Relativistic effects in the breakup of weakly-bound nuclei at intermediate energies are studied by means of the continuum-discretized coupled-channels method with eikonal approximation. Nuclear coupling potentials with Lorentz contraction are newly included and those effects on breakup cross sections are investigated. We show that relativistic corrections lead to larger breakup cross sections. Coupled-channel effects on the breakup cross sections are also discussed.
The scattering of $^9$Be on $^{208}$Pb is addressed within a four-body Continuum-Discretized Coupled-Channels (CDCC) framework, considering a three-body $α+ α+ n$ 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.
The development of a continuum-bins scheme of discretization for three-body projectiles is necessary for studies of reactions of Borromean nuclei such as 6He 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 6He + 208Pb reaction at 22 MeV, where an accurate treatment of both …
Background: Microscopic description of the projectile, based on an effective NN interaction, in a microscopic CDCC (MCDCC) model [PRL 111, 082701 (2013)] has been successfully applied to the 7Li+208Pb scattering. Purpose: The MCDCC method is applied to the low energy elastic scattering of 17F, 17O and 19F on 58Ni and 208Pb targets. The goal of the calculations is twofold - to test the adequacy and the accuracy of the MCDCC model for the heavier projectiles, and to study the contribution of various channels to the elastic scattering cross sections. Methods: The elastic scattering cross sections…
We investigate the elastic scattering of $^9$Be on $^{208}$Pb at beam energies above (50 MeV) and below (40 MeV) the Coulomb barrier. The reaction is described within a four-body framework using the Continuum-Discretized Coupled-Channels (CDCC) method. The $^9$Be projectile states are generated using the analytical Transformed Harmonic Oscillator (THO) basis in hyperspherical coordinates. Our calculations confirm the importance of continuum effects at low energies.
Four-body continuum-discretized coupled-channels calculations using a transformed harmonic oscillator basis
讨论了弱束缚三体系统与目标的散射问题。一个变换的谐振子基础是用来提供一个适当的离散和有限的基础处理的连续部分的频谱的弹丸。连续离散耦合通道框架用于散射计算。该方法适用于不同的反应，6 He +12 C在229.8 MeV，6 He +64 Zn在10和13.6 MeV，和6 He +208 Pb在22 MeV，由Borromean核6 He诱导。库仑和核与目标的相互作用都被考虑在内。
Breakup reactions are one of the main tools for the study of exotic nuclei, and in particular of their continuum. In order to get valuable information from measurements, a precise reaction model coupled to a fair description of the projectile is needed. We assume that the projectile initially possesses a cluster structure, which is revealed by the dissociation process. This structure is described by a few-body Hamiltonian involving effective forces between the clusters. Within this assumption, we review various reaction models. In semiclassical models, the projectile-target relative motion is …
The effect of core excitation in the elastic scattering and breakup of a two-body halo nucleus on a stable target nucleus is studied. The structure of the weakly-bound projectile is described in the weak-coupling limit, assuming a particle-rotor model. The eigenfunctions and the associated eigenvalues are obtained by diagonalizing this Hamiltonian in a square-integrable basis (pseudo-states). For the radial coordinate between the particle and the core, a transformed harmonic oscillator (THO) basis is used. For the reaction dynamics, an extension of the Continuum-Discretized Coupled-Channels (C…
Sensitivity of 8B breakup cross section to projectile structure in CDCC calculations
鉴于$^7$ Be $(p,γ)^8$ B的天体物理学意义，已经有几个实验应用库仑解离方法提取捕获率。密歇根州立大学的测量主要由$E1$贡献，但有一个小的$E2$组件。另一方面，在巴黎圣母院的较低能量测量具有更强的$E2$贡献。预期这两个测量值将束缚$E2$，从而允许准确提取与捕获过程相关的$E1$。这份简要报告的目的是表明，裂解反应中的$E2$因子不会转化为相应捕获反应中$E2$贡献的比例。我们表明，$^8$ B单粒子参数的变化，这是直接相关的$E2$组件在捕获反应，不影响相应的分裂反应，使用本反应理论。
It was recently suggested that the odd-even staggering of reaction cross sections is an evidence of the pair- ing anti-halo effect on projectile radii. We define the dimensionless staggering parameters, Γrds and ΓR, for projectile radii and reaction cross sections, respectively, and analyze the relation between Γrds and ΓR for the scattering of 14,15,16C from a 12C target at 83 MeV/A by taking account of projectile-breakup and nuclear- medium effects newly with the microscopic version of the continuum discretized coupled-channels method. The value of ΓR is deviated from that of Γrds by the pro…
We investigate the low-energy $^9$Be elastic scattering on two different targets (heavy, light) within a four-body framework using the Continuum-Discretized Coupled-Channels (CDCC) method. The $^9$Be projectile is described in a $α+ α+ n$ three-body model using the analytical transformed harmonic oscillator (THO) basis in hyperspherical coordinates. We show that continuum couplings are important to describe the elastic cross section, especially at low energies and on heavy targets. The dipolar contribution to the elastic cross section at energies around the Coulomb barrier is important but sma…
The role of the breakup process and one neutron stripping on the near barrier fusion are investigated for the weakly bound projectile $^{9}$Be on $^{28}$Si, $^{89}$Y, $^{124}$Sn, $^{144}$Sm and $^{208}$Pb targets. Continuum-discretized coupled channels (CDCC) calculations for the breakup with a $^{8}$Be + n model of the $^{9}$Be nucleus and coupled reactions channels (CRC) calculations for the one neutron stripping to several single particle states in the target are performed for these systems. A good description of the experimental fusion cross sections above the Coulomb barrier is obtained f…
We construct a microscopic optical potential including breakup effects for elastic scattering of weakly-binding projectiles within the Glauber model, in which a nucleon-nucleus potential is derived by the $g$-matrix folding model. The derived microscopic optical potential is referred to as the eikonal potential. For $d$ scattering, the calculation with the eikonal potential reasonably reproduces the result with an exact calculation estimated by the continuum-discretized coupled-channels method. As the properties of the eikonal potential, the inaccuracy of the eikonal approximation used in the …
We use the Coulomb dissociation (CD) method to calculate the rate of the $^{36}$Mg(${n,γ}$)$^{37}$Mg radiative capture reaction. The CD cross sections of the $^{37}$Mg nucleus on a $^{208}$Pb target at the beam energy of 244 MeV/nucleon, for which new experimental data have recently become available, were calculated within the framework of a finite range distorted wave Born approximation theory that is extended to include the projectile deformation effects. Invoking the principle of detailed balance, these cross sections are used to determine the excitation function and subsequently the rate o…
We present a novel method of smoothing discrete breakup cross sections calculated by the method of continuum-discretized coupled-channels. The method based on the complex scaling method is tested with success for $^{58}$Ni($d$, $pn$) reaction at 80 MeV as an example of a three-body breakup reaction, and applied to $^{12}$C($^6$He, $nn^4$He) reaction at 229.8 MeV as a typical example of a four-body breakup reaction. The new method does not need to derive continuum states of the projectile in order to evaluate the breakup cross section as a smooth factor of the excitation energy of the projectil…
We investigate projectile breakup effects on 6Li+209Bi elastic scattering near the Coulomb barrier with the four-body version of the continuum-discretized coupled-channel method (four-body CDCC). This is the first application of four-body CDCC to 6Li elastic scattering. The elastic scattering is well described by the p+n+4He+209Bi four-body model. We propose a reasonable three-body model for describing the four-body scattering, clarifying four-body dynamics of the elastic scattering.
A novel method to extract the $B(E1)$ strength of a weakly bound nucleus from experimental Coulomb dissociation data is proposed. The method makes use of continuum discretized coupled channels (CDCC) calculations, in which both nuclear and Coulomb forces are taken into account to all orders. This is a crucial advantage with respect to the standard procedure based on the Equivalent Photon Method (EPM) which does not properly take into account nuclear distortion, higher order coupling effects, or Coulomb-nuclear interference terms. The procedure is applied to the $^{11}$Be nucleus using two sets…
The virtual photon theory (VPT), which is based on first-order Coulomb dissociation restricted to the electric dipole ($E1$), has been successfully used to explain the breakup data for several cases. Our aim is to study the role of various higher-order processes that are ignored in the VPT, such as the nuclear breakup, interference between nuclear and Coulomb amplitudes, and multistep breakup processes mainly due to strong continuum-continuum couplings in the breakup of two-body projectiles on a heavy target at both intermediate and higher incident energies. For the purpose of numerical calcul…
A new method for calculating the Coulomb breakup of unstable neutron-rich isotopes at high energies is presented. The calculations employ the eikonal approximation and use a new Coulomb dynamical polarization potential (CDPP), calculated by solving the Schrödinger equation for the entire motion of the exotic projectile as a two-body cluster structure using the adiabatic approximation and incorporating excitations to the continuum. Calculations for some exotic isotopes are compared with Coulomb dissociation cross section data and found to be in good agreement.
Breakup reactions are one of the favored probes to study loosely bound nuclei, particularly those in the limit of stability forming a halo. In order to interpret such breakup experiments, the continuum discretized coupled channel method is typically used. In this study, the first Bayesian analysis of a breakup reaction model is performed. We use a combination of statistical methods together with a three-body reaction model (the continuum discretized coupled channel method) to quantify the uncertainties on the breakup observables due to the parameters in the effective potential describing the l…
The Continuum-Discretized Coupled-Channels (CDCC) has been successfully employed to describe elastic and breakup of nuclear reactions induced by weakly bound projectiles. In this contribution, we review some other, less widespread applications of the CDCC wavefunction, some of them in combination with other reaction formalisms, which are being currently employed in the analysis of reactions involving three or more fragments in the initial or final state.
Background: Large $α$ yields have been reported over the years in reactions with $^{6}$Li and $^{7}$Li projectiles. Previous theoretical analyses have shown that the elastic breakup (EBU) mechanism (i.e., projectile breakup leaving the target in its ground state) is able to account only for a small fraction of the total $α$ inclusive breakup cross sections, pointing toward the dominance of non-elastic breakup (NEB) mechanisms. Purpose: We aim to provide a systematic study of the $α$ inclusive cross sections observed in nuclear reactions induced by $^{6}$Li projectiles. In addition to estimatin…
In many instances, single nucleon removal reactions from neutron-proton asymmetric projectile nuclei populate final states in the residual nuclei that are very weakly bound. Familiar examples include neutron removal reactions from neutron-rich $^{11}$Be and $^{12}$Be, the latter populating the well-known $1/2^+$ halo ground-state and $1/2^-$ excited-state of $^{11}$Be - both states less than 1 MeV from the first neutron-decay threshold. Numerous additional examples arise in reactions of asymmetric $p$- and $sd$-shell nuclei. The importance of this weak residue binding upon calculated single-nu…
With the recent availability of state-of-the-art heavy-ion stable and radioactive beams, there has been a renew interest in the investigation of nuclear reactions with heavy ions. I first present the role of inelastic and transfer channel couplings in fusion reactions induced by stable heavy ions. Analysis of experimental fusion cross sections by using standard coupled-channel calculations is discussed. The role of multi-neutron transfer is investigated in the fusion process below the Coulomb barrier by analyzing $^{32}$S+$^{90,96}$Zr as benchmark reactions. The enhancement of fusion cross sec…
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 fra…
We employ a microscopic continuum-discretized coupled-channels reaction framework (MCDCC) to study the elastic angular distribution of the $^7$Li$=α+t$ nucleus colliding with $^{12}$C and $^{28}$Si targets at $E_{\text{Lab}}$=350 MeV. In this framework, the $^7$Li projectile is described in a microscopic cluster model and impinges on non-composite targets. The diagonal and coupling potentials are constructed from nucleon-target interactions and $^7$Li microscopic wave functions. We obtain a fair description of the experimental data, in the whole angular range studied, when continuum channels a…
We study anti-halo effects on reaction cross sections $σ_{\rm R}$ for $^{14,15,16}$C scattering from a $^{12}$C target at 83~MeV/nucleon, using the $g$-matrix double-folding model. $^{15}$C is described by the $^{14}$C~+~$n$ two-body model that reproduces the measured large s-wave spectroscopic factor, i.e., the shell inversion that the 1s$_{1/2}$ orbital is lower than the 0d$_{5/2}$ orbital in energy. $^{16}$C is described by the $^{14}$C~+~$n$~+~$n$ three-body model with the phenomenological three-body force (3BF) that explains the measured small s-wave spectroscopic factor. The 3BF allows t…
We use the breakup absorption model to simultaneously describe the measured cross-sections of the Complete fusion (CF), Incomplete fusion (ICF), and Total fusion (TF) in nuclear reactions induced by weakly bound nuclei $^{6,7}$Li on $^{209}$Bi and $^{198}$Pt targets. The absorption cross-sections are calculated using the Continuum Discretized Coupled Channels (CDCC) method with different choices of short range imaginary potentials to get the ICF, CF and TF cross-sections. It is observed that the cross-sections for deuteron-ICF/deuteron-capture are of similar magnitude as the $α$-ICF/$α$-captur…
The electric quadrupole transitions between $0^+$, $2^+$, and $4^+$ states in $^{12}$C are investigated in a $3α$ model. The three-body wave functions are obtained by means of the hyperspherical adiabatic expansion method, and the continuum is discretized by imposing a box boundary condition. Corresponding expressions for the continuum three-body ($3α$) bremsstrahlung and photon dissociation cross sections are derived and computed for two different $α-α$ potentials. The available experimental energy dependence is reproduced and a series of other cross sections are predicted. The transition str…
With the recent availability of state-of-the-art radioactive ion beams, there has been a renew interest in the investigation of nuclear reactions with heavy ions near the Coulomb barrier. The role of inelastic and transfer channel couplings in fusion reactions induced by stable heavy ions can be revisited. Detailed Analysis of recent experimental fusion cross sections by using standard coupled-channel calculations is first discussed. Multi-neutron transfer effects are introduced in the fusion process below the Coulomb barrier by analyzing 32S+90,96Zr as benchmark reactions. The enhancement of …
