-
The scattering angles were obtained by extrapolation of the target position and the positions of particles hitting DSSDs, as shown in Fig. 4. For each scattered particle, the incident track
$ \overrightarrow{AC}$ was determined by a combination of the hit positions in the SiA and SiB detectors,$ \overrightarrow{AB} $ , which was extrapolated to the hit position C in the target. The particle is then scattered by the target and hits point D of DSSDs. The track$ \overrightarrow{CD} $ defines the outgoing path of the scattered particle. The angle between the incident direction$ \overrightarrow{AC} $ and the scattered direction$ \overrightarrow{CD} $ is the scattering angle$ \theta_{\rm Lab} $ , which was calculated on the event-by-event basis. The beam spot on the target was large ($ \approx $ 30 mm) and asymmetrical. Therefore, a Monte Carlo simulation, taking into account the detector geometry and the beam distribution in the target, was used to evaluate the absolute differential cross-section.Another important issue considered was the contamination of the data from particles scattered by DSSDs (SiA and SiB). In previous experiments, we used Parallel-Plate Avalanche Counters (PPACs) to measure the position and direction of the beam particles [40-42]. Compared to silicon detectors, PPAC introduces almost no disturbance of RIB. In this experiment, we used two thin DSSDs replacing PPACs, which caused some contamination of the data from scattering in the detectors. The events coming from the detector system were measured with the target moved out. To check the impact of the target-out events on the data, we performed a simulation assuming that the particles are scattered in the silicon detectors before scattering in the 208Pb target. Only scattered particles in SiB were considered in the simulation. The incident direction of the beam is calculated after scattering in SiA, and thus particles scattered by SiA do not affect the data. Since
$ \mathit{\rm Tel1} $ was placed at a large angle, particles that do not originate in the target can hardly hit it. Thus, the target-out events mostly come from$ {\mathit {\rm Tel2}} $ and$ {\mathit {\rm Tel3}} $ . The results of the simulations for the 9Be and 10Be beams as a function of the angle$ \theta $ , with (blue line) and without (red line) scattered particles in Si$ _{B} $ , are shown in Fig. 5. The scattering events in Si$ _{B} $ for both 9Be and 10Be beams account for less than 5% of all scattering events. We conclude that the contribution of the target-out events affects the data very little, basically in the forward angles, which can be considered negligible in our experiment. Also, the effect of the small contribution at forward angles was diluted by the normalization method.Figure 5. (color online) Simulation results for 9Be (left) and 10Be (right) as a function of
$ \theta $ with and without scattering in SiB.The elastic scattering differential cross-section as the ratio to the Rutherford cross-section is obtained by:
$ \begin{aligned}\frac{\sigma(\theta)}{\sigma_{\rm Ruth}(\theta)}=\frac{{\rm d}\sigma(\theta)/{\rm d}\Omega}{{\rm d}\sigma_{\rm Ruth}(\theta)/{\rm d}\Omega}= \frac{\dfrac{N(\theta)_{\rm exp}}{N_{\rm in}N_{\rm target}{\rm d}\Omega}}{\dfrac{N(\theta)_{\rm Ruth}}{N_{\rm in}N_{\rm target}{\rm d}\Omega}}= C\times\frac{N(\theta)_{\rm exp}}{N(\theta)_{\rm Ruth}}, \end{aligned} $
(1) where
$ \mathit{C} $ is the normalization constant,$ N_{\rm in} $ is the number of incident particles,$ N_{\rm target} $ is the number of target nuclei per unit area,$ N(\theta)_{\rm exp} $ and$ N(\theta)_{\rm Ruth} $ are the yields at a given angle in the data and from the simulations, respectively. The normalization constant$ \mathit{C} $ for the 9Be angular distribution was obtained by normalizing the experimental cross-section to the simulation results for angles below 20°, where the elastic scattering is assumed to be pure Rutherford scattering. This overall normalization was also applied to the cross-section of the 10Be + 208Pb system. With this method, the cross-sections are obtained in a straight forward way, and the influence of the systematic errors of the measured total number of incident particles, target thickness and solid angle determination was avoided. To minimize the systematic errors, small corrections of the detector misalignment were also performed. The details of this procedure can be found in Ref. [43].It is important to mention that, in principle, the elastic and inelastic scattering from the excited states of the lead target nuclei could not be discriminated and the data are quasi-elastic in nature. However, the contributions from the excited states of the lead target were found to be negligible in several other experiments with a similar energy and angular range [13]. For this reason, we consider in the present work that the data are for elastic scattering.
The differential cross-sections for elastic scattering were normalized to the differential cross-section of Rutherford scattering, and are plotted as a function of scattering angle. The elastic scattering angular distributions for 9Be + 208Pb at the energy
$ E_{\rm Lab} $ = 88 MeV, and for 10Be + 208Pb at the energy$ E_{\rm Lab} $ = 127 MeV, are shown in Fig. 6. As can be seen, the ratio$ \sigma \ / \sigma_{\rm Ruth} $ for 9Be is close to unity since the Rutherford scattering is dominant within the measured angular range. In the angular distribution of 10Be + 208Pb, shown in Fig. 6(b), the typical Fresnel diffraction can be observed. -
Optical model analysis of the elastic scattering differential cross-section data was performed. All the calculations were performed with the code FRESCO [44]. We first considered the complex Woods-Saxon (WS) potential, which has six parameters, namely, the real (imaginary) potential depth V (W), radius
$ r_{v} $ ($ r_{w} $ ) and the diffuseness$ a_{v} $ ($ a_{w} $ ). The reduced radii have to be multiplied by the mass term$ (A_{P}^{1/3} + A_{T}^{1/3}) $ , where$ A_{P} $ = 10 and$ A_{T} $ = 208, to give the radii of the real and imaginary potentials. The WS potential for 10Be + 208Pb was obtained by adjusting the six parameters to best reproduce the elastic scattering data. The parameters from the fit procedure are listed in Table 1, and were obtained with the minimum$ \chi^2 $ criteria given by:parameters V/MeV rv/fm av/fm W/MeV rw/fm aw/fm χ2 10Be + 208Pb 18.33 1.251 0.636 20.27 1.255 0.744 0.493 Table 1. The Woods-Saxon parameters obtained by fitting the experimental data.
$ \chi^2 = \frac{1}{N}\sum\limits_{i = 1}^N\frac{[\sigma_{i}^{\rm exp}-\sigma_{i}^{\rm th}]^2}{\Delta\sigma_{i}^2}, $
(2) in which N is the number of data points,
$ \sigma_{i}^{\rm exp} $ and$ \sigma_{i}^{\rm th} $ are the experimental and the calculated differential cross-sections, and$ \Delta\sigma_{i} $ is the uncertainty of the experimental cross-section. The results of the OM analysis with the WS potential are shown in Fig. 7 by the black dashed line. As can be seen, the agreement with the data is good, in particular at the Fresnel peak. The total reaction cross-section obtained with the WS potential is 3370 mb. However, since the experimental data were obtained in a relatively limited angular range, the potential parameters are not unique.Figure 7. (color online) Elastic scattering angular distribution for the 10Be + 208Pb system at 127 MeV. The lines are the results of the optical model analysis with the WS potential and the double-folding SPP.
To decrease the number of free parameters, and thus the ambiguities in the potentials in the OM analysis, folding potentials have been developed. The results with the double-folding São Paulo Potential (SPP) in the OM analysis are shown in Fig. 7 by the red solid line. The total reaction cross-section obtained with SPP is 3240 mb. SPP is a "folding-type" effective nucleon-nucleon interaction with a fixed parametrized nucleon density distributions in the projectile and target. It can be used in association with OM, with
$ N_{\rm R} $ and$ N_{\rm I} $ as normalizations of the real and imaginary parts [45]. From a large set of systematic values,$ N_{\rm R} $ = 1.00 and$ N_{\rm I} $ = 0.78 were proposed [46]. With the standard values of the normalization, we could reproduce well the data in the measured angular region with SPP, as can be seen in Fig. 7. SPP fit is more sensitive to the backward angles, where the influence of the absorption of the flux from direct reactions is more important. A measurement at more backward angles for 10Be would be highly desirable for a more rigorous test of this potential.We considered another global nucleus-nucleus potential, which was obtained from a systematic optical potential analysis by Xu and Pang (X&P) [47]. This global potential can reasonably reproduce the elastic scattering and total reaction cross-sections for projectiles with mass numbers up to
$ A\lesssim40 $ , including the stable and unstable nuclei, and at energies above the Coulomb barrier. It is obtained by folding the semi-microscopic Bruyères Jeukenne-Lejeuue-Mahaux (JLMB) nucleon-nucleus potential with the nucleon density distribution of the projectile nucleus [47]. The JLMB potential itself employs single-folding of the effective nucleon-nucleon interaction with the nucleon density distribution of the target nucleus [48]. Hence, the X&P potential is single folding in nature, but it also requires nucleon density distributions in both the projectile and target nuclei. The results with JLMB are close to SPP, but in some cases it may overestimate the differential cross-section for large angles. These may be caused by the special consideration of the Pauli nonlocality in SPP, which is important at low incident energies [47]. The density distribution in the projectile can be deduced from the observed interaction and total reaction cross-sections using the Glauber model [49], or using the Hartree-Fock calculation [50]. The proton and neutron density distributions in the target nuclei are obtained from the Hartree-Fock calculation with the SkX interaction [47]. Optical model results with this global nucleus-nucleus potential, using different density distributions in 10Be, and the comparison with the data for 10Be + 208Pb are shown in Fig. 8. The root-mean-square (RMS) radii of proton, neutron and nuclear matter distributions used in these calculations are summarized in Table 2. The first row in Table 2 gives the RMS radii derived from the Glauber model with harmonic oscillator distributions which result in the RMS radius$ R_{\rm HO} = 2.299 $ fm for 10Be [49]. In row 2, the RMS radius$ R_{\rm Liatard} = 2.479 $ fm was determined from the Glauber model analysis of the total reaction cross-section of 10Be on a carbon target by Liatard et al. [11].$ R_{\rm Liatard2} $ and$ R_{\rm Liatard3} $ are two artificial density distributions, obtained by stretching the distribution proposed by Liatard et al., so that the radius of 10Be is increased by 10% and 20%, respectively. The elastic scattering angular distributions calculated with the X&P potential using these density distributions are shown in Fig. 8 together with the experimental data. From these results, it can be concluded that a change of the RMS radius by 10% induces a shift of the angular distribution by about 0.7 degrees at angles where$ \sigma/\sigma_{\rm Ruth} = 0.5 $ . In other words, a precision of the angular distribution measurement of 0.1 degrees, which is feasible with modern techniques, allows to determine the RMS radius (of a nucleus like 10Be) with a precision of around 1.4%. Given that there are quite large uncertainties of RMS radius of light heavy ions (see, e.g., the compilation of RMS radii of light heavy ions in Ref. [49]), it might be interesting to measure the RMS radii of these nuclei in elastic scattering experiments. Of course, more effort needs to be made both at the experimental and theoretical level, to fully understand the precision of this method. Obviously, better statistics, which requires higher beam intensities and a high-performance detector arrays, would be needed.Table 2. The RMS radii of the proton, neutron and nuclear matter distributions used, in units of fm. The total reaction cross-sections and references are listed.
Figure 8. (color online) Comparison of the experimental data and the optical model calculations using nucleus-nucleus potentials with different density distributions for 10Be. The angular distribution in green dashed line was calculated with
$R_{\rm HO}$ [49]. The angular distribution in black dotted line, red solid line and orange dash-dotted line were calculated with density distributions$R_{\rm Liatard}$ [11],$R_{\rm Liatard2}$ and$R_{\rm Liatard3}$ , respectively.Comparison of the experimental data and the results of optical model calculations using the SPP and X&P potentials was also made for the 9Be + 208Pb system. The results are shown in Fig. 9. The Coulomb dominance is clearly seen in the range of scattering angles of our experiment. For this reason, a similar analysis as for 10Be was not made. For stable nuclei like 9Be, both SPP and X&P reproduce the elastic scattering data quite well.
Experimental study of the elastic scattering of 10Be on 208Pb at the energy of around three times the Coulomb barrier
- Received Date: 2019-09-03
- Available Online: 2020-02-01
Abstract: Elastic scattering of 10Be on a 208Pb target was measured at