Elastic scattering of ⁷Be

The elastic-scattering data for ⁷Be on light targets of ⁷Li, ¹⁰B, ¹²C, ¹⁴N, and ²⁷Al were reviewed in Ref. [7, 8]. Data on medium- and heavy-target systems are emphasized here.

2.1.1

⁷Be+⁵⁸Ni

The elastic scattering of ⁷Be+⁵⁸Ni at an energy of 21.5 MeV has been measured by Mazzocco et al. [18]. The resulting angular distribution is presented in Fig. 1 with the data obtained by Aguilera et al. [19], which are represented by diamond symbols. Notably, the first excited state of ⁷Be (E_ex.=0.43 MeV) cannot be identified; thus, these data must be considered as quasi-elastic scattering. Optical model calculation utilizing the potential parameters of ⁷Li values from Cook et al. [20] is presented in Fig. 1 using a solid curve. This model is devoid of free parameters and presents an appropriate description of the experimental data. Additionally, a CDCC calculation was performed using a ³He-⁴He two-body structure for ⁷Be. In the CDCC calculation, the potential parameters for the ⁴He-target interaction were derived by fitting the elastic-scattering data of ⁴He and ⁵⁸Ni at 12 MeV [21]. The parameters for the ³He-⁵⁸Ni interaction were derived from Fick et al. [22]. The interaction between ³He and ⁴He were characterized using the parameters reported by Buck et al. [23]. Similar to the optical model, the CDCC calculation provided a satisfactory representation of the data, indicating minor coupling effects to the continuum states of ⁷Be.

Fig. 1Full-screen View

Quasi-elastic scattering angular distribution of ⁷Be+⁵⁸Ni at 21.5 MeV. The circles and diamonds represent experimental data taken from Refs. [18, 19]. The solid curve denotes the optical-model calculation result. The figure is sourced from Ref. [18]

2.1.2

⁷Be+¹²⁰Sn

We experimentally investigated ⁷Be+¹²⁰Sn using the Radioactive Ion Beam Line in Lanzhou (RIBLL1), a facility at the Institute of Modern Physics, Chinese Academy of Sciences [24, 25]. The secondary beam of ⁷Be, which was purified by RIBLL1, was sent to a self-supporting isotopically enriched ¹²⁰Sn target with a thickness of 2.78 mg/cm². The beam was characterized by an energy of 48.05±0.9 MeV in the middle of the secondary target ¹²⁰Sn. Throughout the experiment, the beam intensity was maintained at a typical rate of 1×10⁵ particles per second (pps), with a purity of approximately 90%.

The angular distribution of the elastic scattering of ⁷Be+¹²⁰Sn is shown in Fig. 2. Notably, the total energy resolution of our measurements was insufficient for the distinct differentiation of the first excited states of ⁷Be. Consequently, the scattering data must be interpreted as quasi-elastic scattering.

Fig. 2Full-screen View

Elastic-scattering angular distribution of ⁷Be+¹²⁰Sn at 48.05 MeV. The solid and dashed curves represent the CDCC-R calculations with and without the couplings to the unbound states

We employed a novel framework known as CDCC with Regularization (CDCC-R), as outlined by Chen et al. [26], to analyze the scattering data. This CDCC-R approach leverages the highly efficient regularized Lagrange–Laguerre Mesh (RLLM) technique [27] to construct pseudo-states. A significant innovation within this methodology is the integration of RLLM with the modified Numerov algorithm [28], which not only accelerates the computational process, but also provides the convenient handling of closed channels within the CDCC-R framework. In the CDCC-R calculations, ⁷Be is described using a ³H+⁴He two-body model. The nuclear interaction between these two clusters is encapsulated in a Gaussian form, which is applied to both the central and spin-orbit coupling terms. The CDCC-R method also encompasses the bound 1/2^- first excited state of ⁷Be, thereby enabling a comprehensive analysis of the system. The framework adeptly handles continuous spaces, including two l=3 resonances (7/2^- and 5/2^- at excitation energies of E_ex.=4.57 and 6.73 MeV with widths of Γ of 0.18 and 1.2 MeV, respectively). In addition, a nonresonant continuum extending from l=0 to l=4 was incorporated into the calculations. The discretization of the momentum space was set uniformly with bin intervals Δk of 0.1 fm^-1. To ensure the accuracy of the calculations, the Legendre-polynomial expansion was extended to the eighth order. The optical potentials of the interactions of ⁴He and ³He with ¹²⁰Sn were derived from Refs. [29, 30].

The CDCC-R results are presented in Fig. 2, where the solid and dashed curves correspond to the results with and without the couplings to the continuum states, respectively. A comparative analysis revealed that the influence of the continuum states of ⁷Be on elastic scattering is relatively minor. In addition, the calculated contribution of the elastic breakup cross-section of ⁷Be to the total reaction cross section, was merely 2%, revealing the minimal role played by this process in the total reaction process. Furthermore, an examination of the inelastic scattering channel leading to the excitation of the first excited state of ⁷Be reveals a contribution that is effectively negligible.

2.1.3

⁷Be+²⁰⁸Pb

The elastic scattering of ⁷Be+²⁰⁸Pb was measured by Mazzocco [31] at energies of 37.4, 40.5, and 42.2 MeV. The angular distributions derived from these measurements are shown in Fig. 3. Notably, these measurements yielded results similar to those obtained in previous studies, which are classified as quasi-elastic owing to the indistinguishability of the excitation to the first excited state of ⁷Be from elastic-scattering events.

Fig. 3Full-screen View

Quasi-elastic-scattering angular distributions of ⁷Be+²⁰⁸Pb at (a) 42.2, (b) 40.5, and (c) 37.4 MeV. The solid and dashed curves represent the full CDCC and two-channel calculation results, respectively. The dotted curve in (c) denotes the CDCC calculation for 37.4 MeV with the same ³He+²⁰⁸Pb potential parameters as those at 42.2 and 40.5 MeV. The figures are sourced from Ref. [31]

The results of the CDCC calculations are shown in Fig. 3 using solid curves. For a comparative analysis, the dashed curves in the figure illustrate the results of the two-channel calculations, which account for the ground-state reorientation and coupling to the first excited state of ⁷Be. CDCC calculations provide a proper description of quasi-elastic scattering data. Furthermore, the subtle discrepancies observed between the CDCC and two-channel calculations are indicative of the modest influence that the coupling to the continuum states of ⁷Be exerts on elastic scattering.

Sensitivity analysis of the calculations for the choice of the ⁴He+²⁰⁸Pb interaction potential indicated a relatively low dependency. To reveal the importance of fine-tuning the ³He+²⁰⁸Pb optical potential, a comparative analysis of the quasi-elastic scattering derived from the CDCC calculation at 37.4 MeV is shown in Fig. 3 (c). This calculation utilizes the identical ³He+²⁰⁸Pb optical potential employed at the higher energies of 42.2 and 40.5 MeV. Notably, this approach results in a significant overestimation of the differential cross-section at backward angles, thereby highlighting the necessity of a potential adjustment to achieve congruence with the experimental data at a lower energy.

The total reaction and breakup cross-sections were derived using two-channel and CDCC calculations. The results revealed that the breakup cross-sections constitute a relatively modest proportion of the total reaction cross-section, typically on the order of 10%.

Breakup of ⁷Be

To date, the measurements of direct reactions involving ⁷Be are limited, and only a few inclusive measurements have been reported. Mazzocco et al. conducted an experiment using ⁷Be and ⁵⁸Ni at an energy of 22 MeV [18]. However, owing to limitations in the detection efficiency, no coincidence between the breakup fragments of ³He and ⁴He was observed. The calculation of the CDCC yielded an elastic breakup cross-section for the ⁷Be+⁵⁸Ni system of a mere 10.8 mb. This result reveals the relatively minor contribution of the breakup channel to the total reaction cross-section.

The ⁷Be+¹²C system was experimentally investigated at an incident energy of 34 MeV [32]. Few ³He-⁴He coincident events were observed, which is inadequate to set an upper limit for the breakup cross-section. Pronounced selectivity was observed for the ¹²C(⁷Be,³He) reaction, populating the α-cluster states in ¹⁶O. Furthermore, the measured angular distributions of the emitted ³He particles were predominantly forward-peaked. These two features suggested that the direct α-transfer mechanism is the predominant process responsible for the observed yield of ³He particles, rather than the breakup process.

Fusion of ⁷Be

The fusion reactions induced by ⁷Be were measured using the targets of ²⁷Al [33], ⁵⁸Ni [34], and ²³⁸U [35]. For the ⁷Be+²⁷Al system, the fusion cross-section was determined by subtracting the measured one-proton stripping cross-section from the total reaction cross section. Importantly, this approach yields an upper limit for the total fusion cross-section, with values of 635±76, 858±94, and 922±92 mb at laboratory energies of E_lab=17, 19, and 21 MeV, respectively. For the ⁷Be+⁵⁸Ni system, the fusion cross-sections were deduced by measuring the fusion-evaporation protons at the center-of-mass energies E_c.m. of 13.9, 15.0, 16.6, 17.4, 17.9, and 19.0 MeV. The resultant total fusion cross-sections were 26±3, 61±7, 165±32, 246±78, 292±46, and 395±116 mb, respectively. Intriguingly, an enhancement relative to the predictions of the one-dimensional barrier-penetration model was observed, even for energies above the barrier. For the ⁷Be+²³⁸U case, the fusion-fission events were measured. In addition, several coincident events of ^3,4He-fission fragments were observed to evaluate the possibility of complete fusion.

A comparative analysis of the fusion reactions induced by ⁷Be interacting with ²⁷Al, ⁵⁸Ni, and ²³⁸U targets was conducted in Ref. [8] and is shown in Fig. 4. An enhancement in the fusion cross-sections was observed for each of the three datasets at energies below the Coulomb barrier. This consistent observation across different target nuclei suggests that the enhancement is predominantly attributable to the intrinsic properties of the ⁷Be projectiles rather than the specific characteristics of the target nuclei employed in these experiments.

Fig. 4Full-screen View

Reduced total fusion cross-sections for ⁷Be+²⁷Al, ⁵⁸Ni, and ²³⁸U. For the ⁷Be+²³⁸U system, the results of complete fusion (squares) are shown with the total fusion cross-section deduced from the fusion-fission (triangles). The figure is sourced from Ref. [8]

Elastic scattering of ⁸B

In the energy range of interest, the elastic scattering of ⁸B was measured on targets of ¹²C [37] and ⁵⁸Ni [19]. These results are reviewed in Ref. [8]. In this review, we introduce recent elastic-scattering measurements.

3.1.1

⁸B+⁶⁴Zn

The experimental investigation was performed by Spartà et al. [38] at the HIE-ISOLDE facility of CERN [39] using a post-accelerated beam of ⁸B with an energy of 38.5 MeV. The angular distribution of the elastic scattering of ⁸B with ⁶⁴Zn is shown in Fig. 5. By employing a ⁷Be+p two-cluster structural model for ⁸B, CDCC calculations were performed to analyze the scattering data. The results of these calculations with and without coupling to the continuum states are presented in Fig. 5 by the solid and dashed curves, respectively. The elastic breakup produces a small suppression of the elastic cross-section, which has a modest influence on the Coulomb-nuclear-interference peak.

Fig. 5Full-screen View

Elastic-scattering angular distribution of ⁸B+⁶⁴Zn at 38.5 MeV. The solid and dashed curves represent the CDCC calculations with and without couplings to the continuum states of ⁸B. The figure is sourced from Ref. [38]

The total reaction cross-section for the ⁸B+⁶⁴Zn system was determined to be 1.5 b, as derived from the CDCC calculation. This value is approximately half of the cross-sections observed for the neutron-halo ¹¹Be+⁶⁴Zn system at ratios analogous to the center-of-mass energy E_c.m. to the Coulomb-barrier height V_C. Notably, the total reaction cross-section for the ⁸B+⁶⁴Zn interaction was comparable to that of a stable, weakly bound system of ⁹Be+⁶⁴Zn. This intriguing result implies that despite the presence of a proton-halo structure, ⁸B did not exhibit a pronounced enhancement in reaction activity.

3.1.2

⁸B+⁹⁰Zr

The elastic scattering of ⁸B+⁹⁰Zr was measured by Palli et al. [40] at an energy of 26.5 MeV below the Coulomb barrier. The corresponding experimental results are shown in Fig. 6. The results of the full CDCC calculations and CDCC omitting the breakup couplings (one-channel) are shown in Fig. 6 using the dotted-dashed and dotted curves, respectively. The coupling effects, although nonnegligible, are modest compared with neutron-halo nuclear systems.

Fig. 6Full-screen View

Elastic-scattering angular distribution of ⁸B+^natZr at 26.5 MeV. The solid, dotted-dashed, and dotted curves denote the optical-model calculations with double-folding potentials, and CDCC and one-channel calculations, respectively. The figure is sourced from Ref. [40]

3.1.3

⁸B+²⁰⁸Pb

The elastic scattering of ⁸B+²⁰⁸Pb [41] was measured at 49.4 MeV, and the corresponding angular distribution is shown in Fig. 7. CDCC calculations with and without coupling to the continuum states of ⁸B are presented along with the data by the solid and dashed curves, respectively. In contrast to the ⁸B+⁶⁴Zn and ⁹⁰Zr systems, CDCC calculations with the ⁷Be+p structure model provide a relatively poor description of the experimental data (as shown in Fig. 7). This suggests that the structural configuration may be overly simplistic and the possibility of core excitation cannot be ignored [41]. This discrepancy between the light targets and ²⁰⁸Pb target systems may arise from the greater importance of Coulomb interactions in heavy-target systems.

Fig. 7Full-screen View

Elastic-scattering angular distribution of ⁸B+²⁰⁸Pb system at a beam energy of 50 MeV. The solid and dashed curves denote the results of the CDCC and no-coupling calculations, respectively. The figure is sourced from Ref. [41]

Breakup reactions of ⁸B

To date, only a few inclusive measurements have been reported for the breakup reactions of ⁸B.

The first experiment was performed in 2000 [42] for ⁸B+⁵⁸Ni at an incident energy of 25.8 MeV. A subsequent experiment was performed by Aguilera et al. [43] at energies of 25.0, 26.9, and 28.4 MeV. CDCC calculations provide a good description of the inclusive ⁷Be angular distributions, indicating the predominance of the elastic breakup (EBU) during the direct reaction.

The inclusive measurements of ⁸B+⁶⁴Zn [38] and ⁸B+²⁰⁸Pb [44] were performed at an energy of approximately 1.5 times the Coulomb barrier and at a deep sub-barrier energy, respectively. The inclusive angular distributions of ⁷Be were successfully reproduced by CDCC calculations. For the ⁸B+⁶⁴Zn system, the energy distribution of ⁷Be was extracted to gain insight into the intricate dynamics of the breakup processes and post-acceleration effects. However, the complexity of the breakup dynamics of ⁸B is evident using the CDCC and distorted-wave Born approximation calculations. These results indicate that various factors may influence the ⁷Be energy distributions such as different Coulomb multipoles in the breakup process and their interference, as well as nuclear effects. For ⁸B+²⁰⁸Pb, the breakup cross-section was determined to be 326±84 mb, which exhausts the total reaction cross-section for this system at an energy well below the barrier.

Fusion reactions of ⁸B

The total fusion reaction cross-section of ⁸B+⁵⁸Ni was determined by measuring the fusion-evaporated protons at ten energies near and below the Coulomb barrier [46]. Subsequently, the results for ⁸B+²⁸Si were reported for four energies above the Coulomb barrier by measuring the fusion-evaporated alpha particles [47]. The data for ⁸B+⁵⁸Ni showed a strong fusion enhancement at all energies, even above the Coulomb barrier. This contrasts with the results of neutron-halo systems, as well as the data for ⁸B+²⁸Si, whose total fusion cross-sections were suppressed above the barrier.

Recently, direct measurements of the total fusion cross-section for ⁸B+⁴⁰Ar were performed by Zamora et al. [45] using an active-target technique. The fusion cross-sections were determined by directly measuring the fusion-evaporation residue directly using a gas-filled time-projection chamber. The reduced fusion-excitation function of ⁸B+⁴⁰Ar and the results for ⁸B+²⁸Si and ⁵⁸Ni are shown in Fig. 8. The data for ⁸B+⁴⁰Ar were systematically consistent with the results for ⁸B+²⁸Si, showing a suppression of the total fusion cross-section above the Coulomb barrier. Further theoretical and experimental studies are required to understand the observed fusion enhancement above this barrier for the ⁸B+⁵⁸Ni system.

Fig. 8Full-screen View

Reduced total fusion cross-sections of ⁸B+⁴⁰Ar (circles), ²⁸Si (squares), and ⁵⁸Ni (triangles). The solid curve (universal fusion function, UFF) corresponds to the prediction from the one-dimensional barrier-penetration model. The figure is sourced and modified from Ref. [45]

Elastic scattering of ¹⁷F

Several datasets on the elastic scattering of ¹⁷F at near-barrier energies have been reported, including ¹⁷F+¹²C at 60.0 MeV [49], ¹⁷F+⁵⁸Ni at 54.1 and 58.5 MeV [50], and ¹⁷F+²⁰⁸Pb at 90.4 MeV [51]. The results were reviewed in Ref. [8]. In this section, we focus on two elastic-scattering measurements of ¹⁷F+⁸⁹Y at 50.0 and 59.0 MeV [52] and ¹⁷F+²⁰⁸Pb at 94.5 MeV [53] performed by our group in recent years.

4.1.1

¹⁷F+⁸⁹Y

The elastic scattering of ¹⁷F and ⁸⁹Y was measured at laboratory energies of 50 and 59 MeV using RIBLL1 [52]. The experimental setup included two multiwire proportional chambers to track the beam trajectory and an array of four sets of ΔE-E silicon-detector telescopes symmetrically installed along the beam line. The angular distributions of elastic scattering are shown in Fig. 9, where the error bars consider only statistical uncertainties. Notably, the energy resolution of the detectors was insufficient to distinguish between the first excited state and ground state of ¹⁷F. Consequently, the collected data must be considered as quasi-elastic scattering data.

Fig. 9Full-screen View

Angular distributions of elastic scattering of ¹⁷F on ⁸⁹Y at (a) E_lab = 50 MeV and (b) 59 MeV. The solid circles show the experimental data. The solid curves represent calculation results with the CDCC approach, while the dashed curves denote the results without continuum–continuum couplings. The figures are sourced from Ref. [52]

CDCC calculations were performed to evaluate the influence of the coupling effects on the continuum states. The results are shown by the solid curves in Fig. 9. In Fig. 9 (a), the CDCC prediction is in good agreement with the data at 50 MeV. In Fig. 9 (b), the CDCC calculation provides a good description of the data for the forward angles. However, a slight discrepancy was observed in the backward angles. For comparison, the results that exclude any coupling to the continuum states are illustrated by the dashed curves in Fig. 9. The analysis indicated that the effect of coupling to the continuum states on scattering was insignificant. Furthermore, at backward angles, the coupling introduced an amplification that was considered negligible within the observed scattering-angular distributions.

To address the observed divergence between the elastic-scattering data at 59 MeV and predictions of the CDCC calculations, the potential impact of the one-proton-transfer reaction ⁸⁹Y(¹⁷F,¹⁶O)⁹⁰Zr on the angular distribution of elastic scattering was investigated. This investigation was performed using the coupled reaction channels (CRC) approach, and the results are shown in Fig. 10. In this figure, the CRC results are compared with those derived from coupled-channel (CC) calculations in which the coupling to the one-proton-transfer channels were omitted. The results indicated that the one-proton-transfer reaction does not significantly affect the elastic scattering within the energy range.

Fig. 10Full-screen View

Comparison of CRC (dashed curve) and CC (solid curve) results for the elastic-scattering angular distributions for E_lab(¹⁷F)= 59 MeV. The figure is sourced and modified from Ref. [52]

4.1.2

¹⁷F+²⁰⁸Pb

The quasi-elastic scattering of ¹⁷F+²⁰⁸Pb was measured at an incident energy of 94.5 MeV at RIBLL1 [53]. The angular distribution of quasi-elastic scattering is shown in Fig. 11. CDCC calculations were performed to theoretically interpret the scattering data. The results are presented in Fig. 11 using a solid curve. For comparison, the calculation results that omit the coupling to the continuum states are indicated by dashed curves. Coupling to the continuum states of ¹⁷F has a negligible effect on elastic scattering.

Fig. 11Full-screen View

Quasi-elastic-scattering angular distribution of ¹⁷F+²⁰⁸Pb at 94.5 MeV. The solid circles are the experimental data. The solid and dashed curves represent the CDCC calculations with and without coupling to the continuum states, respectively. The results are sourced from Ref. [53]

The influences of inelastic scattering and the one-proton-transfer reaction on the elastic-scattering process were investigated using the CC and CRC calculations, respectively. In the CC calculations, consideration was given to the first excited state of ¹⁷F, as well as the excited states 3^- (2.614 MeV), 5^- (3.197 MeV), and 2⁺ (4.085 MeV) of ²⁰⁸Pb. These results indicated that the incorporation of the first excited state of ¹⁷F, which was treated as a collective state within the CC framework, significantly influenced the elastic scattering. Conversely, coupling originating from the target nucleus exhibited a moderate effect. To delve deeper into the impact of the one-proton stripping reaction on elastic scattering, CRC calculations were performed using the São Paulo potential [54]. The results revealed that the influence of the one-proton-transfer channel on the elastic angular distribution was negligible. Expanding the investigation beyond the one-proton transfer reaction, the contributions of one-neutron-, two-neutron-, and alpha-transfer channels to the quasi-elastic scattering were also evaluated within the CC and CRC frameworks. These results suggest that transfer channels did not affect the measured scattering data significantly.

Breakup reactions of ¹⁷F

Liang et al. [55] performed coincident measurements to elucidate the breakup mechanisms of ¹⁷F+⁵⁸Ni and ²⁰⁸Pb systems at an incident energy of 10 MeV per nucleon. For the ¹⁷F+²⁰⁸Pb system, theoretical analyses concluded that proton stripping is the primary direct reaction procedure. Additionally, the experimental data indicated that the breakup cross-section of ¹⁷F is relatively modest. This raises the question of its potential to significantly influence the fusion process, which requires further in-depth investigation. In a subsequent study, Kucuk and Moro [17] performed CDCC calculations and compared their results with experimental data, as shown in Fig. 12. The analysis revealed that for the ¹⁷F+⁵⁸Ni system, the CDCC calculation is in good agreement with the experimental data. However, for the ¹⁷F+²⁰⁸Pb system, pronounced divergence was observed between the theoretical predictions and experimental angular distributions, with the theoretical model overestimating the forward-angle data nearly two-fold. This discrepancy in the ¹⁷F+²⁰⁸Pb system may be attributed to the omission of the core nucleus excitation in the theoretical model. The CDCC calculations further suggest that at forward angles, the Coulomb breakup constituted the predominant component, whereas the nuclear-breakup component reached its maximum near the grazing angle. In addition, the calculations revealed that the Coulomb-polarization effect suppressed the breakup probability.

Fig. 12Full-screen View

Nuclear (dashed) and Coulomb (dotted-dashed) contributions to the exclusive breakup cross-section of ¹⁷F+⁵⁸Ni (upper panel) and ²⁰⁸Pb (lower panel). The solid line is the full CDCC calculation, including both nuclear and Coulomb couplings. The figures are sourced from Ref. [17]

Fusion reactions of ¹⁷F

Fusion cross-section measurements for the ¹⁷F+²⁰⁸Pb system [56] were performed. This fusion cross-section was determined through the coincident measurement of fission fragments at four energies ranging from 87 to 99 MeV. The reduced fusion-excitation function for the ¹⁷F+²⁰⁸Pb system is illustrated in Fig. 13 and is compared with the corresponding data for the ¹⁹F+²⁰⁸Pb system. The fusion behaviors of the two systems are strikingly similar, implying that the weakly bound characteristics of ¹⁷F has a minimal impact on the fusion process.

Fig. 13Full-screen View

Reduced fusion-excitation functions of ¹⁷F (full circles) and ¹⁹F (hallow circles)+²⁰⁸Pb. The solid line indicates the cross-sections measured for ¹⁹F+²⁰⁸Pb. The figure is sourced from Ref. [56]

A recent experimental investigation by Asher et al. [57] successfully measured the fusion-reaction cross-section of ¹⁷F+¹²C utilizing a 69.1 MeV ¹⁷F beam. The total fusion-reaction cross section for this system was determined by measuring the fusion-evaporation residues, by employing the Encore active-target detector [58]. In a comparative study, the fusion cross-sections of stable systems ¹⁶O+¹²C and ¹⁹F+¹²C were also measured. The reduced total fusion-reaction cross-sections for these three systems are presented in Figs. 14 and 15. ¹⁷F exhibits a fusion behavior similar to that of its stable counterparts ¹⁶O and ¹⁹F. This observation is particularly noteworthy, as it suggests that neither the weak binding nor the halo-like characteristics of the first excited state of ¹⁷F significantly influence the total fusion-excitation function above the Coulomb barrier.

Fig. 14Full-screen View

Comparisons of reduced total fusion cross-sections for the ^17,19F+¹²C systems. The lines denote the results of the no-coupling barrier-penetration calculations using double-folded real potentials for the corresponding systems. The arrows represent the heights of the fusion barriers. The figure is sourced from Ref. [57]

Fig. 15Full-screen View

Similar to Fig. 14, but for the systems of ¹⁷F and ¹⁶O+¹²C. The figure is sourced from Ref. [57]

Detector arrays

Detector arrays with high detection efficiency are essential for performing complete kinematics measurements, particularly for radioactive ion beams with low intensities. Two detector arrays, the silicon telescope array of the Silicon Telescopic Array for Reactions induced by Exotic nuclei (STARE) [61] and the ionization chamber array of the Multilayer Ionization-chamber Telescope Array (MITA) [62], were designed for the measurements of ⁸B and ¹⁷F, respectively. Homemade preamplifiers [63] have been employed to further improve the signal-to-noise ratio.

For the ⁸B+¹²⁰Sn measurements, the reaction products were identified using silicon telescopes. Therefore, a silicon detector array (STARE) was constructed to cover a large solid angle. STARE is composed of ten silicon telescopes arranged in a spherical shape with an approximate radius of 70 mm. A schematic of STARE is shown in Fig. 16. Each telescope of STARE has three stages of silicon detectors: the first layer is a 40/60 μm double-sided silicon-strip detector (DSSD) with an effective area of 50 mm×50 mm, followed by two layers of quadrant silicon detectors (QSDs) with thicknesses of 1000/1500 μm. STARE covers approximately 40% of the total solid angle, offering an angular coverage spanning 24°-158° with an angular resolution of approximately ±1.23°.

Fig. 16Full-screen View

Schematic of the silicon-detector array STARE used for the ⁸B+¹²⁰Sn measurement

Silicon telescopes are unsuitable for the ¹⁷F+⁵⁸Ni system, characterized by relatively heavy reaction products with low energies because of the potential energy loss of the heavy reaction products within the first detector layer, which impedes accurate particle identification. Consequently, MITA, an array based on an ionization chamber, was used for the ¹⁷F+⁵⁸Ni measurement. MITA also comprises ten telescopic units, each composed of four detection layers: an ionization chamber, a 40 (or 60)-μm DSSD, and two QSDs with thicknesses of 300 μm and 1000 (or 1500) μm. The schematic of the structure of each unit of MITA is shown in Fig. 17.

Fig. 17Full-screen View

(a) Schematic of the structure of each unit of MITA. (b) Photograph of one unit. These figures are sourced from Ref. [62]

Both STARE and MITA have excellent particle-identification capabilities, as demonstrated in Refs. [59, 62, 60].

⁸B+¹²⁰Sn measurement

The experiment was performed at the low-energy radioactive ion beam facility, the Radioactive Ion Beam Separator (CRIB) [64] of the Center for Nuclear Study, University of Tokyo. Secondary ⁸B beams were produced with two energies (37.8±0.5 and 46.1±0.6 MeV) around the Coulomb barrier with an intensity of approximately 10⁴ pps and a purity of ～20%.

The angular distributions of elastic scattering relative to Rutherford scattering for the two distinct measured energies are depicted in Fig. 18(a) and (b), as denoted by the squares. The calculation results from the CDCC approach are represented by solid curves in the same figure. These theoretical curves are in good agreement with the experimental data. Furthermore, Fig. 18 also presents the results of the one-channel calculations, denoted by dashed curves. In these calculations, couplings to the continuum states were excluded. A comparative analysis of these results with the CDCC calculations revealed that the influence of the continuum states on elastic scattering is insignificant.

Fig. 18Full-screen View

Angular distributions of elastic scattering and inclusive and exclusive breakup at (a) 38.7 and (b) 46.1 MeV, which are denoted by squares, diamonds, and stars, respectively. The elastic-scattering and breakup data are respectively related to the left and right axes. CDCC calculations for elastic scattering and EBU are represented by solid curves. The one-channel calculations for the elastic scattering are shown by the dashed curves. The dotted lines correspond to the IAV model results for NEB contributions. The dash-dotted lines denote the sum of EBU and NEB. The figures are sourced and modified from Ref. [59]

Angular distributions for the exclusive and inclusive measurements of ⁷Be are shown in Fig. 18, which are distinguished by stars and diamonds, respectively. These datasets revealed a concordance between them, well within the bounds of experimental uncertainty. This congruence, observed for the first time, offers clear experimental evidence that the production of ⁷Be is predominantly dictated by breakup processes rather than proton-transfer mechanisms.

Calculations employing the CDCC and Ichimura, Austern, and Vincent (IAV) models [65] were performed to evaluate the individual contributions of the elastic breakup (EBU) and non-elastic breakup (NEB) for direct reactions. The CDCC and IAV results are shown in Fig. 18, represented by the solid and dotted curves, respectively. The sum of the CDCC and IAV calculations, referred to as the total breakup (TBU), is shown in Fig. 18 as a dotted-dashed curve. The inclusive breakup angular distribution can be properly described using the TBU curve. The cross-sections for EBU and NEB, as derived from the calculations, were determined to be σ_EBU=351.5 (420.5) mb and σ_NEB=78.3 (91.4) mb at an energy of 38.7 (46.1) MeV, respectively. These results clearly demonstrate that while the NEB contribution is non-negligible, it represents a relatively modest fraction, approximately 18%, of the total ⁷Be yield.

The reconstructed relative-energy distribution E_rel of the breakup fragments resulting from the ⁷Be+ p reaction at 38.7 MeV is shown in Fig. 19 (a). The solid curves represent the simulation based on the outputs of the CDCC, which reproduced the experimental data well. A peak in the E_rel spectrum, located at an energy of approximately 0.6 MeV, has been observed. This energy corresponds closely to the first resonance state of ⁸B (E_ex.=0.77 MeV, Jπ=1⁺, Γ=35.6 keV). Using a CDCC-based simulation, the contribution of this 1⁺ resonance to the breakup process was evaluated and quantified as (4.4±2.0)% at 38.7 MeV. At a higher energy of 46.1 MeV, the contribution of this resonance was determined to be (3.8±2.5)%. Considering the lifetime of this resonance, which is on the order of 10^-20 s, it is sufficiently prolonged to ensure that the breakup occurring via this state is predominantly along the outgoing trajectory, thereby moving away from the target nucleus. However, a modest proportion of the 1⁺ resonance suggests that the prompt component significantly governs the breakup mechanism.

Fig. 19Full-screen View

(a) Measured E_rel distribution and (b) angular correlation for breakup fragments ⁷Be and p from the ⁸B+¹²⁰Sn system at 38.7 MeV. Circles denote the experimental data. The solid and dashed curves in (a) represent the simulated distributions of E_rel and the contribution of the p-wave 1⁺ state based on the detailed CDCC outputs. The squares in (b) represent the CDCC-based simulation results, and the solid curve denotes the expected β-θ₁₂ correlation assuming asymptotic breakup from the 1⁺ resonance of ⁸B. Figures are sourced and modified from Ref. [59]

Reconstructed correlations of the laboratory opening angle (θ₁₂) and orientation of the relative momentum of the breakup fragments (β) in their center-of-mass frame at 38.7 MeV are shown in Fig. 19(b). The dashed line in the figure represents the theoretical correlation anticipated between θ₁₂ and β for the asymptotic breakup via the 1⁺ resonant state of ⁸B [66]. Events that fall along the dashed line indicate the asymptotic nature of the breakup. This implies that the fragments moved beyond the influence of the target Coulomb field, indicative of a distant breakup event. Conversely, breakup events occurring in close proximity to the target nucleus exhibit a perturbed θ₁₂-β correlation, which is attributed to the effects of Coulomb post-acceleration [66]. As depicted in Fig. 19(b), a significant divergence of events from the asymptotic limit is observed. This is indicative of the predominance of a near-target breakup. Simulations based on CDCC calculations offer a reasonable description of the θ₁₂-β correlations represented by the squares in the figure. In terms of θ₁₂, the majority of events are concentrated at small angles, with a pronounced peak around 30°. This distinct forward peak of θ₁₂ implies that most breakup events, while prompt, occur along the outgoing trajectory. This breakup mechanism can potentially exert a minor impact on the complete fusion cross-section of ⁸B. Notably, similar results are observed for the case of 46.1 MeV, reinforcing the consistency of the observed phenomena across the different energy regimes.

¹⁷F+⁵⁸Ni measurement

We performed the first complete kinematics measurements of ¹⁷F interacting with a medium-mass target ⁵⁸Ni at CRIB [60]. Four distinct energies, 43.6±0.7, 47.5±0.7, 55.7±0.8, and 63.1±0.9 MeV, were achieved in the middle of the target with typical intensities of 6-10×10⁵ pps and a purity of approximately 85%. MITA was used for the measurements. Owing to the excellent particle-identification capabilities of MITA, particles as light as protons, deuterons, and alpha particles, and heavier ions, such as ¹⁶O and ¹⁷F, were unambiguously discerned.

CDCC calculations were performed to interpret the quasi-elastic scattering data. The results indicated that the influence of breakup coupling on elastic scattering was relatively minor, suggesting that these couplings do not significantly influence elastic scattering. The angular distributions of ¹⁶O generated by the reaction of ¹⁷F+⁵⁸Ni were subjected to theoretical analysis using both CDCC and IAV model calculations. The results of these calculations demonstrated that exclusive ¹⁶O data were reproduced using the CDCC approach. Moreover, the sum contributions of the EBU and NEB, collectively denoted as TBU, provided a good description of both the amplitude and structural features of the inclusive data. The results also indicated that the NEB is the predominant constituent of the total inclusive ¹⁶O yield.

The total fusion (TF) cross-sections, represented as σ_TF, were determined through an analysis of fusion-evaporation protons and alpha particles, as documented in [62]. A comparative examination of the reduced total-reaction cross-sections σ_R and σ_TF are performed for both ¹⁷F [60, 67] and ^16,17O interacting with a ⁵⁸Ni target [68-70]. This comparison is presented in Figs. 20 (a) and (b). As shown in the figures, the cross-sections of ¹⁷F, both σ_R and σ_TF, exhibit a similar behavior with those of ¹⁶O at energies above the Coulomb barrier. However, a noteworthy divergence was observed in the sub-barrier energy domain, where ¹⁷F demonstrates a significant enhancement in the cross sections.

Fig. 20Full-screen View

Comparison of the reduced (a) σ_R and (b) σ_TF between ¹⁷F+⁵⁸Ni and oxygen systems. The σ_R of ¹⁷F+⁵⁸Ni, denoted by stars, are from our work [60]. The results of ¹⁷F, ¹⁷O, and ¹⁶O +⁵⁸Ni, represented by filled diamonds, empty circles, and empty diamonds, are sourced from Refs. [67-69], respectively. The dashed curve in (a) shows the trend of the behavior of ^16,17O+⁵⁸Ni. The filled and empty stars in (b) represent the σ_TF of ¹⁷F+⁵⁸Ni deduced from fusion- evaporated protons and alpha particles, respectively. The empty triangles are the results for ¹⁶O +⁵⁸Ni sourced from Ref. [70]. The solid curve in (b) represents the benchmark UFF [71]. The figures are sourced and modified from Ref. [60]

Excitation functions for the ¹⁷F+⁵⁸Ni system, including the total reaction, exclusive and inclusive breakup processes, and TF, are shown in Fig. 21. Superimposed on this representation is the theoretical predictions of the corresponding reaction channels. The sum of the cross-sections for the inclusive breakup, TF, and excitations to the first excited states of ¹⁷F and ⁵⁸Ni nearly saturated the total-reaction cross-section, leaving minimal space for additional, unaccounted reaction channels.

Fig. 21Full-screen View

Excitation functions of total reaction (squares), exclusive (diamonds) and inclusive (triangles) breakups, and the total fusion (TF) deduced from fusion-evaporation protons (circles) of ¹⁷F+⁵⁸Ni. The solid curve denotes the CC calculations for the σ_R. The dash-dotted and dashed curves represent the TF derived from CDCC calculations with and without the continuum couplings, respectively. The dash-dotted and dotted curves are the theoretical results of CDCC plus the IAV model and CDCC, corresponding to the inclusive and exclusive breakup, respectively. The figure is sourced and modified from Ref. [60]

CDCC calculations were performed to evaluate the influence of breakup on fusion-reaction dynamics. The results of the CDCC calculations are shown in Fig. 21, where the dashed-dotted curve corresponds to the calculations including couplings to the continuum states, and the dashed curve represents calculations exclusive to such couplings. At energies above the Coulomb barrier, the theoretical results are essentially indistinguishable from one another regardless of whether they incorporate couplings to the continuum states. These theoretical predictions were observed to be in good agreement with the experimental fusion cross-sections. By contrast, within the sub-barrier energy region, the experimental data exhibited good agreement with the theoretical description when the couplings to the continuum states were included in the calculations. Calculations that exclude these couplings are prone to a substantial underestimation of the experimental data, thereby highlighting the strong coupling effects of the breakup in fusion reactions below the Coulomb barrier.