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Journal of Physics B Atomic and Molecular Physics

2P0resonances in He-

B. R. Junker

Published Dec 28, 1982• 18 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/15/24/007

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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.
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.
In his interesting Reference Frame piece (Physics Today, August 2004, page 12), Daniel Kleppner relates Herman Feshbach’s reaction to the term “Feshbach resonance” and gives a capsule account of the topical and vital role this phenomenon plays in today’s explorations of cold collisions of fermions and bosons, and of condensates of atoms and molecules. Readers may be interested in the earlier history and why atomic physicists began to use a term that Feshbach himself dismissed as jargon.In the Los Alamos experiment of Howard Bryant’s group on photo-absorption by the negative ion of hydrogen, two prominent resonances were seen at photon energies of about 10.95 eV. This energy marks the position of the first excited state of the hydrogen atom above the single bound state of the negative ion.One narrow resonance occurs just below the threshold of that excitation, whereas the second, broad one occurs just above. Their physical mechanisms are different and, following the paradigm set by that historic experiment, it has become customary to refer to the two types of resonance, both common in atomic and molecular physics, as Feshbach and shape resonance, respectively. That is, a resonant coupling just below the relevant threshold, leading to narrow profiles, is distinguished from those lying just above the threshold to which they are most strongly coupled, and typically broad.The two types of resonances are most naturally viewed in state and coordinate space, respectively. Feshbach resonances are thought of in terms of the superposition of two quantum states, one with discrete and one with continuum character, leading to a temporary quasi-bound state that manifests itself as a resonance in that continuum channel. The independent and almost simultaneous work in nuclear physics by Feshbach using the language of projection operators and in atomic physics by Ugo Fano using the language of superposition of wave functions serves for their technical description.Shape resonances, on the other hand, may be associated with the shape of a trapping potential in real, coordinate space. Most often, the trapping potential is an angular momentum potential that combines with the internal one to create intervening barriers to low-energy particles. The resulting pictures often display “two-valley” potentials. In the case of the negative ion of hydrogen, such a barrier has been described in terms of the hyperspherical coordinates of that three-body system. Examples of such shape resonances abound in low-energy scattering of electrons from molecules.For more discussion and references, including to recent applications in mesoscopic condensed matter systems, see my pedagogical mini-review in Physica Scripta, volume 69, page C10, 2004.© 2005 American Institute of Physics.
Seiler, Oberoi, and Callaway first located resonances in the positron-hydrogen system below the n=2n=2n=2 inelastic threshold, using the algebraic close-coupling method without including the open positronium channel. Several other calculations including both open channels have failed to find any resonances. In an attempt to clarify the situation, a calculation has been performed which is an extension of the work of Seiler, Oberoi, and Callaway. A wave function including the 1s1s1s, 2s2s2s, and 2p2p2p eigenstates of hydrogen and the 1s−\stackrel{-}{1s}1s− ground state of positronium was used in a calculation that is a simplification of the usual close-coupling technique and that resembles the optical-potential method. In this approximation it is found that the lowest resonance of Seiler, Oberoi, and Callaway is destroyed by opening the 1s−\stackrel{-}{1s}1s− channel. It is concluded that any resonances which may occur in this system must lie closer to the n=2n=2n=2 threshold than do the corresponding electron-hydrogen resonances.
By using a saddle-point technique, the 1s2s2p2P1s2s2p^{2}P1s2s2p2P resonance of ${\mathrm{He}}^{\ensuremath{-}}$ 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.