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Physical Review A

Adiabatic hyperspherical study of the helium trimer

Brett D. Esry, C. D. Lin, Chris H. Greene

Published Jul 1, 1996• 214 citations
Publisher

Abstract

The adiabatic hyperspherical method is used to investigate the spectrum of the helium trimer and to explore effects on the spectrum of varying the helium masses as well as the interaction strength. When a realistic helium dimer potential is adopted, only two isotopic combinations are observed to have three-body bound states: $^{4}\mathrm{He}_{3}$ and $^{4}\mathrm{He}_{2}^{3}$He. By artificially adjusting the dimer potential, we also determine the range of interaction strengths for which halo states can result. The existence of Efimov states for both physical and unphysical systems is also examined. \textcopyright{} 1996 American Institute of Physics.

DOI
10.1103/physreva.54.394

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The Journal of Physical Chemistry A•Dec 15, 2016•7 citations
The adiabatic hyperspherical method is used to investigate the spectrum of the helium trimer and to explore effects on the spectrum of varying the helium masses as well as the interaction strength. When a realistic helium dimer potential is adopted, only two isotopic combinations are observed to have three-body bound states: 4He3^{4}\mathrm{He}_{3}4He3​ and 4He23^{4}\mathrm{He}_{2}^{3}4He23​He. By artificially adjusting the dimer potential, we also determine the range of interaction strengths for which halo states can result. The existence of Efimov states for both physical and unphysical systems is also examined. \textcopyright{} 1996 American Institute of Physics.
The 4He3^{4}\mathrm{He}_{3}4He3​ 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 4He^{4}\mathrm{He}4He system. The bound state energies of the 4He^{4}\mathrm{He}4He trimer are computed as well as the 4He+4He2^{4}\mathrm{He}+^{4}\mathrm{He}_{2}4He+4He2​ 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 4He^{4}\mathrm{He}4He and one 3He^{3}\mathrm{He}3He atoms, and compute the spectrum of the isotopic trimer 4He23He^{4}\mathrm{He}_{2}^{3}\mathrm{He}4He23​He, the 3He+4He2^{3}\mathrm{He}+^{4}\mathrm{He}_{2}3He+4He2​ 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.
Exact solutions for the low-temperature helium dimer and trimer, 4He2^{4}\mathrm{He}_{2}4He2​ and 4He3^{4}\mathrm{He}_{3}4He3​, are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the SSS-wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and r0{r}_{0}r0​ are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.
When studying the structure formation and fragmentation of complex atomic and nuclear systems it is preferable to start with simple systems where all details can be explored. Some of the knowledge gained from studies of atomic dimers can be generalised to more complex systems. Adding a third atom to an atomic dimer gives a first chance to study how the binding between two atoms is affected by a third. Few-body physics is an intermediate area which helps us to understand some but not all phenomena in many-body physics. Very weakly bound, spatially very extended quantum systems with a wave function reaching far beyond the classical forbidden region and with low angular momentum are characterized as halo systems. These unusual quantum systems, first discovered in nuclear physics may also exist in systems of neutral atoms. Since the first clear theoretical prediction in 1977, of a halo system possessing an Efimov state, manifested in the excited state of the bosonic van der Waals helium trimer 42He3, small helium and different spin-polarised halo hydrogen clusters and their corresponding isotopologues have been intensively studied the last three decades. In the work presented here, the existence of the spin-polarized tritium trimer ground state, 31H3, is demonstrated, verifying earlier predictions, and the system's properties elucidated. Detailed analysis has found no found evidence for other bound states and shape resonances in this system. The properties of the halo helium trimers, 42He3 and 42He2-32He have been investigated. Earlier predictions concerning the ground state energies and structural properties of these systems are validated using our three-dimensional finite element method. In the last part of this work we present results on the bound states and structural properties of the van der Waals bosonic atomic trimers Ne3 and Ar3. We believe to be the first to find evidence of a possible shape resonance just above the three-body dissociation limit of the neon trimer.
We have searched for bound states of helium trimers with nonzero angular momenta. Including only pair interactions between helium atoms, we solved the Schrödinger equation in hyperspherical coordinates in the adiabatic approximation. From the resulting potential curves - which are mostly repulsive for nonzero angular momenta - we conclude that there are no bound rotational excited states in any of the isotopes of the helium trimer. The symmetry properties of the trimer wavefunctions in the body-fixed frame are also analysed.
We search for the existence of the weakly bound He-He-K molecules where the binding energies are expected to be of the order of degrees Kelvin. By employing the best empirical interaction between each pair of particles, the Schr\"odinger equation for the triatomic systems is solved using hyperspherical coordinates in the adiabatic approximation. A bound state is found for each of the $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}\text{\ensuremath{-}}^{39}\mathrm{K}$, $^{4}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}^{39}\mathrm{K}$, and $^{3}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}^{39}\mathrm{K}$ trimers, respectively. The bound state for the $^{3}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}^{39}\mathrm{K}$ molecule is the halo state found in realistic molecular systems for there are no bound states in both the $^{3}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}$ and $^{3}\mathrm{He}\text{\ensuremath{-}}^{39}\mathrm{K}$ dimers.
The universality of quantum halo states enables a comparison of systems from different fields of physics, as demonstrated in two- and three-body clusters. In the present work, we studied weakly bound helium tetramers in order to test whether some of these four-body realistic systems qualify as halos. Their ground-state binding energies and structural properties were thoroughly estimated using the diffusion Monte Carlo method with pure estimators. Helium tetramer properties proved to be less sensitive on the potential model than previously evaluated trimer properties. We predict the existence of realistic four-body halo 4 He 2 3 He 2, whereas 4 He 4 and 4 He 3 3 He are close to the border and thus can be used as prototypes of quasi-halo systems. Our results could be tested by the experimental determination of the tetramers’ structural properties using a setup similar to the one developed for the study of helium trimers.