Three-body Continuum Research Articles

Three-body breakup within the fully discretized Faddeev equations

A novel approach is developed to find the three-body breakup amplitudes and cross sections within the modified Faddeev equation framework. The method is based on the lattice-like discretization of the three-body continuum with a three-body stationary wave-packet basis in momentum space. The approach makes it possible to simplify drastically all the three- and few-body breakup calculations due to discrete wave-packet representations for the few-body continuum and simultaneous lattice representation for all the scattering operators entering the integral equation kernels. As a result, the few-body breakup can be treated as a particular case of multi-channel scattering in which part of the channels represents the true few-body continuum states. As an illustration for the novel approach, an accurate calculations for the three-body breakup process $n+d\to n+n+p$ with non-local and local $NN$ interactions are calculated. The results obtained reproduce nicely the benchmark calculation results using the traditional Faddeev scheme which requires much more tedious and time-consuming calculations.

Comparative Study of Three-Body Models for Continuum States

We discuss properties of three-body continuum states in the hyperspherical formalism. The associated coupled-channel system is solved on a Lagrange basis, and is complemented by the R-matrix theory for the treatment of scattering states. Three-body α + n + n phase shifts are determined under various conditions. Two approximations of the continuum are considered: the complex scaling and the pseudostate methods. Both are briefly outlined, and compared with the R-matrix approach for the dipole-strength distribution of 6 He. A fair agreement is obtained between the three methods. We also discuss the influence of the α + n and n + n interactions, as well as the effect of the Pauli principle. Some differences are obtained, but all calculations predict a broad 1 − resonance near 1 MeV.

FOUR-BODY CONTINUUM-DISCRETIZED COUPLED-CHANNEL CALCULATIONS APPLIED TO 6He REACTIONS AROUND THE COULOMB BARRIER

The scattering of a weakly bound three-body system by a target is studied within the four-body continuum-discretized coupled-channels (4b-CDCC) framework. Two different methods, the transformed harmonic oscillator (THO) method and the binning procedure, are used for discretizing the three-body continuum. The formalism is applied to different reactions induced by the Borromean nucleus 6 He at energies around the Coulomb barrier: 6 He +64 Zn at 13.6 MeV, 6 He +120 Sn at 17.4 MeV, and 6 He +208 Pb at 22 MeV. Elastic cross section distributions are presented for these reactions comparing both discretization methods, THO and binning, as the mass of the target increases.

Few-body Decay and Recombination in Nuclear Astrophysics

Three-body continuum problems are investigated for light nuclei of astrophysical relevance. We focus on three-body decays of resonances or recombination via resonances or the continuum background. The concepts of widths, decay mechanisms and dynamic evolution are discussed. We also discuss results for the triple $\alpha$ decay in connection with $2^+$ resonances and density and temperature dependence rates of recombination into light nuclei from $\alpha$-particles and neutrons.

Strength Function in Continuum with a Square Integrable Basis

We present a new approach to describe the three-body continuum with a square integrable basis and show some examples to assess its efficiency. The formulation is done elegantly in the hyperspherical formalism and the tail behavior is appropriately corrected with use of the Green’s function. The efficiency of this approach is shown in a simple solvable problem by calculating the electric dipole strength function of a three-body system and comparing it to that obtained with the complex scaling method, which is widely used for calculating the strength function.

The triple alpha reaction rate and the [formula omitted] resonances in 12C

The triple alpha rate is obtained from the three-body bound and continuum states computed in a large box. The results from this genuine full three-body calculation are compared with standard reference rates obtained by two sequential two-body processes. The fairly good agreement relies on two different assumptions about the lowest 2+ resonance energy. With the same 2+ energy the rates from the full three-body calculation are smaller than those of the standard reference. We discuss the rate dependence on the experimentally unknown 2+ energy. Substantial deviations from previous results appear for temperatures above 3 GK.

Structure and three-body decay ofBe9resonances

The complex-rotated hyperspherical adiabatic method is used to study the decay of low-lying $^9$Be resonances into one neutron and two $\alpha$-particles. We investigate the six resonances above the break-up threshold and below 6 MeV: $1/2^\pm$, $3/2^\pm$ and $5/2^\pm$. The short-distance properties of each resonance are studied, and the different angular momentum and parity configurations of the $^8$Be and $^5$He two-body substructures are determined. We compute the branching ratio for sequential decay via the $^8$Be ground state which qualitatively is consistent with measurements. We extract the momentum distributions after decay directly into the three-body continuum from the large-distance asymptotic structures. The kinematically complete results are presented as Dalitz plots as well as projections on given neutron and $\alpha$-energy. The distributions are discussed and in most cases found to agree with available experimental data.

Green's Function Method for Strength Function in Three-Body Continuum

Practical methods to compute dipole strengths for a three-body system by using a discretized continuum are analyzed. New techniques involving Green's function are developed, either by correcting the tail of the approximate wave function in a direct calculation of the strength function or by using a solution of a driven Schrödinger equation in a summed expression of the strength. They are compared with the complex scaling method and the Lorentz integral transform, also making use of a discretized continuum. Numerical tests are performed with a hyperscalar three-body potential in the hyperspherical-harmonics formalism. They show that the Lorentz integral transform method is less practical than the other methods because of a difficult inverse transform. These other methods provide in general comparable accuracies.

Three-body continuum states in a microscopic cluster model

We propose a microscopic approach to three-body continuum states in the hyperspherical formalism. The $A$-nucleon wave functions are defined in the generator coordinate method, based on a three-cluster approximation of the system. We develop a semianalytical treatment of the matrix elements, limited to $s$ clusters. This allows a fast calculation, as well as a direct interpretation of the wave functions, in terms of nucleon exchanges. We apply the method to the $\ensuremath{\alpha}+n+n$ three-body phase shifts and confirm the existence of broad structures in the ${0}^{+}$ and ${1}^{\ensuremath{-}}$ partial waves at low energies.

Integral Relations for Three-Body Continuum States with the Adiabatic Expansion

The application of the hyperspherical adiabatic expansion to describe three-body scattering states suffers from the problem of very slow convergence. Contrary to what happens for bound states, a huge number of hyper-radial equations has to be solved, and even if done, the extraction of the scattering amplitude is problematic. In this Letter we show how to obtain accurate scattering phase shifts using the hyperspherical adiabatic expansion. To this aim two integral relations, derived from the Kohn variational principle, are used. The convergence of this procedure is as fast as for bound states.

Cluster models of light nuclei and the method of hyperspherical harmonics: Successes and challenges

Hyperspherical-harmonics method to investigate the lightest nuclei having three-cluster structure is discussed together with recent experiments. Properties of bound states and methods to explore three-body continuum are presented. The challenges created by large neutron excess and halo phenomena are highlighted. Astrophysical aspects of the 7Li + n → 8Li + γ reaction and the solar-boron-neutrinos problem are analyzed. Three-cluster structure of highly excited states in 8Be is shown to be responsible for extreme isospin mixing. Progress in studies of 6He- and 11Li-induced inclusive and exclusive nuclear reactions is demonstrated, providing information on the nature of continuum structures of Borromean nuclei.

COUPLED-CHANNELS ANALYSES OF 6He BREAKUP REACTIONS

We present analyses of breakup effects of 6 He on the elastic scattering by the continuum-discretized coupled-channels method, in which the reaction system is described as a four-body model, n+n+4 He +target. In this analysis, three-body breakup continuum of 6 He is discretized by daiagonalizing the internal Hamiltonian of 6 He in a space spanned by the Gaussian basis functions. The calculated elastic cross sections are in good agreement with the experimental data, which shows that nuclear and Coulomb breakup effects are significant.

Efimov states embedded in the three-body continuum

We present analytical solutions for the three-body problem with multichannel interactions and identify a class of quasibound Efimov states that can be viewed as three-body Fano-Feshbach resonances. Our method employs a multichannel generalization of the Fermi pseudopotential to model low-energy atom-atom interactions near a magnetically tunable Fano-Feshbach resonance. We discuss the conditions under which quasibound Efimov states may be supported and identify the interaction parameters that limit the lifetimes of these states. We speculate that it may be possible to observe these states using spectroscopic methods, perhaps allowing for the measurement of multiple Efimov resonances.

Full coupled-channel description of three-body and four-body breakup reactions

Recent development of the continuum-discretized coupled-channels method (CDCC) is reported with introducing the eikonal CDCC method (E-CDCC) and four-body CDCC. The former is applied to analysis of 8B breakup by 208Pb at 52 A MeV and the astrophysical factor S 17 ( 0 ) for the 7Be( p , γ ) 8B reaction is extracted with the asymptotic normalization coefficient method. We obtain S 17 ( 0 ) = 20.9 −1.9 +2.0 eV b , which is significantly larger than the previous one, S 17 ( 0 ) = 18.9 ± 1.8 eV b , determined from an analysis with the virtual photon theory. The difference between the two values is found to be due to the contributions from nuclear breakup and higher-order processes. The latter, four-body CDCC, is applied to 6He+ 209Bi scattering near the Coulomb barrier energy and found to reproduce the elastic cross sections well, in contrast to the standard three-body CDCC. Accurate treatment of the three-body continuum states of 6He turns out to be essential to correctly describe the 6He+ 209Bi scattering.

Decay of low-lying 12C resonances within a 3α cluster model

We compute energy distributions of three α-particles emerging from the decay of 12C resonances by means of the hyperspherical adiabatic expansion method combined with complex scaling. The large distance continuum properties of the wave functions are crucial and must be accurately calculated. The substantial changes from small to large distances determine the decay mechanisms. We illustrate by computing the energy distributions from decays of the 1+ and 3−-resonances in 12C. These states are dominated by direct and sequential decays into the three-body continuum respectively.

Are sequential and direct decays distinguishable?

We compute the momentum distributions of the particles emerging in three-body decay of a nuclear resonance. We discuss if it is possible to distinguish between sequential decay proceeding through non-observable intermediate configurations and direct decay into the three-body continuum. Detailed investigation of the decay of the 9Be(5/2−) resonance is used as a theoretical prototype for explaning the general principles.

Three-body continuum energy correlations in Borromean halo nuclei. III. Short-range external fields

Energy correlations in transitions from the bound state to the three-body continuum of Borromean halo nuclei are considered. A core$+n+n$ three-body cluster model which reproduces the experimentally known properties of $^{6}\mathrm{He}$ and $^{11}\mathrm{Li}$ has been used to study the low-lying resonances and soft modes. An analysis of the correlated responses in $^{6}\mathrm{He}$ shows that in the case of the narrow three-body ${2}_{1}^{+}$ resonance the transition energy correlations are the same as in the intrinsic correlated structure in $3\ensuremath{\rightarrow}3$ scattering. They differ significantly for wide ${2}_{2}^{+},{1}_{1}^{+}$ resonances, and also for the soft dipole and monopole modes, where, due to the transition operators, the intertwining of the ground state and the three-body continuum plays a significant role.

Wave-packet discretization of a continuum: Path toward practically solving few-body scattering problems

A new method for discretizing a three-body continuum with the aid of the L 2 basis of stationary wave packets is considered within the problem of three-body scattering. Substantial advantages of employing this basis in solving problems of few-body scattering are demonstrated. Specific applications of this approach are exemplified by exploring the problem of scattering of a composite particle on a heavy nucleus with allowance for the excitation of this particle to continuum states. This is done within two alternative approaches: a direct wave-packet discretization of a three-body continuum and a method that is based on the Feshbach projection formalism. It is shown explicitly that the resulting scattering amplitudes are convergent as the number of wave-packet states that are taken into account is increased. The results obtained here are compared with the results of other authors whose treatment was based on alternative methods for discretizing a continuum.

Elastic scattering on a nucleus and the breakup of the composite projectile via wave-packet continuum discretization

The scattering of a composite particle off a nucleus with account of the breakup effect is solved via the original formalism of the stationary wave packets. The approach is based on the integral equation formalism of the scattering theory with a complete discretization of the three-body continuum. This allows one to find the elastic and breakup amplitudes from one nonsingular matrix equation and also to use a very convenient ${L}_{2}$ basis in numerical calculations. The efficiency of the approach is demonstrated by a realistic example of deuteron scattering on a ${}^{58}\mathrm{Ni}$ target.

Influence of protons on the capture of electrons byBe7in the Sun

The capture of electrons by the nucleus {sup 7}Be from the three-body initial state p+e{sup -}+{sup 7}Be in the continuum is studied. On the basis of the expansion of the three-body continuum wave function in the small parameter {epsilon}{approx_equal}(m{sub e}/m{sub p}){sup 1/2} [m{sub e} (m{sub p}) is the electron (proton) mass], the role of the protons on the electron capture is considered. The results are compared with the traditional treatment of the electron capture by the nucleus {sup 7}Be. For stars with density and temperature like in the center of the Sun the studied mechanism can make a non-negligible contribution to the capture rate.