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Physical Review

Three-Body Bound State in He4

John M. Blatt, James N. Lyness, Sigurd Yves Larsen

Published Sep 1, 1963• 7 citations
Publisher

Abstract

The binding energy of three ${\mathrm{He}}^{4}$ atoms, subject to pair forces, is investigated numerically for a number of proposed intermolecular potentials and for a series of square wells. For some potentials a 3-body bound state is found, for others not. However, a direct correlation is found between 2-body and 3-body binding. The need for more accurate experimental data is discussed.

DOI
10.1103/physrev.131.2131

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The binding energy of three He4{\mathrm{He}}^{4}He4 atoms, subject to pair forces, is investigated numerically for a number of proposed intermolecular potentials and for a series of square wells. For some potentials a 3-body bound state is found, for others not. However, a direct correlation is found between 2-body and 3-body binding. The need for more accurate experimental data is discussed.
Using the method of binary collisions, the third virial coefficient of He4{\mathrm{He}}^{4}He4 at low temperatures is calculated for a potential consisting of an attractive square well and a repulsive core. The shape of this potential is chosen so as to optimize the fit of the second virial coefficient with experimental data. From a comparison of the results with the third virial data the existence of a three-body bound state is inferred. A binding energy of 0.26\ifmmode^\circ\else\textdegree\fi{}K, calculated from the potential, brings the adjusted results and the experimental data into agreement.
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.
The quantum theory of the third virial coefficient CCC is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of CBE{C}_{\mathrm{BE}}CBE​ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for He4{\mathrm{He}}^{4}He4 and He3{\mathrm{He}}^{3}He3, by means of a comparison of the corresponding expansion for the second virial coefficient BBB with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to BBB. Then the leading term of CCC is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for CCC is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
The bound state of a He3{\mathrm{He}}^{3}He3 atom trapped in the free surface of He4{\mathrm{He}}^{4}He4 is studied, using a trial function suggested by Lekner. The effective potential which determines the bound-state wave function is calculated from first principles, and results in a binding energy 20% larger than the observed value.
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.
The possible existence of bound states in small helium clusters containing atomic and molecular hydrogen has been investigated by means of three completely independent sets of calculations. The first set employs the Jacobi–Davidson filtering procedure recently implemented in the Distributed Gaussian Functions (DGF) approach, the second one is based on the solution of the Faddeev equation by means of the hyperspherical adiabatic expansion method and the third follows the Quantum Monte Carlo approach. The partners within each complex are taken to interact via two-body potentials chosen among the most accurate existing in the literature. All the present quantum treatments show that no bound states are found for H(4He)2 and for D(4He)2 while only one very diffuse bound state is present in T(4He)2, where H, D and T are the three isotopes of hydrogen. The substitution of the atomic dopant with its molecular counterpart, H2, also gives rise to only one bound state whose spatial attributes are reported and analysed, underlining again the ultraweak nature of the interaction forces and the need to implement highly specific tools for the investigation of the ensuing bound states.