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Physical Review A

Nuclear-excited Feshbach resonances in e+HCl scattering

J.P. Gauyacq, Arvid Herzenberg

Published Jun 1, 1982• 94 citations
Publisher

Abstract

A virtual state of an electron interacting with a molecule indicates that a new bound state with a large $s$-wave component is about to appear. In the picture of fixed nuclei, the bound state is caused to appear or disappear by a small displacement of the nuclei. When the nuclear vibrations are taken into account, it is shown, for the example of $e+\mathrm{HCl}$, that nuclear-excited Feshbach resonances occur at energies just below the vibrational thresholds.

DOI
10.1103/physreva.25.2959

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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.

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Cold and ultracold chemical reactions of F+HCl and F+DCl

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We report quantum dynamics calculations of F((2)P)+HCl(v,j)-->HF(v('),j('))+Cl((2)P) and F+DCl(v,j)-->DF(v('),j('))+Cl reactions at cold and ultracold temperatures. The effect of rotational and vibrational excitations of the HCl molecule on the reactivity is investigated. It is found that, in the ultracold regime, vibrational excitation of the HCl molecule from v=0 to v=2 enhances the reactivity by four orders of magnitude. The rotational excitation from j=0 to j=1 decreases the reactivity while the rotational excitation from j=0 to j=2 increases the reactivity. The overall effect of rotational excitation was found to be much smaller than vibrational excitation. The reactivity of the F+DCl system is significantly lower than that of the F+HCl case indicating the importance of quantum tunneling at low energies. For both reactions, Feshbach resonances corresponding to Fcdots, three dots, centered HCl or Fcdots, three dots, centeredDCl triatomic states occur at low energies. We also explored the validity of the coupled-states approximation for cold collisions taking the F+HCl(v=0,j=0) reaction as an illustrative example. It is found that the coupled-states approximation is generally valid for the background scattering even at low energies but it is inadequate to accurately describe the rich resonances in the energy dependence of the cross section resulting from the decay of van der Waals complexes. It is further shown that the coupled-states approximation cannot be used for scattering in the Wigner threshold regime when the molecule is initially in a rotationally excited level.

The Journal of Chemical Physics•Jun 10, 2008•40 citations
A virtual state of an electron interacting with a molecule indicates that a new bound state with a large sss-wave component is about to appear. In the picture of fixed nuclei, the bound state is caused to appear or disappear by a small displacement of the nuclei. When the nuclear vibrations are taken into account, it is shown, for the example of e+HCle+\mathrm{HCl}e+HCl, that nuclear-excited Feshbach resonances occur at energies just below the vibrational thresholds.
We present a two-channel model to describe the quantum state of two atoms with finite-range interaction near a Feshbach resonance. This model provides a simple picture to analytically derive the wave function and the binding energy of the molecular bound state. The results agree excellently with the measurements and multichannel calculations. For small binding energies, the system enters a threshold regime in which the Feshbach molecules are identical to long range atom pairs in single channel. According to their threshold behavior, we find Feshbach resonances can be classified into two types.
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.
A detailed study is made of the nature of rotational Feshbach resonances in the elastic scattering of extremely-low-energy electrons by diatomic polar molecules. The bound-state properties of an electron in the field of a dipole rotator, determined previously, are used in establishing the resonance behavior of elastic-scattering phase shifts in the energy region below the first rotational excitation threshold. Feshbach resonances of very narrow widths are found in partial-wave scattering channels with total angular momenta greater than zero when the strength of the interaction potentials is somewhat less than that necessary to yield bound states of the same symmetry type, and very much less than those necessary to yield a "bound state" of the projected Hamiltonian QHQ\mathrm{QHQ}QHQ. The latter implies a non-negligible value of the resonance energy shift for polar molecules.
It is well-known that cold atoms near sss-wave Feshbach resonances have universal properties that are insensitive to the short-range details of the interaction. What is less known is that atoms near higher partial-wave Feshbach resonances also have remarkable universal properties. We illustrate this with a single atom interacting resonantly with two fixed static centers. At a Feshbach resonance point with orbital angular momentum $L\ensuremath{\ge}1$, we find 2L+12L+12L+1 shallow bound states whose energies behave like 1/R2L+11/{R}^{2L+1}1/R2L+1 when the distance RRR between the two centers is large. We then compute corrections to the binding energies due to other parameters in the effective range expansions. For completeness we also compute the binding energies near sss-wave Feshbach resonances, taking into account the corrections. Afterwards we turn to the bound states at large but finite scattering volumes. For ppp-wave and higher partial-wave resonances we derive a simple formula for the energies in terms of a parameter called ``proximity parameter.'' These results are applicable to a free atom interacting resonantly with two atoms that are localized to two lattice sites of an optical lattice, and to one light atom interacting with two heavy ones in free space. Modifications of the low energy physics due to the long range van der Waals potential are also discussed.
Measurements of positron-on-molecule annihilation have established that positrons bind to a variety of molecules via vibrational Feshbach resonances. Data for deeply bound states in benzene and 1-chlorohexane and for positronically excited (i.e., second) bound states in alkanes are used to establish the dependence of annihilation rates on the binding energy and incident positron energy. With this dependence removed, annihilation rates for a broad class of molecules lie on a universal curve as a function of the number of molecular vibrational degrees of freedom. The implications of these results for theoretical models are discussed.
We report quantum dynamics calculations of F((2)P)+HCl(v,j)-->HF(v('),j('))+Cl((2)P) and F+DCl(v,j)-->DF(v('),j('))+Cl reactions at cold and ultracold temperatures. The effect of rotational and vibrational excitations of the HCl molecule on the reactivity is investigated. It is found that, in the ultracold regime, vibrational excitation of the HCl molecule from v=0 to v=2 enhances the reactivity by four orders of magnitude. The rotational excitation from j=0 to j=1 decreases the reactivity while the rotational excitation from j=0 to j=2 increases the reactivity. The overall effect of rotational excitation was found to be much smaller than vibrational excitation. The reactivity of the F+DCl system is significantly lower than that of the F+HCl case indicating the importance of quantum tunneling at low energies. For both reactions, Feshbach resonances corresponding to Fcdots, three dots, centered HCl or Fcdots, three dots, centeredDCl triatomic states occur at low energies. We also explored the validity of the coupled-states approximation for cold collisions taking the F+HCl(v=0,j=0) reaction as an illustrative example. It is found that the coupled-states approximation is generally valid for the background scattering even at low energies but it is inadequate to accurately describe the rich resonances in the energy dependence of the cross section resulting from the decay of van der Waals complexes. It is further shown that the coupled-states approximation cannot be used for scattering in the Wigner threshold regime when the molecule is initially in a rotationally excited level.