Efimov effect in 2-neutron halo nuclei

This paper presents an overview of our theoretical investigations in search of Efimov states in light 2-neutron halo nuclei. The calculations have been carried out within a three-body formalism, assuming a compact core and two valence neutrons forming the halo. The calculations provide strong evidence for the occurrence of at least two Efimov states in ²⁰C nucleus. These excited states move into the continuum as the two-body (core-neutron) binding energy is increased and show up as asymmetric resonances in the elastic scattering cross-section of the n-¹⁹C system. The Fano mechanism is invoked to explain the asymmetry. The calculations have been extended to ³⁸Mg, ³²Ne and a hypothetical case of a very heavy core (A = 100) with two valence neutrons. In all these cases the Efimov states show up as resonances as the two-body energy is increased. However, in sharp contrast, the Efimov states, for a system of three equal masses, show up as virtual states beyond a certain value of the two-body interaction.     

Analysis of the low-energy ηNN-dynamics within a three-body formalism

The interaction of an η-meson with two nucleons is studied within a three-body approach. The major features of the ηNN-system in the low-energy region are accounted for by using a s-wave separable ansatz for the two-body ηN and NN amplitudes. The calculation is confined to the (J ^π;T) = (0^-;1) and (1^-;0) configurations which are assumed to be the most promising candidates for virtual or resonant ηNN-states. The eigenvalue three-body equation is continued analytically into the nonphysical sheets by contour deformation. The position of the poles of the three-body scattering matrix as a function of the ηN-interaction strength is investigated. The corresponding trajectory, starting on the physical sheet, moves around the ηNN three-body threshold and continues away from the physical area giving rise to virtual ηNN-states. The search for poles on the nonphysical sheets adjacent directly to the upper rim of the real energy axis gives a negative result. Thus no low-lying s-wave ηNN-resonances were found. The possible influence of virtual poles on the low-energy ηNN-scattering is discussed. Received: 27 June 2000 / Accepted: 3 August 2000     

Effective Field Theory and the Gamow Shell Model

We combine halo/cluster effective field theory (H/CEFT) and the Gamow shell model (GSM) to describe the 0⁺ ground state of ⁶He as a three-body halo system. We use two-body interactions for the neutron-alpha particle and two-neutron pairs obtained from H/CEFT at leading order, with parameters determined from scattering in the p_3/2 and s₀ channels, respectively. The three-body dynamics of the system is solved using the GSM formalism, where the continuum states are incorporated in the shell model valence space. We find that in the absence of three-body forces the system collapses, since the binding energy of the ground state diverges as cutoffs are increased. We show that addition at leading order of a three-body force with a single parameter is sufficient for proper renormalization and to fix the binding energy to its experimental value.     

Kaon absorption from kaonic atoms and formation spectra of kaonic nuclei

We considered the kaon absorption from atomic states into the nucleus. We found that the nuclear density probed by the atomic kaon significantly depends on the kaon orbit. Then, we re-examined the meanings of the observed strengths of one-body and two-body kaon absorption, and investigated the effects to the formation spectra of kaon bound states by in-flight (K ^-, p) reactions. As a natural consequence, if the atomic kaon probes a smaller nuclear density, the ratio of the two-body absorption at nuclear center is larger than the observed value in kaonic atoms, and the depth of the imaginary potential is deeper even at smaller kaon energies as in kaonic nuclear states because of the large phase space for the two-body processes. This deeper imaginary potential makes the signals of kaonic nucleus formation more unclear in the (K ^-, p) spectra.     

Investigation of auto-ionizing states in xenon by resonantly enhanced multi-photon ionization

We have examined the autoionization spectrum of xenon by resonantly enhanced three-photon ionization (2 + 1 REMPI) involving intermediate states 5p ⁵6p[J = 0, 2]. The properties of the observed autoionization resonances change significantly with the choice of the intermediate state. For ionization via an intermediate state with predominantly 5p ⁵(²P_3/2) core character, a strong continuum with embedded window-type 5p ⁵(²P _1/2)nd'-autoionization resonances is observed. For intermediate states, predominantly with 5p ⁵(²P_1/2) core character, both 5p ⁵(²P _1/2)nd'- as well as 5p ⁵(²P _1/2)ns'-resonances are present in the spectrum as overlapping, nearly symmetric peaks on a rather weak continuum. Calculations confirm the significant dependence of the spectral lineshapes upon the excitation pathway to the autoionizing state. The ionization data are compared with spectra obtained by monitoring third-harmonic generation via autoionizing states without resonant excitation of intermediate states. These spectra also exhibit the signature of both the nd'- and ns'-resonances. Received 30 September 2002 Published online 28 January 2003 RID="a" ID="a"Permanent address: Rostov State University of Transport Communication, 344038, Rostov-on-Don, Russia. RID="b" ID="b"e-mail: halfmann@physik.uni-kl.de     

On the Uniqueness of the Solution to the Three-Body Problem with Zero-Range Interactions

We consider the uniqueness of the solution to a three-body problem with zero-range Skyrme interactions in configuration space. With the lowest, k⁰, two-body term alone the problem is known to have no unique solution as the system collapses – the variational estimate of the energy tends towards negative infinity, the size of the system towards zero. We argue that the next, k², two-body term removes the collapse and the three-body system acquires finite ground-state energy and size. The three-body interaction term is thus not necessary to provide a unique solution to the problem.     

Shell model structures in the N = 46 isotones 86Zr, 87Nb, 88Mo, 89Tc, 90Ru and 91Rh

The known level energies, electromagnetic moments and decay probabilities of high-spin states in the N = 46 isotones ⁸⁶Zr, ⁸⁷Nb, ⁸⁸Mo, ⁸⁹Tc, and ⁹⁰Ru are interpreted within the shell model. The single-particle space was truncated to the p _1/2 and g _9/2 orbits (relative to the ⁸⁸Sr core) and the single-particle energies and empirical two-body matrix elements derived by Gross and Frenkel were used in the calculations. Based on the generally good success of this approach, energies and decay properties of the yrast spectra in ⁹⁰Ru and ⁹¹Rh are predicted. Received: 31 July 2000 / Accepted: 18 December 2000     

Relative wavefunctions, vertex functions and coupling constants for the virtual decay of 4He

The d + d, t + p and h + n relative wavefunctions and their asymptotic normalizations are considered in the framework of the generator coordinate method (GCM) and compared with ATMS (amalgamation of two-body correlation into multiple scattering processes) method which used the realistic Reid soft core interaction. The asymptotic normalization of relative wavefunctions provide various coupling constants, the cluster probability amplitude (the so-called Z ^1/2-factor) and matter RMS radii. These wavefunctions are also used to obtain ⁴He − d − d, ⁴He − t − p and ⁴He − h − n vertex functions in the virtual decay of ⁴He. The extrapolation of vertex functions for negative values of q ² upto the corresponding poles provide the vertex constants which are comparable with other estimates. It is noticed that in GCM the coupling constants C ² for ⁴He − d − d vertex is less than 2 as has been obtained in the forward dispersion relation technique.     

Three-body model for stripping nuclear reactions

A three-body model for the deuteron stripping nuclear reactions is presented. A set of three integral equations is obtained for the wave functions of the three-body problem by introducing a decomposition into angular momentum states into the Lippmann-Schwinger equation. Simple two-particle interactions with separable potentials are used. These separable potentials reduce the three-body problem to the solution of coupled sets of one-dimensional Fredholm integral equations. The angular distributions for²⁸Si(d,p)²⁹Si and⁴⁰Ca(d, p)⁴¹Ca stripping reactions are calculated. From the extracted spectroscopic factors, good agreement with the experimental measurements is obtained.     

Random matrix structure of nuclear shell model Hamiltonian matrices and comparison with an atomic example

We present a comprehensive analysis of the structure of Hamiltonian matrices based on visualization of the matrices in three dimensions as well as in terms of measures for GOE, banded and two-body random matrix ensembles (TBRE). We have considered two nuclear shell model examples, ²²Na with J^p T = 2⁺0\ensuremath J^{\pi} T = 2^+0 and ²⁴Mg with J^p T = 0⁺0\ensuremath J^{\pi} T = 0^+0 and, for comparison we have also considered the SmI atomic example. It is clearly established that the matrices are neither GOE nor banded. For the TBRE structure we have examined the correlations between diagonal elements and eigenvalues, fluctuations in the basis states variances and structure of the two-body part of the Hamiltonian in the eigenvalue basis. Unlike the atomic example, nuclear examples show that the nuclear shell model Hamiltonians can be well represented by TBRE.     

On the negative point spectrum of quantum mechanical three-body systems

We consider quantum mechanical three-body systems interacting with two-body potentials that are sufficiently regular locally, decrease faster than r^?(2+?) at infinity, and are such that either one two-body subsystem (at least) has a negative energy bound state, or no two-body subsystem has a zero energy bound state or resonance. (This assumption excludes the possibility of the Efimov effect.) We then prove that (1) the discrete spectrum is finite, (2) the point spectrum has no negative accumulation points. (These results have been proved earlier by Sigal and Yafaev, respectively, by different methods and with slightly different assumptions.)     

Two-body and three-body effective interactions in nuclei

Finite-range two-body and zero-range three-body effective forces for use in spin unsaturated systems are determined so as to reproduce the total binding energies, rms radii, and single-particle energies of ¹⁶O, ⁴⁰Ca, ⁴⁸Ca and ⁹⁰Zr; the saturation of nuclear matter; and experimental two-body matrix elements extracted by Schiffer and True. In addition to the Skyrme three-body force which acts only in spatially even states, a spatially odd force is introduced to obtain sufficient generality. The Landau parameters and the effective mass specified by this force are also discussed.     

A class of exactly solvable three-body quantum mechanical problems and the universal low energy behavior

Three-body systems with two-body point interactions are studied. These systems are the universal low energy limits of three-body problems with short-range two-body forces. Hence if there are infinitely many spherically symmetric three-body bound states with energies E_n then lim_n→∞E_n/E_n+1 = e^2λσ, where σ is explicitly computed.     

Universal properties and structure of halo nuclei

The universal properties and structure of halo nuclei composed of two neutrons (2n) and a core are investigated within an effective quantum mechanics framework. We construct an effective interaction potential that exploits the separation of scales in halo nuclei and treat the nucleus as an effective three-body system. The uncertainty from higher orders in the expansion is quantified through theoretical error bands. First, we investigate the possibility to observe excited Efimov states in 2n halo nuclei. Based on the experimental data, ²⁰C is the only halo nucleus candidate to possibly have an Efimov excited state, with an energy less than 7 keV below the scattering threshold. Second, we study the structure of ²⁰C and other 2n halo nuclei. In particular, we calculate their matter form factors, radii, and two-neutron opening angles.     

Properties of proton-rich nuclei in a three-body model

Using a three-body model and realistic two-body potentials, we investigate the properties of the nuclei ¹⁸Ne and ²⁸S near the proton dripline. We figure out the two-proton separation energies, occupation of the valence protons, root-mean-square radii of matter and the valence protons. Besides, the spatial correlation densities are displayed to reflect the correlation between the two valence protons. The first excited 0⁺ state of ¹⁸Ne is most likely to be a halo state according to our calculation. Turning off the Coulomb interactions among the three-body systems, we get the two-neutron separation energies and configuration of the valence neutrons of their corresponding mirror nuclei. The results indicate that the three-body model is proper to describe some proton-rich nuclei and can be used to deduce reliable information.     

Isospin mixing and energy distributions in three-body decay

The structure of the second ²⁺$2^{+}$ resonance in ⁶Li is investigated with special emphasis on its isospin 0 components. The wave functions are computed in a three-body model (α+n+p$\alpha +n+p$) using the hyperspherical adiabatic expansion method combined with complex scaling. In the decay into three free particles the symmetry conserving short-range interaction dominates at short distance whereas the symmetry breaking Coulomb interaction dominates at intermediate and large distances resulting in substantial isospin mixing. We predict the mixing and the energy distributions of the fragments after decay. Computations are consistent with available experiments. We conjecture that nuclear three-body decays frequently produce such large isospin mixing at large distance where the energy distributions are determined.     

Kinematically complete final state investigations of molecular photodissociation: two- and three-body decay of laser-prepared H 3 3 s ? 2 A 1 '

We have performed kinematically complete investigations of molecular photodissociation of triatomic hydrogen in a fast beam translational spectrometer recently built in Freiburg. The apparatus allows us to investigate laser-induced dissociation of neutral molecules into two, three, or more neutral products. The fragments are detected in coincidence and their vectorial momenta in the center-of-mass frame are determined. We demonstrate the potential of the method at the fragmentation of the 3 s ² A ₁ ^′ ( N = 1, K = 0) state of triatomic hydrogen. In this state, three-body decay into ground state hydrogen atoms H+H+H, two-body predissociation into H+H ₂ (v , J), and photoemission to the H ₃ ground state surface with subsequent two-body decay are competing channels. In the case of two-body predissociation, we determine the rovibrational population in the H ₂ (v , J) fragment. The vibrational distribution of H ₂ is compared with approximate theoretical predictions. For three-body decay, we measure the six-fold differential photodissociation cross-section. To determine accurate final state distributions, the geometric collection efficiency of the apparatus is calculated by a Monte Carlo simulation, and the raw data are corrected for apparatus efficiency. The final state momentum distribution shows pronounced correlation patterns which are characteristic for the dissociation mechanism. For a three-body decay process with a discrete kinetic energy release we have developed a novel data reduction procedure based on the detection of two fragments. The final state distribution determined by this independent method agrees extremely well with that observed in the triple-coincidence data. In addition, this method allows us to fully explore the phase space of the final state and to determine the branching ratios between the two- and three-body decay processes. Received 29 March 2001     

Supersymmetry for nuclear cluster systems

A supersymmetry scheme is proposed for nuclear cluster systems. The bosonic sector of the superalgebra describes the relative motion of the clusters, while its fermionic sector is associated with their internal structure. An example of core + α configurations is discussed in which the core is a p-shell nucleus and the underlying superalgebra is U(4|12). The α-cluster states of the nuclei ²⁰Ne and ¹⁹F are analyzed and correlations between their spectra, electric quadrupole transitions, and one-nucleon transfer reactions are interpreted in terms of U(4|12) supersymmetry. Received: 14 September 2001 / Accepted: 26 October 2001     

Tests of the discretized-continuum method in three-body dipole strengths

We investigate the ⁶He dipole distribution in a three-body α+n+n model. Two approaches are used to describe the three-body ¹⁻ continuum: the discretized-continuum method, where the scattering wave functions are approximated by square-integrable functions, and the R-matrix formalism, where their asymptotic behaviour is taken into account. We show that some ambiguity exists in the pseudostate method, owing to the smoothing technique, necessary to derive continuous distributions. We show evidence for the important role of the halo structure in the E1 dipole strength. We also address the treatment of Pauli forbidden states in the three-body wave functions.