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Physical Review Letters

Ground-State Binding Energies of Three Identical Particles Which Interact via Three Mutual Morse Potentials

H. Krüger

Published Jan 10, 1972• 14 citations
Publisher

Abstract

Within the framework of Faddeev theory we calculate the binding energy of the lowest bound state of three identical particles. Excluding three-body potentials, the particles are assumed to interact mutually via a two-body Morse potential. Finally, our model calculations are applied to a system formed of three helium atoms. Using a Morse potential which is fitted to the He-He interaction potential calculated by Bertoncini and Wahl, our calculation predicts a stable triatomic helium molecule with a binding energy of 0.4\ifmmode^\circ\else\textdegree\fi{}K, even though a diatomic helium molecule does not exist for the same well depth.

DOI
10.1103/physrevlett.28.71

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Journal of Physics B Atomic Molecular and Optical Physics•Oct 7, 1999•22 citations
Within the framework of Faddeev theory we calculate the binding energy of the lowest bound state of three identical particles. Excluding three-body potentials, the particles are assumed to interact mutually via a two-body Morse potential. Finally, our model calculations are applied to a system formed of three helium atoms. Using a Morse potential which is fitted to the He-He interaction potential calculated by Bertoncini and Wahl, our calculation predicts a stable triatomic helium molecule with a binding energy of 0.4\ifmmode^\circ\else\textdegree\fi{}K, even though a diatomic helium molecule does not exist for the same well depth.
We study the existence of stable bound states of triatomic molecules containing two helium atoms and one alkali atom X (X = Li, Na) and examine the dependence of their binding energies on the mass of the constituent atoms. The three-body systems are solved within the adiabatic approximation in hyperspherical coordinates using the He - He and He - X pair van der Waals interactions that are available in the literature. The binding energies of these systems are shown to be of less than 1 K for some isotope combinations. For the lighter isotopes it is shown that no bound states exist. We found no evidence of the existence of halo states or Efimov states for these systems.
The three-body Schrdinger equation is approximately solved in the hyperspherical coordinates and the binding energies of the three-body weakly bound systems are calculated with the purpose to find if He-He-Ba trimers could exist. Using the special feature of the B-spline function like the flexible and highly localized properties, hypersphercial potentials are obtained by modifying the knots distribution of the B-spline basis of different weakly bound three-atom systems. Employing the best empirical interaction potentials between each pair of particles, we obtain that in the ground state binding energies of the weakly bound typical three-atom systems, the bindings of the molecules, 4He-4He-138Ba, 4He-3He-138Ba and 3He-3He-138Ba are possible. The binding energies of these systems are shown in the order of 1 Kelvin, each system could support only one bound state. These weakly bound molecules can exist only in a very cold environment. To get insight into the geometry of the molecules, the features of the channel functions associated with the hyperspherical potential curves of each system are investigated.
We found universal laws for the spectrum of two-dimensional three-body systems, composed by two identical particles and a distinct one (AAB). These universal laws appear when the potential range (r_0) is much smaller than the size of the system. In two dimensions this condition is formulated as (E2 is the two-body energy and ? is the reduced mass). The zero range model, which is very appropriated to establish the universal laws, is introduced through the ?-Dirac potential. We derive the corresponding two-dimensional Faddeev equations for the three-body system and solve them numerically in momentum space. Our results showed that the three-body binding energy monotonically increases with the two-body binding energy, and such dependence is more pronounced than the mass variations. We found that the three-body energy depends logarithmic on the two-body energy for large values. Furthermore, been m=mB/mA the ratio between the masses of the B and A particles, the three-body energy is mass-independent for m ? ? and increase without bounds for m ?0. The limit of two non-interacting identical particles is also studied in the AAB system. We found that the two-dimensional three-body system always support at least two bound states and more bound states appear for m<0.22. Finally, we analyze the particular limit of m ?0 using the adiabatic approximation. This approximation can be used to study the three-body system in two-dimensions with an accuracy better than 10% compared to the solutions of the Faddeev equations, for m ? 0.01.
Close-coupling calculations have been performed to investigate the formation of three-particle bound states in a model collinear system. A generalization of Levinson's theorem (1966) is employed to determine the critical potential strengths necessary for the existence of the triatomic states. The system consists of an atom interacting through a Morse potential with a diatomic molecule bound by a harmonic potential. Comparisons are made with classical phase-space and S-matrix theory and with quantum-mechanical variational and perturbation calculations. When only a few bound states are involved it is shown that the quantal approximations are more accurate than the classical results.
We searched for the existence of weakly bound He-He-H, He-He-H - and He-H-H 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, we solved the Schrödinger equation for the triatomic systems using hyperspherical coordinates in the adiabatic approximation. We found no bound states for 4 He- 4 He- 1 H, three bound states for 4 He- 4 He- 1 H - and one bound state for 4 He- 1 H- 1 H. Bound states for other isotope combinations are also examined.