the antihydrogen loss (either through the rearrangement reaction, resulting in formation of protonium and positronium according to $\mathrm{H}+\overline{\mathrm{H}}\ensuremath{\rightarrow}p\overline{p}{+e}^{+}{e}^{\ensuremath{-}}$, or via annihilation in flight) are calculated for the first time in a fully quantum mechanical approach. Implications for experiments intending to trap and cool antihydrogen are discussed.