Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

AIP conference proceedings

Hyperspherical Theory of Three-Body Recombination in Cold Collisions

James Sternberg

Published Jan 1, 2003• 1 citation
Publisher

Abstract

Three‐body recombination in cold atomic collisions has been successfully modeled using the hyperspherical close‐coupling theory. This representation maps many‐body fragmentation channels onto simple radial equations similar to those for two‐body processes. Computation of hyperspherical adiabatic energy eigenvalues is a major task using this theory. We have developed asymptotic representations of the adiabatic eigenvalues based upon energy dependent zero‐range potentials. Our asymptotic expressions are compared with ab initio calculations as well as hyperspherical adiabatic calculations using zero range potentials with constant M that have appeared in the literature.

DOI
10.1063/1.1619682

Related papers

Hyperspherical adiabatic eigenvalues for zero-range potentials

G. Gasaneo, J. H. Macek

The scattering length a associated with two-body interactions is the relevant parameter for near threshold processes in cold atom-atom collisions. For this reason zero-range potentials are traditionally used to model collective behaviour of dilute collections of bosons. The model is also used to compute three-body recombination rates, where it gives an a 4 law. In this paper we examine the applicability of the zero-range model to real physical systems. Hyperspherical adiabatic potentials obtained from the zero-range model are compared with published potentials based on realistic two-body interactions. From these comparisons it is possible to determine the regions where the model applies.

Journal of Physics B Atomic Molecular and Optical Physics•May 9, 2002•25 citations

Adiabatic hyperspherical representation for the three-body problem in two dimensions

José Paulo D'Incao, Brett D. Esry

We explore the three-body problem in two dimensions using the adiabatic hyperspherical representation. We develop the main equations in terms of democratic hyperangular coordinates and determine several symmetry properties and boundary conditions for both interacting and noninteracting solutions. From the analysis of the three-body effective potentials, we determine the threshold laws for low-energy three-body recombination, collision-induced dissociation, as well as inelastic atom-diatom collisions in two dimensions. Our results show that the hyperspherical representation can offer a simple and conceptually clear physical picture for three-body process in two dimensions which is also suitable for calculations using finite-range two-body interactions supporting a number of bound states.

arXiv:1404.0324•Physical Review A•Oct 16, 2014•17 citations

Three-body recombination of helium atoms at cold collision energies

Hiroya Suno

Abstract Three-body recombination (TBR) of helium atoms at cold collision energies is studied using the hyperspherical adiabatic method with different realistic interaction potentials. By considering zero angular momentum J = 0 states as well as J > 0 states, we calculate the 4He TBR rates up to about 0.1 Kelvin, and analyze the influence of modern potentials on the recombination rates. We also calculate and analyze the collision induced dissociation rates.

Journal of Physics B Atomic Molecular and Optical Physics•Mar 19, 2025•1 citation

Three-body recombination in cold helium–helium–alkali-metal-atom collisions

Hiroya Suno, Brett D. Esry

Three-body recombination in helium-helium--alkali-metal collisions at cold temperatures is studied using the adiabatic hyperspherical representation. The rates for the three-body recombination processes $^{4}\text{H}\text{e}+^{4}\text{H}\text{e}+X\ensuremath{\rightarrow}^{4}\text{H}\text{e}+^{4}\text{H}\text{e}X$ and $^{4}\text{H}\text{e}+^{4}\text{H}\text{e}+X\ensuremath{\rightarrow}{^{4}\text{H}\text{e}}_{2}+X$, with $X=^{7}\text{L}\text{i}$, $^{23}\text{N}\text{a}$, $^{39}\text{K}$, $^{85}\text{R}\text{b}$, and $^{133}\text{C}\text{s}$, are calculated at nonzero collision energies by including not only zero total angular momentum, $J=0$, states but also $J>0$ states. The three-body recombination rates show a relatively weak dependence on the alkali-metal species, differing from each other only by about one order of magnitude, except for the $^{4}\text{H}\text{e-}^{4}\text{H}\text{e-}^{23}\text{N}\text{a}$ system.

Physical Review A•Dec 2, 2009•51 citations

Cold three-body recombination in helium-helium-silver-atom collisions using the hybrid slow-variable-discretization–adiabatic hyperspherical R -matrix approach

Hiroya Suno

We study three-body recombination in the helium-helium-silver system at cold collision energies. The three-body Schr\"odinger equation is solved combining the slow variable discretization and adiabatic approaches with the $R$-matrix propagation method, in order to calculate the product-state-selective total and partial recombination rates in the $^{4}\mathrm{He}+^{4}\mathrm{He} +$ Ag system. Recombination at collision energies less than 10 mK is found to take place dominantly to the product molecule $^{4}\mathrm{He}\mathrm{Ag}$ in its least bound state, while the three atoms recombine mainly to the $^{4}\mathrm{He}_{2}$ molecule in its ground state at higher collision energies.

Physical Review A•Apr 11, 2024•3 citations

Asymptotic Adiabatic Hyperspherical Potential Curves for Three Body Systems.

Zheng Zhen

The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.

•Jan 1, 1988•0 citations
Three‐body recombination in cold atomic collisions has been successfully modeled using the hyperspherical close‐coupling theory. This representation maps many‐body fragmentation channels onto simple radial equations similar to those for two‐body processes. Computation of hyperspherical adiabatic energy eigenvalues is a major task using this theory. We have developed asymptotic representations of the adiabatic eigenvalues based upon energy dependent zero‐range potentials. Our asymptotic expressions are compared with ab initio calculations as well as hyperspherical adiabatic calculations using zero range potentials with constant M that have appeared in the literature.
The scattering length a associated with two-body interactions is the relevant parameter for near threshold processes in cold atom-atom collisions. For this reason zero-range potentials are traditionally used to model collective behaviour of dilute collections of bosons. The model is also used to compute three-body recombination rates, where it gives an a 4 law. In this paper we examine the applicability of the zero-range model to real physical systems. Hyperspherical adiabatic potentials obtained from the zero-range model are compared with published potentials based on realistic two-body interactions. From these comparisons it is possible to determine the regions where the model applies.
We explore the three-body problem in two dimensions using the adiabatic hyperspherical representation. We develop the main equations in terms of democratic hyperangular coordinates and determine several symmetry properties and boundary conditions for both interacting and noninteracting solutions. From the analysis of the three-body effective potentials, we determine the threshold laws for low-energy three-body recombination, collision-induced dissociation, as well as inelastic atom-diatom collisions in two dimensions. Our results show that the hyperspherical representation can offer a simple and conceptually clear physical picture for three-body process in two dimensions which is also suitable for calculations using finite-range two-body interactions supporting a number of bound states.
Three-body recombination in helium-helium--alkali-metal collisions at cold temperatures is studied using the adiabatic hyperspherical representation. The rates for the three-body recombination processes $^{4}\text{H}\text{e}+^{4}\text{H}\text{e}+X\ensuremath{\rightarrow}^{4}\text{H}\text{e}+^{4}\text{H}\text{e}X$ and $^{4}\text{H}\text{e}+^{4}\text{H}\text{e}+X\ensuremath{\rightarrow}{^{4}\text{H}\text{e}}_{2}+X$, with X=7LiX=^{7}\text{L}\text{i}X=7Li, 23Na^{23}\text{N}\text{a}23Na, 39K^{39}\text{K}39K, 85Rb^{85}\text{R}\text{b}85Rb, and 133Cs^{133}\text{C}\text{s}133Cs, are calculated at nonzero collision energies by including not only zero total angular momentum, J=0J=0J=0, states but also J>0J>0J>0 states. The three-body recombination rates show a relatively weak dependence on the alkali-metal species, differing from each other only by about one order of magnitude, except for the 4He-4He-23Na^{4}\text{H}\text{e-}^{4}\text{H}\text{e-}^{23}\text{N}\text{a}4He-4He-23Na system.
We study three-body recombination in the helium-helium-silver system at cold collision energies. The three-body Schr\"odinger equation is solved combining the slow variable discretization and adiabatic approaches with the RRR-matrix propagation method, in order to calculate the product-state-selective total and partial recombination rates in the 4He+4He+^{4}\mathrm{He}+^{4}\mathrm{He} +4He+4He+ Ag system. Recombination at collision energies less than 10 mK is found to take place dominantly to the product molecule 4HeAg^{4}\mathrm{He}\mathrm{Ag}4HeAg in its least bound state, while the three atoms recombine mainly to the 4He2^{4}\mathrm{He}_{2}4He2​ molecule in its ground state at higher collision energies.
The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.