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Physical Review A

Zero-energy scattering and bound states in the He4 trimer and tetramer

S. Nakaichi-Maeda, T. K. Lim

Published Aug 1, 1983• 46 citations
Publisher

Abstract

Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the $^{4}\mathrm{He}$ atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the $s$-wave two-body $t$ matrix to a separable form through the unitary-pole approximation and did the same for the $s$-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.

DOI
10.1103/physreva.28.692

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The Journal of Chemical Physics•Oct 1, 1986•100 citations
Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the 4He^{4}\mathrm{He}4He atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the sss-wave two-body ttt matrix to a separable form through the unitary-pole approximation and did the same for the sss-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.
Faddeev-Yakubovski equations are solved numerically for 4He^{4}\mathrm{He}4He tetramer and trimer states using realistic helium-helium interaction models. We describe the properties of ground and excited states, and we discuss with a special emphasis the $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}_{3}$ low energy scattering.
We calculated, using seven realistic $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials in the literature, the Efimov spectra of the 4He^{4}\mathrm{He}4He trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schr\"odinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the 4He^{4}\mathrm{He}4He trimer and tetramer [Phys. Rev. A 85, 022502 (2012); Phys. Rev. A 85, 062505 (2012)]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length rvdW{r}_{\mathrm{vdW}}rvdW​, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius ${R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius ${R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$ makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the 4He^{4}\mathrm{He}4He trimer with the realistic potentials.
The momentum-space structure of the Faddeev-Yakubovsky (FY) components of weakly bound tetramers is investigated at the unitary limit using a renormalized zero-range two-body interaction. The results, obtained by considering a given trimer level with binding energy B3{B}_{3}B3​, provide further support to a universal scaling function relating the binding energies of two successive tetramer states. The correlated scaling between the tetramer energies comes from the sensitivity of the four-boson system to a short-range four-body scale. Each excited NNNth tetramer energy B4(N){B}_{4}^{(N)}B4(N)​ moves as the short-range four-body scale changes, while the trimer properties are kept fixed, with the next excited tetramer B4(N+1){B}_{4}^{(N+1)}B4(N+1)​ emerging from the atom-trimer threshold for a universal ratio ${B}_{4}^{(N)}/{B}_{3}={B}_{4}^{(N)}/{B}_{4}^{(N+1)}\ensuremath{\simeq}4.6$, which does not depend on NNN. We show that both channels of the FY decomposition [atom-trimer (KKK type) and dimer-dimer (HHH type)] present high-momentum tails that reflect the short-range four-body scale. We also found that the HHH channel is favored over the KKK channel at low momentum, when the four-body momentum scale largely overcomes the three-body scale.
Four-body systems are studied using an effective-field theory with two- and three-body contact interactions. A method to systematically address deep trimers (three-body bound states that are more tightly bound than four-body bound or resonant states) in four-body calculations is developed using a diagrammatic approach. Previous calculations were limited by the existence of deep trimers, which this work overcomes. For cold 4He^{4}\mathrm{He}4He atoms, binding energies of 526.1(5) and 128.517(1) mK are obtained at leading order for the tetramer ground and excited states, respectively, where errors come from the truncation of three-body partial waves. Tetramer binding energies and decay widths are also computed approaching the unitary limit. In the unitary limit, there are two tetramers associated with each trimer of binding energy B3(0){B}_{3}^{(0)}B3(0)​. The binding energy and decay width for the associated tetramer ground state are E4(0)=4.60(1)B3(0){E}_{4}^{(0)}=4.60(1){B}_{3}^{(0)}E4(0)​=4.60(1)B3(0)​ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(0)}/2=0.0160(1){B}_{3}^{(0)}$, respectively, and for the associated tetramer excited state, E4(1)=1.0022(3)B3(0){E}_{4}^{(1)}=1.0022(3){B}_{3}^{(0)}E4(1)​=1.0022(3)B3(0)​ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(1)}/2=2.57(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{B}_{3}^{(0)}$, respectively. This calculation is a gateway to higher-order and/or more-body calculations in nuclear and atomic systems.
The system of three 4He atoms interacting through the intermolecular potential HFDHE2 is solved exactly within the Faddeev scheme. We find that the trimer has two bound states at E=−0.11 K and E=−0.16(−2) K, where the excited state shows the property of an Efimov state. It is overrun by the two-body fragmentation threshold for an artificially enhanced overall interaction strength. Uncertainties in the potential parameters, possibly combined with solvent effects, may allow a slightly weaker force, which supports a second Efimov state. In any case dynamical effects of the Efimov states will be the same whether it lies close to E=0 on the physical or unphysical sheet.