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Journal of Physics B Atomic and Molecular Physics

An investigation of the critical potential strengths for binding in a model collinear triatomic system

A P Clark, A S Dickinson

Published Nov 11, 1976• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0022-3700/9/16/025

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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.
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 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.
The behaviour of two bodies that are just bound or nearly bound is discussed. A class of potentials is given for which Schrödinger's equation has exact solutions at critical binding (zero binding energy). This class includes the known solution for the 6–10 potential. For a general potential characterized by a coupling parameter α, it is shown that the bound state energy tends to zero as -(α - α0)2, where α0 is the critical value of the coupling parameter. Small energy scattering of atoms which are near critical binding (e.g. helium atoms) is examined. It is shown that determination of the total cross-section up to terms of order k 2 is in principle sufficient to distinguish between bound and virtual states of the diatomic molecule.
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 one-dimensional form of the correspondence principle for strongly coupled states of Percival and Richards (1970) is modified to include, approximately, the perturbation of the bound particle orbit. The modification will only be significant when the expectation value of the interaction potential depends on the state of the bound system. The excitation of a linear harmonic oscillator by a time dependent potential q2F(t) and the collinear collision of an atom and an anharmonic diatomic molecule are considered as examples. The results obtained for both these systems are in good agreement with the exact quantum mechanical transition probabilities even for low order transitions.