arXiv:0908.1482 · Physical Review A

Spin-orbit corrections of order mα6 to the fine structure of the (37,35) state in the H4e p¯ antiprotonic helium atom

V. I. Korobov, Zhen-Xiang Zhong

Published Oct 14, 2009• 8 citations

Abstract

Precise numerical calculation of radiofrequency intervals between hyperfine sublevels of the (37,35) state of the antiprotonic helium-4 atom is presented. Theoretical consideration includes the QED corrections of order $m{\ensuremath{\alpha}}^{6}$ to the electron spin-orbit interaction. Only the leading order contribution in the electron-to-antiproton and electron-to-$\ensuremath{\alpha}$ particle mass ratios is considered, so the $\ensuremath{\alpha}$ particle and antiproton are treated nonrelativistically. The effective Hamiltonian is derived using the formalism of the nonrelativistic quantum electrodynamics.

DOI: 10.1103/physreva.80.042506

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arXiv:0908.1482 · Physical Review A

Spin-orbit corrections of order mα6 to the fine structure of the (37,35) state in the H4e p¯ antiprotonic helium atom

V. I. Korobov, Zhen-Xiang Zhong

Published Oct 14, 2009• 8 citations

arXiv HTML PDF

Abstract

m{\ensuremath{\alpha}}^{6}

to the electron spin-orbit interaction. Only the leading order contribution in the electron-to-antiproton and electron-to-

\ensuremath{\alpha}

particle mass ratios is considered, so the

\ensuremath{\alpha}

particle and antiproton are treated nonrelativistically. The effective Hamiltonian is derived using the formalism of the nonrelativistic quantum electrodynamics.

DOI: 10.1103/physreva.80.042506

Related papers

Coupled rearrangement channel calculation of the fine and hyperfine structures of the antiprotonic helium atom

Nobuhiro Yamanaka, Yasushi Kino, Hiroshi Kudo +1 more

We precisely calculate the fine and hyperfine structure splittings of the $(v,J)=(1,35)$ state in the antiprotonic helium atom, where $v$ is the vibrational quantum number and $J$ is the total orbital angular momentum. The coupled rearrangement channel method is employed to efficiently describe correlation effects in the three-body system with atomic and molecular characters. An accuracy of better than ${\ensuremath{\alpha}}^{2}\ensuremath{\approx}50\mathrm{ppm}$ has been achieved. The result obtained is in good agreement with the previous calculation within $\ensuremath{\sim}200 \mathrm{ppm}.$ We also examine the correlation effects, in particular orbital polarization effects.We precisely calculate the fine and hyperfine structure splittings of the $(v,J)=(1,35)$ state in the antiprotonic helium atom, where $v$ is the vibrational quantum number and $J$ is the total orbital angular momentum. The coupled rearrangement channel method is employed to efficiently describe correlation effects in the three-body system with atomic and molecular characters. An accuracy of better than ${\ensuremath{\alpha}}^{2}\ensuremath{\approx}50\mathrm{ppm}$ has been achieved. The result obtained is in good agreement with the previous calculation within $\ensuremath{\sim}200 \mathrm{ppm}.$ We also examine the correlation effects, in particular orbital polarization effects.

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Jan 1, 19981 citation

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Noriaki Yamanaka, Yasushi Kino, Hiroshi Kudo +1 more

Physical Review AJan 1, 20016 citations

Hyperfine Structure of Antiprotonic Helium Revealed by a Laser-Microwave-Laser Resonance Method

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Using a newly developed laser-microwave-laser resonance method, we observed a pair of microwave transitions between hyperfine levels of the $(n,L)=(37,35)$ state of antiprotonic helium. This experiment confirms the quadruplet hyperfine structure arising from the interaction of the antiproton orbital angular momentum, the electron spin and the antiproton spin as predicted by Bakalov and Korobov. The measured frequencies of ${\ensuremath{\nu}}_{\mathrm{H}\mathrm{F}}^{+}=12.895\text{ }96\ifmmode\pm\else\textpm\fi{}0.000\text{ }34\text{ }\mathrm{G}\mathrm{H}\mathrm{z}$ and ${\ensuremath{\nu}}_{\mathrm{H}\mathrm{F}}^{\ensuremath{-}}=12.924\text{ }67\ifmmode\pm\else\textpm\fi{}0.000\text{ }29\text{ }\mathrm{G}\mathrm{H}\mathrm{z}$ agree with recent theoretical calculations on a level of $6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$.Using a newly developed laser-microwave-laser resonance method, we observed a pair of microwave transitions between hyperfine levels of the $(n,L)=(37,35)$ state of antiprotonic helium. This experiment confirms the quadruplet hyperfine structure arising from the interaction of the antiproton orbital angular momentum, the electron spin and the antiproton spin as predicted by Bakalov and Korobov. The measured frequencies of ${\ensuremath{\nu}}_{\mathrm{H}\mathrm{F}}^{+}=12.895\text{ }96\ifmmode\pm\else\textpm\fi{}0.000\text{ }34\text{ }\mathrm{G}\mathrm{H}\mathrm{z}$ and ${\ensuremath{\nu}}_{\mathrm{H}\mathrm{F}}^{\ensuremath{-}}=12.924\text{ }67\ifmmode\pm\else\textpm\fi{}0.000\text{ }29\text{ }\mathrm{G}\mathrm{H}\mathrm{z}$ agree with recent theoretical calculations on a level of $6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$.

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arXiv:1209.5142Physical Review AAug 10, 20123 citations

m{\ensuremath{\alpha}}^{6}

to the electron spin-orbit interaction. Only the leading order contribution in the electron-to-antiproton and electron-to-

\ensuremath{\alpha}

particle mass ratios is considered, so the

\ensuremath{\alpha}

particle and antiproton are treated nonrelativistically. The effective Hamiltonian is derived using the formalism of the nonrelativistic quantum electrodynamics.

We precisely calculate the fine and hyperfine structure splittings of the

(v,J)=(1,35)

state in the antiprotonic helium atom, where

v

is the vibrational quantum number and

J

is the total orbital angular momentum. The coupled rearrangement channel method is employed to efficiently describe correlation effects in the three-body system with atomic and molecular characters. An accuracy of better than ${\ensuremath{\alpha}}^{2}\ensuremath{\approx}50\mathrm{ppm}$ has been achieved. The result obtained is in good agreement with the previous calculation within $\ensuremath{\sim}200 \mathrm{ppm}.$ We also examine the correlation effects, in particular orbital polarization effects.

Using a newly developed laser-microwave-laser resonance method, we observed a pair of microwave transitions between hyperfine levels of the

(n,L)=(37,35)

state of antiprotonic helium. This experiment confirms the quadruplet hyperfine structure arising from the interaction of the antiproton orbital angular momentum, the electron spin and the antiproton spin as predicted by Bakalov and Korobov. The measured frequencies of ${\ensuremath{\nu}}_{\mathrm{H}\mathrm{F}}^{+}=12.895\text{ }96\ifmmode\pm\else\textpm\fi{}0.000\text{ }34\text{ }\mathrm{G}\mathrm{H}\mathrm{z}$ and ${\ensuremath{\nu}}_{\mathrm{H}\mathrm{F}}^{\ensuremath{-}}=12.924\text{ }67\ifmmode\pm\else\textpm\fi{}0.000\text{ }29\text{ }\mathrm{G}\mathrm{H}\mathrm{z}$ agree with recent theoretical calculations on a level of $6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$.

With the use of an accurate nonrelativistic wave function, the relativistic effects of the ${\mathrm{He}}^{\ensuremath{-}}$

1s2s2p^{4}P

state are computed. These include the mass-correction term, Darwin term, and retardation and mass-polarization effects. The calculated result enables us to significantly reduce the uncertainty of the electron affinity given by Bunge and Bunge. To assist future experimental measurement on the fine structure of this ion, the expectation values of the spin-orbit, spin-other-orbit, and spin-spin operators are also calculated.

The Land\'e

g

factor describes the response of an atomic energy level to an external perturbation by a uniform and constant magnetic field. In the case of many-electron systems, the leading term is given by the interaction ${\ensuremath{\mu}}_{B}\phantom{\rule{0.16em}{0ex}}(\stackrel{P\vec}{L}+2\phantom{\rule{0.16em}{0ex}}\stackrel{P\vec}{S})\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{B}$, where $\stackrel{P\vec}{L}$ and $\stackrel{P\vec}{S}$ are the orbital and spin angular momentum operators, respectively, summed over all electrons. For helium, a long-standing experimental-theoretical discrepancy for

P

states motivates a re-evaluation of the higher order terms which follow from relativistic quantum theory and quantum electrodynamics (QED). The tensor structure of relativistic corrections involves scalar, vector, and symmetric and antisymmetric tensor components. We perform a tensorial reduction of these operators in a Cartesian basis, using an approach which allows us to separate the internal atomic from the external degrees of freedom (magnetic field) right from the start of the calculation. The evaluation proceeds in a Cartesian basis of helium eigenstates, using a weighted sum over the magnetic projections. For the relativistic corrections, this leads to a verification of previous results obtained using the Wigner-Eckhart theorem. The same method, applied to the radiative correction (Bethe logarithm term) leads to a spin-dependent correction, which is different for singlet versus triplet

P

states. Theoretical predictions are given for singlet and triplet

2P

and triplet

3P

states and compared to experimental results where available.