Physical Review A

Variational expansion for antiprotonic helium atoms

V. I. Korobov, D. Bakalov, Hendrik J. Monkhorst

Published Feb 1, 1999• 61 citations

Abstract

A very accurate variational expansion is suggested for calculation of nonrelativistic energies of the metastable antiprotonic helium atoms ${\mathrm{He}}^{+}\overline{p}.$ 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.

DOI: 10.1103/physreva.59.r919

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Physical Review A

Variational expansion for antiprotonic helium atoms

V. I. Korobov, D. Bakalov, Hendrik J. Monkhorst

Published Feb 1, 1999• 61 citations

Publisher

Abstract

A very accurate variational expansion is suggested for calculation of nonrelativistic energies of the metastable antiprotonic helium atoms

{\mathrm{He}}^{+}\overline{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.

DOI: 10.1103/physreva.59.r919

Related papers

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Physical Review ASep 1, 199691 citations

Metastable States of Antiprotonic Helium Atoms

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, 19981 citation

Auger transition rates for metastable states of antiprotonic helium He+p¯

V. I. Korobov, Isao Shimamura

The rates of Auger transitions of antiprotonic helium ${}^{4}$ He ${}^{+}\overline{p}$ and ${}^{3}$ He ${}^{+}\overline{p}$ are calculated by using a variational scattering method with an elaborate three-body trial function. The shift of the energy levels of these systems due to the coupling with the Auger-decay channels is also obtained from this calculation. Thus an improvement is made over the previously calculated nonrelativistic energies. Together with the relativistic corrections calculated elsewhere, the theoretical transition wavelength between the states $(n,l)=(37,34)$ and $(38,33)$ of ${}^{4}$ He ${}^{+}\overline{p}$ , for example, is significantly improved to be 713.593 nm, which compares well with the experimental value 713.578$\ifmmode\pm\else\textpm\fi{} $0.006 nm. The calculated Auger lifetime of the level (38,33) is 3.2 ps, as compared with the value 4.1$ \ifmmode\pm\else\textpm\fi{}$0.2 ps deduced from the measured line broadening. Auger rates are calculated also using Fermi's golden-rule formula with a regular Coulomb function with no phase shift for the Auger electron. The results agree fairly well with the accurate variational results.The rates of Auger transitions of antiprotonic helium ${}^{4}$He${}^{+}\overline{p}$ and ${}^{3}$He${}^{+}\overline{p}$ are calculated by using a variational scattering method with an elaborate three-body trial function. The shift of the energy levels of these systems due to the coupling with the Auger-decay channels is also obtained from this calculation. Thus an improvement is made over the previously calculated nonrelativistic energies. Together with the relativistic corrections calculated elsewhere, the theoretical transition wavelength between the states $(n,l)=(37,34)$ and $(38,33)$ of ${}^{4}$He${}^{+}\overline{p}$, for example, is significantly improved to be 713.593 nm, which compares well with the experimental value 713.578$\ifmmode\pm\else\textpm\fi{}$0.006 nm. The calculated Auger lifetime of the level (38,33) is 3.2 ps, as compared with the value 4.1$\ifmmode\pm\else\textpm\fi{}$0.2 ps deduced from the measured line broadening. Auger rates are calculated also using Fermi's golden-rule formula with a regular Coulomb function with no phase shift for the Auger electron. The results agree fairly well with the accurate variational results.

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The long-range interaction between an antiprotonic helium atom $\overline{p}{\mathrm{He}}^{+}$ and helium atom in its ground state is studied. We calculate the dispersion coefficients ${C}_{6}$ 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 ${C}_{6}$ may be used to estimate the collisional shift and broadening of transition lines in a low-density precision spectroscopy of the antiprotonic helium.The long-range interaction between an antiprotonic helium atom $\overline{p}{\mathrm{He}}^{+}$ and helium atom in its ground state is studied. We calculate the dispersion coefficients ${C}_{6}$ 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 ${C}_{6}$ may be used to estimate the collisional shift and broadening of transition lines in a low-density precision spectroscopy of the antiprotonic helium.

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Formation of antiprotonic helium p¯He+ and ionization in low-energy collisions of p¯ with He in the ground 1S1 and metastable 2S3 and 2S1 states

Kazuhiro Sakimoto

Calculations of low-energy collisions of antiprotons $\overline{p}$ with helium atoms in the ground $(1\phantom{\rule{4pt}{0ex}}^{1}\mathit{S})$ and metastable $(2\phantom{\rule{4pt}{0ex}}^{3}\mathit{S}$ and $2\phantom{\rule{4pt}{0ex}}^{1}\mathit{S})$ states are carried out by using a semiclassical method and by further assuming that one of the two electrons is passive. Antiproton capture to form antiprotonic helium $(\ensuremath{\rightarrow}\overline{p}{\mathrm{He}}^{+}+e)$ and breakup ionization $(\ensuremath{\rightarrow}\overline{p}+{\mathrm{He}}^{+}+e)$ are investigated. The energy dependence of the capture and ionization cross sections and the distributions of the product states are presented and compared with previous theoretical calculations. For the metastable helium targets, most of the $\overline{p}{\mathrm{He}}^{+}$ atoms produced in the capture have long lifetimes against subsequent Auger decays and can be observed experimentally. For the ground-state targets, however, the formation of such long-lived states is rather a rare event at low collision energies.Calculations of low-energy collisions of antiprotons $\overline{p}$ with helium atoms in the ground $(1\phantom{\rule{4pt}{0ex}}^{1}\mathit{S})$ and metastable $(2\phantom{\rule{4pt}{0ex}}^{3}\mathit{S}$ and $2\phantom{\rule{4pt}{0ex}}^{1}\mathit{S})$ states are carried out by using a semiclassical method and by further assuming that one of the two electrons is passive. Antiproton capture to form antiprotonic helium $(\ensuremath{\rightarrow}\overline{p}{\mathrm{He}}^{+}+e)$ and breakup ionization $(\ensuremath{\rightarrow}\overline{p}+{\mathrm{He}}^{+}+e)$ are investigated. The energy dependence of the capture and ionization cross sections and the distributions of the product states are presented and compared with previous theoretical calculations. For the metastable helium targets, most of the $\overline{p}{\mathrm{He}}^{+}$ atoms produced in the capture have long lifetimes against subsequent Auger decays and can be observed experimentally. For the ground-state targets, however, the formation of such long-lived states is rather a rare event at low collision energies.

Physical Review AApr 9, 20159 citations

A very accurate variational expansion is suggested for calculation of nonrelativistic energies of the metastable antiprotonic helium atoms

{\mathrm{He}}^{+}\overline{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.

A variational calculation is presented of energy levels for the nonrelativistic Hamiltonian of the metastable antiprotonic helium atom

{\overline{p}}^{3, 4}{\mathrm{He}}^{+}

. This calculation is based on a molecular expansion of the wave function. We estimate the precision of the results to be about ${10}^{\ensuremath{-}7}$ a.u.

The rates of Auger transitions of antiprotonic helium

{}^{4}

{}^{+}\overline{p}

and

{}^{3}

{}^{+}\overline{p}

are calculated by using a variational scattering method with an elaborate three-body trial function. The shift of the energy levels of these systems due to the coupling with the Auger-decay channels is also obtained from this calculation. Thus an improvement is made over the previously calculated nonrelativistic energies. Together with the relativistic corrections calculated elsewhere, the theoretical transition wavelength between the states

(n,l)=(37,34)

and

(38,33)

{}^{4}

{}^{+}\overline{p}

, for example, is significantly improved to be 713.593 nm, which compares well with the experimental value 713.578$\ifmmode\pm\else\textpm\fi{}

0.006 nm. The calculated Auger lifetime of the level (38,33) is 3.2 ps, as compared with the value 4.1

\ifmmode\pm\else\textpm\fi{}$0.2 ps deduced from the measured line broadening. Auger rates are calculated also using Fermi's golden-rule formula with a regular Coulomb function with no phase shift for the Auger electron. The results agree fairly well with the accurate variational results.

Metastable states of antiprotonic helium

\overline{p}{\mathrm{He}}^{+}

whose energies lie in the range of the

2S

and

2P

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

\overline{p}{\mathrm{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 long-range interaction between an antiprotonic helium atom

\overline{p}{\mathrm{He}}^{+}

and helium atom in its ground state is studied. We calculate the dispersion coefficients

{C}_{6}

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

{C}_{6}

may be used to estimate the collisional shift and broadening of transition lines in a low-density precision spectroscopy of the antiprotonic helium.

Calculations of low-energy collisions of antiprotons

\overline{p}

with helium atoms in the ground

(1\phantom{\rule{4pt}{0ex}}^{1}\mathit{S})

and metastable

(2\phantom{\rule{4pt}{0ex}}^{3}\mathit{S}

and

2\phantom{\rule{4pt}{0ex}}^{1}\mathit{S})

states are carried out by using a semiclassical method and by further assuming that one of the two electrons is passive. Antiproton capture to form antiprotonic helium $(\ensuremath{\rightarrow}\overline{p}{\mathrm{He}}^{+}+e)$ and breakup ionization $(\ensuremath{\rightarrow}\overline{p}+{\mathrm{He}}^{+}+e)$ are investigated. The energy dependence of the capture and ionization cross sections and the distributions of the product states are presented and compared with previous theoretical calculations. For the metastable helium targets, most of the

\overline{p}{\mathrm{He}}^{+}

atoms produced in the capture have long lifetimes against subsequent Auger decays and can be observed experimentally. For the ground-state targets, however, the formation of such long-lived states is rather a rare event at low collision energies.