arXiv:1211.1802 · Physical Review A

Integral relations and the adiabatic expansion method for 1+2 reactions above the breakup threshold: Helium trimers with soft-core potentials

Eric Garrido, Carolina Romero-Redondo, Alejandro Kievsky, M. Viviani

Published Nov 26, 2012• 15 citations

Abstract

The integral relations formalism introduced in [P. Barletta et al., Phys. Rev. Lett. 103, 090402 (2009); C. Romero-Redondo et al., Phys. Rev. A 83, 022705 (2011)] and designed to describe $1+N$ reactions is extended here to collision energies above the threshold for the target breakup. These two relations are completely general, and in this work they are used together with the adiabatic expansion method for the description of $1+2$ reactions. The neutron-deuteron breakup, for which benchmark calculations are available, is taken as a test of the method. The $s$-wave collision between the ${}^{4}$He atom and ${}^{4}$He${}_{2}$ dimer above the breakup threshold and the possibility of using soft-core two-body potentials plus a short-range three-body force will be investigated. Comparisons to previous calculations for the three-body recombination and collision dissociation rates will be shown.

DOI: 10.1103/physreva.86.052709

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arXiv:1211.1802 · Physical Review A

Integral relations and the adiabatic expansion method for 1+2 reactions above the breakup threshold: Helium trimers with soft-core potentials

Eric Garrido, Carolina Romero-Redondo, Alejandro Kievsky, M. Viviani

Published Nov 26, 2012• 15 citations

arXiv HTML PDF

Abstract

The integral relations formalism introduced in [P. Barletta et al., Phys. Rev. Lett. 103, 090402 (2009); C. Romero-Redondo et al., Phys. Rev. A 83, 022705 (2011)] and designed to describe

1+N

reactions is extended here to collision energies above the threshold for the target breakup. These two relations are completely general, and in this work they are used together with the adiabatic expansion method for the description of

1+2

reactions. The neutron-deuteron breakup, for which benchmark calculations are available, is taken as a test of the method. The

s

-wave collision between the

{}^{4}

He atom and

{}^{4}

{}_{2}

dimer above the breakup threshold and the possibility of using soft-core two-body potentials plus a short-range three-body force will be investigated. Comparisons to previous calculations for the three-body recombination and collision dissociation rates will be shown.

DOI: 10.1103/physreva.86.052709

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Elena A. Kolganova, Alexander Konstantinovich Motovilov, S. A. Sofianos

We present a mathematically rigorous method suitable for bound-state and scattering process calculations for various three-atomic or molecular systems where the underlying forces are of a hard-core nature. We employed this method to calculate the binding energies and the ultra-low-energy scattering phase shifts below as well as above the breakup threshold for the three-He-atom system. The method has proved to be highly successful and suitable for solving the three-body bound-state and scattering problem in configuration space and thus it paves the way to study various three-atomic systems, and to calculate important quantities such as the cross sections, recombination rates, etc.We present a mathematically rigorous method suitable for bound-state and scattering process calculations for various three-atomic or molecular systems where the underlying forces are of a hard-core nature. We employed this method to calculate the binding energies and the ultra-low-energy scattering phase shifts below as well as above the breakup threshold for the three-He-atom system. The method has proved to be highly successful and suitable for solving the three-body bound-state and scattering problem in configuration space and thus it paves the way to study various three-atomic systems, and to calculate important quantities such as the cross sections, recombination rates, etc.

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Alejandro Kievsky, M. Gattobigio

We investigate universal behavior in elastic atom-dimer scattering below the dimer breakup threshold calculating the atom-dimer effective-range function $ak\mathrm{cot}\ensuremath{\delta}$. Using the He-He system as a reference, we solve the Schr\"odinger equation for a family of potentials having different values of the two-body scattering length $a$ and we compare our results to the universal zero-range form deduced by Efimov, $ak\mathrm{cot}\ensuremath{\delta}={c}_{1}(ka)+{c}_{2}(ka)\mathrm{cot}[{s}_{0}\mathrm{ln}({\ensuremath{\kappa}}_{*}a)+\ensuremath{\phi}(ka)]$, for selected values of the three-body parameter ${\ensuremath{\kappa}}_{*}$. Using the parametrization of the universal functions ${c}_{1},{c}_{2},\ensuremath{\phi}$ given in the literature, a good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections. Furthermore, we show that the same parametrization describes a very different system: nucleon-deuteron scattering below the deuteron breakup threshold. Our analysis confirms the universal character of the process, and relates the pole energy in the effective-range function of nucleon-deuteron scattering to the three-body parameter ${\ensuremath{\kappa}}_{*}$.We investigate universal behavior in elastic atom-dimer scattering below the dimer breakup threshold calculating the atom-dimer effective-range function $ak\mathrm{cot}\ensuremath{\delta}$. Using the He-He system as a reference, we solve the Schr\"odinger equation for a family of potentials having different values of the two-body scattering length $a$ and we compare our results to the universal zero-range form deduced by Efimov, $ak\mathrm{cot}\ensuremath{\delta}={c}_{1}(ka)+{c}_{2}(ka)\mathrm{cot}[{s}_{0}\mathrm{ln}({\ensuremath{\kappa}}_{*}a)+\ensuremath{\phi}(ka)]$, for selected values of the three-body parameter ${\ensuremath{\kappa}}_{*}$. Using the parametrization of the universal functions ${c}_{1},{c}_{2},\ensuremath{\phi}$ given in the literature, a good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections. Furthermore, we show that the same parametrization describes a very different system: nucleon-deuteron scattering below the deuteron breakup threshold. Our analysis confirms the universal character of the process, and relates the pole energy in the effective-range function of nucleon-deuteron scattering to the three-body parameter ${\ensuremath{\kappa}}_{*}$.

arXiv:1212.3457Physical Review AMay 31, 201340 citations

The integral relations formalism introduced in [P. Barletta et al., Phys. Rev. Lett. 103, 090402 (2009); C. Romero-Redondo et al., Phys. Rev. A 83, 022705 (2011)] and designed to describe

1+N

1+2

reactions. The neutron-deuteron breakup, for which benchmark calculations are available, is taken as a test of the method. The

s

-wave collision between the

{}^{4}

He atom and

{}^{4}

{}_{2}

a

of the He-He interaction and a three-body parameter

{a}_{*}

as input, where

{a}_{*}

is associated with the positive scattering length at which an interference minimum in the TBR rate occurs at the threshold. This universal property has been reinvestigated by considering various post-Born-Oppenheimer (post-BO) effects on the relevant scattering process. It is found that a linear correction to

1/{a}_{*}

versus

1/a

should be introduced to give a better description of these considered TBR cases because of the finite-range effects. Particularly, such a correction may be ascribed not only to the general scattering properties associated with the Efimov channel at the short hyperradial distance, but also to the tunneling probability in the lowest incident channel. In addition, a successful fit of the modified universal function that includes correctly the finite-range effects [E. Garrido, M. Gattobigio, and A. Kievsky, Phys. Rev. A 88, 032701 (2013)] or the zero-range universal function with a linearly variational

1/{a}_{*}

versus

1/a

to numerically accurate data also indicates a clear manifestation of the van der Waals universality. Finally, the post-BO effects on the TBR process for the partial wave with the total angular momentum

J>0

are also discussed.

We have developed a method for calculating the rates for three-body recombination of cold atoms. This method allows us to treat not only zero total angular momentum,

J=0,

states but also

J>0

states, so that recombinations at nonzero collision energies can be considered. Our method is applied to ground-state helium atoms

{}^{4}\mathrm{He},

using a realistic interaction potential. In addition, we obtain the rates for collision induced dissociation.

The

^{4}\mathrm{He}_{3}

system is studied using the adiabatic hyperspherical representation. We adopt the current state-of-the-art helium interaction potential including retardation and the nonadditive three-body term to calculate all low-energy properties of the triatomic

^{4}\mathrm{He}

system. The bound state energies of the

^{4}\mathrm{He}

trimer are computed as well as the

^{4}\mathrm{He}+^{4}\mathrm{He}_{2}

elastic scattering cross sections, the three-body recombination, and collision induced dissociation rates at finite temperatures. We also treat the system that consists of two

^{4}\mathrm{He}

and one

^{3}\mathrm{He}

atoms, and compute the spectrum of the isotopic trimer

^{4}\mathrm{He}_{2}^{3}\mathrm{He}

, the

^{3}\mathrm{He}+^{4}\mathrm{He}_{2}

elastic scattering cross sections, the rates for three-body recombination, and the collision induced dissociation rate at finite temperatures. The effects of retardation and the nonadditive three-body term are investigated. Retardation is found to be significant in some cases, while the three-body term plays only a minor role for these systems.

a

and we compare our results to the universal zero-range form deduced by Efimov, $ak\mathrm{cot}\ensuremath{\delta}={c}_{1}(ka)+{c}_{2}(ka)\mathrm{cot}[{s}_{0}\mathrm{ln}({\ensuremath{\kappa}}_{*}a)+\ensuremath{\phi}(ka)]$, for selected values of the three-body parameter ${\ensuremath{\kappa}}_{*}$. Using the parametrization of the universal functions ${c}_{1},{c}_{2},\ensuremath{\phi}$ given in the literature, a good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections. Furthermore, we show that the same parametrization describes a very different system: nucleon-deuteron scattering below the deuteron breakup threshold. Our analysis confirms the universal character of the process, and relates the pole energy in the effective-range function of nucleon-deuteron scattering to the three-body parameter ${\ensuremath{\kappa}}_{*}$.