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Physical Review A

Structure changes along the lowest rotational band of the antiprotonic helium atom

D. Baye, Jérémy Dohet-Eraly, P. Schoofs

Published Feb 21, 2019• 16 citations
Publisher

Abstract

Along its lowest rotational band, the antiprotonic helium atom undergoes several changes of structure. It evolves from an hydrogenlike atom with broader and broader resonant states, to a quasistable molecularlike structure with narrow resonances, and then to a quasistable Rydberg pseudoatom. The antiprotonic helium atom is studied with high accuracy as a nonrelativistic three-body Coulomb system in the framework of the Lagrange-mesh method. Its metastable states are first determined from $L=0$ to $L=80$ by searching conditions of calculation for which energies are stationary. Mean values of distances between the particles are then easily deduced. They show the evolution of the structure of the system. Broad resonances are also analyzed with the complex scaling method. Physical interpretations are provided.

DOI
10.1103/physreva.99.022508

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V. I. Korobov

Abstract New variational calculations of the nonrelativistic energies with precision of ∼ 10 −10 a.u. and relativistic and QED corrections to the energy levels of order mα 5 are presented.

•Jan 1, 1998•1 citation
Along its lowest rotational band, the antiprotonic helium atom undergoes several changes of structure. It evolves from an hydrogenlike atom with broader and broader resonant states, to a quasistable molecularlike structure with narrow resonances, and then to a quasistable Rydberg pseudoatom. The antiprotonic helium atom is studied with high accuracy as a nonrelativistic three-body Coulomb system in the framework of the Lagrange-mesh method. Its metastable states are first determined from L=0L=0L=0 to L=80L=80L=80 by searching conditions of calculation for which energies are stationary. Mean values of distances between the particles are then easily deduced. They show the evolution of the structure of the system. Broad resonances are also analyzed with the complex scaling method. Physical interpretations are provided.
A very accurate variational expansion is suggested for calculation of nonrelativistic energies of the metastable antiprotonic helium atoms He+p‾.{\mathrm{He}}^{+}\overline{p}.He+p​. This expansion reflects the dual atomic-molecular nature of the system. Convergence of the results as a function of the increasing sets of basis functions shows an accuracy better than ${10}^{\ensuremath{-}10}\mathrm{a}.\mathrm{u}.,$ two orders of magnitude better than in our previous calculations.
The long-range interaction between an antiprotonic helium atom p‾He+\overline{p}{\mathrm{He}}^{+}p​He+ and helium atom in its ground state is studied. We calculate the dispersion coefficients C6{C}_{6}C6​ using the complex coordinate rotation (CCR) formalism in order to comply with the resonant nature of metastable states of the antiprotonic helium. We present as well numerical data on static dipole polarizabilities of antiprotonic helium states. The obtained coefficients C6{C}_{6}C6​ may be used to estimate the collisional shift and broadening of transition lines in a low-density precision spectroscopy of the antiprotonic helium.
Accurate three-body quantal calculations of the system composed of a proton, an antiproton, and an electron are performed in perimetric coordinates with the Lagrange-mesh method, an approximately variational calculation with the simplicity of a calculation on a grid. Quasibound states with respect to the p‾+H(n=2\overline{p} + \mathrm{H}(n=2p​+H(n=2) threshold are obtained for $L=60\ensuremath{-}73$ for various vibrational excitations. Their energies have accuracies up to about ${10}^{\ensuremath{-}14}$ atomic units, while less precise energies are determined for L=56−−59L=56--59L=56−−59 broader resonances. Their structure is analyzed with the help of mean distances between the particles. These mean distances indicate that the proton-electron subsystem is in excited states, mostly n=2n=2n=2, as predicted by Sakimoto [Phys. Rev. A 98, 042503 (2018)] with the Born-Oppenheimer approximation. A comparison performed with this approximation provides the accuracies of its energies and of its proton-antiproton mean distances.
Metastable states of antiprotonic helium p‾He+\overline{p}{\mathrm{He}}^{+}p​He+ whose energies lie in the range of the 2S2S2S and 2P2P2P excited states of He are discussed and their energies are calculated using the hyperspherical method within the "symmetric top" approximation. These states can be populated via collisions of antiprotons with excited He atoms. They represent another group of the potentially observable metastable states of p‾He+\overline{p}{\mathrm{He}}^{+}p​He+, which are characterized by the total angular momentum $L\ensuremath{\sim}70$, in contrast to the recently discovered metastable states having $L\ensuremath{\sim}35$.
The exotic helium atom, consisting of a helium nucleus He2+{\mathrm{He}}^{2+}He2+, an electron ${\mathit{e}}^{\mathrm{\ensuremath{-}}}$, and a massive, negatively charged particle ${\mathit{X}}^{\mathrm{\ensuremath{-}}}$, such as an antiproton (p\ifmmode\bar\else\textasciimacron\fi{}) or a kaon (${\mathit{K}}^{\mathrm{\ensuremath{-}}}$), is studied theoretically. This exotic atom is more like a polar molecule with two nuclei He2+{\mathrm{He}}^{2+}He2+ and ${\mathit{X}}^{\mathrm{\ensuremath{-}}}$, if ${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ is in a low-lying orbital and ${\mathit{X}}^{\mathrm{\ensuremath{-}}}$ is in a highly excited orbital. Such states are generally formed in the capture of ${\mathit{X}}^{\mathrm{\ensuremath{-}}}$ by helium. The Born-Oppenheimer (BO) separation of the ${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ motion and the He2+{\mathrm{He}}^{2+}He2+-${\mathit{X}}^{\mathrm{\ensuremath{-}}}$ motion is a good approximation for these states, and affords a transparent, unified perspective of exotic helium with different ${\mathit{X}}^{\mathrm{\ensuremath{-}}}$ and different isotopes of He2+{\mathrm{He}}^{2+}He2+. A propensity rule that favors small transition energies (except for the smallest one) is found for radiative emission by low vibrational and high rotational levels. This rule resembles the selection rule for infrared emission by the usual diatomic polar molecules, and contradicts the conventional expectation that radiative transitions in exotic atoms favor larger transition energies. The radiative lifetimes \ensuremath{\tau} of highly excited states formed immediately after the capture of ${\mathit{X}}^{\mathrm{\ensuremath{-}}}$ are of the order of \ensuremath{\mu}sec for 4He2+^{4}\mathrm{He}^{2+}4He2+-p\ifmmode\bar\else\textasciimacron\fi{}-${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ and 3He2+^{3}\mathrm{He}^{2+}3He2+-p\ifmmode\bar\else\textasciimacron\fi{}-${\mathit{e}}^{\mathrm{\ensuremath{-}}}$, and those for 4He2+^{4}\mathrm{He}^{2+}4He2+-${\mathit{K}}^{\mathrm{\ensuremath{-}}}$-${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ are about half the values for the antiprotonic helium.The effect of the electronic charge polarization by ${\mathit{X}}^{\mathrm{\ensuremath{-}}}$ on \ensuremath{\tau} is explained quantitatively in a simple picture in terms of the molecular dipole moment. Low vibrational, high rotational states are described as circular or near-circular states in the atomic picture. They are stable against Auger decay, which the nonadiabatic perturbation on the BO states induces. An infinite series of bound \ensuremath{\Sigma} electronic states of exotic helium negative ions (He2+{\mathrm{He}}^{2+}He2+-${\mathit{X}}^{\mathrm{\ensuremath{-}}}$-${\mathit{e}}^{\mathrm{\ensuremath{-}}}$)-${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ exists for any He2+{\mathrm{He}}^{2+}He2+-${\mathit{X}}^{\mathrm{\ensuremath{-}}}$ distances R beyond 0.819 a.u. in the BO approximation. An infinite \ensuremath{\Pi} series exists for R>3.811 a.u., and an infinite \ensuremath{\Delta} series for R>9.532 a.u.