Binding Energies and Scattering Observables in the 4He3 and 3He4He2 Three-Atomic Systems

The asymmetric ³He⁴He₂ three-atomic system is studied on the basis of a hard-core version of the Faddeev differential equations. Binding energy and scattering observables for this system are compared to previous results for the symmetric ⁴He₃ problem.     

Binding energies and scattering observables in the 4He3 atomic system

The ⁴He₃ system is investigated using a hard-core version of the Faddeev differential equations. Realistic ⁴He-⁴He interactions are employed, among them the LM2M2 potential by Aziz and Slaman and the recent TTY potential by Tang, Toennies and Yiu. We calculate the binding energies of the ⁴He trimer, but concentrate in particular on scattering observables. The scattering lengths and the atom-diatom phase shifts are calculated for center of mass energies up to 2.45 mK. It is found that the LM2M2 and TTY potentials, although of quite different structure, give practically the same bound-state and scattering results. Received 19 June 2000     

Efimov states in asymmetric three-body atomic clusters

The work is devoted to the investigation of the weakly bound three-body atomic clusters. The calculations on the van der Waals trimer ⁷Li⁴He₂ are carried out using the differential Faddeev equations, which allows us to give accurate binding energies for both the ground and the exited state of the system. The results obtained indicate the Efimov nature of the excited state in this system.     

Bound-State Calculations for Three Atoms Without Explicit Partial Wave Decomposition

A method to calculate the bound states of three atoms without resorting to an explicit partial wave decomposition is presented. The differential form of the Faddeev equations in the total angular momentum representation is used for this purpose. The method utilizes Cartesian coordinates combined with the tensor trick preconditioning for large linear systems and Arnoldi’s algorithm for eigenanalysis. As an example, we consider the He₃ system in which the interatomic force has a very strong repulsive core requiring the inclusion of a large number of partial waves to achieve convergence. To improve convergence and stability in the Arnoldi’s algorithm, we modify the system of equations by introducing a background potential that removes the large number of unphysical eigenvalues stemming from the hard core. The introduction of such a modifying potential does not affect the physical spectrum of the system. The results obtained by solving the modified, three-dimensional, Faddeev equations compare favorably with other results in the field.     

Comparison of Brueckner theory with “exact” results for H3 and He4 nuclei

The consistency of the second-order Brueckner-Baranger approximation to the ground-state energyE of H³ and He⁴ nuclei is confirmed by the agreement ofE with the recent Faddeev and Faddeev-Yakubovsky equation results for Reid local potential, by the dependence ofE on different phase-equivalent transforms (= off-shell changes) of the Reid potential which agrees with Faddeev equation and nuclear matter calculations, and by the Coester line well reproduced for a set of realistic potentials in He⁴. The influence of approximations in solving Bethe-Goldstone equation is also discussed.     

Zum Drei- und Vierkörperproblem der Kernphysik

The independent pair model has been applied to the nuclear three- and four-body problem. The equations of the model have been solved by a perturbation approach in the case of spin- and charge-independent central two-body forces with hard-core. In lowest order of perturbation theory the nuclear average potential has been approximated by an oscillator potential including an effective mass, the parameters of which were so chosen as to give as nearly a self-consistent potential as possible. For a sqare well two-body interaction the lowest order equations have been solved exactly, the corrections are shown to be small. Some tests for the internal consistency of the approach have been developped and are shown to be well fulfilled by our solutions. According to a proposal byLipkin we succeeded in separating off the energy of the center of mass motion. The results for the binding energy of He⁴(27,9 MeV), the binding energy of H³ (7,1 MeV), the energy difference between H³ and He³ (0,735 MeV) and the RMS-radius of He⁴ (1,75f) are in rather good agreement with experimental results.     

Three-body scattering in configuration space

The asymptotic forms of the wavefunction and Faddeev components in configuration space are shown to determine uniquely the solutions of the Schrödinger or Faddeev differential equations for 2 → (2, 3) and 3 → (2, 3) processes. An antisymmetrized form of the Faddeev differential equation for three equivalent fermions is given and its angular analysis is performed in the general case of local potentials with tensor interaction for neutron-deuteron scattering. We describe a numerical method for solving the corresponding boundary value problem and apply it to scattering and break-up at E_n^1ab = 14.4 MeV in the doublet S state for the four local potentials of Malfliet and Tjon, Reid, de Tourreil and Sprung, and de Tourreil, Rouben and Sprung. For the three realistic potentials, elastic scattering amplitudes differ by 5%, and amplitudes for break-up in the two-neutron state ¹S₀ differ by less than 4%.     

Helium trimer calculations with a public quantum three-body code

We present an illustration of using a quantumthree-body code being prepared for public release. The code is based on iterative solving of the three-dimensional Faddeev equations. The code is easy to use and allows users to perform highly-accurate calculations of quantum three-body systems. The previously known results for He₃ ground state are well reproduced by the code.     

Ultracold helium trimers

We summarize the results on binding energies and scattering observables for ⁴He-⁴He₂ and ³He-⁴He₂ collisions obtained with the Tang-Toennies-Yiu potential.     

α8Be cluster model for the 0 2 + resonance in the 12C nucleus

The 0 ₂ ⁺ resonance in the ¹²C nucleus is treated on the basis of the α+⁸Be two-cluster model. An equation for the function describing the relative motion of the clusters is derived by using the s-wave differential Faddeev equations for the 3α system and by relying on the simplest version of the resonating-group method. A phenomenological potential is taken to simulate the pair αα interaction. A three-body potential binding three alpha particles together gives rise to a resonance in two-cluster α+⁸Be scattering. The calculated resonance features and the calculated parameters of the wave function of the system are compared with the results obtained by other authors.     

Exchange and nuclear symmetry interference in He+-He scattering

A method is described to determine the potential difference, V _g-V _u, that will accurately reproduce the exchange and nuclear symmetry oscillations observed by Aberth et al. in their experimental work on He⁺-He differential scattering. It is demonstrated that at higher energies and angles the experimental results for ⁴He⁺-⁴He, ⁴He⁺-³He and ³He⁺-³He appear incompatible with one another when investigated in terms of a two-state scattering theory. This is an indication that at these energies and angles the theory may require modification to include the effects of coupling to higher states.     

New approach for numerical solution of configuration-space Faddeev equations

A new computational scheme for solving the bound state configuration-space Faddeev equations is applied. The scheme is based on the spline-approximation and the adiabatic limit of Faddeev equations. An ordering of variables being in agreement with the limit was chosen. As a result the matrix of the eigenvalue problem has a sparse block structure. Calculations of the bound states of µpp, µdd, µtt mesic molecules and ¯pdd, ¯ptt antiprotonic ones, were performed. To check the method, calculations of the binding energies for such systems as the positronium ion Ps_–,³H and³He were carried out. The results are compared with the best results of other authors.     

On the π3He3He coupling

A critical discussion of different methods of estimation of the coupling constant f_π³He³He² is given. It is claimed that at present the best way of obtaining information on this coupling constant is to extrapolate certain differential cross sections, in particular, of the elastic ³He³H scattering, to the pion pole. The resulting value f_π³He³He² = 0.055 ± 0.020 is smaller than the corresponding pion-nucleon value. It supports the expected shadowing effects inside the nucleus but disagrees with other more model-dependent estimations.     

Neutron–Triton Elastic Scattering

The Kohn variational principle and the hyperspherical harmonics technique are applied to study the n ? ³H elastic scattering at low energies. In this contribution the first results obtained using a non-local realistic interaction derived from the chiral perturbation theory are reported. They are found to be in good agreement with those obtained solving the Faddeev–Yakubovsky equations. The calculated total and differential cross sections are compared with the available experimental data. The effect of including a three-nucleon interaction is also discussed.     

Stability of 3He2 4He M and 3He3 4He M L = 0,1 Clusters

We have studied the stability of mixed ³He/⁴He clusters in L = 0 and L = 1 states by the diffusion Monte Carlo method, employing the Tang-Toennies-Yiu (TTY) He-He potential. The clusters ³He⁴He_M ( ) and ³He₂⁴He_M (L = 0, S = 0) are stable for M > 1, while to bind two ³He in a triplet state the minimum number of ⁴He is four. Considering clusters with three ³He, ³He₃⁴He₄ is the smallest stable system in the L = 1 state, while ³He₃⁴He₈ is the smallest stable system in the L = 0 state.     

Theoretical study of charge-exchange and excitation processes in collisions of He<Superscript>+</Superscript> ions with C<Superscript>5+</Superscript>, N<Superscript>6+</Superscript>, and O<Superscript>7+</Superscript> hydrogen-like ions

Data on the cross sections for single-electron charge exchange and excitation in collisions of He⁺ ions with C⁵⁺, N⁶⁺, and O⁷⁺ ions in the He⁺ ion energy range of 0.2–3.0 MeV are obtained for the first time. The cross sections for the single-electron charge transfer into the singlet and triplet 1snl states of C⁴⁺, N⁵⁺, and O⁶⁺ (2≤n≤5) ions and for the 1s → 2p _{0, ±1} electronic excitation of He⁺(1s) ions are calculated. The calculations were performed by solving close-coupling equations on the basis of ten two-electron quasi-molecular states.     

He5 scattering states in the refined cluster function representation

As a first application of the refined cluster model reaction theory we calculated the scattering states of He⁵ using a specific ansatz for the nucleon-nucleon forces. He⁴-n and H³-d channels are considered only. Our results agree qualitatively with the experimental values if available; in particular we were able to explain the narrow resonance in theD _3/2 He⁴-n channel as a consequence of the strong coupling to theS _3/2 T-d channel.     

ASYMMETRY PARAMETERS AND DIFFERENTIAL CROSS SECTIONS FOR PHOTOIONIZATION-EXCITATION OF HELIUM

Asymmetry parameter β₂ for photoionization of helium leaving the He⁺ ion in the n=2 level and differential cross sections(DCSs) in the n=2,3,4 levels with photon emission angle θ=90⁰ are provided at photon energies 69-76.8 eV employing the R-matrix method with a 20-term target representation, in which five polarization orbitals \bar{6}l are included. The asymmetry parameter β₂ and the DCS, in the n=2 level in the region of 69-73 eV, are in good agreement with available experimental and theoretical results. Above the n=3 threshold, we present new theoretical results along with measurements. No theoretical and experimental results can be used to compare with the present calculations of the DCSs for photoionization into the He⁺ ion in the n=3,4 levels.     

The double-Λ hypernucleus ΛΛ 11 Be

The ground-state energy of the double-Λ hypernucleus _ΛΛ ¹¹ Be is calculated within the frame-work of the five-body model ααnΛΛ. The five-body nucleus is described using the Faddeev formalism for the unequal mass clusters involved and the potential harmonics expansion of the corresponding amplitudes. The resulting coupled differential equations depend only on the inter-cluster interactions, ΛΛ, Λn, Λα, nα, and αα, which are fairly well known from previous investigations of hypernuclei. The results obtained are in excellent agreement with the recent KEK-E373 experiment as well as with the theoretical results obtained within the Gaussian expansionmethod by Hiyama et al. The ΛΛnn double hypernucleus is also discussed.     

Scattering Length for Helium Atom-Diatom Collision

We present results on the scattering lengths of ⁴He-⁴He₂ and ³He-⁴He₂ collisions. We also study the consequence of varying the coupling constant of the atom-atom interaction.