Elastic scattering for the B 8 and Be 7 + Pb 208 systems at near-Coulomb barrier energies

PHYSICAL REVIEW C 100, 024602 (2019) Elastic scattering for the 8Band7Be +208Pb systems at near-Coulomb barrier energies M. Mazzocco ,1,2,*N. Keeley,3A. Boiano,4C. Boiano,5M. La Commara,6,4C. Manea,2,†C. Parascandolo,4 D. Pierroutsakou,4C. Signorini,1,2E. Strano,1,2D. Torresi,1,2,‡H. Yamaguchi,7D. Kahl,7,§L. Acosta,8,9,P. Di Meo,4 J. P. Fernandez-Garcia,9,¶T. Glodariu,10,#J. Grebosz,11 A. Guglielmetti,12,5Y. Hirayama,13 N. Imai,7,13 H. Ishiyama,13 N. Iwasa,14 S. C. Jeong,13,15 H. M. Jia,16 Y. H. Kim,13 S. Kimura,13,** S. Kubono,7,17 G. La Rana,18,4C. J. Lin,16 P. Lotti,2 G. Marquínez-Durán,8I. Martel,8,19 H. Miyatake,13 M. Mukai,13 T. Nakao,7M. Nicoletto,2A. Pakou,20 K. Rusek,21 Y. Sakaguchi,7A. M. Sánchez-Benítez,22,23 T. Sava,10 O. Sgouros,20,‡V. Soukeras,20,‡F. Soramel,1,2E. Stiliaris,24 L. Stroe,10 T. Teranishi,25 N. Toniolo,26 Y. Wakabayashi,17 Y. X. Watanabe,13 L. Yang,16,7Y. Y. Yang,27 and H. Q. Zhang16 1Dipartimento di Fisica e Astronomia, Università di Padova, via F. Marzolo 8, I-35131 Padova, Italy 2INFN-Sezione di Padova, via F. Marzolo 8, I-35131 Padova, Italy 3National Centre for Nuclear Research, ul. Andrzeja Sołtana 7, 05-400 Otwock, Poland 4INFN-Sezione di Napoli, via Cintia, I-80126 Napoli, Italy 5INFN-Sezione di Milano, via Celoria 16, I-20133 Milano, Italy 6Dipartimento di Farmacia, Università di Napoli “Federico II”, via D. Montesano, I-80131 Napoli, Italy 7Center for Nuclear Study (CNS), The University of Tokyo, Wako Branch, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan 8Departamento de Física Aplicada, Universidad de Huelva, Campus de El Carmen, E-21071 Huelva, Spain 9INFN-Sezione di Catania, via Santa Sofia 64, I-95123 Catania, Italy 10NIPNE, 30 Reactorului Street, 077125 Magurele, Romania 11Institute of Nuclear Physics Polish Academy of Science, ul. Radzikowskiego 152, 31-342 Kraków, Poland 12Università degli Studi di Milano, Dipartimento di Fisica, Via Celoria 16, I-20133 Milano, Italy 13KEK, Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan 14Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan 15Institute for Basic Science (IBS), 55, Expo-ro, Yuseong-gu, 34126 Daejeon, Korea 16China Institute of Atomic Energy, P.O. Box 275, Beijing 102413, China 17RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan 18Dipartimento di Fisica, Università di Napoli “Federico II”, via Cintia, I-80126 Napoli, Italy 19Department of Physics, University of Liverpool, Liverpool L69 7ZE, United Kingdom 20Department of Physics and HINP, University of Ioannina, 45110 Ioannina, Greece 21Heavy Ion Laboratory, University of Warsaw, ul. Pasteura 5a, 02-093 Warsaw, Poland 22Nuclear Physics Center, University of Lisbon, P-1649-003 Lisbon, Portugal 23Centro de Estudios Avanzados en Física, Matemáticas y Computación (CEAFMC), Department of Integrated Sciences, University of Huelva, E-21071 Huelva, Spain 24Institute of Accelerating Systems and Applications and Department of Physics, University of Athens, Athens, Greece 25Department of Physics, Kyushu University, Motooka 744, Fukuoka 819-0395, Japan 26INFN-LNL, viale dell’Università 2, I-35020 Legnaro (PD), Italy 27Institute of Modern Physics, Chinese Academy of Sciences, 509 Nanchang Road, Lanzhou 730000, China (Received 7 June 2019; published 1 August 2019) The elastic scattering of the weakly bound radioactive nuclei 8Band7Be from a 208Pb target was measured for the first time in the energy range around the Coulomb barrier. The data were analyzed using the optical model and the continuum discretized coupled channels (CDCC) formalisms. The reaction cross sections extracted from the optical model fits clearly indicate a remarkably enhanced reaction probability for the very weakly bound 8B (Sp=137.5 keV) compared to similar mass nuclei interacting with the same target nucleus. CDCC calculations assuming a 3He +4He cluster model well described the 7Be experimental elastic scattering angular distributions, whereas the use of a 7Be +pcluster model of 8B with an inert 7Be core gave a relatively poor description of the corresponding experimental data, suggesting that this model may be too simplistic and the possibility of core excitation should not be ignored. DOI: 10.1103/PhysRevC.100.024602 *[email protected] †Present address: INFN-TIFPA, Via Sommarive 14, I-38123 Povo (TN), Italy. ‡Present address: INFN-Laboratori Nazionali del Sud (LNS), via Santa Sofia 62, I-95123 Catania, Italy. 2469-9985/2019/100(2)/024602(13) 024602-1 ©2019 American Physical Society M. MAZZOCCO et al. PHYSICAL REVIEW C 100, 024602 (2019) I. INTRODUCTION The investigation of the reaction dynamics of light weakly bound nuclei in the energy range around the Coulomb barrier has attracted the interest of the low-energy nuclear physics community for many years now. Several review articles have been published on this subject, e.g., Refs. [1–9]. Light weakly bound nuclei are characterized by very low nucleon separation energies, typically one order of magnitude smaller than for stable nuclei located along the valley of βstability, and, in most cases, by well pronounced cluster structures. These features can generate nuclei with exotic shapes, such as the “halo” nuclei, which may be described as a well-bound inert core surrounded by a halo of rarefied nuclear matter. Depending on the type and the number of nucleons in the nuclear halo, we may have 1n-halo nuclei like 11Be (Sn= 0.501 MeV), 2n-halo nuclei such as 6He (S2n=0.972 MeV) and 11Li (S2n=0.369 MeV), or possible 1p-halo nuclei, for instance the proton-halo candidate 8B(Sp=0.1375 MeV). The low binding energy and the presence of the nuclear halo contribute to enlarging the variety of nuclear reaction mechanisms which can be triggered by these exotic projectiles. The situation is particularly remarkable in the energy range around the Coulomb barrier, where several reaction channels open up. Being very weakly bound, the projectile can more easily break up while in the proximity of the target Coulomb and nuclear fields, or one of its constituent clusters may be readily transferred or fuse with the target. All these processes are strongly interconnected to each other. Depending on how the breakup channel was modeled, either enhancement or hindrance of the sub-barrier fusion cross section was predicted. Despite earlier measurements [10,11], it was soon realized that breakup related effects increase the total reaction cross section rather than the fusion probability (Ref. [1] and references therein) and that the enhanced reactivity of weakly bound nuclei was essentially due to direct processes, especially at sub-barrier energies. The investigation then moved to understanding what direct processes were mostly responsible for the observed enhancement of the reaction cross section. A series of measurements performed with the 2n-halo nucleus 6He [12–21], the neutron skin nucleus 8He [22–25], and 8Li [26,27] indicated neutron transfer channels as the most dominant reaction mechanisms at near-barrier energies. More recently, experiments performed with the 2n-halo 11Li [28,29] and the 1n-halo 11Be §Present address: School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK. Present address: Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Cd. Mx. 01000, Mexico. ¶Present address: Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, E-41080 Seville, Spain. #Deceased. **Present address: RIKEN, 2-1 Hirosawa, Wako, Saitama 351- 0198, Japan. [30,31] also pointed out the relevance of the breakup process in this energy range. These strong channels clearly have an impact on the elastic scattering process and, in fact, large deviations from Rutherford scattering were observed even at deep sub-barrier energies for many systems—see, for instance, Refs. [28,31]— together with the disappearance of the Coulomb-nuclear interference peak. In this respect a measurement of the elastic scattering process, which for very low intensity Radioactive Ion Beams (RIBs) is sometimes the only feasible one, has a double utility: (1) It provides gross information concerning the reactivity of an exotic projectile via the extraction of the total reaction cross section. (2) It allows the investigation of the influence of nonelastic reaction channels (breakup, transfer, fusion, core excitations, etc.) on the elastic scattering angular distribution. In this context we have performed the first measurement of the elastic scattering of the 1p-halo candidate 8B from the heavy target 208Pb at Coulomb barrier energies. 8Bisavery interesting nucleus: it is very weakly bound (Sp=137.5keV), it is the lightest particle stable boron isotope, and it plays a fundamental role in the production of high-energy neutrinos in the solar fusion model [32]. The reaction dynamics of this exotic projectile have already been investigated for a variety of targets, from 12Cto58Ni, in the energy range around the Coulomb barrier. The use of different targets helps exploration of the influence on the reaction dynamics of the different nuclear and Coulomb interactions and of transfer channels with different Qvalues. Before describing the experiment, we briefly summarize the main results of these studies. Recently, Morcelle and coworkers measured the elastic scattering for the 8B+27Al system at 15.3 and 21.7 MeV [33]. The experiment was partly performed at the Twinsol facility (University of Notre Dame, USA) [34] and partly at RIBRAS (São Paolo, Brazil) [35,36]. The authors noted that the breakup channel had a small, but not negligible, effect on the elastic scattering differential cross section and that it slightly damped the Coulomb rainbow peak originating from the interference between the nuclear and Coulomb interactions. As a result the 8B reaction cross sections (in reduced units) were enhanced with respect to those for other light weakly bound projectiles interacting with a 27Al target and were much larger than those for the doubly magic projectile 16O. Pakou and collaborators measured the α-particle production cross section for the 8B+28Si system [37] and deduced the fusion cross section at four above-barrier energies via statistical model calculations. The experiment was performed at the EXOTIC facility (LNL, Italy) [38] by means of the active target technique. The trend of the experimental data was fairly well reproduced by the predictions of the Universal Fusion Function (UFF) [39]. By far the most studied system involving a 8B projectile is 8B+58Ni, investigated in a series of experiments performed at TwinSol, beginning with the pioneering work of Guimarães and coworkers, who measured the 7Be yield, 024602-2 ELASTIC SCATTERING FOR THE 8B AND 7Be+… PHYSICAL REVIEW C 100, 024602 (2019) essentially equivalent to the 8B→7Be +pbreakup cross section [40,41]. In the following years, Aguilera et al. [42] measured the elastic scattering at five energies around the Coulomb barrier. The extracted reaction cross section was as large as for the 2n-halo nucleus 6He and turned out to be exhausted by the sum of the 7Be +58Ni total reaction cross section (measured in the same experiment) and the 8B breakup cross section. Two years later the same group published a measurement of the fusion cross section for the same system at about ten near-barrier energies [43] and observed that the sum of the cross sections for the fusion and breakup processes exhausted the total reaction cross section. A simultaneous analysis, performed by Gomez-Camacho et al. [44], of the elastic scattering angular distributions and the fusion cross sections suggested that the so-called Breakup Threshold Anomaly (BTA) [45,46] occurred in this system. As the fusion data generated a controversy [47], the original data for the 8B+28Si and 58Ni systems were critically reanalyzed [48] and the two fusion cross sections, when suitably reduced, were found to be in agreement within the error bars at the only overlapping energy point. We may conclude that, for a 8B projectile breakup, related effects tend to increase with the target mass as a consequence of the increasing predominance of the long-range Coulomb interaction with respect to the nuclear part of the potential. For this reason, we decided to investigate the elastic scattering process for the 8B+208Pb system at Coulomb barrier energies. The only existing measurements for this system, performed at Lanzhou (China), are at 170.3 [49] and 178 MeV [50], i.e., more than three times the Coulomb barrier. As expected for this bombarding energy [7], very small breakup effects on the elastic scattering differential cross section were observed. Our study was complemented by the first measurement of the elastic scattering process for the 7Be +208Pb system in the energy range around the Coulomb barrier, which was not only necessary to fix the core-target interaction in a 7Be +pcluster model of 8B but also of considerable interest in itself (see Ref. [51]), especially in relation to the similarities of 7Be with the binding energy of 6Li and the nuclear structure of 7Li, as underlined in the recent works by Sgouros and collaborators [52,53]. A quite comprehensive overview of previous experiments aimed at studying the 7Be-induced reaction dynamics has been already given in Ref. [54]. The paper is organized as follows: Section II describes the 8B and 7Be RIB production and the detector setup used in the experiments. Section III illustrates the data reduction procedure required to obtain the elastic scattering angular distributions for both systems. The results of an optical model analysis are presented in Sec. IV and continuum discretized coupled channels (CDCC) calculations are described and compared with the data in Sec. V. The results are discussed in Sec. VI and some concluding remarks are made in Sec. VII. II. EXPERIMENTS A. 8B secondary beam production The 8B RIB was produced by the in-flight technique with the CNS Radioactive Ion Beams (CRIB) facility [55] belonging to the Center for Nuclear Study (CNS) of the University of Tokyo, located on the RIKEN campus at Wako (Saitama, Japan). A 6Li3+primary beam with an energy of 11.2 MeV/nucleon and an intensity of about 1 pμAwas delivered by the AVF cyclotron and impinged on an 8-cm-long gas cell, doubly walled with 2.5-μm-thick Havar foils. The gas target was filled with 3He gas at a pressure of 1 bar and kept at a cryogenic temperature (≈90 K) [56]. The 8B beam was produced by means of the 3He(6Li,8B)n two-body reaction (Q=−1.97 MeV). The secondary beam intensity on target was about 104pps and the purity ≈20%, with 7Be, 6Li,and 3He being the most intense contaminants. Before reaching the final focal plane the 8B beam crossed two x-ysensitive Parallel Plate Avalanche Counters (PPACs) [57], located (for timing and beam reconstruction purposes) 669 and 439 mm upstream from the secondary target, and two collimators, displaced 306 mm (diameter 22 mm) and 180 mm (diameter 24 mm) upstream of the 208Pb target. The 8B beam energy on target was 50 ±1MeV. B. 8B+208Pb experimental setup Charged particles produced by the interaction between the 8B secondary beam and a 2.2-mg/cm2-thick 208Pb target (enrichment 99.57%) evaporated onto a 1.5-μm-thick mylar foil were detected by six modules of the EXPADES [58] detector array. Each module consisted of a double-sided silicon strip detector (DSSSD) E-Eres telescope. The thicknesses of the Eand Eres stages were 43–57 and 300 μm, respectively. Each detector had an active area of 64.0mm×64.0 mm and the front and rear sides were segmented into 32 strips in order to achieve a 2.0mm×2.0 mm pixel resolution. However, to reduce the cost of the readout electronics, the strips of the E layers were short circuited, two by two. Additional details on the detectors and the related electronics may be found in Ref. [58]. The EXPADES modules were located around the target position in a cylindrical configuration. The polar angle coverage of each telescope was θlab =[+8◦,+40◦] (telescope A), [+50◦,+84◦](B),[−55◦,−85◦](C),[+95◦,+128◦](D), [−95◦,−125◦] (E), and [−134◦,−166◦] (F). Positive and negative polar angles θlab refer to detectors displaced to the left and right, respectively, of the beam axis in a downstream view of the experimental setup. The central positions of the Elayers of telescopes A and F were located at a distance of 120 mm from the center of the target, while all other telescopes were at 110 mm. This configuration ensured a solid angle coverage of about 1.92 sr, corresponding to about 15% of 4π. The detectors were calibrated in energy in the range ≈3–6 MeV by means of standard αcalibration sources containing the following radioisotopes: 148Gd, 237Np, 241Am, and 244Cm. Typical energy resolutions were about 0.9% and 0.8% for Eand Eres layers, respectively. The data acquisition system trigger condition employed was the AND of the anode signals of the two PPACs and the OR of the silicon detectors. Trigger rates of about 10–15 Hz were typically recorded during the experiment. The effective beam on target period was about 90 h. The energy loss in the rather thick 208Pb target gives a mid-target beam energy of 49.35 MeV, within the uncertainty in the nominal beam energy 024602-3 M. MAZZOCCO et al. PHYSICAL REVIEW C 100, 024602 (2019) (50 ±1 MeV), so a value of 50 MeV was used in the optical model and CDCC analyses. C. 7Be secondary beam production The experiment with the 7Be RIB was performed at the EXOTIC facility [38], installed at the Laboratori Nazionali di Legnaro (LNL) of the Istituto Nazionale di Fisica Nucleare (INFN), in Italy. The 7Be beam was produced using the 1H(7Li,7Be)ntwo-body reaction (Q=−1.64 MeV), following a procedure similar to that described in Ref. [54]. A 7Li3+primary beam, delivered by the LNL XTU-Tandem accelerator with an energy of 48.8 MeV and an intensity of 50–60 p nA, impinged on a gas target filled with H2gas. The gas cell was 5 cm long and the entrance and exit windows were made of 2.2-μm-thick Havar foils. In this experiment the beam also crossed two x-ysensitive PPACs (described in detail in Ref. [58]), located 900 and 365 mm upstream of the final focal plane of EXOTIC, before reaching the secondary target. The first 7Be energy of 40.5±0.4 MeV was achieved by operating the target station at a gas pressure of 1 bar and at liquid nitrogen temperature (90 K). A second energy of 37.4±0.5 MeV was obtained by inserting a 12.5-μm-thick aluminum degrader immediately behind the first PPAC. Finally, a third measurement at 42.2±0.4 MeV was performed after warming up the target cell to room temperature and keeping the same hydrogen gas pressure inside. The secondary beam intensities were (2–3) ×105pps at the intermediate energy and about 105pps and (7–8) ×104pps at the lowest and highest energies, respectively. The beam purity was about 99% at all energies. D. 7Be +208Pb experimental setup The detector setup already described in Sec. II B was also used for the measurement of the charged particles produced by the interactions between the 7Be projectiles and a self-supporting 1-mg/cm2-thick 208Pb target (enrichment 99.57%). The six modules of EXPADES were also arranged in a cylindrical configuration around the target position in this experiment. The polar angle coverage of the telescopes was as follows: θlab =[+52◦,+85◦](A),[+95◦,+128◦](B), [+134◦,+171◦](C),[−52◦,−85◦](D),[−95◦,−128◦](E), and [−138◦,−167◦] (F). The central positions of the Elayers of detectors A–E were at an average distance of 111.5 mm from the center of the target. Telescope F was located a bit further away than the others at a distance of 141.5 mm. This arrangement guaranteed an overall solid angle coverage of 1.85 sr. The energy calibration was performed in the range ≈5–6 MeV using triple sources with the following long-lived αemitters: 239Pu, 241Am, and 244Cm. The energy resolutions were on average 1.1% for both Eand Eres telescope stages. The trigger for the data acquisition system was provided by the AND condition between the cathode signals of both PPACs and the OR of the silicon detectors. Trigger rates of about 10 Hz were typical throughout the experiment. The acquisition times for data collection were about 24, 48, and 8 h for the lowest, intermediate and highest energies, respectively. (MeV) res E 0 5 10 15 20 25 30 35 40 E (MeV)Δ 0 2 4 6 8 10 12 14 16 18 20 22 1 10 2 10 (a) 8 Be 7 He 4 He 3 p,d B (MeV) res E 0 5 10 15 20 25 30 35 40 E (MeV)Δ 0 2 4 6 8 10 12 14 16 18 20 22 1 10 2 10 B 8(b) FIG. 1. (a) E-Eres correlation plot for the 8B+208Pb reaction at 50 MeV collected by telescope A, covering the polar angle range θlab =[+8◦,+40◦]. The 7Be and 3He contaminant beams are clearly visible on the plot. (b) Same as (a) after imposing an RF gate on the incoming 8B beam (see text for additional details). We again used the nominal values for the beam energies in the optical model and CDCC analyses since the energy losses in the overall target thickness (≈400 keV) were within the uncertainties in the 7Be secondary beam energies. III. DATA ANALYSIS A. 8B+208Pb elastic scattering The upper panels of Figs. 1and 2show two typical E-Eres correlation plots collected by telescope A, located at forward angles, and telescope F, at backward angles, respectively. The events displayed correspond to those where only one detector strip on both the Evertical and horizontal sides fired (with the two energy deposits within an uncertainty of ±200 keV) and only one Eres horizontal strip. An energy threshold of approximately 250 keV was imposed on both telescope layers in the data reduction procedure, while charge sharing events, i.e., events releasing energy in the 40-μm-wide detector area in between two adjacent strips, were not considered in the present analysis. Due to gain instabilities it was not possible to use the energy information collected by the Eres vertical strips. The loci of the elastic scattering processes for the 8B, 7Be, and 3He beams are clearly visible in Fig. 1(a) and are still distinguishable in Fig. 2(a) in spite of the much smaller statistics. In addition, the kinematic locations of other reaction products, such as 4He, 2H, and 1H, can also be observed in both figures. As a remark, we observe that due to the excellent rejection capabilities of the CRIB facility only very few nuclei of the 1 p μA6Li primary beam were able to reach the final focal plane. 024602-4 ELASTIC SCATTERING FOR THE 8B AND 7Be+… PHYSICAL REVIEW C 100, 024602 (2019) (MeV) res E 0 5 10 15 20 25 30 35 40 E (MeV)Δ 0 2 4 6 8 10 12 14 16 18 20 22 0 1 2 3 4 5 6 7 8 9 (a) B 8 Be 7 He 4 He 3 p,d (MeV) res E 0 5 10 15 20 25 30 35 40 E (MeV)Δ 0 2 4 6 8 10 12 14 16 18 20 22 -2 10 -1 10 1 (b) B 8 Be 7 He 4 p,d FIG. 2. As in Fig. 1, but for telescope F, located in the backward hemisphere (θlab =[−134◦,−166◦]). The lower panels of Figs. 1and 2show the same correlation plots as the upper panels after requiring a coincidence with the radio-frequency (RF) signal corresponding to the Time- of-Flight (ToF) of the 8B secondary beam through CRIB. We can clearly see that the elastic peaks due to 3He and 7Be contaminant beams were suppressed to a large extent by this selection, while the remaining events should essentially originate from the interaction of the incoming 8B beam with the 208Pb target. The two-dimensional plot at backward angles [Fig. 2(b)] is rather clean and the surviving 7Be should mainly originate from the 8B→7Be +pbreakup channel, as we will discuss later. Figure 3presents the elastic scattering angular distribution for the 8B+208Pb system at a beam energy of 50 MeV. To avoid the limited efficiency introduced by the shade of the mechanical support of the telescopes on the detector active areas, especially for marginal trajectories, events collected in the strips at the detector edges were not processed. Moreover, to compensate for statistical fluctuations, events from independent triggers gathered in adjacent Evertical strips were grouped together and, where possible, the weighted averages of the differential cross sections computed by telescopes located at analogous polar angles in opposite hemispheres were employed. Of course, as can be seen in Fig. 3, this procedure is performed at the cost of a reduced angular resolution, especially at backward angles. We finally remark that the error bars displayed on Fig. 3include the statistical uncertainty only. In order to evaluate the elastic scattering differential cross section, we coded a Monte Carlo simulation following a procedure similar to those already employed in two previous experiments [54,59]. The code took into account the kinematics of the elastic scattering process, the Rutherford differential 0 20 40 60 80 100 120 140 160 180 θc.m. (deg) 0.0 0.2 0.4 0.6 0.8 1.0 Ratio to Rutherford 50.0 MeV 8B+208Pb FIG. 3. Elastic scattering angular distribution for the 8B+208Pb system at a beam energy of 50 MeV. Error bars are statistical only. The curve represents an optical model fit to the data; see text for more details. cross section, the beam spot size on target and the beam divergence (reconstructed by means of the two PPACs), and the detector geometry. In the simulation a random interaction point within the target thickness was assumed and the energy loss prior to and after the scattering process was computed using a proper parameterization of the SRIM stopping power tables [60]. Theoretical calculations, described in detail in the following section, predicted an essentially pure (within the statistical accuracy of our measurement) Rutherford scattering process for polar angles smaller than 40◦. Therefore, the experimental data were normalized, requiring the average value of the ratios to the Rutherford cross section evaluated for telescope A to be equal to unity. The systematic uncertainty introduced by the normalization procedure was estimated to be 2.5–3%. According to our analysis, no evidence for a Coulomb-nuclear interference peak appeared in the evaluation of the elastic scattering angular distribution, even if the value at θc.m.≈70◦ clearly does not follow the trend described by the neighboring data points. This behavior is present in both individual evaluations obtained from telescopes B and C. After the rather steep decrease around 90◦, the differential cross section tends to saturate to about one third of the Rutherford cross section for scattering angles larger than approximately 110◦. B. 7Be +208Pb elastic scattering Figure 4displays three E-Eres correlation plots for the 7Be +208Pb reaction at 40.5 MeV for events collected by telescope A at forward angles (top panel), telescope B at backward angles (middle panel), and telescope F at very backward angles (bottom panel). In all plots the peaks related to the elastic scattering process are clearly visible, obviously with decreasing statistics as the scattering angle increases from forward to backward polar angles. The kinematic loci for isotopes of hydrogen, helium, and (at backward angles) 024602-5 M. MAZZOCCO et al. PHYSICAL REVIEW C 100, 024602 (2019) (MeV) res E 0 5 10 15 20 25 30 E (MeV)Δ 0 5 10 15 20 25 1 10 (a) Be 7 He 4 He 3 p,d (MeV) res E 0 5 10 15 20 25 30 E (MeV)Δ 0 5 10 15 20 25 1 10 2 10 (b) Be 7 Li 7 He 4 He 3 p,d (MeV) res E 0 5 10 15 20 25 30 E (MeV)Δ 0 5 10 15 20 25 1 10 2 10 (c) Be 7 Li 7 He 4 He 3 p,d FIG. 4. E-Eres correlation plots for the 7Be +208Pb reaction at 40.5 MeV registered by telescope A, θlab =[+52◦,+85◦](a), telescope B, θlab =[+95◦,+128◦] (b), and telescope F, θlab = [−138◦,−167◦] (c). In addition to the elastic scattering peaks the kinematic loci related to the isotopes of hydrogen and helium produced by the interactions between the beam and the target are clearly visible. lithium produced by the interactions between the 7Be RIB and the 208Pb target can also be distinguished in Fig. 4. The data reduction followed a procedure very similar to that of the 8B+208Pb elastic scattering events. Again, we did not consider events registered in either the detector marginal strips or interstrip areas, and we selected only multiplicity-1 events for both Evertical and horizontal strips and the Eres horizontal strips. Since the tandem accelerator provided a continuous primary beam no radio-frequency information was available for this experiment. However, since the secondary beam had a purity of 99%, the residual contamination had a negligible influence on the evaluation of the elastic scattering differential cross sections. To compensate for the large statistical fluctuations, events collected in adjacent Evertical strips were grouped together. In this respect four differential cross section values per telescope were obtained for the data set at 40.5 MeV, whereas for the highest and lowest energies we employed only two multistrips, consisting of seven adjacent Evertical strips, per telescope. 40 60 80 100 120 140 160 180 θc.m. (deg) 0.0 0.2 0.4 0.6 0.8 1.0 Ratio to Rutherford 42.2 MeV 40.5 MeV 37.4 MeV 7Be+208Pb FIG. 5. Quasielastic differential cross section for 7Be +208Pb at beam energies of 42.2 MeV (red triangles), 40.5 MeV (black circles), and 37.4 (blue squares). Errors bars are statistical only. The curves represent optical model fits to the data; see text for details. Figure 5presents the quasielastic scattering differential cross sections for the 7Be +208Pb reaction at 40.5 MeV (black circles), 37.4 MeV (blue squares), and 42.2 MeV (red triangles). Due to the secondary beam energy spread, target thickness, and detector energy resolution it was not possible to separate pure elastic scattering events from those leading to the excitation of the 7Be first excited state at 0.429 MeV. Therefore, the data presented in Fig. 5are “quasielastic,” consisting of the sum of these two processes. The errors plotted on Fig. 5are statistical only. As with the 8B data, the normalization procedure was performed by means of a Monte Carlo simulation. We required that at all measured energies the differential cross sections for polar angles θcm smaller than 75–80◦be equal (on average) to the Rutherford cross section since, according to theoretical calculations described later, pure Rutherford scattering was expected at these angles. The systematic uncertainty introduced by this procedure was estimated to be about 3% for the lowest and middle energies and up to 7% for the highest energy. We also note that for the 7Be +208Pb reaction our data do not show any evidence of a Coulomb-nuclear interference peak before the sharp drop of the differential cross section due to the opening of nuclear reaction mechanisms. However, we emphasize that at the highest energy this peak, if present, would occur in the angular region where the target frame is positioned and therefore would not be covered by our detector setup. IV. OPTICAL MODEL ANALYSIS As a first step the data were analyzed using the optical model with volume Woods-Saxon potentials, principally to extract total reaction cross sections, σR. The data were fitted by χ2minimization using SFRESCO, the searching version of the FRESCO code [61]. 024602-6 ELASTIC SCATTERING FOR THE 8B AND 7Be+… PHYSICAL REVIEW C 100, 024602 (2019) TABLE I. Optical potential parameters obtained by fitting the 8Band7Be +208Pb elastic scattering data. Total reaction cross sections (σR) are also reported. Radii follow the convention Ri= ri×At1/3fm and rC=1.3fm. Projectile Vr VaVWr WaWσR(mb) 8B (50.0 MeV) 109.5 1.326 0.811 15.00 1.234 1.85 1112 7Be (42.2 MeV) 134.3 1.326 0.811 14.42 1.534 0.884 371 7Be (40.5 MeV) 128.1 1.326 0.811 15.71 1.534 0.884 253 7Be (37.4 MeV) 194.8 1.326 0.811 19.05 1.534 0.884 121 The starting parameters for the 50 MeV 8B+208Pb search were taken from the global 6Li optical potential of Cook [62]. The resulting parameters are given in Table I, although, given the sparse nature of the data points and the relatively large uncertainties, the exact values have little physical significance beyond the observation of a definite preference for an imaginary potential with a large (>1.0) diffuseness parameter, aW. The data were found to be relatively insensitive to the real part of the optical potential, therefore these parameters were held fixed at the values of the Cook global 6Li potential and the parameters of the imaginary part only were varied. The resultant fit is shown by the solid curve on Fig. 3. The 7Be +208Pb quasielastic scattering data were fitted in a similar fashion. The slight error introduced into the fitting procedure by treating the data as pure elastic scattering was ignored since this was estimated to be within the statistical accuracy of the data. Test calculations found that the global 6Li parameters of Cook [62] gave a better description of the 7Be data than the 7Li set, as predicted by the calculations presented in Ref. [51]. The 6Li parameters were therefore used as a starting point for the fitting procedure. Again, given the sparse nature of the points and their limited statistics we did not attempt full parameter space searches. It was found that varying the real and imaginary well depths was sufficient to obtain good descriptions of the data at each energy. However, the exact values of these parameters are again without much physical significance, particularly at the lowest beam energy, mainly due to the sparseness of the data points. Overall, we estimate the total reaction cross sections extracted in this way to have uncertainties of the order of 20%, typical for RIB data of this type. V. CONTINUUM DISCRETIZED COUPLED CHANNELS CALCULATIONS The data were further analyzed by means of CDCC calculations, explicitly taking into account the 8B→7Be +p and 7Be →4He +3He breakup processes. The calculations are fully described in the following two subsections. All calculations were carried out with the FRESCO code [61]. A. The 8B+208Pb case The CDCC calculations for the 50 MeV 8B+208Pb data assumed a 7Be +pcluster model of 8B with an inert 7Be core. The model was further simplified by setting the spin of the 7Be core to zero, in order to avoid the prohibitive 0 20 40 60 80 100 120 140 160 180 θc.m. (deg) 0.0 0.2 0.4 0.6 0.8 1.0 Ratio to Rutherford Full CDCC No coupling 8B+208Pb FIG. 6. Elastic scattering angular distribution for the 8B+208Pb system at a beam energy of 50 MeV. The solid curve denotes the result of the full CDCC calculation while the dashed curve denotes that of the no-coupling case. computational overhead of calculations including the full spin. Tests confirmed that the omission of the core spin made a negligible difference to the results. Diagonal and transition potentials were calculated using the cluster folding model [63]. The 7Be +pbinding potential was taken from Ref. [64] (including the spin-orbit term, since the proton spin was retained in our model) and the proton was considered to be in a pure p3/2state. The cluster folding model further requires optical model potentials for the core +target and cluster +target at the same incident energy per nucleon as the projectile, in this case 43.75 and 6.25 MeV for 7Be +208Pb and p+208Pb, respectively. The 7Be +208Pb optical potential employed the parameters of the fit to the 42.2 MeV data given in Table Iand the p+208Pb optical potential used the 9 MeV parameters from Table I of Ref. [65]. The 7Be +pcontinuum was divided up into bins in momentum (k) space of width k=0.1fm −1up to a maximum value kmax =0.7fm −1, corresponding to a maximum energy above the breakup threshold of 11.7 MeV. Angular momenta of the 7Be +prelative motion L=0–5 ¯hwere included in the binning scheme with couplings (including continuum-continuum couplings) up to multipolarity λ=5. This continuum space was found to give adequate convergence of both the elastic scattering and the breakup cross sections. The resulting elastic scattering angular distribution is compared with the data in Fig. 6, together with the no-coupling result. The total reaction cross section given by the full CDCC calculation is 1020 mb and the total 8B→7Be +p breakup cross section is 619 mb. Of this, approximately 46% is due to dipole breakup, where in order to estimate the dipole component we have assumed that the continuum bins with relative p+7Be angular momenta L=0 and 2 are solely populated by direct L=1 excitation from the ground state (in reality there will also be some contribution from other L’s via the continuum-continuum couplings, which we 024602-7 M. MAZZOCCO et al. PHYSICAL REVIEW C 100, 024602 (2019) assume to be small.) The no-coupling calculation yields a total reaction cross section of 558 mb. Figure 6shows that the breakup coupling effect, while not negligible, is hardly commensurate with the large breakup cross section. This is consistent with previous CDCC calculations for the interaction of 8B with 208Pb [4] and 58Ni [4,66] targets. It is also apparent that the full calculation gives a relatively poor description of the elastic scattering data. A good description of the 8B data can be obtained by multiplying the imaginary part of the cluster folding potential, including the transition potentials, by a factor of 2.5, possibly compensating for the influence of some effect missing from the simplified model of 8B employed here. Since the similar CDCC calculations of Ref. [66]forthe8B+58Ni system give a satisfactory description of the data (although there are hints of similar problems at some energies), this may be linked with the larger Coulomb field of the 208Pb target. B. The 7Be +208Pb case The 7Be +208Pb CDCC calculations employed the 3He + 4He cluster model of 7Be of Ref. [67]. In addition to the bound 1/2−first excited state (Eex =0.429 MeV), the continuum space consisted of the two L=3¯hresonances (7/2−and 5/2−at Eex =4.57 and 6.73 MeV, respectively) plus the L=0–4 ¯hnonresonant continuum. All couplings, including continuum-continuum couplings, up to a multipolarity of λ=4 were included in the calculations. The continuum was divided into bins in kspace of width k=0.1fm −1up to maximum values of kmax =0.9fm −1, corresponding to a maximum energy above the breakup threshold of 9.88 MeV, for the calculations at beam energies of 42.2 and 40.5 MeV, and kmax =0.8fm −1, corresponding to a maximum energy above the breakup threshold of 7.81 MeV, for the calculation at a beam energy of 37.4 MeV. This scheme was suitably modified in the presence of the two L=3¯hresonances, which were treated as bins of width 0.4 and 2.0 MeV for the 7/2−and 5/2−states, respectively, to avoid double counting. This continuum space gave adequate convergence of both the elastic scattering and breakup cross sections. The diagonal and transition potentials were again calculated using the cluster folding model. The 3He +4He binding potential was taken from Ref. [67] and the 4He +208Pb optical potentials were calculated using the global parameters of Ref. [68]. The 3He +208Pb optical potentials for the 42.2 and 40.5 MeV calculations were obtained by refitting the 20 MeV 3He +208Pb elastic scattering data of Ref. [69], yielding values of V=28.4MeV, rv=1.50 fm, av=0.60 fm, W= 9.17 MeV, rw=1.50 fm, and aw=0.60 fm, with the form factors being of volume Woods-Saxon type and the radii given by Ri=ri×2081/3fm. The Coulomb radius parameter was rC=1.50 fm. As the calculation at 37.4 MeV required a 3He +208Pb optical potential for a 3He energy of 16 MeV, significantly lower than 20 MeV, and no suitable data are available at this energy, the real and imaginary depths of the 3He +208Pb optical potential were tuned to obtain the best description of the 7Be +208Pb elastic scattering at this energy by the full CDCC calculation. This was achieved by multiplying both by a factor of 2.0. 0.0 0.2 0.4 0.6 0.8 1.0 Full CDCC 2 chan 0.2 0.4 0.6 0.8 1.0 Ratio to Rutherford 40 60 80 100 120 140 160 180 θc.m. (deg) 0.6 0.8 1.0 42.2 MeV 40.5 MeV 37.4 MeV (a) (b) (c) FIG. 7. Quasielastic scattering angular distributions for the 7Be +208Pb system at beam energies of 42.2 (a), 40.5 (b), and 37.4 (c) MeV. The solid curves denote the results of the full CDCC calculation while the dashed curves denote those of the two-channel case; see text for details. The dotted curve at 37.4 MeV denotes the quasielastic scattering predicted by a full CDCC calculation using the same 3He +208Pb optical model potential as at 42.2 and 40.5 MeV. The resulting quasielastic scattering angular distributions are compared with the data in Fig. 7. Also shown on the figure are the quasielastic scattering angular distributions obtained from two-channel calculations including ground state reorientation and coupling to the bound 1/2−state of 7Be only. These results are shown in preference to the no-coupling cases since they represent the simplest coupling scheme that will yield a quasielastic scattering cross section. The description of the data at 42.2 and 40.5 MeV by the CDCC calculations is good, especially when it is recalled that no tuning of parameters was involved at these energies. The description of the 37.4 MeV data is also good, although here the 3He +208Pb optical potential was tuned to yield a better description. Tests with different 3He and 4He +208Pb optical potentials found that the choice of a realistic 3He + 208Pb potential, i.e., one that described scattering data at 024602-8 ELASTIC SCATTERING FOR THE 8B AND 7Be+… PHYSICAL REVIEW C 100, 024602 (2019) TABLE II. Total reaction (σR) and breakup (σbu) cross sections for 7Be +208Pb obtained from the CDCC and two-channel calculations; see text for details. Energy Two-channel calculation Full CDCC 42.2 MeV σR=475 mb σR=463 mb σbu =38 mb 40.5 MeV σR=339 mb σR=337 mb σbu =30 mb 37.4 MeV σR=222 mb σR=225 mb σbu =20 mb the appropriate energy (3/7ofthe7Be beam energy) was essential to a good description of the 42.2 and 40.5 MeV data. The calculations were found to be relatively insensitive to the choice of 4He +208Pb potential. These findings provide the justification for tuning the 3He +208Pb potential for the calculation at 37.4 MeV. To further emphasize the need to tune the 3He +208Pb optical potential in order to obtain a good description of the 37.4 MeV data, we plot in Fig. 7(c) the quasielastic scattering from a full CDCC calculation at 37.4 MeV using the same 3He +208Pb optical potential as at 42.2 and 40.5 MeV. It significantly overpredicts the backward angle data points. The total reaction cross sections for the two-channel and full CDCC calculations and the total 7Be →4He +3He breakup cross sections for the full CDCC calculations are given in Table II. Note that the breakup cross sections are a relatively small fraction of the total reaction cross section (of the order of 10%). Approximately 13% of the breakup is dipole at 42.2 MeV, increasing to about 34% at 40.5 and 37.4 MeV (the dipole contributions were estimated by summing the breakup cross sections for continuum bins where the 3He +4He relative angular momenta were L=0 and 2, so will again contain some contribution from other Lvalues). The reduction in the total reaction cross section at 42.2 and 40.5 MeV caused by the addition of coupling to breakup is by no means unusual and has already been remarked on; see, e.g., Ref. [70]. VI. DISCUSSION The optical model fits described in Sec. IV yield a much larger total reaction cross section for the 8B+208Pb system than for 7Be +208Pb. This might be aprioriexpected due to the order-of-magnitude lower breakup threshold for 8B (0.1375 MeV for the 8B→7Be +pprocess) compared to that for 7Be (1.5866 MeV for the 7Be →4He +3He process). The CDCC calculations of Sec. Vappear to confirm this expectation since they predict a 8B breakup cross section over an order of magnitude larger than those for 7Be. However, breakup is seldom the dominant direct contribution to the total reaction cross section for systems involving weakly bound projectiles, so it is of interest to compare these values with those obtained for similar systems. In Fig. 8we plot the total reaction cross sections for the 8B+208Pb and 7Be +208Pb systems derived from the FIG. 8. Total reaction cross sections for the 8B, 7Be, 6Li, 7Li, and 8Li +208Pb systems extracted from optical model fits plotted on the dimensionless scale according to the formalism of Canto et al. [73]. present optical model fits with those for the 6,7Li +208Pb [71] and 8Li +208Pb [72] systems on the dimensionless scale according to the formalism of Canto et al. [73]. The Coulomb barrier radii, heights, and curvatures required for the reduction procedure were calculated using the São Paulo potential [74] and are given in Table III. The formalism of Canto et al. provides a convenient means of eliminating the gross effects of the different projectile charges and what might be termed the “geometrical” contribution to the total reaction cross section due to the different projectile radii. This method, originally developed for the comparison of the fusion cross sections, can still be meaningfully employed, even if with the severe limitations described in Ref. [75], for reaction cross section data, especially in this case where we are comparing projectiles with very similar masses and all interacting with the same target nucleus. Figure 8confirms the exceptionally large value of the total reaction cross section for 8B, as noted for the lighter 58Ni target [42]. It also shows that, after the reduction procedure, the lowest total reaction cross sections are for the 7Li + 208Pb system, with the 6Li +208Pb and 7Be +208Pb systems TABLE III. Coulomb barrier radius RB, height VB, and curvature ¯hωfor the systems plotted in Fig. 8calculated using the São Paulo potential [74]. System RB(fm) VB(MeV) ¯hω(MeV) 8B+208Pb 11.16 49.40 4.49 7Be +208Pb 11.19 39.45 5.09 6Li +208Pb 11.25 29.46 4.77 7Li +208Pb 11.43 29.04 4.39 8Li +208Pb 11.59 28.68 4.08 024602-9