Continuum-discretized Coupled-channels Method Research Articles

Continuum-discretized coupled-channels calculations for Li6 fusion reactions with closed channels

Fusion reactions induced by the weakly bound nucleus $^{6}\mathrm{Li}$ with targets $^{28}\mathrm{Si}, ^{64}\mathrm{Ni}, ^{144}\mathrm{Sm}$, and $^{209}\mathrm{Bi}$ at energies around the Coulomb barrier are investigated within a three-body model where $^{6}\mathrm{Li}$ is described with an $\ensuremath{\alpha}+d$ cluster model. The total fusion (TF) cross sections are calculated with the continuum-discretized coupled-channels (CDCC) method and the complete fusion (CF) cross sections are extracted through the sum-rule model. The calculations demonstrate that (i) for the TF cross-section calculations, the continuum states up to 40 MeV are found to be necessary, which corresponds to the inclusion of closed channels for light and medium mass targets, such as $^{28}\mathrm{Si}, ^{59}\mathrm{Co}$, and $^{144}\mathrm{Sm}$; (ii) the converged CDCC results for TF cross section at energies above the Coulomb barrier are almost the same as single-channel results in which the continuum coupling effect is neglected; and (iii) the continuum coupling strongly influences partial-wave fusion cross sections and the closed channels play a significant role in the improvement of the description of the CF cross sections at energies below the Coulomb barrier for the $^{6}\mathrm{Li}+^{28}\mathrm{Si}, ^{59}\mathrm{Co}$, and $^{144}\mathrm{Sm}$ systems.

Cluster scattering in the non-local model

Scattering of weakly bound nuclei with a pronounced cluster structure is strongly affected by their breakup. Usually, this mechanism is accounted for in a three-body model with pairwise potentials. The interaction potentials between complex systems are non-local due to the existence of excitation channels and antisymmetrization. However, a common practice is to use local optical potentials in cluster scattering studies. To assess the validity of replacing non-local optical potentials by their local equivalents, we extend the local-equivalent continuum-discretized coupled-channel (LECDCC) approach proposed by us for deuteron scattering in [Phys. Rev. C98, 011601(R) (2018)] to the case of cluster scattering. We consider the case of 6Li + 120Sn at 27 and 60 MeV, and compare the angular distributions and reaction cross sections for elastic and breakup cross sections with those obtained in the standard continuum-discretized coupled-channel (CDCC) method with local equivalents of non-local potentials. We found that while elastic scattering is not significantly affected by non-locality, the breakup observables could be affected by up to 20% depending on kinematical conditions of their observation.

Analysis of 6Li+16O system using different theoretical methods

In this paper, we report on the [Formula: see text] system to which the crystal model (CM), the coupled reaction channel method (CRC), and continuum discretized coupled channel (CDCC) method were applied in addition to the distorted waves-Born approximation (DWBA), which was also applied to the [Formula: see text] system. Also, CM is folded with inelastic scattering. The analysis of the experimental data was performed simply with elastic and inelastic scattering in a first step. Then transfers have been studied like pp, nn, dd, [Formula: see text] and [Formula: see text]B using the CRC method. The CRC method was used to study many suggested combinations of transfers with elastic scattering. The analyses were performed for 6Li elastically scattered from [Formula: see text]O in the energy range of 13–50[Formula: see text]MeV. The cluster structure of [Formula: see text] has been examined during the CDCC method in addition to the breakup of the weakly bound structure of the same nucleus. A dynamic polarization potential (DPP) was used at [Formula: see text][Formula: see text]MeV. A comparison between different models was performed. A general agreement was found between the experimental data and the proposed theoretical methods.

Elastic scattering of 6Li from different target nuclei using continuum discretized coupled channels

In this study, the elastic scattering angular distributions (ADs) for the weakly stable 6Li nucleus from [Formula: see text]Ca, [Formula: see text]Ni and [Formula: see text]Zr targets are theoretically studied at various energies. The ADs for the following systems are considered: The [Formula: see text] system in the energy range of 32–240[Formula: see text]MeV; the [Formula: see text] system in the energy range of 9.9–210[Formula: see text]MeV; and the [Formula: see text] system in the energy range of 29.93–240[Formula: see text]MeV. The sophisticated continuum discretized coupled channels (CDCC) method is used to reanalyze the considered data. Reasonable agreements between the data considered and the theoretical calculations within the CDCC method are obtained. Values for the reaction cross-sections of the several systems under consideration are extracted, and they are compared to those previously reported ones.

Core-valence absorption in breakup and stripping reactions and its isospin dependence

In this work, the effect of nucleon-core absorption on nucleon removal reaction is explored through the use of complex nucleon-core interactions. Results are presented for exclusive breakup reactions, where absorption is explored through the use of a binormal basis in the continuum-discretized coupled-channel method, and for stripping nucleon knockout reactions, where absorption is considered through the application of an effective density in the eikonal approximation. Both methods show an increased effect of nucleon-core absorption when removing a deeply-bound nucleon, which leads to smaller cross sections, a reduction that is larger than in the weakly-bound case.

Importance of deuteron breakup in the deuteron knockout reaction

Background: An isoscalar $pn$ pair is expected to emerge in nuclei that have similar proton and neutron numbers and it may be a candidate for a deuteron ``cluster.'' There is, however, no experimental evidence for it.Purpose: The purpose of this paper is to construct a new reaction model for the $(p,pd)$ reaction including the deuteron breakup in the elementary process and the deuteron reformation by the final-state interactions. How these processes contribute to the observables of the reaction is investigated.Methods: The distorted wave impulse approximation is extended twofold. The elementary processes of the $(p,pd)$, i.e., the $p\text{\ensuremath{-}}d$ elastic scattering and the $d(p,p)pn$ reaction, are described with an impulse picture employing a nucleon-nucleon effective interaction. The three-body scattering waves in the final state of the $(p,pd)$ reaction are calculated with the continuum-discretized coupled-channels method. The triple-differential cross section (TDX) of the $(p,pd)$ reaction is calculated with the new model.Results: The elementary processes are described reasonably well with the present model. As for the $(p,pd)$ reaction, the deuteron reformation can either increase or decrease the TDX height depending on the interference between the elastic and breakup channels of deuteron, while the back-coupling effect always decreases it.Conclusions: It is shown that the deuteron reformation significantly changes the TDX of the $(p,pd)$ reaction through the interference. It is important to include this process to quantitatively discuss the $(p,pd)$ cross sections in view of the deuteron formation in nuclei. For more quantitative discussion regarding the experimental data, further improvement will be necessary.

Coherent description of elastic scattering and fusion at near-barrier energies for the Be9+Pb208 and Be9+Au197 reactions

Elastic scattering, total reaction, and fusion cross sections of the systems $^{9}\mathrm{Be}+^{208}\mathrm{Pb}$ and $^{9}\mathrm{Be}+^{197}\mathrm{Au}$ at near-barrier energies, are coherently analyzed into two theoretical contexts. In a phenomenological approach, a BDM3Y1 interaction is used in an optical model potential analysis of the elastic-scattering data and the energy dependence of the potential is mapped around the Coulomb barrier. In a second approach, the elastic scattering is investigated within a four body framework by using the continuum-discretized coupled-channels method. In both scenarios, total reaction cross sections are determined and compared with phenomenological predictions. Moreover, an effective potential is obtained and in a one barrier penetration model analysis, the fusion excitation function is determined and compared with experimental data, further validating our analysis.

Investigation of multistep effects for proton inelastic scattering to the 21+ state in He6

Multistep effects among bound, resonant, and nonresonant states have been investigated by the continuum-discretized coupled-channels method (CDCC). In the CDCC, a resonant state is treated as multiple states fragmented in a resonance energy region, although it is described as a single state in usual coupled-channels calculations. For such the fragmented resonant states, one-step, and multistep contributions to the cross sections should be carefully discussed because the cross sections obtained by the one-step calculation depend on the number of those states, which corresponds to the size of the model space. To clarify the role of the multistep effects, we propose the one-step calculation without model-space dependence for the fragmented resonant states. Furthermore, we also discuss the multistep effects among the ground, ${2}_{1}^{+}$ resonant, and nonresonant states in $^{6}\mathrm{He}$ for proton inelastic scattering.

Three-body description of C9 : Role of low-lying resonances in breakup reactions

The $^9$C nucleus and related capture reaction, ${^8\mathrm{B}}(p,\gamma){^9\mathrm{C}}$, have been intensively studied with an astrophysical interest. Due to the weakly-bound nature of $^9$C, its structure is likely to be described as the three-body (${^7\mathrm{Be}}+p+p$). Its continuum structure is also important to describe reaction processes of $^9$C, with which the reaction rate of the ${^8\mathrm{B}}(p,\gamma){^9\mathrm{C}}$ process have been extracted indirectly. We perform three-body calculations on $^9$C and discuss properties of its ground and low-lying states via breakup reactions. We employ the three-body model of $^9$C using the Gaussian-expansion method combined with the complex-scaling method. This model is implemented in the four-body version of the continuum-discretized coupled-channels method, by which breakup reactions of $^9$C are studied. The intrinsic spin of $^7$Be is disregarded. By tuning a three-body interaction in the Hamiltonian of $^9$C, we obtain the low-lying $2^+$ state with the resonant energy 0.781 MeV and the decay width 0.137 MeV, which is consistent with the available experimental information and a relatively high-lying second $2^+$ wider resonant state. Our calculation predicts also sole $0^+$ and three $1^-$ resonant states. We discuss the role of these resonances in the elastic breakup cross section of $^9$C on $^{208}$Pb at 65 and 160 MeV/A. The low-lying 2$^+$ state is probed as a sharp peak of the breakup cross section, while the 1$^-$ states enhance the cross section around 3 MeV. Our calculations will further support the future and ongoing experimental campaigns for extracting astrophysical information and evaluating the two-proton removal cross-sections.

An investigation of the elastic scattering of 17O projectiles by different target nuclei using the CDCC method

The Continuum Discretized Coupled Channels (CDCC) method is a convenient method that was developed in order to examine weakly bound nuclei. For this purpose, the elastic scattering data of 17 O projectile for 90 Zr, 124 Sn and 208 P b target nuclei were investigated at 340 MeV using the CDCC method. In calculations using this method, 17 O projectiles were taken to be .Optical potentials were selected as the interaction potentials. It was seen that the results obtained were compatible with the experimental data. The effects of excited channels in all three systems were also determined.

Effect of deuteron breakup on the deuteron- Ξ correlation function

The hadron-deuteron correlation function has attracted many interests as a potential method to access the three-hadron interactions. However, the weakly-bound nature of deuteron has not been considered in the preceding studies. In this study, the breakup effect of deuteron on the deuteron-$\Xi^-$ ($d$-$\Xi^-$) correlation function $C_{d\Xi^-}$ is investigated. The $d$-$\Xi^-$ scattering is described by a nucleon-nucleon-$\Xi$ three-body reaction model. The continuum-discretized coupled-channels method, which is a fully quantum-mechanical and non-perturbative reaction model, is adopted. $C_{d\Xi^-}$ turns out to be sensitive to the strong interaction and enhanced by the deuteron breakup effect by 6--8 % for the $d$-$\Xi^-$ relative momentum below about 70 MeV/$c$. Low-lying neutron-neutron continuum states are responsible for this enhancement. Within the adopted model, the deuteron breakup effect on $C_{d\Xi^-}$ is found to be appreciable but not very significant. Except for the enhancement by several percent, studies on $C_{d\Xi^-}$ without the deuteron breakup effect can be justified.

Practical method for decomposing discretized breakup cross sections into components of each channel

Background: In the continuum-discretized coupled-channel method, a breakup cross section (BUX) is obtained as an admixture of several components of different channels in multi-channel scattering. Purpose: Our goal is to propose an approximate way of decomposing the discretized BUX into components of each channel. This approximation is referred to as the "probability separation (P-separation)". Method: As an example, we consider $^{11}$Be scattering by using the three-body model with core excitation ($^{10}\mathrm{Be}+n+\mathrm{T}$, where T is a target). The structural part is constructed by the particle-rotor model and the reaction part is described by the distorted wave Born approximation (DWBA). Results: The validity of the P-separation is tested by comparing with the exact calculation. The approximate way reproduces the exact BUXs well regardless of the configurations and/or the resonance positions of $^{11}$Be. Conclusion: The method proposed here can be an alternative approach for decomposing discretized BUXs into components in four- or five-body scattering where the strict decomposition is hard to perform.

Investigation of contributions of the 22+ resonance in He6 via analysis of the He6(p, p′) reaction

We investigate the contribution of the $2^{+}_{2}$ resonance in $^6$He to observables via analysis of the $^6$He($p,p'$) reaction by using the continuum-discretized coupled channels method combined with the complex-scaling method. In this study, we obtain the $2^{+}_{2}$ state with the resonant energy 2.25 MeV and the decay width 3.75 MeV and analyse contributions of resonances and nonresonant continuum states to the cross section separately. It is found that the $2^{+}_{2}$ state plays an important role in the energy spectrum. Furthermore, contributions of nonresonant continuum states are also important to clarify the properties of the $2^{+}_{2}$ state.

Effect of the repulsive core in the proton-neutron potential on deuteron elastic breakup cross sections

The role of the short-range part (repulsive core) of the proton-neutron ($pn$) potential in deuteron elastic breakup processes is investigated. A simplified one-range Gaussian potential and the Argonne V4' (AV4') central potential are adopted in the continuum-discretized coupled-channels (CDCC) method. The deuteron breakup cross sections calculated with these two potentials are compared. The repulsive core is found not to affect the deuteron breakup cross sections at energies from 40 MeV to 1 GeV. To understand this result, an analysis of the peripherality of the elastic breakup processes concerning the $p$-$n$ relative coordinate is performed. It is found that for the breakup processes populating the $pn$ continua with orbital angular momentum $\ell$ different from 0, the reaction process is peripheral, whereas it is not for the breakup to the $\ell=0$ continua (the s-wave breakup). The result of the peripherality analysis indicates that the whole spatial region of deuteron contributes to the s-wave breakup.

Microscopic optical potentials including breakup effects for elastic scattering

Abstract We construct a microscopic optical potential including breakup effects for the elastic scattering of weakly binding projectiles within the Glauber model, in which a nucleon–nucleus potential is derived using the $g$-matrix folding model. The derived microscopic optical potential is referred to as the eikonal potential. For $d$ scattering, calculation with the eikonal potential, i.e., the EP model, reasonably reproduces the result of the exact calculation with the continuum-discretized coupled-channels method at intermediate energies. From the properties of the EP model, the inaccuracy of the eikonal approximation used in the Glauber model is partially excluded. Also analyzed with the eikonal potential is the scattering of $^6$He from $^{12}$C, and its applicability to scattering with many-body projectiles is shown.

Perey-effect in continuum-discretized coupled-channel description of (d, p) reactions

The Perey effect in two-body channels of (d, p) reactions has been known for a long time. It arises when the nonlocal two-body deuteron-target and/or proton-target problem is approximated by a local one, manifesting itself in a reduction of the scattering channel wave functions in the nuclear interior. However, the (d, p) reaction mechanism requires explicit accounting for three-body dynamics involving the target and the neutron and proton in the deuteron. Treating nonlocality of the nucleon-target interactions within a three-body context requires significant effort and demands going beyond the widely-used adiabatic approximation, which can be done using a continuum-discretized coupled-channel (CDCC) method. However, the inclusion of nonlocal interactions into the CDCC description of (d, p) reactions has not been developed yet. Here, we point out that, similarly to the two-body nonlocal case, nonlocality in a three-body channel can be accounted for by introducing the Perey factors. We explain this procedure and present the first CDCC calculations to our knowledge including the Perey effect.

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.

Examination of the adiabatic approximation for (d,p) reactions

Background: Deuteron-induced one-neutron transfer reactions have been used to extract single-particle properties of nuclei, and the adiabatic (AD) approximation is often used to simply treat the deuteron breakup states. Purpose: The primary goal is to examine the validity of the AD approximation for the ($d,p$) reaction systematically. We clarify also the role of the closed-channels often ignored in the description of breakup reactions. Methods: We calculate the ($d,p$) cross sections with the continuum-discretized coupled-channels method (CDCC) for 128 reactions systems and compare the results with those obtained by the CDCC calculation with the AD approximation. Effect of the closed channels are investigated by ignoring the closed channels in CDCC. Results: The AD approximation affects in general the ($d,p$) cross section by less than 20 %, but some exceptional nonadiabatic cases for which the AD approximation breaks down are found. The closed channels turn out to give significant effects on the cross section at deuteron energies less than about 10 MeV. Conclusions: The use of the AD approximation in the description of the ($d,p$) reaction can be justified in many cases, with the uncertainty of less than about 20 %. The existence of some nonadiabatic cases nevertheless should be realized. The neglect of the closed channels without confirming the convergence of the CDCC result is not recommended.

Four- and three-body breakup mechanism of 6Li elastic scattering

We investigate a breakup mechanism of 6Li elastic scattering on heavy targets (T = 209Bi or 208Pb) with the four-body version of the continuum-discretized coupled-channels method (four-body CDCC). Four-body CDCC successfully reproduces measured elastic cross sections with no adjustable parameter, and we can then clearly discuss the four-body dynamics. Our analysis shows that dα breakup (6Li + T → d + α + T) is much more essential than npα breakup (6Li + T → n + p + α + T) in 6Li scattering.

Extracting three-body breakup observables from continuum-discretized coupled-channels calculations with core excitations

Core-excitation effects in the scattering of two-body halo nuclei have been investigated in previous works. In particular, these effects have been found to affect in a significant way the breakup cross sections of neutron-halo nuclei with a deformed core. To account for these effects, appropriate extensions of the continuum-discretized coupled-channels (CDCC) method have been recently proposed. We aim to extend these studies to the case of breakup reactions measured under complete kinematics or semi-inclusive reactions in which only the angular or energy distribution of one of the outgoing fragments is measured. We use the standard CDCC method as well as its extended version with core excitations (XCDCC), assuming a pseudo-state basis for describing the projectile states. Two- and three-body observables are computed by projecting the discrete two-body breakup amplitudes, obtained within these reaction frameworks, onto two-body scattering states with definite relative momentum of the outgoing fragments and a definite state of the core nucleus. The presented method provides a tool to compute double and triple differential cross sections for outgoing fragments following the breakup of a two-body projectile, and might be useful to analyze breakup reactions with other deformed weakly-bound nuclei, for which core excitations are expected to play a role. We have found that, while dynamical core excitations are important for the proton target at intermediate energies, they are very small for the Zn target at energies around the Coulomb barrier.