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.
Low-energy antiproton capture, $\overline{p}+{\mathrm{He}}^{+}\ensuremath{\rightarrow}\overline{p}{\mathrm{He}}^{2+}+e$, is investigated using a wave-packet-propagation full-quantum-mechanical method or a quantum-classical hybrid (i.e., semiclassical) approximation. If the adiabatic picture (i.e., Born-Oppenheinmer separation) is a good approximation, the antiproton capture, which is a typical nonadiabatic process, can rarely occur. It is found that the probability of the nonadiabatic transition is small (at most $\ensuremath{\sim}0.1$). However, even in the zero energy limit $(E\ensuremath{\rightarrow}0)$, the antiproton capture always occurs for the total angular momenta $J\ensuremath{\lesssim}40$. Correspondingly, the cross section for the antiproton capture diverges as $E\ensuremath{\rightarrow}0$. This happens due to the peculiarity that the Coulomb force range is longer than the centrifugal one. The importance of a Rydberg series of resonances in the capture process is emphasized. On the low-energy dynamical feature, some interesting similarities to the
e+ion system are also found.
The interaction between atoms behaves as $\ensuremath{-}\ensuremath{\alpha}{/r}^{n}$ at large distances and, owing to the large reduced mass \ensuremath{\mu} of the collision pair, allows a semiclassical treatment within the potential well. As a result, the low-energy scattering is governed by two large parameters: the asymptotic parameter $\ensuremath{\gamma}=\sqrt{2\ensuremath{\mu}\ensuremath{\alpha}}/\ensuremath{\Elzxh}\ensuremath{\gg}{a}_{0}^{(n\ensuremath{-}2)/2}$ (
a0 is the Bohr radius) and the semiclassical zero-energy phase $\ensuremath{\Phi}\ensuremath{\gg}1.$ In our previous work [Phys. Rev. A 48, 546 (1993)] we obtained an analytical expression for the scattering length
a, which showed that it has 75% preference for positive values for
n=6, characteristic of collisions between ground-state neutral atoms. In this paper we calculate the effective range and show that it is a function of
a, ${r}_{e}{=F}_{n}\ensuremath{-}{G}_{n}{/a+H}_{n}{/a}^{2},$ where
Fn, Gn, and
Hn depend only on \ensuremath{\gamma}. Thus, we know the
s-phase shift at low momenta $k\ensuremath{\ll}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ from the expansion $k\mathrm{cot}{\ensuremath{\delta}}_{0}\ensuremath{\simeq}\ensuremath{-}1/a+\frac{1}{2}{r}_{e}{k}^{2}.$ At $k\ensuremath{\gg}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ the phase shift is obtained semiclassically as ${\ensuremath{\delta}}_{0}=\ensuremath{\Phi}+\ensuremath{\pi}/4\ensuremath{-}{I}_{n}{\ensuremath{\gamma}}^{2/n}{k}^{(n\ensuremath{-}2)/n},$ where ${I}_{n}=[n/(n\ensuremath{-}2)]\ensuremath{\Gamma}((n\ensuremath{-}1)/n)\ensuremath{\Gamma}((n+2)/2n)/\sqrt{\ensuremath{\pi}}.$ Therefore, \ensuremath{\gamma} and \ensuremath{\Phi} determine the
s-wave atomic scattering in a wide range of momenta, as well as the positions of upper bound states of the diatomic molecule.