Three-body Continuum Research Articles

Continuum spectroscopy of Borromean two-neutron halo nuclei

Energy and angular correlation distributions of the three fragments in $^{6}\mathrm{He}$ breakup on $^{208}\mathrm{Pb}$ at a collision energy of 240 MeV/nucleon are discussed within the microscopic four-body distorted wave model and compared with experimental data. The nuclear structure of the ground state and low-energy three-body continuum of $^{6}\mathrm{He}$ is calculated by the method of hyperspherical harmonics within the three-body cluster model. Reflections of the fundamental permutation symmetry of the halo neutrons in angular and energy correlations are pointed out. The calculations describe the experimental data for fragment correlations near breakup threshold rather well, and the physics is contained in a few elementary modes; but with increasing excitation energy of $^{6}\mathrm{He}$, some striking deviations from experimental distributions are encountered. Possible reasons for this are discussed.

Coulomb breakup effects on the elastic cross section ofHe6+Bi209scattering near Coulomb barrier energies

We accurately analyze the $^6$He+$^{209}$Bi scattering at 19 and 22.5 MeV near the Coulomb barrier energy, using the continuum-discretized coupled-channels method (CDCC) based on the $n$+$n$+$^4$He+$^{209}$Bi four-body model. The three-body breakup continuum of $^6$He is discretized by diagonalizing the internal Hamiltonian of $^6$He in a space spanned by the Gaussian basis functions. The calculated elastic and total reaction cross sections are in good agreement with the experimental data, while the CDCC calculation based on the di-neutron model of $^6$He, i.e., the $^2n$+$^{4}$He+$^{209}$Bi three-body model, does not reproduce the data.

Three-body continuum energy correlations in Borromean halo nuclei. II

The general properties of intrinsic energy correlations in the three-body continuum of Borromean halo nuclei are considered. A model that describes the system as a three-body {alpha}+n+n cluster structure and reproduces the experimentally known properties of {sup 6}He and {sup 6}Li is used to study low-lying resonances and soft modes. The intrinsic correlated structure of the {sup 6}He continuum reveals a unique structure for three-body 2{sub 1}{sup +},2{sub 2}{sup +}, and 1{sub 1}{sup +} resonances and a lack of resonant structure in soft dipole and monopole modes.

Three-body continuum states on a Lagrange mesh

Three-body continuum states are investigated with the hyperspherical method on a Lagrange mesh. The R-matrix theory is used to treat the asymptotic behaviour of scattering wave functions. The formalism is developed for neutral as well as for charged systems. We point out some specificities of continuum states in the hyperspherical method. The collision matrix can be determined with a good accuracy by using propagation techniques. The method is applied to the 6He ( = α + n + n ) and 6Be ( = α + p + p ) systems, as well as to 14Be ( = Be 12 + n + n ). For 6He, we essentially recover results of the literature. Application to 14Be suggests the existence of an excited 2 + state below threshold. The calculated B(E2) value should make this state observable with Coulomb excitation experiments.

Origin of three-body resonances

We expose the relation between the properties of the three-body continuum states and their two-body subsystems. These properties refer to their bound and virtual states and resonances, all defined as poles of the $S$-matrix. For one infinitely heavy core and two non-interacting light particles, the complex energies of the three-body poles are the sum of the two two-body complex pole-energies. These generic relations are modified by center-of-mass effects which alone can produce a Borromean system. We show how the three-body states evolve in $^6$He, $^6$Li, and $^6$Be when the nucleon-nucleon interaction is continuously switched on. The schematic model is able to reproduce the main properties in their spectra. Realistic calculations for these nuclei are shown in detail for comparison. The implications of a core with non-zero spin are investigated and illustrated for $^{17}$Ne ($^{15}$O+p+p). Dimensionless units allow predictions for systems of different scales.

Three-body continuum discretization in a basis of transformed harmonic oscillator states

The inclusion of the continuum in the study of weakly bound three-body systems is discussed. A transformed harmonic oscillator basis is introduced to provide an appropriate discrete and finite basis for treating the continuum part of the spectrum. As examples of the application of the method the strength functions corresponding to several operators that couple the ground state to the continuum are investigated, for $^{6}\mathrm{He}$, and compared with previous calculations. It is found that the energy moments of these distributions are accurately reproduced with a small basis set.

Three-body resonant radiative capture reactions in astrophysics

We develop the formalism based on the S-matrix for $3 \to 3$ scattering to derive the direct three-body resonant radiative capture reaction rate. Within this formalism the states, which decay only/predominantly directly into three-body continuum, should also be included in the capture rate calculations. %The results obtained indicate that some theoretical calculations are %incomplete. Basing on the derivation, as well as on the modern experimental data and theoretical calculations concerning $^{17}$Ne nucleus, we significantly update the reaction rate for $^{15}$O($2p$,$\gamma$)$^{17}$Ne process in explosive environment. We also discuss possible implementations for the $^{18}$Ne($2p$,$\gamma$)$^{20}$Mg, $^{38}$Ca($2p$,$\gamma$)$^{40}$Ti, and $^4$He($n\alpha$,$\gamma$)$^{9}$Be reactions.

Photo double ionization of helium 100 eV and 450 eV above threshold: I. Linearly polarized light

We present a joint experimental and theoretical study for the fully differential cross section of the photo double ionization (PDI) of helium with linearly polarized light at the excess energies Eexc = 100 eV and 450 eV above the threshold. The fully differential cross section is obtained by measuring the three-dimensional momentum vectors of one electron and the He2+ ion in coincidence using the COLTRIMS method. We give an overview of the momentum distribution of the three-body continuum 100 eV above the threshold. We show angular distributions for both electrons for various energy sharings at Eexc = 100 eV and 450 eV. The experimental results are well reproduced by a set of convergent close-coupling (CCC) calculations.

The three-body rigid rotator and multipole expansions of the three-body continuum

Two approaches in the derivation of multipole expansions of the three-body continuum wavefunction are considered. These approaches are determined by the basis of angular functions used. The two angular bases considered are bipolar harmonics and Wigner D-functions. In the asymptotic region of large inter-particle separations the continuum wavefunction tends to be a product of two plane waves. It is shown that the coefficients in the multipole expansion of the product of two plane waves in terms of D-functions are equivalent to the wavefunctions of the three-body rigid rotator in momentum representation. Possible applications of the developed formalism to the theory of angular distributions in molecular photo and predissociation processes are discussed. Several novel relations between integrals with two Bessel functions are discovered during the calculations.

Continuum-discretized coupled-channels method for four-body nuclear breakup inHe6+C12scattering

We propose a fully quantum-mechanical method of treating four-body nuclear breakup processes in scattering of a projectile consisting of three constituents, by extending the continuum-discretized coupled-channels method. The three-body continuum states of the projectile are discretized by diagonalizing the internal Hamiltonian of the projectile with the Gaussian basis functions. For $^{6}\mathrm{He}+^{12}\mathrm{C}$ scattering at 18 and $229.8\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$, the validity of the method is tested by convergence of the elastic and breakup cross sections with respect to increasing the number of the basis functions. Effects of the four-body breakup and the Borromean structure of $^{6}\mathrm{He}$ on the elastic and total reaction cross sections are discussed.

Structure of theLi11continuum from breakup on proton target

Inelastic scattering of $^{11}\mathrm{Li}$ on proton targets at energy of $68\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}∕\text{nucleon}$ has been studied in the microscopic four-body distorted-wave model. The ground state and three-body continuum excitations of $^{11}\mathrm{Li}$ were calculated in the $^{9}\mathrm{Li}+n+n$ cluster model using the method of hyperspherical harmonics. The theory describes the experimental data well and clarifies contributions from the three-body monopole resonance in $^{11}\mathrm{Li}$ and from its soft dipole excitation. Various aspects of the $^{11}\mathrm{Li}$ excitations and their nature are discussed in terms of correlation plots.

Anatomy of three-body decay I: schematic models

Sequential three-body decay proceeds via spatially confined quasi-stationary two-body configurations. Direct three-body decay populates the three-body continuum without intermediate steps. The relative importance of these decay modes is discussed in a schematic model employing only Coulomb or centrifugal barrier potentials. Decisive dimensionless charge, mass and energy ratios are derived. Sequential decay is usually favored for charged particles. Small charge and small mass of high energy is preferably emitted first. Without Coulomb potential the sequential decay is favored except when both resonance energy and intermediate two-body energy are large.

Loosely bound three-body nuclear systems in the J-matrix approach

We discuss the extension of the oscillator-basis J-matrix formalism on the case of true A-body scattering. The formalism is applied to loosely bound 11Li and 6He nuclei within three-body cluster models 9 Li+n+n and α+ n+ n. The J-matrix formalism is used not only for the calculation of the three-body continuum spectrum wave functions but also for the calculation of the S-matrix poles associated with the 11Li and 6He ground states to improve the description of the binding energies and ground state properties. The effect of the phase equivalent transformation of the n– α interaction on the properties of the 6He nucleus is examined.

Three-body continuum spatial correlations in Borromean halo nuclei

Spatial correlations in the three-body continuum of Borromean (having no bound binary subsystems) three-body systems are discussed. The hyperspherical harmonics method is used to investigate low-lying resonances and the soft dipole mode in the two-neutron halo nucleus $^{6}\mathrm{He}$, which has only the $\ensuremath{\alpha}+n+n$ continuum for excitation energies below $13\phantom{\rule{0.3em}{0ex}}\text{MeV}$. The spatial correlations reveal characteristic structures for true three-body resonances, a moderate amplification in the interior region for above-barrier resonances and long-range correlations in the cases of three-body ${1}^{\ensuremath{-}}$ virtual and ${0}^{+}$ continuum states.

Three-body Coulomb breakup of 11Li in the complex scaling method

Coulomb breakup strengths of 11Li into a three-body 9Li+n+n system are studied in the complex scaling method. We decompose the transition strengths into the contributions from three-body resonances, two-body “10Li+n” and three-body “9Li+n+n” continuum states. In the calculated results, we cannot find the dipole resonances with a sharp decay width in 11Li. There is a low energy enhancement in the breakup strength, which is produced by both the two- and three-body continuum states. The enhancement given by the three-body continuum states is found to have a strong connection to the halo structure of 11Li. The calculated breakup strength distribution is compared with the experimental data from MSU, RIKEN and GSI.

Few-body effects in nuclear halos

Nuclear halos are spatially extended quantum states with large cluster components. Characteristic observable properties of two- and three-body halo structures are discussed in general terms suited for generalization of the concept. Specific properties of known nuclear halos are used as examples. The bound-state structure is basically understood in contrast to the three-body continuum structure. The high-energy reaction mechanism is roughly understood but not accurately worked out and tested in details. Open questions and challenges are indicated.

Photo double ionization of He by circular andlinear polarized single-photon absorption

We have measured absolute fully differential cross sections for photodouble ionization of helium by circularly and linearly polarized light20 eV above threshold. The data have been obtained by measuring incoincidence the momentum vector of the He2+ ion and one of theelectrons, covering 4π solid angle for all particles. We give anoverview over the momentum distribution in the three-body continuum andshow fivefold differential cross sections. We find a swirl in the electronmomentum space for double ionization by circularly polarized light. Thepresent data supersede earlier data from our group(V mergel et al 1998 Phys. Rev. Lett. 805301).

Analytic properties of three-body continuum Coulomb wave functions

We study the analytic properties of the functions known as C3 and ${\ensuremath{\Phi}}_{2},$ used in atomic collision theory for the description of the three-body continuum state. We analyze the bound states for both models obtained by analytic continuation in the case of ion-atom collision. The C3 wave function is an uncorrelated model represented by the product of two-body Coulomb functions and the bound states are found for negative relative energies of electron-target or electron-projectile pairs. On the other hand, the ${\ensuremath{\Phi}}_{2}$ model is based on a two-variable hypergeometric function that correlates the electron motion relative to both the target and projectile. We found that only decaying bound states are allowed and the atomic spectra becomes continuous. The bound states of the ${\ensuremath{\Phi}}_{2}$ model have a complex energy due to the action of the projectile. Expressions for the wave functions in the different thresholds are given and studied.

Nuclear structure of5Hin a three-body3H+n+nmodel

Complete dynamical investigation of the extremely neutron-rich ${}^{5}\mathrm{H}$ nucleus is performed in a three-body ${}^{3}\mathrm{H}+n+n$ continuum. A three-body resonance enhancement is found for the ``ground state'' ${J}^{\ensuremath{\pi}}{=1/2}^{+}$ at about 2.5--3.0 MeV. The broad structures in the production cross sections correspond to wide ${3/2}^{+}$ and ${5/2}^{+}$ levels in the three-body continuum.

Long-range interactions and artificial poles

The structure of the three-body continuum of the 6He nucleus within the hyperspherical harmonic calculation scheme has been the subject of extensive study. An approximate treatment of the long-range couplings inherent in such models gives rise to unphysical poles in the deduced S-matrix at complex energies. We consider here a simple two-channel example, which includes long-range interactions, and demonstrates the existence of these artificial S-matrix poles. This is also illustrated in a physical context by considering the extreme adiabatic approximation of the hyperspherical harmonics technique for 6He. A correct treatment of the long-range (1/) behaviour of the channel potentials is shown to remove the artificial poles from the physically significant region of the complex energy plane.