Applicability of ⁹Be global optical potential to description of ^8,10,11B elastic scattering

In the last few years, reactions involving ^8,10,11B isotopes have increasingly been attracting intense experimental and theoretical attention. The optical model potential (OMP) plays an important role in the investigation of these reactions. Theoretical studies have already been performed on this subject using both phenomenological and microscopic approaches. In this study, the phenomenological OMP are discussed with the aim to describe elastic scattering. Since the global phenomenological OMP is achieved by fitting large quantities of experimental data in a certain range of energy and mass, the basical elastic scattering observables can be reliably predicted using it in the region where no experimental measurement data exist [1]. Thus far, the experimental data of elastic scattering involving ^8,10,11B projectiles are relatively scarce, because the radioactive beams are not produced at sufficiently high intensities [2]. Therefore, it is difficult to achieve reliable global OMP on the basis of existing experimental data.

In our previous work, the elastic scattering observable for ^8,10,11B isotopes has been predicted using the global phenomenological OMP of the ⁷Li projectile. Reasonable agreement is obtained between the predictions and corresponding experimental data for ^8,10B. However, the global OMP of ⁷Li cannot provide a good description for backward-angle area for ¹¹B, and the radius parameter of the real part potential was adjusted to improve the fit for ¹¹B on the basis of ⁷Li global OMP [3]. Recently, the global phenomenological OMP of ⁹Be was achieved by simultaneously fitting the experimental data of elastic-scattering angular distributions and total reaction cross-sections below 200 MeV in the range of target mass from 24 to 209 [4]. Moreover, the stable weakly bound projectile ⁹Be is adjacent to the ^8,10,11B isotopes. Within this context, we intend to apply the obtained global phenomenological OMPs of ⁹Be to perform a systematic study involving the elastic scattering of ^8,10,11B isotopes impinging on different targets, which can further study the nuclear reaction and structure properties for ^8,10,11B projectiles.

This paper is constructed as follows. In Sec. 2, the phenomenological OMP formula and methods used in this work are described for the elastic scattering of ^8,10,11B projectiles. The elastic scattering observables describing the reactions induced by ^8,10,11B are predicted using the global OMP of ⁹Be, which are further discussed by comparison with the existing experimental data. Finally, the main conclusions of this work are summarized in Sec. 3.

As previously outlined [4], the optical model potential of the Woods-Saxon type,

and Coulomb potential of a uniform charged sphere with radius $ R_{C} $ were used in OM calculations. $ V_{R}(E) $, $ W_{S}(E) $, and $ W_{V}(E) $ are the energy-dependent potential depths, and they are respectively expressed as

The radial functions are given by

where $ A $ depicts the target mass number. $ r_{R} $, $ r_{S} $, $ r_{V} $, and $ r_{C} $ are the radius parameters of real, surface, volume imaginary, and Coulomb potentials, respectively. $ a_{R} $, $ a_{S} $, and $ a_{V} $ are the corresponding diffuseness parameters. The radius parameters of the real potential is expressed by

We achieved a set of ⁹Be global OMP parameters on the basis of experimental data of elastic-scattering angular distributions and total reaction cross-sections in the mass number range from 24 to 209 below 100 MeV [4]. The parameters of global OMP are listed in Table 1.

parameter	value	unit
$ V_{0} $	268.0671	MeV
$ V_{1} $	−0.180
$ V_{2} $	−0.0009
$ W_{0} $	52.149	MeV
$ W_{1} $	−0.125
$ U_{0} $	2.965	MeV
$ U_{1} $	0.286
$ r_{R_{0}} $	1.200	fm
$ r_{R_{1}} $	0.0273	fm
$ r_{S} $	1.200	fm
$ r_{V} $	1.640	fm
$ r_{C} $	1.556	fm
$ a_{R} $	0.726	fm
$ a_{S} $	0.843	fm
$ a_{V} $	0.600	fm

Table 1.  Global phenomenological OMP parameters for ⁹Be.

In what follows, we apply the obtained ⁹Be global OMP to predict elastic scattering observables for ^8,10,11B projectiles and compared this with the available experimental data.

The radioactive nucleus ⁸B is a lighter nucleus far from the β-stability valley, which is widely discussed as a candidate for a first proton drip line nucleus with a proton halo [5]. The proton separation energy is only 137 keV. Thus far, there were various reports on ⁸B in the literature studying its properties and the respective influences on different reaction mechanisms, because of the relevance of ⁸B in astrophysics, nuclear structure, and reaction theories [6]. However, the elastic scattering data with this projectile remain scarce because of the extreme difficulty to obtain reasonably intense beams [7]. To date, the elastic-scattering angular distributions and the total reaction cross-sections of ⁸B have been measured for ¹²C, ²⁷Al, ²⁸Si, ⁵⁸Ni, and ²⁰⁸Pb targets [6, 8-16]. These observables are predicted using obtained global OMP of ⁹Be and compared with those predicted using global OMP of ⁷Li [3]. Further, since ⁸Li is the mirror nucleus of ⁸B, they are also predicted using our global OMP of ⁸Li [17]. These predictions are compared with existing experimental data.

Figure 1 presents the comparison of elastic-scattering angular distributions with the experimental data [6] for the ⁸B + ²⁷Al system at incident energies of 15.3 MeV and 21.7 MeV. The figure shows that all of these global OMPs can reasonably predict the corresponding experimental data, and these predictions are relatively closed. The elastic-scattering angular distributions for the ⁸B + ⁵⁸Ni system are predicted and compared with the experimental data [8] from 20.7 to 29.3 MeV, which is shown in Fig. 2. Although there is some divergence among these results predicted using the different global OMPs, all of them can reasonably generate the experimental data within the error range.

Figure 1.  Comparisons of ⁸B elastic-scattering angular distributions calculated using ⁹Be, ⁷Li and ⁸Li global OMPs with corresponding experimental data for ²⁷Al.

Figure 2.  Same as Fig. 1, but for ⁵⁸Ni.

For the ⁸B + ²⁰⁸Pb system, elastic-scattering angular distributions are also measured at incident energies of 50.0, 170.3, and 178.0 MeV [9-11]. They are further predicted using different global OMPs. The results are in good agreement with these existing experimental data [10, 11] at 170.3 MeV and 178.0 MeV. As for the incident energy of 50.0 MeV, the global OMP of ⁸Li can provide a more satisfactory description of the experimental data [9] compared with the global OMPs of ⁹Be and ⁷Li at backward angles. These results are shown in Fig. 3.

Figure 3.  Same as Fig. 1, but for ²⁰⁸Pb.

Further, elastic-scattering angular distributions are also predicted using different global OMPs for the lighter target ¹²C. Comparing with experimental data [12, 13], these predictions seem inaccurate, as there is some divergence at extreme values. The result is shown in Fig. 4. Since the reactions of the lighter targets (A < 24) induced by different weakly bound nuclei ⁹Be, ⁷Li, and ⁸Li were not included in the process of adjusting the global OMP parameters, they should be studied using the local OMP.

Figure 4.  Same as Fig. 1, but for ¹²C.

To date, the total reaction cross-sections had been only measured for the ⁸B + ²⁸Si system [14–16], and most of them are above 200 MeV. The comparison between the predictions and data from different experiments below 250 MeV are shown in Fig. 5, which exhibits some divergences between them.

Figure 5.  Comparisons of ⁸B total reaction cross-sections calculated using ⁹Be, ⁷Li and ⁸Li global OMPs with corresponding experimental data for ²⁸Si.

From the above comparisons, the theoretical results predicted using the global OMPs of ⁹Be and ⁷Li can provide a reasonable description of the reactions induced ⁸B within the allowed error range, although it seems that the predictions of global OMP ⁸Li are more consistent with the few existing experimental data.

In the case of the ¹⁰B projectile, the elastic-scattering angular distributions and total reaction cross-sections are predicted using the global OMPs of ⁹Be and ⁷Li.

Figures 6 and 7 present the comparisons of elastic-scattering angular distributions between theoretical predictions and experimental data [18] for ²⁷Al and ^28,30Si targets at the bombarding energies from 33.7 MeV to 50 MeV. Figure 6 shows slight oscillations in the angular distributions appearing in the angular range from 50° to 80°, while agreement between the predictions of ⁹Be global OMP and experimental data is rather good.

Figure 6.  Comparisons of ¹⁰B elastic-scattering angular distributions calculated using ⁹Be and ⁷Li global OMPs with corresponding experimental data for ²⁷Al.

Figure 7.  Same as Fig. 6, but for ^28,30Si.

For the target ⁵⁸Ni, the angular distributions are predicted using global OMPs of ⁹Be and ⁷Li at incident energies from 19.0 MeV to 35.0 MeV. In comparison with the experimental data [19], the predictions provide a good description of these data, which is shown in Fig. 8. The elastic-scattering angular distributions for ¹⁰B on ¹²⁰Sn were measured at the bombarding energies of 31.5, 33.5, 35.0, and 37.5 MeV [20]. Global OMPs of the ⁹Be and ⁷Li were used to describe the experimental data. The ⁹Be global OMP provides a better description of experimental data. The result is displayed in Fig. 9.

Figure 8.  Same as Fig. 6, but for ⁵⁸Ni.

Figure 9.  Same as Fig. 6, but for ¹²⁰Sn.

Figure 10 presents the theoretical results of angular distributions along with experimental measurements [21-23] for different targets. The theoretical results predicted using global OMPs of ⁹Be and ⁷Li give a satisfactory description for ⁴⁰Ca and ²⁰⁸Pb. There are some discrepancies between them for lighter targets ¹⁶O and ²⁰Ne, while the results of ⁹Be global OMP are more consistent with the corresponding measurements.

Figure 10.  Same as Fig. 6, but for ¹⁶O, ²⁰Ne, ⁴⁰Ca, and ²⁰⁸Pb.

For the other lighter targets, the elastic-scattering angular distributions are also measured by different experiments. These reactions are further predicted using different global OMPs. Figure 11 presents the comparisons between them for ⁹Be. The discrepancy observed in Fig. 11 between theory and experiment [24, 25] indicates that the addition of coupled channels effects is needed in the backward-angle area for some lighter targets.

Figure 11.  Same as Fig. 6, but for ⁹Be.

For the total reaction cross-sections of ¹⁰B, there are only the experimental data for ^nat.Si. We compare the predicted results of total reaction cross-sections to the experimental data [16, 26, 27] for ²⁸Si. The result is shown in Fig. 12. The results of the ⁹Be global OMP are completely in agreement with all of the experimental data within the error range. For ²⁰⁸Pb, the data of total reaction cross-sections [23] was derived in terms of the optical model by analyzing the elastic scattering data at different incident energies. The predictions of the global OMP of ⁹Be and ⁷Li are also compared with the data, which is shown in Fig. 13. All of them are in good agreement.

Figure 12.  Comparison of ¹⁰B total reaction cross-sections calculated using ⁹Be and ⁷Li global OMPs with corresponding experimental data for ²⁸Si.

Figure 13.  Same as Fig. 12, but for ²⁰⁸Pb.

For the ¹¹B projectile, we obtained the global OMP [3] by adjusting the radius parameters of the real part potential on the basis of global OMP of ⁷Li. Although the predictions gave a reasonable description of the ¹¹B elastic scattering for most of targets, the OMP cannot provide a satisfactory agreement with the experimental data in the backward-angle area for a few targets. In this section, the elastic scattering observables are predicted using the global OMPs of ⁹Be and ¹¹B. The predictions are further compared with the existing experimental data.

Figure 14 presents the comparisons of elastic-scattering angular distributions between theoretical predictions and experimental data [18] for ^28,30Si targets at the bombarding energies from 33.7 MeV to 50 MeV. The fits are generally reasonable with no apparent systematic nor major discrepancy of the data. However, the results predicted using the global OMP of ⁹Be are more consistent with the experimental data.

Figure 14.  Comparisons of ¹¹B elastic-scattering angular distributions calculated using ⁹Be and ¹¹B global OMPs with corresponding experimental data for ^28,30Si.

The elastic-scattering angular distributions for ⁵⁸Ni are measured at incident energies from 19.0 MeV to 35.0 MeV [28]. The comparison between the predictions and experimental data is shown in Fig. 15. The figure shows that the predictions using global OMPs of ⁹Be and ¹¹B are almost consistent and in good agreement with the data, except for 35.0 MeV, where the prediction using the global OMP ¹¹B is more consistent with the experimental data. Moreover, the elastic angular distributions for ⁵⁸Ni are measured at the same incident angle with different incident energies [29]. Comparisons of the predictions of elastic-scattering angular distributions from the global OMP of ⁹Be and ¹¹B show that all of them are identical and can reproduce the data, which is shown in Fig. 16.

Figure 15.  Same as Fig. 14, but for ⁵⁸Ni.

Figure 16.  Calculated elastic-scattering angular distributions in Rutherford ratio at same incident angles compared with experimental data for ⁵⁸Ni target.

For ⁴⁰Ca and ²⁰⁸Pb, the elastic angular distributions are measured at incident energies of 51.5 MeV and 69.0 MeV [21, 30]. Figure 17 presents the comparisons between the predictions and experimental data. The predictions of ⁹Be and ¹¹B are in good agreement with the experimental data. For ²⁰⁹Bi, the angular distributions for elastic scattering calculated using the ¹¹B global OMP were larger than the experimental data [30, 31] at backward angles. Hence, the radius of the real part for the ¹¹B global phenomenological OMP was added by 0.15 to improve the fit with the data [3]. Figure 18 presents the theoretical results predicted using the global OMPs of ⁹Be and ¹¹B together with the experimental data. From the figure, one can see that the calculations of ⁹Be global OMP are in excellent agreement with the experimental data at all energies. Compared to those of the global OMP ¹¹B, the global OMP of ⁹Be can give a better prediction for the ¹¹B + ²⁰⁹Bi reaction. One of the reasons may be that the ¹¹B global OMP was obtained by only adjusting the radius parameters of the real part potential on the basis of the global OMP of ⁷Li [3], since the existing experimental data on elastic scattering is scarce for the reactions induced by the ¹¹B projectile. Moreover, this may be the influence of target shell effects for the doubly closed shell ²⁰⁸Pb nucleus and one proton outside the closed shell ²⁰⁹Bi nucleus. Meanwhile, the radius parameter of the real potential of the global OMP ⁹Be was further defined by $ r_{R_{0}}+r_{R_{1}}A^{\frac{1}{3}} $ as compared to $ r_{R_{0}} $ of the global OMP ¹¹B, which may compensate the influence of target shell effects.

Figure 17.  Same as Fig. 14, but for ⁴⁰Ca and ²⁰⁸Pb.

Figure 18.  Same as Fig. 14, but for ²⁰⁹Bi.

Similarly, the elastic-scattering angular distributions for some lighter targets are predicted using the global OMPs of ⁹Be and ¹¹B. Figure 19 presents the comparisons for ^12,13C. The discrepancy observed in Fig. 19 between theory and experiment [32-34] shows that it possibly requires the addition of coupled channels effects in the backward-angle area for some lighter targets.

Figure 19.  Same as Fig. 14, but for ^12,13C.

There are no measurements of the total reaction cross-sections for the reactions induced by ¹¹B. The existing data of total reaction cross-sections for ²⁰⁹Bi were extracted from the experimental elastic scattering data [31]. The predictions of total reaction cross-sections are further compared with the data for ²⁰⁹Bi, which is shown in Fig. 20. The predictions are in satisfactory agreement with the data extracted from the measured elastic-scattering angular distributions.

Figure 20.  Comparisons of ¹¹B total reaction cross-sections calculated using ⁹Be and ¹¹B global OMPs with corresponding data for ²⁰⁹Bi.

We predicted elastic scattering observables involving ^8,10,11B projectiles by applying the obtained global OMP of ⁹Be. We compared these predictions with those of the other global OMPs, and investigated and analyzed them in detail. The theoretical results predicted using the global OMP of ⁹Be give a more satisfactory description of the elastic scattering for ^8,10,11B projectiles for targets from ²⁷Al to ²⁰⁹Bi. For the lighter targets, we made a tentative prediction. All of the results predicted using different global OMPs are not in good agreement with the experimental data in the backward-angle area for lighter targets. The reason is that the other reaction mechanisms have to be considered in the calculations for such light systems, such as transfer or breakup. The present work shows that the global OMP of ⁹Be is useful to systematically investigate the reactions involving ^8,10,11B projectiles.