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Physical Review A

1s2s2pP2 resonance of He−

Kwong T. Chung

Published Mar 1, 1981• 34 citations
Publisher

Abstract

By using a saddle-point technique, the $1s2s2p^{2}P$ resonance of ${\mathrm{He}}^{\ensuremath{-}}$ is found to be a Feshbach resonance. It is formed by the strong coupling of [$(1s2s)^{1}S$,$2p$] and [$(1s2p)^{1}P$,$3d$] configurations. By using a seven-partial-wave 54-term wave function and neglecting the open channels, the energy is -2.149 045 a.u. This is higher than the -2.175 23 a.u. $(1s2s)^{3}S$ threshold, but significantly below the -2.145 97 a.u. $(1s2s)^{1}S$ threshold. A quasi-projection-operator method is also carried out. Using the same wave function, the energy is -2.150 54 a.u. These results generally agree with that of experiment.

DOI
10.1103/physreva.23.1079

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Our unexpounded writing in the first paragraph of page L24 of the letter may give a false impression to some about what was really meant. We would like, therefore, to clarify it. We simply intended to stress that the deficiency of the polarizability of H(1s) and of Ps(1s) can be seen more easily at low energies in higher-partial-wave phase shifts of e + -H scattering and in higher-partial-wave elastic cross sections of Ps(1s)-p scattering, and that for the purpose of determining the sequences of S, P, D Feshbach resonances (within the assumption of a degenerate threshold limit at -0.25 Ryd, i.e. when the threshold splitting due to fine-structure and QED corrections is ignored), the use of the E6S coupling scheme is appropriate since this coupling scheme contains the 2s and 2p states of H essential for the degenerate dipole interaction. It is therefore quite obvious that for the calculations in a very narrow range of energy of less than 1 × 10 -2 Ryd beneath the degenerate threshold (resonance region), the use of the E6PS coupling scheme may not be able to yield reliable results for S, P, D scattering, since the scheme cannot handle the resonances properly, due to the absence of the 2s and 2p H states essential for the degenerate dipole interaction. This is, however, of little concern, since the range of energies that we have considered for our various accurate calculations (such as E6PS, E8PS, E6S, ME6S, E8S) is further below the degenerate threshold. Rigorously speaking, even if the 2s and 2p H states are included in the coupling scheme, care concerning the nondegeneracy of threshold might still have to be taken for calculations in the closest vicinity of the threshold.

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Photodetachment study of the 1s3s4s4S resonance in He −

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A Feshbach resonance associated with the ${1s3s4s}^{4}S$ state of He${}^{\ensuremath{-}}$ has been observed in the He${(1s2s}^{3}S)$ + ${e}^{\ensuremath{-}}(\ensuremath{\epsilon}s)$ partial photodetachment cross section. The residual He${(1s2s}^{3}S)$ atoms were resonantly ionized and the resulting He${}^{+}$ ions were detected in the presence of a small background. A collinear laser-ion beam apparatus was used to attain both high resolution and sensitivity. We measured a resonance energy ${E}_{\mathrm{r}}=2.959255(7)$ eV and a width $\ensuremath{\Gamma}=0.19(3)$ meV, in agreement with a recent calculation.

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By using a saddle-point technique, the 1s2s2p2P1s2s2p^{2}P1s2s2p2P resonance of He\ensuremath−{\mathrm{He}}^{\ensuremath{-}}He\ensuremath− is found to be a Feshbach resonance. It is formed by the strong coupling of [(1s2s)1S(1s2s)^{1}S(1s2s)1S,2p2p2p] and [(1s2p)1P(1s2p)^{1}P(1s2p)1P,3d3d3d] configurations. By using a seven-partial-wave 54-term wave function and neglecting the open channels, the energy is -2.149 045 a.u. This is higher than the -2.175 23 a.u. (1s2s)3S(1s2s)^{3}S(1s2s)3S threshold, but significantly below the -2.145 97 a.u. (1s2s)1S(1s2s)^{1}S(1s2s)1S threshold. A quasi-projection-operator method is also carried out. Using the same wave function, the energy is -2.150 54 a.u. These results generally agree with that of experiment.
The 2P^{2}P2P (2s2s2p2s2s2p2s2s2p) and 2D^{2}D2D (2s2p2p2s2p2p2s2p2p) resonance structure observed in the electron-helium scattering experiments has been investigated using a saddle-point technique. For 2P^{2}P2P we obtain for the energy --- 0.796 93 a.u. with an 82-term wave function. This agrees well with the experimental result. In the past, the 2D^{2}D2D structure was thought to be a Feshbach resonance. In this calculation, we show that the 2s2p2s2p2s2p 3P^{3}P3P target is repulsive to the third ppp electron in the sense that the variational energy of the ${\mathrm{He}}^{\ensuremath{-}}$ system is monotonically increased as the ppp electron approaches the target. Hence, the Feshbach resonance cannot be formed for this state. Some comments will be made concerning earlier theoretical calculations in the literature.
Low-energy positron scattering from singlet and triplet metastable He(1s2s1s2s1s2s) is investigated using the RRR-matrix propagation method in hyperspherical coordinates. Elastic and positronium-formation cross sections are reported, and near-threshold resonance structures are analyzed in terms of eigenphase sums and the time-delay matrix. For the triplet target, the calculated cross sections are in agreement with available convergent-close-coupling results and display the expected threshold behavior. Beyond the known SSS-wave features, Feshbach resonance series in higher partial waves extending up to highly excited atomic thresholds are systematically identified. The time-delay matrix further uncovers hidden resonances that produce obvious structures in the positronium-formation cross sections.
Our unexpounded writing in the first paragraph of page L24 of the letter may give a false impression to some about what was really meant. We would like, therefore, to clarify it. We simply intended to stress that the deficiency of the polarizability of H(1s) and of Ps(1s) can be seen more easily at low energies in higher-partial-wave phase shifts of e + -H scattering and in higher-partial-wave elastic cross sections of Ps(1s)-p scattering, and that for the purpose of determining the sequences of S, P, D Feshbach resonances (within the assumption of a degenerate threshold limit at -0.25 Ryd, i.e. when the threshold splitting due to fine-structure and QED corrections is ignored), the use of the E6S coupling scheme is appropriate since this coupling scheme contains the 2s and 2p states of H essential for the degenerate dipole interaction. It is therefore quite obvious that for the calculations in a very narrow range of energy of less than 1 × 10 -2 Ryd beneath the degenerate threshold (resonance region), the use of the E6PS coupling scheme may not be able to yield reliable results for S, P, D scattering, since the scheme cannot handle the resonances properly, due to the absence of the 2s and 2p H states essential for the degenerate dipole interaction. This is, however, of little concern, since the range of energies that we have considered for our various accurate calculations (such as E6PS, E8PS, E6S, ME6S, E8S) is further below the degenerate threshold. Rigorously speaking, even if the 2s and 2p H states are included in the coupling scheme, care concerning the nondegeneracy of threshold might still have to be taken for calculations in the closest vicinity of the threshold.
We analyze, within the generalized multichannel quantum defect theory framework, atomic binary collision cross sections over the $1\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{K}$ to 10mK10\phantom{\rule{0.3em}{0ex}}\mathrm{mK}10mK energy range just above the entrance channel threshold, a domain where the Wigner threshold law should apply. By adjusting the strength of a constant external magnetic field a Feshbach resonance is tuned at will over this energy range. In the threshold regime, the quasibound state interacts with an opening continuum whose wave function presents a strong energy dependence which reflects the breakdown of the WKB approximation. The effective discrete-continuum interactions become very sensitive to the energy. The consequences of these variations have been investigated by including threshold effects in the analysis of the Fano configuration-interaction theory [Phys. Rev. 124, 1866 (1961)] in terms of quantum defect theory quantities proposed by [Lecomte J. Phys. B 20, 3645 (1987)]. This analysis shows that the energy variations are in general so important that it becomes meaningless to associate a width with a Feshbach resonance. However, it is still possible to define the resonance energy, as long as the energy variation of the shift of the resonance, induced by the effective discrete-continuum interaction, remains linear over an energy range corresponding to the magnitude of the shift.
Reports a number of calculations for the 2 P 0 resonances in He - below the 2 3 P 0 state of He. These calculations indicate that there is no 2 P 0 shape resonance associated with the 2 3 S state of helium that the Feshbach resonance associated with the 2 1 S state of He (as calculated by Chung (1981), and Bhatia and Temkin (1981)) is shifted down to the vicinity of the resonance computed by Burke et al. (1969) and Oberoi and Nesbet (1973), and that there appears to be no second 2 P 0 resonance between the 2 1 S and 2 3 P 0 states of He.
A Feshbach resonance associated with the 1s3s4s4S{1s3s4s}^{4}S1s3s4s4S state of He${}^{\ensuremath{-}}$ has been observed in the He(1s2s3S){(1s2s}^{3}S)(1s2s3S) + ${e}^{\ensuremath{-}}(\ensuremath{\epsilon}s)$ partial photodetachment cross section. The residual He(1s2s3S){(1s2s}^{3}S)(1s2s3S) atoms were resonantly ionized and the resulting He+{}^{+}+ ions were detected in the presence of a small background. A collinear laser-ion beam apparatus was used to attain both high resolution and sensitivity. We measured a resonance energy Er=2.959255(7){E}_{\mathrm{r}}=2.959255(7)Er​=2.959255(7) eV and a width $\ensuremath{\Gamma}=0.19(3)$ meV, in agreement with a recent calculation.