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Physical Review A

Muon transfer and elastic scattering int+dμ collisions at low energies

Akinori Igarashi, Nobuyuki Toshima, Toshizo Shirai

Published Dec 1, 1994• 30 citations
Publisher

Abstract

Muon transfer and elastic scattering in low-energy collisions of t+d\ensuremath{\mu}(1s) are studied by the hyperspherical coupled-channel method in which many closed channels are explicitly coupled to obtain converged cross sections. A hybrid procedure of the diabatic-by-sector and the traditional adiabatic-basis-expansion methods is developed for solving a larget set of coupled equations influenced by the nearly singular nature of couplings induced by numerous avoided crossings. It is shown that the adiabatic states belonging to the n=2 excited states in the separated-atom limit contribute significantly to the muon transfer. The discrepancy between the previous hyperspherical calculation of Fukuda et al. [Phys. Rev. A 41, 145 (1990)] and other existing theoretical calculations has been resolved by taking into account the couplings with these excited states.

DOI
10.1103/physreva.50.4951

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Muon transfer and elastic scattering in low-energy collisions of t+d\ensuremath{\mu}(1s) are studied by the hyperspherical coupled-channel method in which many closed channels are explicitly coupled to obtain converged cross sections. A hybrid procedure of the diabatic-by-sector and the traditional adiabatic-basis-expansion methods is developed for solving a larget set of coupled equations influenced by the nearly singular nature of couplings induced by numerous avoided crossings. It is shown that the adiabatic states belonging to the n=2 excited states in the separated-atom limit contribute significantly to the muon transfer. The discrepancy between the previous hyperspherical calculation of Fukuda et al. [Phys. Rev. A 41, 145 (1990)] and other existing theoretical calculations has been resolved by taking into account the couplings with these excited states.
We apply the hyperspherical close-coupling method to the elastic and spin-flip processes in $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}+\mathrm{Ps}$ (or equivalently, e++Ps){e}^{+}+\mathrm{Ps})e++Ps) scattering at low collision energies. The spin-spin interactions VS{V}_{S}VS​ are included explicitly in the Hamiltonian, producing the correct hyperfine-splitting energy $\ensuremath{\Delta}\ensuremath{\epsilon}$ in the separated-atom limit. A comparison is made with a simpler method in which the effect of VS{V}_{S}VS​ is represented only by the symmetry of the wave functions with respect to the interchange of the two identical particles. Explicit inclusion of VS{V}_{S}VS​ is found to be of vital importance for the calculation of the spin-flip cross sections for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$ for both $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}{(e}^{+})+\mathrm{Ps}$ collisions. The elastic cross sections are less sensitive to VS,{V}_{S},VS​, but those for $p+p\ensuremath{\mu}$ are still affected by VS{V}_{S}VS​ for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$.
Antihydrogen formation cross sections are calculated for low-energy collisions of antiproton with positronium by the hyperspherical coupled-channel method. The scattering wavefunctions are expanded in terms of a large number of sector-adiabatic states associated with the n=1-4 states of antihydrogen and n=1-3 states of positronium in the asymptotic separated-atom region. The present formation cross sections differ considerably from those of the unitarized Born approximation reported by Mitroy and Stelbovics (1994). The disagreement becomes larger with decreasing collision energy owing to the different energy dependence of the cross sections. The authors have confirmed that the formation cross sections from the 2s and 2p states are larger than that from the ground state by one order of magnitude.
Low-energy elastic and inelastic scattering in the Ps(1s1s1s)-Ps(2s2s2s) channel is treated in a four-body hyperspherical coordinate calculation. Adiabatic potentials are calculated for triplet-triplet, singlet-singlet, and singlet-triplet spin symmetries in the spin representation of coupled electrons and coupled positrons, with total angular momentum L=0L=0L=0 and parity equal to +1+1+1. The sss-wave scattering lengths for the asymptotic Ps(1s1s1s)-Ps(2s2s2s) channel are calculated for each spin configuration. Results obtained for the sss-wave scattering lengths are ${a}_{\mathrm{TT}}=7.3(2){a}_{0}\ensuremath{-}i0.02(1){a}_{0}, {a}_{\mathrm{SS}}=13.2(2){a}_{0}\ensuremath{-}i0.9(2){a}_{0}$, and aST=9.7(2)a0{a}_{\mathrm{ST}}=9.7(2){a}_{0}aST​=9.7(2)a0​ for each spin configuration. Spin recoupling is implemented to extract the scattering lengths for collisions of Ps in different spin configurations through properly symmetrized unitary transformations. Calculations of experimentally relevant scattering lengths and cross sections are carried out for Ps atoms initially prepared in different uncoupled spin states.
Collisions between atoms and molecules govern how energy is exchanged and redistributed in the gas phase, underpinning phenomena from spectral line broadening and interstellar molecular excitation to the cooling and trapping of molecules. A quantitative account of these collisions requires connecting the theoretical description of the interaction between the collision partners to measurable quantities such as cross sections. However, scattering theory as presented in quantum mechanics textbooks does not readily translate into numerical methods. Here, we formulate the scattering problem in Dirac notation, as customary in present-day quantum mechanics courses, and show how to build a practical numerical framework from it. We derive the "coupled channel" representation of the scattering problem with correct asymptotic behaviour and introduce Numerov's method to obtain time-independent wavefunctions, from which we calculate cross sections, the observables measured in scattering experiments. Starting from single channel scattering, we build up to the realistic collision of a ground-state atom with a diatomic molecule.
A close coupling quantum-mechanical calculation is performed for rotational energy transfer in a HD+HD collision at very low energy, down to the ultracold temperatures: $T\ensuremath{\sim}{10}^{\ensuremath{-}8}$ K. A global six-dimensional H2{}_{2}2​-H2{}_{2}2​ potential-energy surface is adopted from a previous work [Boothroyd et al., J. Chem. Phys. 116, 666 (2002)]. State-resolved integral cross sections ${\ensuremath{\sigma}}_{ij\ensuremath{\rightarrow}{i}^{\ensuremath{'}}{j}^{\ensuremath{'}}}({\ensuremath{\varepsilon}}_{\text{kin}})$ of different quantum-mechanical rotational transitions $ij\ensuremath{\rightarrow}{i}^{\ensuremath{'}}{j}^{\ensuremath{'}}$ in the HD molecules and corresponding state-resolved thermal rate coefficients ${k}_{ij\ensuremath{\rightarrow}{i}^{\ensuremath{'}}{j}^{\ensuremath{'}}}(T)$ have been computed. Additionally, for comparison, H2{}_{2}2​+H2{}_{2}2​ calculations for a few selected rotational transitions have also been performed. The hydrogen and deuterated hydrogen molecules are treated as rigid rotors in this work. A pronounced isotope effect is identified in the cross sections of these collisions at low and ultracold temperatures.
The binary collision dynamics of 2Dg,3/2{}^{2}{D}_{g,3/2}2Dg,3/2​ ground state scandium atoms is studied from first principles. We employ 30 coupled diabatic ab initio potentials in a coupled-channels study of the scattering dynamics of cold and ultracold scandium atoms in external magnetic fields. Due to the long-ranged magnetic dipolar interaction, the field dependence of the cross section does not follow the threshold laws derived by Volpi and Bohn [Phys. Rev. A 65, 052712 (2002)]. In the field-free case, the near-threshold cross section is independent of the collision energy, and hence the cross section does not follow the well-established Wigner threshold laws. The observed threshold behavior is explained in the Born approximation. For energies above $1\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{K}$, inelastic collisions are driven by the anisotropic nonrelativistic electronic interaction. For energies below $100\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{K}$, the ratio of elastic-to-inelastic collisions is likely to be favorable for evaporative cooling. Both anisotropy in the long-range interaction and in the short-range potential contribute to large cross sections for inelastic collisions at higher energies and lead to a small ratio of elastic-to-inelastic collisions. This is in agreement with the large rates for Zeeman relaxation of submerged-shell atoms observed experimentally. The effect of the uncertainty in the ab initio potential is sampled by scaling the reduced mass and is found to have little influence on the conclusions drawn from this work.