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ORIGINAL RESEARCH article
Front. Phys. , 11 February 2025
Sec. Nuclear Physics
Volume 13 - 2025 | https://doi.org/10.3389/fphy.2025.1518626
Introduction: A phenomenological analysis, based on distances, has been performed for elastic scattering data induced by tightly bound (11B, 12C, and 16O), weakly bound (6Li, 7Li, 7Be, and 9Be), and exotic (6He, 8B, 11Be, and 15C) nuclei on light (27Al), medium (58Ni and 120Sn), and heavy mass (208Pb) targets, respectively, at energies close to the Coulomb barrier.
Methods: The cross-section data on the angular distributions have been converted as a function of the distance of the closest approach.
Results: From a fitting analysis, critical interaction and strong absorption distances were extracted from the data.
Discussion: Correlation was observed with the projectile cluster configuration for the data on the target 208Pb.
The complexity of nuclear structure and reactions is governed by the interplay of the strong and electroweak interactions. The need to understand how these forces act within the atomic nucleus drives experimental and theoretical efforts to explore the limits of nuclear existence. Today, 288 isotopes are known to be stable or long-lived nuclei in a vast landscape that may encompass nearly 7,800 nuclei, according to theoretical models [1, 2]. Some of these nuclei are tightly bound in their ground state and have been well-described as spherical in shape, such as 16O, 58Ni, and 208Pb. Other nuclei, on the other hand, are weakly bound and can exhibit clustering signatures, such as 7Li
Due to the large extended distance of nuclear matter, 11Li is considered a
Nuclear reaction is one of the most commonly employed techniques for exploring the structures exhibited by nuclei across the vast nuclear landscape. To extract structure information on the colliding partners, reliable reaction models and high-quality experimental data are required. In this regard, elastic scattering is the simplest and most studied process among the possible outcomes in collisions between two nuclei. From the analysis of these elastic scattering angular distributions, we can obtain information on both the static (deformation and cluster configuration) and dynamic (couplings to nonelastic reaction channels) effects on the collision. The theoretical description of this process is often based on quantum theories with a model space of the reaction channels. Some of the ingredients in these calculations are the effective optical potential between the projectile and target nuclei, coupling constants, and the structure of the nuclei involved. The shape of the optical potential is usually linked to the overall geometry of the nuclei. By choosing specific targets, it is possible to focus on the properties of the projectile. For example, the peculiar cluster configurations in some light exotic nuclei, such as 6He, 8B, 11Be, and 15C, induce a strong coupling to the continuum states. In turn, this coupling introduces a characteristic dynamic polarization (attractive or repulsive) in the optical potential that is not present in the elastic scattering induced by strongly bound projectiles. A review investigating the elastic scattering data can be found in [9, 10]. A specific review on the elastic scattering of light radioactive projectiles can be found in [11], where the peculiar surface properties (static effects) of exotic weakly bound nuclei are highlighted. The strong coupling effect in elastic scattering has been reviewed and well-discussed in [12].
Although a quantum formulation for elastic scattering is well-grounded on a theoretical basis, adopting a semi-classical approach is useful for complementary phenomenological analysis. In classical mechanics, scattering is described in terms of trajectories and connects the distance of the closest approach
In this work, we present a semiclassical phenomenological analysis based on distances to investigate static and dynamic effects on the elastic scattering of light nuclei, at energies close to the Coulomb barrier. We present new results for the analysis of elastic scattering data on the targets 27Al and 120Sn, which can be considered a sequel of the previous analysis on the targets 58Ni [15] and 208Pb [16]. On account of this, this analysis was inspired by the initial work of Pakou and Rusek, which is presented in [17]. Systematic analysis, in which several data sets can be compared on the same grounds, has been shown to be a powerful tool for investigating general behavior and highlighting the particular properties of some of the nuclei involved.
The cross sections of the angular distributions, listed in Tables 1, 3, were converted from the angular dependence to the distance of the closest approach on a Rutherford trajectory and then to reduced distances. In classical scattering, the distance at the closest approach is related to the incident energy and scattering angle in the center of mass (c.m.) frame as follows:
with
By plotting the data as a function of
Table 1. List of selected energies of the angular distributions with the 27Al target considered in this work.
After converting all the angular distributions to the reduced distance dependence, we observed a common behavior among them. The d
where
We have surveyed the literature for a series of measured angular distributions of elastic scattering involving tightly bound (10B, 11B, 12C, 13C, and 14N), weakly bound (6Li, 7Li, 7Be, and 9Be), and exotic (6He, 8B, 11Be, and 15C) nuclei projectiles on 27Al, 58Ni, 120Sn, and 208Pb targets, at energies close to the Coulomb barrier. In addition to the new analysis of elastic scattering on 27Al and 120Sn targets, we included, in the present work, part of the results of the previous analysis on 58Ni and 208Pb targets [15, 16] and the new analysis of data recently published for 8B [18], 10C [19], 13C [20], and 15C [21] projectiles on 208Pb target, which is not present in the previous work. The targets considered here are among the most common ones used in elastic scattering measurements, mainly because they are tightly bound and not very deformed, and even double magic as in the case of 208Pb target, for which we expected to have very low collectivity or influence of other channels in the elastic (except for the 27Al target, which may have some collective effect). Thus, the dynamic and static effects on the elastic process can mostly rely on the projectile’s properties. The data used in the present analysis are compiled in Tables 1, 3. We have only selected elastic scattering data for light projectiles on 27Al, 58Ni, 120Sn, and 208Pb targets, at energies around the Coulomb barrier. This allows us to explore the collisions mediated by a nuclear interaction, with the 27Al target, all the way through the Coulomb-dominated interaction, with 208Pb and the (possible) nuclear Coulomb interferences with the 58Ni and 120Sn targets. For some systems, the angular distributions have also been measured at several other energies well above the Coulomb barrier. Still, we selected the angular distributions measured close to the Coulomb barrier, where the Fresnel peak is absent or very small.
The selected angular distributions and their corresponding energies and references used in the analysis of elastic scattering on the light 27Al target are listed in Table 1, which includes data induced by tightly bound (11B, 12C, 14N, and 16O), weakly bound (6Li, 7Li, 7Be, and 9Be), and exotic (Borromean 6He and proton-halo 8B) projectiles. The data for the elastic scattering were, actually, extracted from the EXFOR database (https://www-nds.iaea.org/exfor/) [22] and converted to a function of the reduced distance of the closest approach, according to Equations 1, 2. The plots of the cross sections
Figure 1.
Figure 2.
Figure 3.
Table 2. Distances of interaction for 27Al determined from
As the results of the fit, the reduced critical interaction distance,
The phenomenological distance analysis has already been performed for the elastic scattering data induced by some light nuclei as 6He, 6,7,8Li, 7,9,10,11Be, 8,10,11,12B, 9,11Li, 12C, and 16O, on the medium-mass targets 58Ni and 64Zn, reported in [15]. Since this analysis has already been published, we are just resuming the import results. The values of the reduced interaction distance can be divided into three groups. The average value for systems with weakly bound projectiles 6,7,8Li and 7,9Be is
The selected data used in the analysis of elastic scattering on the medium-mass 120Sn target are listed in Table 3. For this target, we also analyzed the elastic scattering data for systems including tightly bound (11B, 12C, and 16O), weakly bound (6Li, 7Li, and 9Be), and exotic (Borromean 6He, proton-halo 8B, and neutron-halo 11Be) projectiles. Furthermore, for this target, the elastic scattering cross-section data were extracted from the EXFOR database (https://www-nds.iaea.org/exfor/) [22] for most of the system and converted to a function of the reduced distance of the closest approach. The corresponding plots of the cross-section ratios
Table 3. List of selected energies of the angular distributions with the 120Sn target considered in this work.
Figure 4.
Figure 5.
Figure 6.
Table 4. Distances of interactions for 120Sn determined from
For this target, the average reduced strong absorption distance for the tightly bound projectiles is
The present phenomenological distance analysis has already been performed for the elastic scattering data induced by some light nuclei such as 6He, 6Li, 7Li, 7Be, 8He, 8Li, 8B 9Li, 9Be, 10Be, 11Li, 12C, 16O, 17F, and 19F on the 208Pb target, reported in [16]. The analysis of the heavy spherical target yielded interesting results related to the dependence of the cluster configuration throughout the separation energy and the critical distance of interaction. By choosing heavy targets with a stronger Coulomb field, all absorption effects can be related to the projectile configuration. We are now expanding the analysis with data recently published on 8B, 10C, 13C, and 15C. The plots for the cross-section ratio as a function of the reduced distances for these nuclei are shown in Figures 6, 7. In particular, we emphasize the importance of the new data for proton-rich nuclei that were not included in the previous analysis. The low binding energy for 8B projectile (
Figure 7.
The critical distance of interaction obtained for the new data on 8B, 10C, 13C, and 15C projectiles were included in the systematic as a function of the separation energy, as performed in [16]. The extended and upgraded plot can be seen in Figure 8. The values used in this plot are listed in Table 5 in the column for 208Pb. The proton halo 8B has the second-largest critical distance of interaction. The neutron halo 15C also has a considerable critical interaction distance. On the other hand, 10C and 13C follow the trend of the weakly and tightly bound nuclei; the weaker the projectile, the more significant the critical interaction distance.
Figure 8. Reduced critical distance of interaction as a function of the separation energy for the nuclei indicated. The dashed curve corresponds to the trend of the data for the weakly and tightly bound nuclei in red. The plot is an upgrade of this figure in [16].
A comparative analysis was also performed by combining all the distances obtained for exotic, weakly bound, and tightly bound projectiles on light (27Al), medium (58Ni and 120Sn), and heavy (208Pb) targets together. The values for all obtained strong absorption and critical interaction distances are listed in Tables 5, 6. A more complete set of data was obtained for 208Pb since several elastic scattering experiments have been performed on this target, including the most recent for 8B, 10C, 13C, and 15C. This makes the comparative analysis more reliable for this target, as shown in the previous section. We can observe in these tables that, although there are some fluctuations in the values for some projectiles, the average value for a reduced strong absorption distance is about the same for all targets analyzed here. As mentioned above, this distance is related somehow to the geometry (radius) of the nucleus. However, we should emphasize that what is obtained are the reduction distances, where a factor 1/(
A semiclassical approach, by plotting the ratio of elastic cross section to the Rutherford value as a function of the distance of the closest approach on a Rutherford trajectory, was performed for some light projectiles on light (27Al), medium (58Ni and 120Sn), and heavy (208Pb) targets. This analysis is of special advantage for investigating angular distributions induced by low-statistics radioactive nuclei because several angular distributions can be grouped in one data set. In this sense, the present analysis is a good approach to check the quality of the data. The reduced critical and strong absorption distances obtained were compared, and the influence of static and dynamic effects on the elastic scattering process was discussed. Although these distances can be somehow related to the size of the nuclei, they are also influenced by the reaction mechanisms. In particular, the critical interaction distance has some correlation with the separation energy of the valence particles or a particular cluster configuration, which may affect the strength of the couplings and the importance of a particular channel. The significantly higher value obtained for exotic nuclei such as 11Li, 6He, 8B, and 15C can be understood as a result of the influence of the large Coulomb dipole polarizability of these projectiles, which induces a higher breakup probability. For a neutron-halo projectile, the Coulomb breakup originates only from the recoil of its core. However, for a proton-halo projectile, the valence proton also feels the effect of the direct Coulomb interaction with the Coulomb field of the target. Therefore, for a proton-halo projectile, the breakup will originate from a combination of three forces: the nuclear interaction with the target, the effective force due to the recoil of the core, and the direct proton–target Coulomb repulsion. The interplay between these three interaction modes is important in describing the angular distribution of the elastic scattering with proton-halo projectiles. These forces act coherently, and their final effects are due to strong interferences at the scattering angles, where the three forces have comparable values. An interesting discussion on the different behavior of the proton and neutron halo projectile in a reaction is presented in [49, 50]. In the present analysis, only the overall effects are observed as a large distance of interaction and, as also observed, are strongly related to the target mass. For the 208Pb target, the extended and upgraded plot of the critical distance of interaction versus the separation energy for the given cluster configuration indicates a clear correlation.
Publicly available datasets were analyzed in this study. These data can be found here: the original data are in the published paper already referred.
VG: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing–original draft, and writing–review and editing. PN: formal analysis, investigation, and writing–review and editing. SO: data curation, formal analysis, investigation, and writing–review and editing. RL: formal analysis, investigation, validation, visualization, writing–original draft, and writing–review and editing. JL: investigation, validation, visualization, and writing–review and editing.
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. The authors acknowledge financial support from the Brazilian Funding Agencies: CNPq (Grant 303769/2021-1), FAPESP (Grants 2016/17612-7, 2022/14052-1 and 2024/02463-2), and INCT-FNA (Instituto Nacional de Ciência e Tecnologia-Física Nuclear e Aplicações) Proc. No. 464898/2014-5 and FAPERJ Proc. No. 210805/2024.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.
The author(s) declare that no Generative AI was used in the creation of this manuscript.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Keywords: nuclear reactions, nuclear structures, elastic scattering, critical interaction distance, heavy ions
Citation: Guimarães V, Nistal PC, Olorunfunmi SD, Linares R and Lubian J (2025) Distance of interaction: a phenomenological analysis of elastic scattering data induced by light projectiles. Front. Phys. 13:1518626. doi: 10.3389/fphy.2025.1518626
Received: 28 October 2024; Accepted: 07 January 2025;
Published: 11 February 2025.
Edited by:
Alan Wuosmaa, University of Connecticut, United StatesReviewed by:
Angela Bonaccorso, National Institute of Nuclear Physics of Pisa, ItalyCopyright © 2025 Guimarães, Nistal, Olorunfunmi, Linares and Lubian. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Valdir Guimarães, dmFsZGlyZ0BpZi51c3AuYnI=
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