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The potential of Woods-Saxon (WS) shape is expressed as:
$ \begin{aligned}[b] U(r, E)=&V_{R}(E)f(r, R_{R}, a_{R})+ {\rm i} W_{V}(E)f(r, R_{V}, a_{V})\\ &+ {\rm i} W_{S}(E)(-4a_{S})\frac{{\rm d} f(r, R_{S}, a_{S})}{{\rm d} r}+V_{C}(r), \end{aligned} $
(1) where
$ V_{R}(E) $ ,$ W_{V}(E) $ , and$ W_{S}(E) $ are the depths of the real, volume-imaginary, and surface-imaginary parts, respectively. f is the WS form factor, given by:$ f(r, R_{i}, a_{i})=-\frac{1}{1+\exp[(r-R_{i})/a_{i}]}. $
(2) The energy dependences of potential depths [6, 7, 18, 19]
$ V_{R}(E) $ ,$ W_{S}(E) $ , and$ W_{V}(E) $ are defined as:$ V_{R}(E)=V_{0}+V_{1}E+V_{2}E^{2}, $
(3) $ W_{S}(E)=\max\{0, W_{0}+W_{1}E\}, $
(4) $ W_{V}(E)=\max\{0, U_{0}+U_{1}E+U_{2}E^{2}\}. $
(5) The Coulomb potential
$ V_{C}(r) $ is written as:$ V_{C}(r)=\left\{ \begin{array}{ll} \dfrac{zZe^{2}}{2R_{C}}\left(3-\dfrac{r^{2}}{R^{2}_{C}}\right) &\quad r<R_{C}, \\ \dfrac{zZe^{2}}{r} & \quad r\geq R_{C}, \end{array} \right. $
(6) where the index
$ i=R, V, S, C $ , which represents the real, imaginary, and Coulomb components,$ R_{i}=r_{i}(A_{T}^{\frac{1}{3}}+A_{P}^{\frac{1}{3}}) $ is the interaction radius, and$ a_{i} $ is the nuclear diffuseness. -
The elastic scattering angular distributions for the reactions of 7Li on 1p-shell nuclei were collected and investigated including targets from 7Li to 16O below 131.8 MeV. All the experimental data involved in the analyzsis are displayed in Table 1.
Target E/MeV Ref. 7Li 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0 [20] 20.0, 25.0 [21] 42.0 [22] 9Be 15.75, 19.0, 24.0, 30.0 [23] 17.7, 21.9 [24] 34.0 [25] 63.0, 130.0 [26] 10B 24.0 [27] 39.0 [28] 11B 9.85, 13.3, 18.3, 23.3, 28.3 [29] 34.0 [30] 12C 4.5, 5.8, 11.0, 13.0 [31] 7.5, 9.0, 12.0, 15.0 [32] 21.1 [27] 34.0 [33] 36.0 [34] 48.0 [35] 63.0, 78.7 [36] 89.0 [37] 131.8 [38] 13C 5.8, 9.0, 13.0, 20.0, 36.0 [31] 34.0 [34] 63.0, 130.0 [26] 15N 28.8 [39] 44.0 [40] 16O 9.0, 13.0 [41] 20.0 [31] 26.0 [42] 36.0 [34] 42.0 [43] 50.0 [44] Table 1. The elastic scattering angular distributions database of 7Li projectile on 1p-shell nuclei. E is the incident energy for different targets in the laboratory system.
The experimental elastic scattering angular distributions for all 1p-shell nuclei were adjusted using improved computer code APMN [45], which can automatically search POP parameters relying on the improved fastest falling method [46] below 300 MeV. In the adjustment process, the optimal POP parameters are determined via minimization of deviation
$ \chi^{2} $ of the results. χ2 is calculated using the optical potential from the experimental data and defined by:$ \chi^{2}=\frac{1}{N}\sum\limits_{i=1}^{N}\left[\frac{\sigma_{i}^{\rm th}-\sigma_{i}^{\rm ex}}{\Delta\sigma_{i}^{\rm ex}}\right]^{2}, $
(7) where N is the number points of experimental data,
$\sigma_{i}^{\rm th}$ and$\sigma_{i}^{\rm ex}$ are the theoretical and experimental values of the elastic scattering angular distributions, and$ \Delta\sigma $ is the experimental error. It should be noted that 10% of experimental uncertainties are uniformly assumed for all data for comparison of the degree of agreement between different data when calculating the$ \chi^{2} $ values in the work. Detailed descriptions for the procedure in the data analysis can be found in Refs. [6, 9].For the reaction of 7Li + 7Li, the corresponding systematics are presumed to be inconsistent with those established for the other 1p-shell nuclei because 7Li has very weakly bound properties with 3H cluster structure. Thus, the potential parameters of the 7Li + 7Li system are searched separately.
The GPOP parameters for the reactions of 7Li induced 1p-shell nuclei are obtained via fitting of the experimental data of elastic scattering angular distributions in the mass number range of 7–16, which are listed in Table 2. The parameters for 7Li target clearly exhibit remarkably different behavior in comparison to the other 1p-shell nuclei. This may relate to the weakly bound nature of these two nuclei and their cluster structures. Generally, the POP parameters for scattering from light target nuclei are more likely to show fluctuations owing to nuclear structure and channel-coupling effects [29, 33]. Thus, one of the parameters for some lighter and deformed targets, such as 9Be, 10B, 11B, and 15N, are slightly adjusted to better fit the present experimental data due to the different structures. For example, 12C and 16O have a pronounced cluster structure and are N-alpha nuclei [47]. However, the 9Be nucleus has an interesting structure known as the Borromean structure, in which the 9Be consists of two alpha particles and one weakly bound neutron. 10B is a weakly bound stable nucleus that may breakup into different partitions, the most energetically favorable being 10B → 6Li + 4He (Q = –4.461 MeV) [2]. The 11B nucleus is a strongly bound nucleus with the binding energy of 8.67 MeV in respect to the 7Li + α channel decay, and has a nonzero ground state deformation indicating a nonspherical symmetric charge distribution [48]. 15N is odd-A nuclei and has the ground state spin of 1/2. Accordingly, the diffuseness parameter of real part
$ a_{R} $ is 0.624 fm for the 9Be target, and the diffuseness parameter of imaginary part surface absorption$ a_{S} $ is 0.582 fm for the 15N target. While the radius parameters of real part$ r_{S} $ are 0.810 and 1.060 fm for 10, 11B targets, respectively.Parameter 7Li 1p Unit $V_{0}$ 49.257 70.668 MeV $V_{1}$ −0.449 −0.0344 $V_{2}$ 0.000580 $W_{0}$ 19.363 18.676 MeV $W_{1}$ −0.187 −0.298 $U_{0}$ −1.201 −1.539 MeV $U_{1}$ 0.0739 0.424 $U_{2}$ −0.00144 $r_{R}$ 0.900 0.845 fm $r_{S}$ 1.312 1.060 fm $r_{V}$ 1.800 1.039 fm $r_{C}$ 1.800 1.800 fm $a_{R}$ 0.842 0.774 fm $a_{S}$ 0.385 0.432 fm $a_{V}$ 1.050 0.850 fm Table 2. The GPOP parameters of 7Li induced 1p-shell nuclei.
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The double folding potential is computed as a Hartree-Fock (HF)-type potential according to the following relation:
$ V_{F}(R)=\iint{\rho_{1}(\vec{r_{1}})\rho_{2}(\vec{r_{2}})v_{NN}(\vec{R}-\vec{r_{1}}+\vec{r_{2}}){\rm d}\vec{r_{1}}{\rm d}\vec{r_{2}}}, $
(8) where
$ \rho_{1} $ and$ \rho_{2} $ are the ground state nuclear matter density distributions for the projectile and target, respectively, and$ v_{NN} $ represents effective$ NN $ interaction.The SPP2 effective interaction is proposed to successfully describe the low energy α + α experimental phase-shifts within the context of the double-folding approach [17], which is a new version of the original SPP [49] that includes a dependence on the relative velocity between two interacting nuclei. This effective interaction is parameterized as follows:
$ v_{NN}^{\rm SPP2}(r)=-U_{0}\exp(-\vec{r}/a)^{2}\exp(-4v^{2}/c^{2}), $
(9) where
$ U_{0} $ = 735.813 MeV and a = 0.5 fm.Accordingly, an extension of the SPP model for the real and imaginary components with different normalization values was successfully applied to the elastic scattering of stable nuclei [50]. The final folded potential is used as a complex optical potential (OP) by scaling it with a complex renormalization factor as:
$ V_{\rm OP}(R)=N_{R}V_{F}(R)+{\rm i} N_{I}V_{F}(R), $
(10) where
$ N_{R} $ and$ N_{I} $ represent the real and imaginary renormalization factors, respectively.
Description of elastic scattering for 7Li-induced reactions on 1p-shell nuclei
- Received Date: 2023-11-27
- Available Online: 2024-02-15
Abstract: The experimental data of elastic scattering angular distributions for 9Be, 10B, 11B, 12C, 13C, 15N, and 16O targets from 4.5 to 131.8 MeV and 7Li target from 8.0 to 42.0 MeV are fitted to realize the global phenomenological optical potentials (GPOPs) for the 7Li-induced reactions on 1p-shell nuclei. Thus, the 7Li elastic scattering from the 1p-shell nuclei can be systematically described using the established GPOPs. The elastic scattering angular distributions are also reanalyzed using a microscopic method within the framework of the new version of double folding São Paulo potential (SPP2). To better describe the elastic scattering at backward angles, the contribution of elastic transfer is further estimated by the distorted wave Born approximation (DWBA) method. Based on the obtained GPOPs, the inelastic scattering angular distributions are also obtained through the coupled channels (CC) method for the different excited states.