CDCC Calculations Research Articles

Elastic scattering measurements of 7Be+natZr at sub- and near – barrier energies

We will present a recent experiment for elastic scattering of 7Be+natZr at five sub – and near- barrier energies, namely at 21.5, 22.5, 24, 27 and 28 MeV, for determining the optical potential. The experiment was performed at the TriSol radioactive beam facility of Notre Dame University, simultaneously with elastic scattering and breakup measurements of 8B+90Zr , the last performed at 28MeV. This optical potential will be used as one of the coupling potentials in our CDCC calculations for 8B. Further more we will look for the energy dependence of the potential and its resemblance to either the one exhibited by 7Li or the one by 6Li, extracting useful conclusions about their structure. Preliminary data are presented and discussed.

Breakup of s-wave neutron-halo systems with near zero binding energy

In this paper, we analyze the Be11+208Pb and Mg37+208Pb breakup reactions in the projectile zero binding energy limit. The objective is to verify whether finite and converged breakup observables can be obtained for an s-wave neutron-halo system when its ground-state binding energy approaches zero. Applying the CDCC formalism, we report that for both projectiles, finite and converged breakup cross sections are obtained despite the extension of the projectile ground-state wave functions to infinity. We argue that this is due to the square-integrability of the bin wave functions, which are involved in CDCC calculations. This is evidenced by the lack of convergence of the elastic scattering calculations, where the radial integral of the matrix elements contains the projectile ground-state density and is independent of the bin wave functions.These results could prove useful in the analysis of future experimental data in the breakup of extremely weakly-bound neutron-halo systems and may contribute towards a better understanding of open quantum systems.

Electric dipole response of low-lying excitations in the two-neutron halo nucleus F29

The neutron-rich $^{28,29}$F isotopes have been recently studied via knockout and interaction cross-section measurements. The $2n$ halo in $^{29}$F has been linked to the occupancy of $pf$ intruder configurations. We investigate bound and continuum states in $^{29}$F, focusing on the $E1$ response of low-lying excitations and the effect of dipole couplings on nuclear reactions. $^{29}\text{F}$ ($^{27}\text{F}+n+n$) wave functions are built within the hyperspherical harmonics formalism, and reaction cross sections are calculated using the Glauber theory. Continuum states and $B(E1)$ transition probabilities are described in a pseudostate approach using the analytical THO basis. The corresponding structure form factors are used in CDCC calculations to describe low-energy scattering. Parity inversion in $^{28}$F leads to a $^{29}$F ground state characterized by 57.5% of $(p_{3/2})^2$ intruder components, a strong dineutron configuration, and an increase of the matter radius with respect to the core radius of $\Delta R=0.20$ fm. Glauber-model calculations for a carbon target at 240 MeV/nucleon provide a total reaction cross section of 1370 mb, in agreement with recent data. The model produces also a barely bound excited state corresponding to a quadrupole excitation. $B(E1)$ calculations into the continuum yield a total strength of 1.59 e$^2$fm$^2$ up to 6 MeV, and the $E1$ distribution exhibits a resonance at $\approx$ 0.85 MeV. Results using a standard shell-model order for $^{28}$F lead to a considerable reduction of the $B(E1)$ distribution. The four-body CDCC calculations for $^{29}\text{F}+^{120}\text{Sn}$ around the Coulomb barrier are dominated by dipole couplings, which totally cancel the Fresnel peak in the elastic cross section. These results are consistent with a two-neutron halo and may guide future experimental campaigns.

Global study of Be9 + p at 2.72A MeV

Background: In our recent experiment, $^9$Be+p at 5.67A MeV, the breakup decay rates to the 3 configurations, $\alpha$+$\alpha$+n, $^8$Be$^*$+n and $^5$He+$^4$He of $^9$Be, were observed and quantified, in a full kinematics approach. Unfolding step by step the accessibility to the above configurations, it will require similar studies at lower and higher energies as well as the interpretation of the data in a theoretical framework. Purpose: Investigate the breakup decay rate of $^9$Be+p at 2.72A MeV, where the $\alpha$+$\alpha$+n configuration is mainly accessible. Compare and interpret data at 2.72A MeV and 5.67A MeV into a 4-body CDCC formalism; Point out and discuss couplings to continuum. Methods: Our experimental method includes an exclusive breakup measurement in a full kinematic approach of $^9$Be incident on a proton target at 2.72A MeV, together with elastic scattering and other reaction channels measurements under the same experimental conditions. The interpretation of the data at 2.72A MeV and 5.67A MeV is considered in a 4-body CDCC approach, using the Transformed Harmonic Oscillator method for the 3-body projectile. Results: An elastic scattering angular distribution at 2.72A MeV is measured, which compares very well with CDCC calculations, indicating a strong coupling to continuum. At the same energy, the measured breakup and total reaction cross sections present good agreement with the calculated values. Further on, the elastic scattering and breakup cross section data at 5.67A MeV are found in very good agreement with the CDCC calculations. The present results support further our 3-body model for the structure of $^9$Be, validating relevant radiative reaction rates obtained previously.

Insights into the dynamics of breakup of the halo nucleus 11Be on a 64Zn target

We reexamine the elastic and breakup observables of the Be11+64Zn reaction at the near-barrier energy of 28.7 MeV. The measured quasi-elastic data are compared with CDCC and extended CDCC (XCDCC) calculations, the latter including the effect of the 10Be deformation.The angular distribution of emitted 10Be fragments, reported in the original analysis of this experiment, along with newly extracted 10Be energy distributions, are compared with calculations for the elastic breakup and non-elastic breakup contributions. Elastic breakup, computed with CDCC, accounts for most of the observed yields. The remaining difference (∼20%) is attributed to non-elastic breakup events (neutron absorption or target excitation) as confirmed by calculations performed with the model of Ichimura, Austern and Vincent (1985) [14].Finally, the effect of post-acceleration in the energy distribution is investigated. This effect is well accounted for by the present CDCC calculations and, according to simple kinematical considerations, is estimated to be of the order of 1 MeV, and roughly independent of the scattering angle.

Effect of breakup and resonance states of6Li on fusion

CDCC calculations of total fusion cross sections are presented for reactions of the weakly bound6Li with targets144Sm and154Sm at energies around the Coulomb barrier. Couplings to the low-lying excited states 2+, 3− of144Sm and 2+, 4+ of154Sm are included in the calculations. In the cluster structure frame of6Li → α + d, short range absorption potentials are considered for the interactions between α- and d- with the target to account for fusion. The effect of the excited states of the target on total fusion is investigated, as well as, that from couplings of resonance states of6Li, namely, l = 2, Jπ = 3+, 2+, 1+. The latter effect is calculated by two different approaches, (a) by considering only resonance states couplings and (b) by omitting these states from the full discretized energy space. Among other things, it is found that resonance and non-resonance continuum couplings give rise to small and similar fusion suppression at the higher energies.

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.

Interplay of projectile breakup and target excitation in reactions induced by weakly bound nuclei

In this work, we reexamine the extension of the CDCC formalism to include target excitation and apply it to a variety of reactions to study the effect of breakup on inelastic cross sections. We use a transformed oscillator basis to discretize the continuum of the projectiles in the different reactions and use the extended CDCC method developed in this work to solve the resulting coupled differential equations. A new code has been developed to perform the calculations. Reactions 58Ni(d, d) 58Ni*, 24Mg(d, d) 24Mg* , 144Sm( 6Li, 6Li) 144Sm* and 9Be( 6Li, 6Li) 9Be* are studied. Satisfactory agreement is found between experimental data and extended CDCC calculations. The studied CDCC method is proved to be an accurate tool to describe target excitation in reactions with weakly-bound nuclei. Moderate effects of breakup on inelastic observables are found for the reactions studied. Cross section magnitudes are not modified much, but angular distributions present smoothing when opposed to calculations without breakup.

Resonance and non-resonance effect of continuum states of 6Li on elastic scattering angular distributions

CDCC calculations of elastic scattering angular distributions for reactions of the weakly bound projectile 6Li with targets 28Si and 58Ni at energies around the Coulomb barrier are presented. Special emphasis is given to account for the effect of couplings from 6Li resonance states l = 2, Jπ = 3+, 2+, 1+. Similarly, the effect produced by non-resonant state couplings is studied. The convergent calculations are carried out with global α-target and d-target interactions. The calculated elastic scattering angular distributions are in general in good agreement with the measurements for the systems considered in this work. It is found that the calculations with only resonance states are very similar to that with all couplings (resonance+non-resonance). So, the absence of these states have a strong effect on elastic scattering (non-resonance states calculation). It is shown that the effects increase as the collision energy increases. An interpretation of the strength of the different effects is given in terms of the polarization potentials that emerge from the different couplings.

Backscattering measurement ofHe6onBi209: Critical interaction distance

An elastic backscattering experiment has been performed at energies below the Coulomb barrier to investigate static and dynamic effects in the interaction of He6 with Bi209. The measured cross sections are presented in terms of the dσ/dσRuth ratio, as a function of the distance of closest approach on a Rutherford trajectory. The data are compared with a three-body CDCC calculation and good agreement is observed. In addition, the critical distance of interaction was extracted. A larger value was obtained for the exotic He6 nucleus as compared with the weakly bound Li6 and Be9 nuclei and the tightly bound He4, C12, and O16 nuclei.

The scattering cross sections for6,7Li +nreactions

We investigate the continuum-discretized coupled-channel analysis to the integrated elastic and inelastic scattering cross sections for 6,7 Li + n reactions. We used the Jeukenne- Lejeune-Mahaux effective nucleon-nucleon interaction and optical model potential at incident neutron energies below and above 10 MeV, respectively. The calculated elastic and inelastic scattering cross sections with observed incident energies are almost in good agreement with experimental and evaluated data. cross sections for 6,7 Li at incident neutron energies below 10 MeV by using optical model potential (OMP) (7,8) and above 10 MeV by using JLM. We adjust the normalization constants for the OMP, because the agreement of the calculated cross sections with the data in very low incident energies of the neutron is insufficient without any adjustments. The energy dependent normalization constants, real part λv and imaginary part λw, of the OMP and JLM are determined explicitly from the integrated elastic cross section data, respectively. Comparing the results of calculations with experimental data, we discuss that the present CDCC calculations, which reproduce the experimental data observed in incident energies higher than 10 MeV with the single folding potential of the JLM and in lower energies with introducing the normalization factors for the cluster folding potential of the OMP.

Theoretical study of the elastic breakup of weakly bound nuclei at near-barrier energies

We have performed CDCC calculations for collisions of $^{7}$Li projectiles on $^{59}$Co, $^{144}$Sm and $^{208}$Pb targets at near-barrier energies, to assess the importance of the Coulomb and the nuclear couplings in the breakup of $^{7}$Li, as well as the Coulomb-nuclear interference. We have also investigated scaling laws, expressing the dependence of the cross sections on the charge and the mass of the target. This work is complementary to the one previously reported by us on the breakup of $^{6}$Li. Here we explore the similarities and differences between the results for the two Lithium isotopes. The relevance of the Coulomb dipole strength at low energy for the two-cluster projectile is investigated in details.

Reaction mechanism ofB8breakup at the Fermi energy

The longitudinal momentum distributions of $^{7}\mathrm{Be}$ fragments in the breakup of $^{8}\mathrm{B}$ on a carbon target have been measured at 36 MeV/u. The different mechanisms have been distinguished by coincidence measurements. The longitudinal momentum distributions of $^{7}\mathrm{Be}$ fragments from both the stripping and the diffraction mechanisms are consistent with the results of noneikonal calculations and CDCC calculations. The full widths at half maximum of the longitudinal momentum distributions are $124\ifmmode\pm\else\textpm\fi{}17$ and $92\ifmmode\pm\else\textpm\fi{}7$ MeV/$c$ for the stripping and diffraction components, respectively. The comparison with the different model calculations is discussed. It is crucial to separate the different reaction mechanisms experimentally to benchmark nuclear reaction theories.

Total reaction cross sections for 8Li + 90Zr at near-barrier energies

Total reaction cross sections for the radioactive nucleus 8Li on 90Zr are reported at the near-barrier energies of 18.5 and 21.5MeV, derived from quasi-elastic scattering measurements. An analysis of the quasi-elastic scattering results is performed within an optical model framework using the BDM3Y1 interaction and total reaction cross sections are deduced. These quantities, appropriately reduced, are compared with previous data obtained in elastic scattering measurements with well and weakly bound projectiles on various targets and a formula for predicting total reaction cross sections with an uncertainty of \( \sim 20\)% is obtained. Further on, the ratios of direct to total reaction cross sections are estimated for 6,8Li on various targets and are compared with CDCC or CRC calculations. The energy dependence of the optical potential is also discussed.

CDCC calculations of elastic scattering for the systems 6Li+144Sm and 6Li+208Pb. Effect of resonances of 6Li on elastic scattering angular distributions

Calculations of elastic scattering angular distributions for reactions of the weakly bound projectile 6Li with targets 144Sm and 208Pb at energies above the barrier, are performed with the Continuum Discretized Coupled-Channel method (CDCC). Ground, resonant and nonresonant continuum states of 6Li are included up to some maximum energy ϵmax for which convergence is achieved. In the three-body system, global interactions are used for the α-target and d - target sub-systems. The effect of continuum resonant states of 6Li, i.e., l = 2, jπ = 3+, 2+ and 1+ on elastic scattering angular distributions is investigated by extracting these states from the continuum space. It is found that the calculated elastic scattering angular distributions are in good agreement with the measurements for most of the cases studied where consideration of couplings to continuum states is essential. It is also found that the resonance character of the continuum states is in some cases important to obtain agreement with the data.

Analysis of a low-energy correction to the eikonal approximation

Extensions of the eikonal approximation to low energy (20 MeV/nucleon typically) are studied. The relation between the dynamical eikonal approximation (DEA) and the continuum-discretized coupled-channels method with the eikonal approximation (E-CDCC) is discussed. When Coulomb interaction is artificially turned off, DEA and E-CDCC are shown to give the same breakup cross section, within 3% error, of $^{15}\mathrm{C}$ on $^{208}\mathrm{Pb}$ at 20 MeV/nucleon. When the Coulomb interaction is included, the difference is appreciable and none of these models agrees with full CDCC calculations. An empirical correction significantly reduces this difference. In addition, E-CDCC has a convergence problem. By including a quantum-mechanical correction to E-CDCC for lower partial waves between $^{15}\mathrm{C}$ and $^{208}\mathrm{Pb}$, this problem is resolved and the result perfectly reproduces full CDCC calculations at a lower computational cost.

Cross Section Calculations of Deuteron-induced Reactions Using the Extended CCONE Code

We have extended the CCONE code to make it possible to calculate cross sections for deuteron-induced reactions. Elastic breakup and stripping reactions to continuum are calculated using another codes based on the Continuum-Discretized Coupled-Channels theory (CDCC) and the Glauber model, respectively, and the calculated results are inputted to the CCONE code as direct reaction components. Statistical decay from compound nuclei formed by nucleon stripping and deuteron absorption is calculated with the exciton and Hauser-Feshbach models implemented in the original CCONE code. The extended CCONE code is applied to analyses of deuteron-induced reactions on 27Al and 58Ni. CDCC calculations for deuteron elastic scattering show good agreement with the experimental data at incident energies of several tens of MeV. The calculated double-differential (d,xp) cross sections reproduce the measured ones at forward angles for incident energies of 56 and 100 MeV fairly well.

Scaling laws for near-barrier Coulomb and nuclear breakup

We investigate the nuclear and the Coulomb contributions to the breakup cross sections of $^6$Li in collisions with targets in different mass ranges. Comparing cross sections for different targets at collision energies corresponding to the same $E/V_{\mathrm{\scriptscriptstyle B}}$, we obtain interesting scaling laws. First, we derive an approximate linear expression for the nuclear breakup cross section as a function of $A_{\mathrm{% \scriptscriptstyle T}}^{1/3}$. We then confirm the validity of this expression performing CDCC calculations. Scaling laws for the Coulomb breakup cross section are also investigated. In this case, our CDCC calculations indicate that this cross section has a linear dependence on the atomic number of the target. This behavior is explained by qualitative arguments. Our findings, which are consistent with previously obtained results for higher energies, are important when planning for experiments involving exotic weakly bound nuclei.

On the near-barrier fusion of the proton-halo 8B + 58Ni system

We have performed two independent calculations, without any free parameter, to predict the near-barrier fusion cross section for the proton-halo 8B + 58Ni system, for which data were recently reported. Standard coupled channel calculations predict fusion cross sections smaller than the data, while CDCC calculations for the absorption cross section (fusion + transfer + inelastic cross sections) agree with the data above the barrier, although transfer cross sections are calculated to have non-negligible cross section at this energy regime. At sub-barrier energies, region where transfer cross sections are particularly important, the CDCC calculations overpredict the data. The fusion data of the 8B + 58Ni system fail to follow the systematics of other weakly bound nuclei and the UFF curve and do not agree with the fusion data of the 8B + 28Si system. We try to explain this anomalous behaviour.

Complete fusion of weakly bound cluster-type nuclei near barrier energies

We consider the influence of breakup channels on the complete fusion of weakly bound cluster-type systems in terms of dynamic polarization potentials. It is argued that the enhancement of the cross section at sub-barrier energies may be consistent with recent experimental observations that nucleon transfer, often leading to breakup, is dominant compared to direct breakup. The main trends of the experimental complete fusion cross sections are analysed in the framework of the Dynamic Polarization Potential approach. The qualitative conclusions are supported by CDCC calculations including a sequential breakup channel, the one-neutron stripping of 7Li followed by the breakup of 6Li.