Continuum-discretized Coupled-channels Calculations Research Articles

Elastic scattering and total reaction cross sections of 6Li examined via a microscopic continuum discretized coupled-channels model* *Supported by the National Natural Science Foundation of China (U2067205)

We present a systematic study of 6Li elastic scattering and total reaction cross sections at incident energies around the Coulomb barrier within the continuum discretized coupled-channels (CDCC) framework, where 6Li is treated in an α+d two-body model. Collisions with 27Al, 64Zn, 138Ba, and 208Pa are analyzed. The microscopic optical potentials (MOP) based on Skyrme nucleon-nucleon interaction for α and d are adopted in CDCC calculations and satisfactory agreement with the experimental data is obtained without any adjustment on MOPs. For comparison, α and d global phenomenological optical potentials (GOP) are also used in CDCC analysis and a reduction of no less than 50% on the surface imaginary part of deuteron GOP is required for describing the data. In all cases, the 6Li breakup effect is significant and provides repulsive correction to the folding model potential. The reduction on the surface imaginary part of GOP of deuteron reveals a strong suppression of the reaction probability of deuteron as a component of 6Li when compared with that of a free deuteron. Further investigation is performed by considering the d breakup process equivalently within the dynamic polarization potential approach, and the results show that d behaves in a manner similar to a tightly bound nucleus in 6Li induced reactions.

Effects of deuteron breakup and nucleon-transfer reactions on d+^{11}B elastic scattering

A set of experimental data consisting of angular distributions of d+^{11}B elastic and inelastic scattering and (d,t) reaction obtained at deuteron energy of 14.5 MeV as well as data sets for (d,p) (E_d = 21.5 MeV), (d,n) (E_d = 15.4 MeV) and (d,^3He) (E_d = 11.8 MeV) nucleon transfer reactions were analysed using the continuum-discretized coupled-channel (CDCC) and coupled-reaction channel (CRC) methods. It was found that at forward scattering angles the elastic scattering data are well described by parameter-free CDCC calculations which include breakup of the deuteron, but at more backward angles virtual effects due to target excitation and neutron-transfer reactions play an important role. The strongest effects are due to the excitation of ^{11}B and the neutron pickup reaction, reflecting their large cross sections. The effects induced by proton-transfer reactions are found to be negligible.

Elastic, inelastic, and one-neutron transfer angular distributions of Li6+Sn120 at energies near the Coulomb barrier

The elastic scattering, first ${2}^{+}$ and ${3}^{\ensuremath{-}}$ target inelastic excitation and one neutron pickup angular distributions for the $^{6}\mathrm{Li}+^{120}\mathrm{Sn}$ reaction have been measured for three bombarding energies (19, 24, and 27 MeV). Data have been analyzed through coupled-channel calculations and continuum-discretized coupled-channel calculations extended to include target excitation. In general, both theoretical models give a reasonable description of the data. For the elastic and inelastic angular distributions taken at ${E}_{\mathrm{lab}}=24$ and 27 MeV, the continuum-discretized coupled-channel results are slightly better in comparison to the coupled-channel predictions. For the elastic and inelastic angular distributions measured at ${E}_{\mathrm{lab}}\phantom{\rule{4pt}{0ex}}=19$ MeV, the effect of the break-up channel seems to be quite important. At this energy, the elastic scattering data can be well explained by coupled channel calculations in which a strong absorptive optical imaginary potential is considered. In particular, the continuum-discretized coupled-channel theoretical results provided the best description of the ${3}^{\ensuremath{-}}$ excitation data at 19 MeV.

Application of the Lagrange-mesh method in continuum-discretized coupled-channel calculations

We apply the Lagrange-mesh method to discretize continuum states of weakly bound nuclei for continuum-discretized coupled-channel (CDCC) calculations of three-body breakup reactions. This discretization method is compared with the bin method, which is regarded as the standard continuum discretization method, for the d and 6Li induced reactions. An improved Numerov algorithm is used to solve the coupled channels equations, which permits a fast integration of equations and a convenient treatment of the closed channels. A new CDCC model code named CDCC-R is developed. In all cases, the combination of the Lagrange-mesh method and the Numerov algorithm shows high efficiency and accuracy for the CDCC calculations of the elastic scattering and breakup reactions. Especially, various numerical and physical aspects are discussed for 6Li induced reactions. The coupling effects of the continuum states with high orbital angular momentum and closed channels are discussed. Moderate effects are found in the calculations for elastic scattering and breakup reaction when F- and G-wave continuum states are included in CDCC model space for 6Li induced reactions at the incident energies well above the Coulomb barrier. The closed channel effect is found to be indispensable for 6Li breakup reaction calculation when the incident energy is around the Coulomb barrier.

Systematic calculations of reactions with exotic and stable nuclei to establish a unified theoretical approach

We report on systematical optical model (OM) and continuum discretized coupled channel (CDCC) calculations applied to describe the elastic scattering angular distributions of exotic and stable nuclei projectiles on heavy targets. Our optical potential (OP) is composed of the nuclear microscopic double folding S\~ao Paulo potential (SPP), derived from the nonlocal nature of the interaction, and the Coulomb dipole polarization (CDP) potential, derived from the semiclassical theory of Coulomb excitation. The OP is compared to the trivial equivalent local potential (TELP), extracted from CDCC calculations. The OM and CDCC predictions corroborate each other and account for important differences in the nuclei reaction mechanisms, which are directly related to their structural properties. Thus, OM and CDCC establish a common basis for analyzing or even predicting exotic and stable nuclei reactions.

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.

An Analysis of Elastic Scattering Angular Distributions of $$^{17}$$F on a $$^{208}$$Pb Target at Different Energies

This study investigated the elastic scattering of a $$^{\mathrm {17}}$$ F $$+^{\mathrm {208}}$$ Pb system at different energies using the Continuum-Discretized Coupled Channels (CDCC) method. A microscopic optical model was selected and double-folding potential was used for the $$^{\mathrm {16}}$$ O $$+$$ target interaction. The real part of the optical model was determined as folding potential and the imaginary part of the optical model was taken as standard volume Woods–Saxon shape. Single-channel and multi-channel CDCC calculations were performed to determine the effect of excited channels. An elastic scattering cross section of the $$^{\mathrm {17}}$$ F $$+^{\mathrm {208}}$$ Pb system was calculated and its compatibility with experimental data was determined.

Progress on reaction mechanisms of the proton drip-line nucleus <sup>17</sup>F at energies around the Coulomb barrier

<p indent=0mm>The reaction mechanism of proton drip-line nuclear systems at energies around the Coulomb barrier is one of the most popular topics in the field of nuclear physics. The present status of reactions induced by ¹⁷F at energies around the Coulomb barrier is reviewed in this paper. To make a further investigation of the reaction mechanisms of ¹⁷F, our group performed the measurements of elastic scattering of ¹⁷F+¹²C, ⁸⁹Y and ²⁰⁸Pb, as well as the complete kinematics of ¹⁷F+⁵⁸Ni at energies around the Coulomb barrier. For the elastic scattering, the excitation to the first excited state of ¹⁷F cannot be distinguished, hence the results have to be considered as the quasi-elastic scattering. The continuum-discretized coupled-channels (CDCC) calculations were performed to investigate the couplings to the continuum states on the elastic scattering. Both the calculations with and without the couplings to the continuum states can describe the quasi-elastic scattering data properly. The results indicate that, in a wide target-mass range, the couplings to the continuum states of ¹⁷F are not significant. For the breakup mechanism, breakup fragments of ¹⁷F+⁵⁸Ni and ²⁰⁸Pb were measured at energies above the Coulomb barrier. CDCC calculation could reproduce the breakup angular distribution of ¹⁷F+⁵⁸Ni well. However, it significantly overestimated the results of ¹⁷F+²⁰⁸Pb. The possible reason is that the excitation of the core, ¹⁶O, was not taken into account in the calculation. However, the underlying physics needs a further study. Within the energy region close to the Coulomb barrier, the breakup reactions of ¹⁷F+⁵⁸Ni were measured in single and in coincidence. The CDCC and the newly developed three-body model (the IAV model) were adopted to analyze the breakup data. The coincident data could be described well by the CDCC calculations. The sum of CDCC and IAV, which is referred to be as the total breakup, can reproduce properly the angular distributions of ¹⁶O. Moreover, the results also indicate that the non-elastic breakup dominates the ¹⁶O production. For the fusion measurement, only the results of ¹⁷F+⁵⁸Ni and ²⁰⁸Pb have been reported so far. The total fusion cross section of ¹⁷F+⁵⁸Ni was derived by measuring the fusion-evaporated protons and alphas. Compared with ¹⁶O+⁵⁸Ni, an obvious enhancement was observed at the energy below the Coulomb barrier. CDCC calculation indicates that such an enhancement is mainly due to the couplings to the continuum states. While for ¹⁷F+²⁰⁸Pb, the total fusion cross section was deduced by measuring the fusion-fission fragments in coincidence. The fusion excitation function of ¹⁷F+²⁰⁸Pb is almost identical with the stable system ¹⁹F+²⁰⁸Pb, indicating that the weakly bound nature of ¹⁷F does not play an important role in the fusion of ¹⁷F+²⁰⁸Pb. The underlying physics of the difference between ¹⁷F+⁵⁸Ni and ²⁰⁸Pb strongly deserves a further investigation. Furthermore, to establish the systematics of the reaction dynamics of ¹⁷F, reaction measurements of ¹⁷F on a range of targets at energies around and below the Coulomb barrier will be valuable. The results of ¹⁷F further indicate that the complete kinematics measurement could be the only promising approach to understand the reaction mechanisms of weakly bound nuclear systems comprehensively and provide the convincing data to promote the development of nuclear reaction theory.

Elastic scattering and breakup reactions of the proton drip-line nucleus B8 on Pb208 at 238 MeV

Elastic scattering and breakup angular distributions of the weakly bound radioactive nucleus B8 on a Pb208 target at an incident energy of 238 MeV, which corresponds to four times the Coulomb barrier, have been measured at the HIRFL-RIBLL facility (Institute of Modern Physics, Lanzhou). The data have been analyzed using the optical model and the continuum discretized coupled channels (CDCC) formalism. The measured and calculated elastic scattering angular distributions do not show any significant Coulomb rainbow suppression. The angular distribution for the breakup reaction was measured for the first time at this energy. The angular distribution of the Be7 fragments could be reproduced considering elastic plus nonelastic breakup contributions, with the former evaluated with the CDCC calculations and the latter with the model of Ichimura, Austern, and Vincent [Phys. Rev. C 32, 431 (1985)]. The comparison of the breakup cross section of B8 with that of Be11 suggests that the Coulomb and centrifugal barriers encountered by the valence proton may suppress the breakup cross section.Received 25 October 2020Revised 9 December 2020Accepted 13 January 2021DOI:https://doi.org/10.1103/PhysRevC.103.024606©2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBreakup reactionsDirect reactionsLow & intermediate energy heavy-ion reactionsNuclear reactionsUnstable nuclei induced nuclear reactionsTechniquesOptical, coupled-channel & distorted wave modelsNuclear Physics

Role of continuum in nuclear direct reactions with one-neutron halo nuclei: A one-dimensional model

We study the evolution of a single-particle wave function during the collision of a one dimensional potential well by another well, which can be regarded as a simple model for the problem of the scattering of a one-neutron halo nucleus by another nucleus. This constitutes an effective three-body problem, whose solution in three dimensions can be extremely complicated, particularly when breakup and rearrangement channels are to be considered. Our one-dimensional model provides the essential three-body nature of this problem, and allows for a much simpler application and assessment of different methods of solution. To simplify further the problem, we assume that the potential well representing the projectile moves according to a predetermined classical trajectory, although the internal motion of the "valence" particle is treated fully quantum-mechanically. This corresponds to a semiclassical approach of the scattering problem. Different approaches are investigated to understand the dynamics involving one-body halo-like systems: the "exact" time-dependent solution of the Schr\"odinger equation is compared to a numerical continuum-discretized coupled-channels (CC) calculation presenting various model cases including different reaction channels. This framework allows us to discuss the reaction mechanism and the role of continuum, whose inclusion in the CC calculation results to be crucial to reproduce the "exact" solution, even when the initial and final states are well bound. We also link each dynamical situation with analogous problem solved in a three dimensional (3D) CC framework, discussing the main challenges experienced in the usual 3D models.

An investigation of effect of different nuclear potentials on CDCC calculations: 11Be + 120Sn reaction

Aim of this study is to examine the effect of various potentials on continuum discretized coupled channel (CDCC) calculations. For this, the elastic scattering cross section of 11Be projectile from 120Sn target is calculated by using ten different potentials. The results are compared with experimental data.

Insight into the reaction dynamics of proton drip-line nuclear system 17F+58Ni at near-barrier energies

The mechanism of reactions with weakly-bound proton-rich nuclei at energies near the Coulomb barrier is a long-standing open question owing to the paucity of experimental data. In this study, a complete kinematics measurement was performed for the proton drip-line nucleus 17F interacting with 58Ni at four energies near the Coulomb barrier. Thanks to the powerful performance of the detector array, exhaustive information on the reaction channels, such as the differential cross sections for quasielastic scattering, exclusive and inclusive breakup, as well as for fusion-evaporation protons and alphas, was derived for the first time. The angular distributions of quasielastic scattering and exclusive breakup can be described reasonably well by the continuum-discretized coupled-channels calculations. The inclusive breakup was investigated using the three-body model proposed by Ichimura, Austern, and Vincent, and results indicate the non-elastic breakup is the dominant component. The total fusion cross sections were determined by the fusion-evaporation protons and alphas. Based on the measured exclusive breakup data, the analysis of the classical dynamical simulation code PLATYPUS demonstrates that the incomplete fusion plays a minor role. Moreover, compared with 16O+58Ni, both the reaction and total fusion cross sections of 17F+58Ni exhibit an enhancement in the sub-barrier energy region, which mainly arises from couplings to the continuum states. This work indicates that the information of full reaction channels is crucially important to comprehensively understand the reaction mechanisms of weakly bound nuclear systems.

The refractive scattering of 17F+12C

The experimental data of the elastic scattering angular distribution of 17F+12C at 170 MeV is analyzed by the continuum-discretized coupled channels (CDCC) method and the optical model (OM). In the CDCC calculation, the unambiguous optical potential of 16O+12C is used as the input to give the coupling potentials. A very refractive feature is found and two evident Airy minima are predicted at large angles. The one-channel calculation is also performed and gives nearly the same result. In the OM calculations, this optical potential of 16O+12C is used again and adjusted to reproduce the angular distribution of 17F+12C. The Airy oscillation appears again in the calculated angular distribution. These results indicate that the elastic scattering of 17F+12C at 170 MeV has the possibility of the nuclear rainbow phenomenon, which is probably due to the contribution from the 16O core.

Strong coupling effect in the elastic scattering of the C10+Ni58 system near barrier

The elastic scattering of the radioactive proton-rich carbon isotope $^{10}\mathrm{C}$ has been investigated. A full differential angular distribution for the elastic scattering of $^{10}\mathrm{C}$ on a $^{58}\mathrm{Ni}$ target has been measured at ${E}_{\mathrm{Lab}}=35.3$ MeV, which is just above the Coulomb barrier for this system (${V}_{B}=27$ MeV). The obtained angular distribution was analyzed in terms of coupled-channels, coupled-reaction-channels, and continuum-discretized coupled-channels calculations. In the coupled-channels calculation, several inelastic transitions of the target and the first (${2}^{+}$) excited state of the $^{10}\mathrm{C}$ projectile were considered. The coupling to the first (${2}^{+}$) excited state of $^{10}\mathrm{C}$ was shown to be very important to describe the rising of the cross section at backward angles. We also performed an analysis with the continuum-discretized coupled channels, considering the $^{9}\mathrm{B}+p$ and $^{6}\mathrm{Be}+\ensuremath{\alpha}$ channels, which indicated a strong cluster configuration for the $^{10}\mathrm{C}$ projectile.

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.

Reactions with Exotic Nuclei at Near and Sub-barrier Energies

We will introduce some new results derived recently by the nuclear reaction group at China Institute of Atomic Energy, on the properties of the optical model potentials of neutron-halo 6He system and reaction mechanisms induced by proton-drip line nucleus 17F at energies around the Coulomb barrier. For the study of optical model potentials of exotic nuclear system, we proposed a novel method, i.e., the transfer reaction method. This method has been applied to extract the optical potentials of neutron-halo 6He+209Bi system by measuring the one-proton transfer reactions induced by 7Li on a 208Pb target. A complete picture of threshold anomaly behavior was obtained in the 6He+209Bi system for the first time, where a decreasing trend of the depth of the imaginary potential is observed in the deep sub-barrier region, and the reaction threshold energy is extracted. Moreover, results show thatthe dispersion relation is not applicable for this exotic nuclear system. Reaction mechanisms of proton-rich nuclear systems 17F+89Y and 58Ni were also studied at energies around the Coulomb barrier. Continuum discretized coupled-channels calculations indicate that the coupling effects of the continuum states of 17F+89Y is not significant. Thanks to the employment of a powerful ionization-chamber based detector array, the reaction products over a large Z in the systems of 17F+58Ni can be identified clearly. The data analysis is undergoing.

Using a quadrupole deformed generalized Woods–Saxon plus spin-orbit potential to describe the polarized interaction 7Li→ +12C at 34MeV

An overall description of the cross-sections and tensor analyzing powers corresponding to the elastic interaction [Formula: see text] at [Formula: see text] has been obtained with a coupled channel analysis using the rotational model. Based on the high electric quadrupole moment of the [Formula: see text] projectile, an optical model potential equal to the sum of a quadrupole deformed generalized Woods–Saxon potential and a standard Spin-Orbit term has been used. The deformed part of the potential induces reorientations and transitions that involve the first four energy states of the [Formula: see text] nucleus. These states are supposed to belong to an inverted [Formula: see text][Formula: see text][Formula: see text] rotational band proposed in the literature. Partial reproductions of the inelastic interaction [Formula: see text] have also been achieved by using the same deformed potential with the same parameter values that describe the elastic interaction. Comparisons are made with previous results obtained from optical model potential and continuum-discretized coupled-channel calculations using the cluster-folding model. The use of a deformed scattering potential introduces a tensor term that allows the reproduction of the tensor analyzing powers.

Elastic scattering, inelastic excitation, and 1n pick-up transfer cross sections for B10 + Sn120 at energies near the Coulomb barrier

The $^{10}\mathrm{B}+^{120}\mathrm{Sn}$ reaction has been systematically studied at laboratory energies around the Coulomb barrier: ${E}_{\mathrm{LAB}}=31.5$, 33.5, 35.0, and 37.5 MeV. Cross sections for the elastic scattering and some reaction processes have been measured: excitation to the ${1}^{+}$ state of $^{10}\mathrm{B}$; excitation to the ${2}^{+}$ and ${3}^{\ensuremath{-}}$ states of $^{120}\mathrm{Sn}$; and the one-neutron pick-up transfer $^{120}\mathrm{Sn}(^{10}\mathrm{B},^{11}\mathrm{B})^{119}\mathrm{Sn}$. Coupled reaction channel (CRC) calculations have been performed in the context of the double-folding S\~ao Paulo potential. The theoretical calculations result in a good overall description of the experimental angular distributions. The effect on the theoretical elastic-scattering angular distributions of couplings to the inelastic and transfer states (through the CRC calculations) and to the continuum states (through continuum-discretized coupled-channels calculations) has been investigated.

Analysis of total reaction cross sections for deuterons on 1p-shell-nuclei

This study aims to analyze the differential cross sections (DCSs) of elastic scattering and total reaction cross sections (TRCSs) of the loosely-bound deuteron projectile impinging on 1 p -shell nuclei, such as $^{9}$Be, $^{12}$C, and $^{16}$O, at incident energies ranging between 10.6 and 171 MeV using the continuum discretized coupled channel (CDCC) method. By fitting the experimental data for the DCSs and TRCSs, energy-dependent renormalization factors for the real and imaginary parts of the nucleon-nucleus optical-model potentials deduced from the studies proposed by Koning and Delaroche (KD02) and by Watson, Singh, and Segel (WSS), are obtained.It is found that with the WSS potential, which was obtained specifically for $1p$-shell nuclei, the CDCC calculations can simultaneously reproduce both the DCSs and the TRCSs. The results show that it is important to choose appropriate optical potentials to describe deuteron-induced reactions.

Energy evolution of channel coupling for the system Li6+Ni64 at near-barrier energies

Background: Different theoretical models have been used to understand the effects of coupling of direct reaction channels on reaction observables in recent years. However, very few attempts have been made to check the consistency of the model calculations in terms of comparing different experimental observables with the model predictions.Purpose: In the present work, coupled channel calculations are performed with transfer and breakup coupling for collision of the weakly bound stable projectile $^{6}\mathrm{Li}$ with the medium-mass target $^{64}\mathrm{Ni}$ at near-barrier energies. The main goal is to find the consistency of the model calculation in simultaneous reproduction of reaction observables like the elastic angular distributions, back-angle elastic excitation function, and fusion excitation function to find the effective energy regions of dominance of these couplings.Method: Both coupled reaction channel (CRC) and continuum discretized coupled channel (CDCC) calculations are performed using the code fresco. Transfer coupling is included through the CRC formalism. The CDCC scheme is used for breakup coupling and to distinguish the effect of resonant and nonresonant couplings.Results: In the case of the CRC calculation, the full coupling including inelastic scattering, $1n$ stripping, and $1p$ stripping gives the best description of the experimental data and particularly reproduces the data in the subbarrier energy region. It is observed that the $1p$-transfer channel has a larger impact on reaction observables compared to $1n$ transfer. On the other hand, for breakup nonresonant breakup has the dominant effect on the reaction observables, although resonant breakup has a larger cross section. The full breakup coupling reproduces the elastic scattering angular distributions and excitation function in the above-barrier region. But the scheme overpredicts these observables in the subbarrier region.Conclusion: CRC and CDCC calculations are complementary to each other. The energy dependencies of the relevant couplings for the $^{6}\mathrm{Li}+^{64}\mathrm{Ni}$ system around the barrier show that inelastic and transfer coupling are important at energies below the Coulomb barrier but continuum coupling is dominant in the above-barrier energy region.