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 R m = 2.58 (6) fm and the rms halo radius R h = 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 ( S p = 0.601 MeV) [6] and ¹⁷Ne ( S 2 p = 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 ( D s ) and the interaction distances ( D i ) for a binary collision. The former corresponds to the distance at which the ratio of elastic scattering to Rutherford scattering (d σ /d σ Ruth ) drops to 0.25. The corresponding angle at which d σ /d σ Ruth = 0.25 is also called the grazing angle ( θ gr ) or the quarter-point angle ( θ 1 / 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, d σ / d σ Ruth , 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.

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 = 1 2 k Z p Z t E c . m . 1 + 1 sin θ / 2 , (1) with k = 1.44 MeV.fm. Z i and A i 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 = D A p 1 / 3 + A t 1 / 3 . (2) By plotting the data as a function of d , we can combine several data sets corresponding to the angular distribution d σ / d σ Ruth v e r s u s θ c.m. measured in different energies into one data set d σ / d σ Ruth v e r s u s d . 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.

After converting all the angular distributions to the reduced distance dependence, we observed a common behavior among them. The d σ /d σ Ruth 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 d s and the reduced distance of critical interaction d i . As mentioned above, the reduced critical interaction distance is defined as the distance for which the ratio d σ / d σ Ruth drops to 0.98, while the reduced strong absorption distance (also associated with the grazing angle or with θ 1 / 4 ) is defined for when d σ / d σ Ruth 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 = p 1 1 + e − p 2 d − p 3 , (3) where y ≡ d σ / d σ Ruth . The parameters p 1 , p 2 , and p 3 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 p 1 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.

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.

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 p 1 / 3 + A t 1 / 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 ( A t 1 / 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.

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.