Three-body Continuum States Research Articles

Three-body continuum states and Efimov physics in noninteger geometry

Continuum structures of three short-range interacting particles in a deformed external one-body field are investigated. We use the equivalent $d$ method employing noninteger dimension, $d$, in a spherical calculation with a dimension-dependent angular momentum barrier. We focus on dimensions close to the critical dimension, $d={d}_{E}$, between 2 and 3, defined by zero two-body energies, where the Efimov effect can occur. We design for this dimension region a schematic, long-distance realistic, square-well based, three-body spherical model, which is used to derive analytic expressions for the wave functions, scattering lengths, phase shifts, and elastic-scattering cross sections. The procedure and the results are universal, valid for all short-range potentials, and for large scattering lengths. We discuss the properties and validity of the derived expressions by means of the simplest system of three identical bosons. The derived expressions are particularly useful for very small energies, where full numerical calculations are often not feasible. For energies where the numerical calculations can be performed, a good agreement with the analytic results is found. These model results may be tested by scattering experiments for three particles in an equivalent external deformed oscillator potential. The cross sections all vanish in the zero-energy limit for $d<3$ with definite $d$-dependent power of energy.

Precise calculation of the triple-αreaction rates using the transmission-free complex absorbing potential method

We study the triple-$\ensuremath{\alpha}$ reaction process at low temperatures, which is known to play an important role in stellar physics. The Schr\odinger equation for three $\ensuremath{\alpha}$ particles is solved by using hyperspherical coordinates, while a complex absorbing potential is introduced in order to describe correctly the three-body continuum states. We use an angular-momentum-independent $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ potential and introduce three-body potentials to reproduce the energies of both the Hoyle state and the first ${2}^{+}$ state. The triple-$\ensuremath{\alpha}$ reaction rate is computed accurately at temperatures from $T=0.01$ to 10 GK and compared with those available in the literature. Our reaction rate is found to be up to three orders of magnitude larger than the NACRE rate at low temperatures $T\ensuremath{\approx}0.01$ GK, while we find a reasonable agreement between them at higher temperatures $T\ensuremath{\gtrsim}0.1$ GK.

Three-α model calculations of low-lying 12C states and the triple-α reaction

Properties of some 12C resonant states at low energies are studied in three-α- particle model. Calculations for breakup reactions of 12 C bound state leading to three-body continuum states are performed at energies up to a few MeV above the breakup threshold by applying the Faddeev three-body formalism in coordinate space. Calculated cross sections reveal some resonant peaks. Decay modes and the structure at interior region of these resonant states as well as their effects on the triple-alpha reaction rate will be reported.

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.

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.

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.

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.

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.

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.