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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

Distance of interaction: a phenomenological analysis of elastic scattering data induced by light projectiles

  • 1Instituto de Física, Universidade de São Paulo, São Paulo, Brazil
  • 2Department of Physics and Engineering Physics, Obafemi Awolowo University, Ile-Ife, Nigeria
  • 3Instituto de Física, Universidade Federal Fluminense, Niterói, Rio de Janeiro, Brazil

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.

1 Introduction

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 (α+t) and 9Be (α+α+n). There are also neutron- and proton-rich nuclei near the boundaries of the driplines with even more exotic configurations. Nuclei such as 8B (7Be+p) and 11Be 10Be+n) have shown an exotic configuration, in which the valence nucleon orbits a core at a large distance. Others, such as 11Li (9Li+n+n), have two valence-orbiting nucleons, forming a structure that resembles Borromean rings and are, therefore, referred to as Borromean nuclei.

Due to the large extended distance of nuclear matter, 11Li is considered a halo nucleus, while 6He is called a skin nucleus. The structure of halo in the nucleus can be considered a threshold effect, arising from the small separation energy of the one- or two-valence nucleons. In a halo structure, the valence nucleon(s) is (are) nearly decoupled from a well-defined inert core. Such a structure makes halo nuclei a suitable object to explore the behavior of open quantum systems (OQS) since the valence particle in a halo nucleus is sensitive to any interaction with continuum states and the external environment. In this case, the valence neutrons would tunnel the potential well with a slowly decaying exponential tail extending beyond the range of the potential. Thus, neutron halo can be considered one of the most exotic phenomena of the quantum tunneling effect of a loosely bound system, which can be considered to be an OQS. The boundary for defining a nucleus as halo is not exactly clear, and there are some other nuclei that are also considered to have a halo structure, such as 11Be and 15C. For the proton-rich nuclei, the situation is even less defined since the Coulomb barrier between the core and the valence proton prevents it from having the same behavior as neutron valence. The radial density distribution deduced for elastic scattering of 8B + p exhibited a clear halo structure with the root-mean-square (rms) matter radius Rm = 2.58 (6) fm and the rms halo radius Rh = 4.24 (25) fm [3]. This result, combined with the large breakup cross sections [4] and the narrow longitudinal momentum distribution for the 7Be core [5], is very strong experimental evidence of a proton-halo structure in 8B nuclei. Despite the lack of experimental evidence, some other proton-halo candidates are 12N (Sp = 0.601 MeV) [6] and 17Ne (S2p = 0.933 MeV) [7, 8]. In summary, several exciting phenomena have emerged from the investigations of the structures of these light nuclei, and several other phenomena remain to be investigated. In particular, whether and/or why halo and cluster structures in light nuclei occur preferentially at thresholds, which is a clear characteristic of OQS, is an actual topic under investigation.

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 D to the asymptotic scattering angle. Therefore, the angular distribution of normalized elastic cross sections can be converted to normalized cross sections as a function of distance D. Under this transformation, it is possible to determine the strong absorption (Ds) and the interaction distances (Di) for a binary collision. The former corresponds to the distance at which the ratio of elastic scattering to Rutherford scattering (dσ/dσRuth) drops to 0.25. The corresponding angle at which dσ/dσRuth = 0.25 is also called the grazing angle (θgr) or the quarter-point angle (θ1/4). The distance of strong absorption is closely related to the radius of the stable nucleus, as discussed in [13, 14]. However, the nuclear radius is not as easily defined for weakly bound and exotic nuclei, which can have an exotic cluster structure and/or a very diffuse density distribution at the surface region. The interaction distance represents the distance at which the nuclear potential (or a long-range Coulomb interaction) starts manifesting itself during the income trajectory in the nuclear collision and the cross-section ratio starts to deviate from unity. In the present work, the critical interaction distance is defined when the elastic cross-section ratio to Rutherford, dσ/dσRuth, is equal to 0.98. This value corresponds to the S-matrix’s absolute value of 0.99, and it is the distance where the flux from elastic scattering starts to be absorbed.

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.

2 Critical distance of interaction

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:

D=12kZpZtEc.m.1+1sinθ/2,(1)

with k=1.44 MeV.fm. Zi and Ai correspond to the atomic number and mass of the nuclei of the projectile (i=p) and the target (i=t), respectively. For a better comparison of the different data sets, involving different projectiles, we consider the reduced distance at the closest approach (d) defined as follows:

d=DAp1/3+At1/3.(2)

By plotting the data as a function of d, we can combine several data sets corresponding to the angular distribution dσ/dσRuthversusθc.m. measured in different energies into one data set dσ/dσRuthversusd. This is very convenient for elastic angular distributions with radioactive projectiles, where cross sections are usually obtained at fewer angles for each energy but at several different energies.

Table 1
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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 dσ/dσRuth ratio for all systems is very close to unity for large distances and begins to decrease rapidly at short distances. The cross sections are dropped because of strong absorption of the elastic flux into nonelastic channels, mostly fusion for very small distances. In the intermediate region, the static (cluster structure) and dynamic (couplings) effects play a relevant role for the tightly, weakly bound, and exotic configuration projectile, which can be related to the reduced distance of strong absorption ds and the reduced distance of critical interaction di. As mentioned above, the reduced critical interaction distance is defined as the distance for which the ratio dσ/dσRuth drops to 0.98, while the reduced strong absorption distance (also associated with the grazing angle or with θ1/4) is defined for when dσ/dσRuth is 0.25. We can extract these distances from the plots of the angular distributions as a function of the reduced distance of the closest approach. The procedure of the present work is also performed in [15, 16], which defines a phenomenological expression that could describe the region where the cross sections fall. The adopted expression is based on a Boltzmann exponential function and is defined as follows:

y=p11+ep2dp3,(3)

where ydσ/dσRuth. The parameters p1, p2, and p3 are free to vary during the data fitting process. The expression itself has no physical meaning and can only be used in the restricted region of the cross-section ratio between 1 and 0.1. The parameter p1 is related to the asymptotic value of y for a large distance d, and thus, it is associated with the normalization of the data. When its value is very close to unity, it indicates a suitable normalization of the data. The whole procedure is very reliable in obtaining the values of the reduced critical interaction and strong absorption distances, in particular for the data at energies close to the Coulomb barrier with no strong Fresnel peak. The errors in the values of the parameters were also obtained and are related to the quality of the cross-section data. This work mainly aims to obtain a reduced critical interaction and strong absorption distances for different projectile types such as exotic, weakly and strongly bound, stable, and radioactive light nuclei on light, medium, and heavy mass targets. Furthermore, the idea is to verify the correlation between these distances and, for instance, the projectile cluster configuration or the separation energies for the given cluster configuration. To perform a systematic and comparative analysis, we consider the reduced distances, where the size and geometric effect of the projectiles associated with mass dependence are somehow disregarded. The remaining geometric effect is associated only with the projectile deformation, cluster, and halo configurations.

3 Data analysis

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.

3.1 Distances for light mass target A = 27

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 versus distances for 6Li, 6He, 9Be, 6Li, 7Li, 7Be, 8B, 11B, 12C, and 14N are shown in Figures 13. The results of the fitting using Equation 3 and the corresponding parameters obtained are listed in Table 2. It is worth highlighting the good quality of the data for elastic scattering induced by 7Li, 9Be, and 12C projectiles. However, some angular distributions have a clear normalization issue. For the present analysis, correct normalization of the angular distributions is important since the critical interaction distance is defined on the basis of it. For example, the cross-section ratios for the angular distribution for 11B + 27Al at ELab = 24.0 MeV and 12C + 27Al at ELab = 21.0 MeV were re-normalized by a factor of 0.95 and 0.98, respectively, so at large distances, the ratios become, on average, equal to 1.0. For the 8B and 7Be + 27Al systems, the error bars for the first two points were artificially reduced to 1% to ensure that the parameters p1 were obtained close to unity. For the 14N + 27Al system, the Fresnel points, indicated as open symbols in the plot, were removed from the fitting. χred2 obtained for the fit is also listed in the Table 2. As can be seen, χred2 obtained for the 7Be + 27Al system is very small due to the large error bars in the cross sections, while for 12C + 27Al, it is small due to the small fluctuation of the data compared to the error bars.

Figure 1
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Figure 1. σ/σRuth as a function of the reduced distance for 6Li, 6He, and 9Be + 27Al systems at the indicated energies. The cross-section data are from references indicated in Table 1.

Figure 2
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Figure 2. σ/σRuth as a function of the reduced distance for 7Li, 7Be, and 8B+ 27Al systems at the indicated energies. The cross-section data are from references indicated in Table 1.

Figure 3
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Figure 3. σ/σRuth as a function of the reduced distance for 11B, 12C, and 14N + 27Al systems at the indicated energies. The cross-section data are from references indicated in Table 1.

Table 2
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Table 2. Distances of interaction for 27Al determined from σ/σRuth as a function of the reduced distance. Sn corresponds to the separation energy of the nucleus for the given cluster structure.

As the results of the fit, the reduced critical interaction distance, di, and the reduced strong absorption distance, ds, could be extracted and are listed in Table 2. The uncertainties in the reduced critical interaction and strong absorption distances were obtained, considering the difference in the distance for the cross-section ratios 0.97–0.99 and 0.24–0.26, respectively. The quality and fluctuation of the data are indirectly included in these uncertainties by the fitting curve. Inspecting the reduced distances listed in Table 2, we can conclude that the average reduced strong absorption distances for the system with exotic and weakly bound projectiles are ds=1.62(8) fm, a little larger than for the tightly bound projectile, ds=1.526(4) fm. For the reduced critical interaction distance, the values are di=2.72(18) fm and di=2.31(10) fm for the two groups, respectively. Although the difference is not large, it is not negligible. The highest value is obtained for the 6He + 27Al system, due to the exotic configuration of the 6He projectile. The small fluctuation among the values of the critical interaction distances might be due to the fact that the long-range Coulomb interaction, which can be quite different for different projectile types, is weak for this light target.

3.2 Distances for medium-mass target A = 58

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 di=2.18(49) fm; for the tightly bound system, the average value is di=1.87(1) fm, while for the exotic projectiles, 6He, 8B, and 11Be, the reduced critical distances are much larger, being in the range of di= 2.5 to 3.0 fm. The differences in the values between these three groups are more pronounced than for light 27Al, indicating stronger dynamic effects since static effects are related to the projectile itself. The extended matter density (static effect) and the lower breakup threshold (dynamic effect) of the exotic projectiles induce the nuclear forces to be felt beyond the classical range, resulting in a strong absorption and early deviation of the dσ/dσRuth ratio from unity. In particular, the extension of the direct interaction region for the 6He and 11Be projectiles is closely related to the importance of long-range Coulomb and/or nuclear interaction for these exotic projectiles. This effect also provokes a strong damping of the Fresnel diffraction peak observed in the corresponding angular distributions compared to those for the tightly bound isotopes of the same elements (4He and 10Be).

3.3 Distances for the medium-mass target A = 120

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 versus distances for 6Li, 6He, 9Be, 10B, 11B, 11Be, and 8B are shown in Figures 46. The parameters and corresponding values of χred2, obtained as the result of the fitting using Equation 3, are listed in Table 4. The data for this target are of much better quality than those for the aluminum target. In particular, we can highlight the good quality of the data for 9Be, 10B, and 11B, obtained in recent years at the Tandar Laboratory in Argentina [23, 24, 25]. By removing data points at the Fresnel peak for the 11B + 120Sn system, the values for the distances did not change, but χred2 drops from 7.87 to 3.00. The good quality for the 9Be + 120Sn system is also reflected in the small χred2 = 0.80. For the 12C + 120Sn data, the obtained large χred2 value is due to the too small error bars reported in EXFOR.

Table 3
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Table 3. List of selected energies of the angular distributions with the 120Sn target considered in this work.

Figure 4
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Figure 4. σ/σRuth as a function of the reduced distance for 6Li, 6He, and 9Be + 120Sn systems at the indicated energies. The cross-section data are from references indicated in Table 1.

Figure 5
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Figure 5. σ/σRuth as a function of the reduced distance for 10B, 11B, and 11Be + 120Sn systems at the indicated energies. The cross-section data are from references indicated in Table 1.

Figure 6
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Figure 6. σ/σRuth as a function of the reduced distance for 8B + 120Sn and 8B + 208Pb systems at the indicated energies. The cross-section data are from references indicated in Table 1.

Table 4
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Table 4. Distances of interactions for 120Sn determined from σ/σRuth as a function of the reduced distance. Sn corresponds to the separation energy of the nucleus for the given cluster configuration.

For this target, the average reduced strong absorption distance for the tightly bound projectiles is ds=1.537(6) fm, which is consistent with the value obtained for the 27Al target. For exotic and weakly bound projectiles, the distance is ds=1.57(4) fm, which is again just a little larger than that for tightly bound projectiles. For the reduced critical interaction distance, the averaged values are di=1.80(4) fm for tightly bound projectiles and di=2.13(4) fm for exotic and weakly bound projectiles. The distance obtained for 11Be was not included in the previous averaging. The 11Be projectile is a neutron-rich isotope that forms a halo configuration with the 10Be core and a weakly bound neutron (Sn=504(6) keV). The quite large value for the critical interaction distance, di=4.41(7) fm, for the 11Be + 120Sn system is clear experimental evidence of the strong effect of the 11Be neutron halo structure on elastic scattering, at energies close to the Coulomb barrier. This large absorption effect has already been observed for the elastic scattering of 11Be on the 64Zn target [26] close to the barrier energy and also on the 208Pb target at higher energies (three times the Coulomb barrier) [27]. The strong absorption in elastic scattering induced by this projectile is due to the strong influence of the break-up channel, related to its loosely bound structure. By comparison, this effect is not as drastic for the proton-rich halo nucleus 8B, with proton separation energy Sp=0.138 MeV for the 7Be+p configuration, indicated by the not-so-large reduced critical distance of interaction for this nucleus, di=2.10(6) fm, shown in Figure 6. The other large critical interaction distance is obtained for the 6He projectile, di=2.49(7) fm. Both 6He (α+n+n) and 9Be (α+α+n) nuclei are considered to have a Borromean configuration, where by removing one of the elements, the other two also dissociate, resembling the Borromean ring. However, 6He is radioactive and weakly bound (S2n=0.973 MeV), while 9Be is a stable bound nucleus with Sn = 1.665 MeV. The difference in their critical interaction distances can be attributed to static and dynamic effects related to the extended matter distribution and the lower breakup threshold, respectively. The extended critical interaction distance for neutron-rich 6He and 11Be projectiles is again clear experimental evidence of the importance of a long-range Coulomb and/or nuclear interaction for these exotic projectiles. Consequently, these give rise to a combination of effects of a large value of Coulomb dipole polarizability and large transfer breakup probabilities already observed experimentally.

3.4 Distances for heavy-mass target A = 208

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 (Sp=0.138 MeV) and the nuclear proton-halo configuration (7Be+p) contribute to the opening of several elastic absorption channels, observed as a large total reaction cross section [18]. In addition, because of the low binding energy, the projectile can easily break up near the target Coulomb and nuclear fields, enhancing the breakup channel, mainly at energies close to the Coulomb barrier. However, since the proton is in the p orbital, the matter density is not as extended far from the core due to the centrifugal barrier. The critical interaction distance obtained for this projectile is di=2.78(4) fm, which is the second largest in the table, just below that for 11Li. The large distance of interaction is not due to the extended matter density (static effect) but due to the lower breakup threshold and stronger coupling to the continuum. The other proton-rich nucleus is 10C. It has a α+α+p+p configuration, and it is called the Brunnian or super Borromean nucleus [28]. This configuration is similar to that for 10Be (α+α+n+n). However, the presence of the neutrons makes 10Be a quite tightly bound nucleus. As noted in [19, 29], the exotic cluster configuration for 10C induces strong absorption, making the critical interaction distance for this nucleus similar to the weakly bound nuclei.

Figure 7
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Figure 7. σ/σRuth as a function of the reduced distance for 10C, 13C, and 15C + 208Pb systems at the indicated energies. The cross-section data are from references indicated in Table 1.

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
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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].

Table 5
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Table 5. Distance of critical interaction for 27Al, 58Ni, 120Sn, and 208Pb.

4 Comparative analysis

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/(Ap1/3+At1/3) has been applied. Therefore, the strong absorption distance (not the reduced one) should be larger for a larger mass target. This can be confirmed just by multiplying each of the average values, only by the target contribution of the reduction factor (At1/3). In this case, the average values turn out to be <4.77>, <5.85>, <7.64>, and <9.02> fm, for 27Al, 58Ni, 120Sn, and 208Pb targets, respectively. As expected, larger values are obtained for the larger-mass target. The reduced critical interaction distances should be larger than the corresponding reduced strong absorption distances. This is expected since not only static but also dynamic effects play a role in the elastic scattering process at distances larger than the strong absorption distance. However, as observed in the tables, the reduced critical interaction distances are smaller for heavier targets. The average values (taking the anomalous value for 11 Li) decrease for heavier targets. Again, this is the effect of the reduction factor (mass influences). The projectiles begin to feel the interaction at shorter distances for the lighter target. The average values after recovering the contribution of the target mass factor become <7.68>, <8.44>, <9.80>, and <11.56> fm, for the targets 27Al, 58Ni, 120Sn, and 208Pb, respectively. We can conclude that more relevant information is obtained when a comparison is performed for different projectile types but on the same target.

Table 6
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Table 6. Distance of strong absorption for 27Al, 58Ni, 120Sn, and 208Pb.

5 Summary

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.

Data availability statement

Publicly available datasets were analyzed in this study. These data can be found here: the original data are in the published paper already referred.

Author contributions

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.

Funding

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.

Conflict of interest

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.

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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 States

Reviewed by:

Angela Bonaccorso, National Institute of Nuclear Physics of Pisa, Italy
Alberto Camaiani, Dipartimento di Fisica e Astronomia, Italy

Copyright © 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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