Four-body Continuum-discretized Coupled-channels Research Articles

Elastic scattering measurements for the C10+Pb208 system at Elab=66 MeV

Background: The influence of halo structure of $^6$He, $^8$B, $^{11}$Be and $^{11}$Li nuclei in several mechanisms such as direct reactions and fusion is already established, although not completely understood. The influence of the $^{10}$C Brunnian structure is less known. Purpose: To investigate the influence of the cluster configuration of $^{10}$C on the elastic scattering at an energy close to the Coulomb barrier. Methods: We present experimental data for the elastic scattering of the $^{10}$C+$^{208}$Pb system at $E_{\rm lab}$ = 66 MeV. The data are compared to the three- and the four-body continuum-discretized coupled-channels calculations assuming $^9$B+$p$, $^6$Be+$\alpha$ and $^8$Be+$p$+$p$ configurations. Results: The experimental angular distribution of the cross sections shows the suppression of the Fresnel peak that is reasonably well reproduced by the continuum-discretized coupled-channels calculations. However, the calculations underestimate the cross sections at backward angles. Couplings to continuum states represent a small effect. Conclusions: The cluster configurations of $^{10}$C assumed in the present work are able to describe some of the features of the data. In order to explain the data at backward angles, experimental data for the breakup and an extension of theoretical formalism towards a four-body cluster seem to be in need to reproduce the measured angular distribution.

Reaction dynamics of exotic and stable weakly-bound nuclei using a four-body continuum-discretized coupled-channels formalism

Reactions induced by four weakly-bound nuclei on a lead target, at energies around the Coulomb barrier, are described using the same four-body continuum-discretized coupled-channels formalism. The nuclei considered are the halo nuclei 11Li and 6He, the stable nucleus 9Be and the proton-rich nucleus 10C. The differences and similarities on the reaction dynamics are addressed.

Soft giant resonance in two neutron halo nucleus 11Li

Lithium-11 is a showcase of neutron halos and has been providing new insight to nuclear structure researches. Recently, an isoscalar resonance at low excitation energy ($ E_\mathrm{x} \sim 1$ MeV) has been confirmed by the development of a low-energy high-intensity beam of 11Li at TRIUMF. By inelastic scattering measurements $ (\mathrm{p}, \mathrm{p}^{\prime})$ and $ (\mathrm{d}, \mathrm{d}^{\prime})$ with inverse kinematics, the first excited state of 11Li has been studied in detail and an isoscalar nature of the transitions has been confirmed and a strong transition amplitude has been observed indicating the soft-resonance of 11Li. Those recent experimental results on the excitation of the soft resonance are described and then recent theoretical developments based on the four-body continuum-discretized coupled-channels 11Li + p system are presented together with a brief history of the resonance search in 11Li.

Four-body extension of the continuum-discretized coupled-channels method

I develop an extension of the continuum-discretized coupled-channels (CDCC) method to reactions where both nuclei present a low breakup threshold. This leads to a four-body model, where the only inputs are the interactions describing the colliding nuclei, and the four optical potentials between the fragments. Once these potentials are chosen, the model does not contain any additional parameter. First I briefly discuss the general formalism, and emphasize the need for dealing with large coupled-channel systems. The method is tested with existing benchmarks on $4\ensuremath{\alpha}$ bound states with the Ali-Bodmer potential. Then I apply the four-body CDCC to the $^{11}\mathrm{Be}+d$ system, where I consider the $^{10}\mathrm{Be}({0}^{+},{2}^{+})+n$ configuration for $^{11}\mathrm{Be}$. I show that breakup channels are crucial to reproduce the elastic cross section, but that core excitation plays a weak role. The $^{7}\mathrm{Li}+d$ system is investigated with an $\ensuremath{\alpha}+t$ cluster model for $^{7}\mathrm{Li}$. I show that breakup channels significantly improve the agreement with the experimental cross section, but an additional imaginary term, simulating missing transfer channels, is necessary. The full CDCC results can be interpreted by equivalent potentials. For both systems, the real part is weakly affected by breakup channels, but the imaginary part is strongly modified. I suggest that the present wave functions could be used in future DWBA calculations.

Be9+Sn120 scattering at near-barrier energies within a four-body model

Cross sections for elastic and inelastic scattering of the weakly-bound $^9$Be nucleus on a $^{120}$Sn target have been measured at seven bombarding energies around and above the Coulomb barrier. The elastic angular distributions are analyzed with a four-body continuum-discretized coupled-channels (CDCC) calculation, which considers $^9$Be as a three-body projectile ($\alpha$ + $\alpha$ + n). An optical model analysis using the S\~ao Paulo potential is also shown for comparison. The CDCC analysis shows that the coupling to the continuum part of the spectrum is important for the agreement with experimental data even at energies around the Coulomb barrier, suggesting that breakup is an important process at low energies. At the highest incident energies, two inelastic peaks are observed at 1.19(5) and 2.41(5) MeV. Coupled-channels (CC) calculations using a rotational model confirm that the first inelastic peak corresponds to the excitation of the 2$_1^+$ state in $^{120}$Sn, while the second one likely corresponds to the excitation of the 3$_1^-$ state.

9Be scattering within a four-body continuum-discretized coupled-channels framework

The scattering of $^9$Be on $^{208}$Pb is addressed within a four-body Continuum-Discretized Coupled-Channels (CDCC) framework, considering a three-body $\alpha + \alpha + n$ projectile plus a structureless target. The projectile states are generated using the analytical Transformed Harmonic Oscillator (THO) basis in hyperspherical coordinates. Both the elastic and breakup channels are described on the same footing. We find a good agreement between our calculations and the experimental data at beam energies around and below the Coulomb barrier.

Four-body effects in the [formula omitted] scattering

We present angular distributions of the He6+Ni58 scattering measured at three bombarding energies above the Coulomb barrier: Elab=12.2 MeV,16.5 MeV,and 21.7 MeV. The angular distributions have been analysed in terms of three- and four-body Continuum-Discretized Coupled-Channels calculations considering the effect of the 6He breakup. A behaviour in the cross section at large angles has been observed which was reproduced only by the four-body Continuum-Discretized Coupled-Channels calculation.

11Li Breakup on 208 at energies around the Coulomb barrier.

The inclusive breakup for the (11)Li + (208)Pb reaction at energies around the Coulomb barrier has been measured for the first time. A sizable yield of (9)Li following the (11)Li dissociation has been observed, even at energies well below the Coulomb barrier. Using the first-order semiclassical perturbation theory of Coulomb excitation it is shown that the breakup probability data measured at small angles can be used to extract effective breakup energy as well as the slope of B(E1) distribution close to the threshold. Four-body continuum-discretized coupled-channels calculations, including both nuclear and Coulomb couplings between the target and projectile to all orders, reproduce the measured inclusive breakup cross sections and support the presence of a dipole resonance in the (11)Li continuum at low excitation energy.

Do Halo Nuclei Follow Rutherford Elastic Scattering at Energies Below the Barrier? The Case ofLi11

The first measurement of the elastic scattering of the halo nucleus 11Li and its core 9Li on 208Pb at energies near the Coulomb barrier is presented. The 11Li+208Pb elastic scattering shows a strong reduction with respect to the Rutherford cross section, even at energies well below the barrier and down to very small scattering angles. This drastic change of the elastic differential cross section observed in 11Li+208Pb is the consequence of the halo structure of 11Li, as it is not observed in the elastic scattering of its core 9Li at the same energies. Four-body continuum-discretized coupled-channels calculations, based on a three-body model of the 11Li projectile, are found to explain the measured angular distributions and confirm that the observed reduction is mainly due to the strong Coulomb coupling to the dipole states in the low-lying continuum of 11Li. These calculations suggest the presence of a low-lying dipole resonance in 11Li close to the breakup threshold.

Experimental study ofHe6+Be9elastic scattering at low energies

New data for the $^{6}\mathrm{He}+^{9}\mathrm{Be}$ reaction at ${E}_{\mathrm{lab}}=16.2$ and 21.3 MeV have been taken and analyzed. The effect of the collective couplings to the excited states of the target has been studied by means of coupled-channels calculations, using a double-folding potential for the bare interaction between the colliding nuclei, supplemented with a phenomenological imaginary part of Woods-Saxon type. In addition, three- and four-body continuum-discretized coupled-channels calculations have been performed to investigate the effect of the projectile breakup on the elastic scattering. Both effects, the coupling to target and projectile excited states, are found to affect significantly the elastic scattering. The trivial local polarization potential extracted from the continuum-discretized coupled-channels calculations indicates that continuum couplings produce a repulsive real part and a long-range imaginary part in the projectile-target interaction.

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.

Simultaneous analysis of elastic, breakup, and fusion channels for6He induced reactions at energies near the Coulomb barrier

We present some recent applications of the Continuum-Discretized Coupled-Channels (CDCC) and Coupled-Reaction-Channels (CRC) methods to the study of reactions induced by weakly-bound nuclei on stable targets. The methods are applied to the scattering of the Borromean nucleus 6 He at energies around the Coulomb barrier. Elastic scattering data are very well described by four-body CDCC calculations (based on a three-body description of the 6 He nucleus) but also by simpler three-body CDCC calculations, using a suitable two-body model for 6 He. By contrast, inclusivedata are better described using the CRC approach, which treats the breakup within the transfer to the continuum picture and, unlike the CDCC method, allows the inclusion of transfer to bound states of the target. We explore also the possibility of calculating the fusion cross sections using the CRC framework.

Long range effects on the optical model of 6He around the Coulomb barrier

We present an optical model (OM) analysis of the elastic scattering data of the reactions 6He + 27Al and 6He + 208Pb at incident energies around the Coulomb barrier. The bare part of the optical potential is constructed microscopically by means of a double folding procedure, using the São Paulo prescription without any renormalization. This bare interaction is supplemented with a Coulomb dipole polarization (CDP) potential, which takes into account the effect of the dipole Coulomb interaction. For this CDP potential, we use an analytical formula derived from the semiclassical theory of Coulomb excitation. The rest of the optical potential is parametrized in terms of Woods–Saxon shapes. In the 6He + 208Pb case, the analysis confirms the presence of long range components, in agreement with previous works. Four-body Continuum-Discretized Coupled-Channels calculations have been performed in order to better understand the features of the optical potentials found in the OM analysis. This study searches to elucidate some aspects of the optical potential of weakly bound systems, such as the dispersion relation and the long range (attractive and absorptive) mechanisms.

Four-body continuum-discretized coupled-channels calculations

The development of a continuum-bin scheme of discretization for three-body projectiles is necessary for studies of reactions of Borromean nuclei such as $^{6}\mathrm{He}$ within the continuum-discretized coupled-channels approach. Such a procedure, for constructing bin states on selected continuum energy intervals, is formulated and applied for the first time to reactions of a three-body projectile. The continuum representation uses the eigenchannel expansion of the three-body $S$ matrix. The method is applied to the challenging case of the $^{6}\mathrm{He}+^{208}\mathrm{Pb}$ reaction at 22 MeV, where an accurate treatment of both the Coulomb and the nuclear interactions with the target is necessary.

Four-body continuum-discretized coupled-channels calculations using a transformed harmonic oscillator basis

The scattering of a weakly bound three-body system by a target is discussed. A transformed harmonic oscillator basis is used to provide an appropriate discrete and finite basis for treating the continuum part of the spectrum of the projectile. The continuum-discretized coupled-channels framework is used for the scattering calculations. The formalism is applied to different reactions, $^{6}\mathrm{He}$$+$$^{12}\mathrm{C}$ at 229.8 MeV, $^{6}\mathrm{He}$$+$$^{64}\mathrm{Zn}$ at 10 and 13.6 MeV, and $^{6}\mathrm{He}$$+$$^{208}\mathrm{Pb}$ at 22 MeV, induced by the Borromean nucleus $^{6}\mathrm{He}$. Both the Coulomb and nuclear interactions with a target are taken into account.