Systematic study of the Woods-Saxon potential parameters between heavy-ions

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Lin GAN, Hong-Li ZHI, Hui-Bin SUN, Shi-Peng HU, Er-Tao LI and Jian ZHONG. Systematic study of the Woods-Saxon potential parameters between heavy-ions[J]. Chinese Physics C.
Lin GAN, Hong-Li ZHI, Hui-Bin SUN, Shi-Peng HU, Er-Tao LI and Jian ZHONG. Systematic study of the Woods-Saxon potential parameters between heavy-ions[J]. Chinese Physics C. shu
Received: 2020-02-02
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Systematic study of the Woods-Saxon potential parameters between heavy-ions

  • 1. Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060
  • 2. China Institute of Atomic Energy, Beijing 102413, China
  • 3. School of Nuclear Science and Technology, University of Chinese Academy of Science, Beijing 101408, China
  • 4. College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China

Abstract: Experimental elastic scattering angular distributions of 11B, 12C and 16O + heavy-ions were used to study the Woods-Saxon potential parameters. The diffuseness parameters were found to have best fitted values for each system, and a linear expression of diffuseness parameters with $A_1^{1/3}+A_2^{1/3}$ was summarized. The correlations of potential depths and radius parameters with $A_1^{1/3}+A_2^{1/3}$ were revealed with the limitation of diffuseness parameter formula. As the incident energies of most analyzed reactions are below and around the Coulomb barrier, the energy dispersion relation between the real and imaginary potentials was taken into consideration to study the ratio of the imaginary and real potential well depths, and the expression of $W/V$ was concluded. With the limitation of volume integrals calculated with S$\tilde{a}$o Paulo potential, parameterized formulas for depths and radius parameters were obtained. The deduced systematic Woods-Saxon potential parameters in present work can reproduce not only the experimental data of elastic scattering angular distributions induced by 11B, 12C and 16O, but also some elastic scattering induced by other heavy-ions.


    • The nucleus-nucleus interaction potential, which includes short-range attractive and absorptive nuclear potential and long-range repulsive Coulomb potential, has always been a major issue in nuclear physics. The Coulomb interaction between two nuclei is well known, but the nuclear component is much more difficult. Over the last decades, optical model potential (OMP) has been used to describe nuclear component widely, and several different potential forms were proposed to reproduce a large number of nuclear reaction data[1-3]. Not only the nuclear reactions induced by light particles, but also the nuclear reactions between heavy ions can be described by optical model [3-6].

      Elastic scattering, the simplest nuclear reaction process, are often used to understand more complicated reaction channels [7]. The optical potential parameters extracted by fitting elastic scattering data can be used to obtain the information in the nucleus. A large amount of systematic optical parameters have been presented to provide reliable optical potential parameters for reactions of different systems and different incident energies [3]. So that one can acquire the optical potential parameters directly without fitting elastic scattering data. Most of these works focus on the reactions induced by light particles, such as proton, neutron, deuteron, helium-4 [8-11]. Because of the strong Coulomb interaction between heavy-ions, systematic researches of optical potential between heavy ions are much more difficult and scarce compare to the reactions induced by light particles. However, several considerable progress have been published in recent years. Based on double folding potential and Pauli nonlocality, Chamon et al [12] proposed the well-known S$ \tilde{a} $o Paulo potential (SPP). This work provide a systematic description on the realistic nucleus-nucleus interaction, and can be applied to various reaction systems including stable and unstable particles. A modified Woods-Saxon potential based on the Skyrme energy-density functional approach was proposed by Wang et al [13, 14]. to provide a global description of nucleus-nucleus interaction. Not only elastic scattering, but also fusion barrier and the fission barrier of fusion-fission reactions can be described. Through analyzing angular distributions of $ ^{6,7}{\rm{Li}} $ elastic scattering from heavy-ions, Xu and Pang proposed a systematic single-folding potential. This systematics can give reasonable account for both elastic scattering and total reaction cross sections [15]. We proposed a systematic six-parameter Woods-Saxon potential with fixed imaginary parameters based on the elastic scattering angular distributions induced by 12C [16]. However, it is not satisfactory for some elastic scattering at the energies around coulomb barriers or higher than 300 MeV. The reason may be due to the fixed imaginary potential parameters.

      In present work, the Woods-Saxon shape for both real and imaginary parts was adopted and the imaginary parameters were also studied as variables. Experimental elastic scattering angular distributions of 11B, 12C and 16O + heavy-ions were used to extract the optical potential parameters.

    • The Woods-Saxon model potential was proposed by Woods and Saxon to approximate the shape of nuclear component of nucleus-nucleus interaction [17]. Although it is a phenomenological approximation, reactions of different incident energies and different projectile-target combinations can be well reproduced with six parameters used in the calculations [7].

      Six-parameter Woods-Saxon potential contains a real and an imaginary component, which is expressed as:

      $ V_n(r) = -\frac{V}{1+e^{(r-R_V)/a_V}}-\frac{iW}{1+e^{(r-R_W)/a_W}}, $


      where V, W are potential depth parameters for real and imaginary potentials, respectively. $ R_i $ = $ r_i(A_1^{1/3}+A_2^{1/3}) $, in which $ i = V, W $, indicate the real and imaginary radius parameters. $ a_V $ and $ a_W $ represent for the real and imaginary diffuseness. $ A_{1} $ and $ A_{2} $ are the mass numbers of the projectile and target nuclei.

      The Coulomb potential is always expressed as:

      $ V_C(r) = \left\{ \begin{array}{l} \dfrac{Z_1Z_2e^2}{2R_C}\left(3-\dfrac{r^2}{R_C^2}\right) \quad r < R_C \\ \dfrac{Z_1Z_2e^2}{r} \quad r > R_C, \\ \end{array}\right. $


      where $ R_C = r_C(A_1^{1/3}+A_2^{1/3}) $ is the radius parameter. It had been proved that, theoretical angular distributions are not sensitive to the changes in the Coulomb radius [18], thus $ r_C $ is fixed equal to 1.0 fm throughout the following processes.

      In present work, 27 sets of elastic scattering angular distribution data were adopted to extract the Woods-Saxon potential parameters. Targets are from 12C to 209Bi, and the incident energies are from 25 MeV to 420 MeV. Most of the data are from National Nuclear Data Center, and the angular distributions of 12C elastic scattering from 90Zr, 91Zr, 96Zr and 116Sn were measured in China Institute of Atomic Energy (CIAE), Beijing. Because the optical potential parameters have large ambiguities, several different sets of parameter combinations can reproduce experimental data well [19]. It has been confirmed that when fitting two or more optical potential parameters at the same time, we are easily trapped in a certain minimum trap. In order to study the intrinsic relationship between these parameters, we set more than 1 million sets of parameter combinations of W, $ R_V $, $ R_W $, $ a_V $ and $ a_W $ in advance, and then took each of them to fit V for all the data. W varied in the ranges of 10 - 300 MeV by step of 2 MeV. The geometric parameters of real and imaginary component set as the same value: $ r_V $ = $ r_W $ varied in the range of 0.5-1.5 fm by step of 0.01 fm, and $ a_V $ = $ a_W $ varied in the range of 0.3 - 1.0 fm by step of 0.01 fm. The analysis process can be divided into four steps:

      1). Take all the parameter combinations into the nuclear reaction code PTOLEMY[20] to fit each elastic scattering data. Thus, the output V and the corresponding $ \chi^2 $ for every calculation were obtained.

      2). Analyze the correlation of $ \chi^2 $ with depths, radius and diffuseness parameters. We found each system exists a suitable value of diffuseness parameter which give the minimum $ \chi^2 $. Fig. 1 shows the $ \chi^2 $ versus diffuseness parameter for the 12C + 90Zr elastic scattering as a typical result. However, the depths and radius parameters can not be extracted in the same way as diffuseness parameter, because there are no best fit values. We listed the extracted diffuseness for each interaction system in Table 1. The analysis of diffuseness uncertainties employed the $ \chi^2 $ envelope method proposed in Ref. [21]. As shown in Fig. 2, a linear formula of diffuseness parameter a versus $ A_1^{1/3}+A_2^{1/3} $ is summarized in Eq. (3), and the uncertainties of slope and intercept are 0.011 fm and 0.063 fm, respectively. The slope is negative, indicating that as the mass of the system increases, the diffuseness parameter tends to decrease.

      Figure 1.  $\chi^2$ versus diffuseness parameter for 12C + 90Zr at 66 MeV. Each dot represents a parameter set. It can be found that, a = 0.59 fm is corresponding to the minimum $\chi^2$.

      Reaction $E_{lab}$ [MeV] a [fm] $\Delta a$ [fm]
      11B+58Ni 25 0.64 0.03
      11B+58Ni 35 0.54 0.03
      12C+12C 158 0.72 0.02
      12C+12C 360 0.75 0.03
      12C+13C 127.2 0.74 0.03
      12C+16O 76.8 0.61 0.02
      12C+19F 40.3 0.67 0.02
      12C+19F 50 0.76 0.02
      12C+19F 60 0.73 0.02
      12C+40Ca 180 0.69 0.03
      12C+40Ca 300 0.69 0.04
      12C+40Ca 420 0.74 0.03
      12C+64Ni 48 0.59 0.06
      12C+90Zr 66 0.59 0.03
      12C+90Zr 120 0.71 0.04
      12C+90Zr 180 0.68 0.04
      12C+90Zr 300 0.66 0.07
      12C+90Zr 420 0.71 0.06
      12C+91Zr 66 0.64 0.03
      12C+96Zr 66 0.58 0.08
      12C+116Sn 66 0.51 0.06
      12C+169Tm 84 0.44 0.04
      12C+208Pb 58.9 0.40 0.07
      12C+209Bi 87.4 0.54 0.04
      16O+64Zn 48 0.66 0.07
      16O+68Zn 52 0.56 0.08
      16O+209Bi 90 0.46 0.05

      Table 1.  The most suitable diffuseness parameters extracted in the analysis.

      Figure 2.  (color online) Diffusion parameter as a function of $A_1^{1/3}+A_2^{1/3}$.

      $ a_V = a_W = -0.0736(A_1^{1/3}+A_2^{1/3})+1.087\ \ {\rm{(fm)}} $


      3). When diffuseness parameters were fixed as Eq. (3), we found a correlation between depths and radius parameters for real and imaginary part respectively. As an example, $ \chi^2 $ versus depths of 12C + 90Zr elastic scattering at 66 MeV is shown in Fig. 3. It is obvious that, the curve of the V is much steeper than that of W, it means the $ \chi^2 $ is much more sensitive to V than W. Each curve corresponding to a specific radius parameter value, and the correlation of the depths and radius parameter combinations of best fit are as following:

      Figure 3.  (color online) $\chi^2$ versus V (a) and W (b) for 12C + 90Zr at 66 MeV. Each curve is corresponding to different radius parameter, and the valleys give almost the same $\chi^2$. The radius parameter for each curve from left to right are 1.07 fm to 1.03 fm, by step of 0.01 fm. The curve of the V (a) is much steeper than that of W (b).

      $ \begin{split} &V\times exp(R_V/a_V) = const1\\ &W\times exp(R_W/a_W) = const2 \end{split} $


      The values of $ const1 $ and $ const2 $ were listed in Table. 2. The errors are from the uncertainties of diffuseness parameters. Since radius parameter and diffuseness parameter are the same for real and imaginary component, the ratio of W and V is equal to the ratio of $ const2 $ and $ const1 $.

      Reaction $E_{lab}\;{\rm{ [MeV]}}$ ${\rm{Vol}} [{\rm{MeV}}\cdot\;{\rm{fm}}^3]$ $const1\;{\rm{ [MeV]}}$ $const2\;{\rm{ [MeV]}}$ V [MeV] R [fm]
      11B+58Ni 25 418.5 3.02$\times 10^6 \pm$1.16$\times 10^6$ 5.79$\times 10^5 \pm$2.40$\times 10^5$ 266.1$\pm$34.8 6.00$\pm$0.30
      11B+58Ni 35 415.6 2.66$\times 10^6 \pm$9.88$\times 10^5$ 1.29$\times 10^6 \pm$5.59$\times 10^5$ 277.0$\pm$36.0 5.89$\pm$0.29
      12C+12C 158 416.7 2.54$\times 10^4 \pm$8.88$\times 10^3$ 1.90$\times 10^4 \pm$7.43$\times 10^3$ 221.5$\pm$54.2 3.56$\pm$0.43
      12C+12C 360 365.4 1.86$\times 10^4 \pm$1.39$\times 10^4$ 1.37$\times 10^4 \pm$9.93$\times 10^3$ 234.4$\pm$97.7 3.28$\pm$0.82
      12C+13C 127.2 422.6 2.58$\times 10^4 \pm$1.03$\times 10^4$ 1.89$\times 10^4 \pm$7.56$\times 10^3$ 254.5$\pm$70.0 3.49$\pm$0.50
      12C+16O 76.8 431.4 5.41$\times 10^4 \pm$3.74$\times 10^4$ 1.76$\times 10^4 \pm$1.14$\times 10^4$ 240.0$\pm$79.9 3.95$\pm$0.68
      12C+19F 40.3 436.9 6.66$\times 10^4 \pm$4.00$\times 10^4$ 4.86$\times 10^4 \pm$3.55$\times 10^3$ 306.8$\pm$95.2 3.87$\pm$0.60
      12C+19F 50 434.1 7.71$\times 10^4 \pm$4.99$\times 10^4$ 4.38$\times 10^4 \pm$3.22$\times 10^4$ 267.8$\pm$81.5 4.07$\pm$0.62
      12C+19F 60 431.3 7.79$\times 10^4 \pm$277$\times 10^4$ 5.05$\times 10^4 \pm$1.42$\times 10^4$ 263.3$\pm$54.8 4.09$\pm$0.39
      12C+40Ca 180 384.4 5.90$\times 10^5 \pm$260$\times 10^5$ 5.70$\times 10^5 \pm$2.21$\times 10^5$ 283.9$\pm$51.3 5.10$\pm$0.38
      12C+40Ca 300 356.3 4.63$\times 10^5 \pm$2.00$\times 10^5$ 4.88$\times 10^5 \pm$2.05$\times 10^5$ 336.1$\pm$64.9 4.83$\pm$0.38
      12C+40Ca 420 330.4 5.53$\times 10^5 \pm$2.23$\times 10^5$ 5.58$\times 10^5 \pm$2.07$\times 10^5$ 234.5$\pm$37.7 5.18$\pm$0.34
      12C+64Ni 48 409.6 7.69$\times 10^6 \pm$4.97$\times 10^6$ 2.96$\times 10^6 \pm$1.50$\times 10^6$ 258.9$\pm$44.1 6.43$\pm$0.43
      12C+90Zr 66 400.9 4.85$\times 10^7 \pm$1.21$\times 10^7$ 1.88$\times 10^7 \pm$6.17$\times 10^6$ 266.3$\pm$16.8 7.13$\pm$0.17
      12C+90Zr 120 387.5 3.83$\times 10^7 \pm$2.60$\times 10^7$ 1.85$\times 10^7 \pm$1.15$\times 10^7$ 274.4$\pm$40.8 6.97$\pm$0.40
      12C+90Zr 180 376.1 2.63$\times 10^7 \pm$1.74$\times 10^7$ 1.32$\times 10^7 \pm$8.29$\times 10^6$ 298.6$\pm$43.9 6.70$\pm$0.39
      12C+90Zr 300 349.1 2.96$\times 10^7 \pm$2.01$\times 10^7$ 3.05$\times 10^7 \pm$2.02$\times 10^7$ 260.8$\pm$38.9 6.85$\pm$0.40
      12C+90Zr 420 324.2 2.06$\times 10^7 \pm$1.32$\times 10^7$ 1.66$\times 10^7 \pm$1.08$\times 10^7$ 264.5$\pm$38.9 6.63$\pm$0.39
      12C+91Zr 66 400.4 5.14$\times 10^7 \pm$3.53$\times 10^7$ 1.80$\times 10^7 \pm$1.12$\times 10^7$ 267.6$\pm$36.8 7.15$\pm$0.39
      12C+96Zr 66 402 8.90$\times 10^7 \pm$6.26$\times 10^7$ 2.73$\times 10^7 \pm$1.71$\times 10^7$ 253.7$\pm$35.9 7.42$\pm$0.40
      12C+116Sn 66 397 2.85$\times 10^8 \pm$2.09$\times 10^8$ 8.03$\times 10^7 \pm$5.31$\times 10^7$ 273.3$\pm$36.1 7.75$\pm$0.39
      12C+169Tm 84 396.3 1.08$\times 10^{10} \pm$7.86$\times 10^9$ 6.78$\times 10^9 \pm$4.88$\times 10^9$ 254.3$\pm$26.2 8.98$\pm$0.34
      12C+208Pb 58.9 402.2 8.95$\times 10^{10} \pm$7.71$\times 10^{10}$ 5.40$\times 10^{10} \pm$4.68$\times 10^{10}$ 275.6$\pm$29.0 9.45$\pm$0.35
      12C+209Bi 87.4 395.2 1.14$\times 10^{11} \pm$9.85$\times 10^{10}$ 3.73$\times 10^{10} \pm$3.03$\times 10^{10}$ 262.0$\pm$14.5 9.56$\pm$0.35
      16O+64Zn 48 409.3 1.98$\times 10^7 \pm$1.09$\times 10^7$ 8.12$\times 10^6 \pm$4.36$\times 10^6$ 303.8$\pm$42.4 6.73$\pm$0.36
      16O+68Zn 48 408.6 3.96$\times 10^7 \pm$2.31$\times 10^7$ 2.39$\times 10^7 \pm$1.33$\times 10^7$ 270.6$\pm$36.2 7.15$\pm$0.36
      16O+209Bi 90 394.9 1.15$\times 10^{12} \pm$1.04$\times 10^{12}$ 1.40$\times 10^7 \pm$1.25$\times 10^12$ 284.6$\pm$28.0 10.28$\pm$0.35

      Table 2.  The Volume integrals (Vol) calculated with SPP, $const1$ and $const2$ in Eq. (4), real depths (V) and radius parameters (R) determined by Volume integrals and $const1$.

      For most of the analyzed nuclear reactions, the incident energies are around and below the Coulomb barrier, the energy dispersion relation between the real and imaginary potentials [22-25] can not be ignored. Due to the scarcity of energy points around the Coulomb barrier for each system analyzed in present work, we can hardly conduct the detailed study on energy dispersion relation. We proposed a rough method to analyze the relationship between the real and imaginary parts near the Coulomb barrier. As Coulomb barrier is proportional to $ Z_1Z_2/ (A_1^{1/3}+A_2^{1/3}) $, we adopted ${\rm{E}} _{\rm{Lab}}\times (A_1^{1/3}+A_2^{1/3})/Z_1Z_2 $ as the energy dispersion relation parameter ($ EDRP $) to study the energy dependence of the ratio of W and V, as shown in Fig. 4. The trend of $ W/V $ versus $ EDRP $ has a turning point near $ EDRP = $ 10. When $ EDRP < $ 10, the value of $ W/V $ decreases rapidly as $ EDRP $ decreases. When $ EDRP \geqslant $ 10, the value of $ W/V $ remains relatively stable. We concluded the expression of $ W/V $ as: $ W/V = (0.0416\pm 0.0220)\times EDRP+(0.4124\pm 0.0879),\ EDRP<10; $ $ W/V = (0.8284\pm 0.1321),\ EDRP\geqslant 10 $.

      Figure 4.  (color online) The value of $W/V$ versus ${\rm{E}}_{\rm{Lab}}\times(A_1^{1/3}+A_2^{1/3})/Z_1Z_2$. The dots represent the value of each system, and one can see that, $W/V$ decreases sharply when ${\rm{E}}_{\rm{Lab}}\times(A_1^{1/3}+A_2^{1/3})/Z_1Z_2$ $<$ 10, and remains relatively stable when ${\rm{E}}_{\rm{Lab}}\times(A_1^{1/3}+A_2^{1/3})/Z_1Z_2$ $\geqslant$ 10. The red line is a simple linear fitting to describe the behavior of $W/V$.

      4). The volume integral is considered as a reliable physical quantity to describe the total strength of a potential [26, 27], and has the expression of:

      $ J_V = \frac{4\pi}{A_1\ A_2} \int V_n(r)r^2\ dr $


      The SPP can describe the realistic potential successfully [12, 28], and was adopted to calculate the volume integrals for every system in this work. The depths and radius parameters were determined by requiring that they satisfy Eq. (4) [with a determined with Eq. (3)] and their corresponding $ J_V $-values to be the same as those given by the SPP systematics. The calculated volume integrals, depths and radius parameters were shown in Table. 2 and in Fig. 5,6,7 respectively.

      Figure 5.  The volume integrals versus $(A_1^{1/3}+A_2^{1/3})$ (a) and versus incident energy (b). We can see the volume integrals of different systems are close to each other, and decrease slightly with increasing incident energy.

      Figure 6.  (color online) The depth of real part determined by $const1$ and volume integral. The values show system independence.

      Figure 7.  (color online) The radius paramters determined by $const1$ and volume integral. There is a very good linear relationship between the radius parameter and $(A_1^{1/3}+A_2^{1/3})$.

      One can see in Fig. 5, the volume integrals of the various systems are relatively close, and decrease slightly as incident energy increases. The depths of real part of various system show system independence, and an insignificant energy dependence. The variations of real depths induced by incident energy are less than 10 MeV for all systems, and the theoretical angular distributions are rarely changed. So that the average value 268.7 $ \pm $ 24.1 MeV were adopted for depths, as shown in Fig. 6. The analysis of the radius parameters is shown in Fig. 7, we can find that the linear expression with $ A_1^{1/3}+A_2^{1/3} $ can describe the trend of the radius parameter very well. The expressions of depths and radius parameters are shown in Eq. (6). Using the expressions of optical potential parameters in Eq. (3) and Eq.(6), the $ const1 $ and $ const2 $ of all systems can be calculated, and the result is exhibited in Fig. 8.

      Figure 8.  (color online) The values of $const1$ (a) and $const2$ (b) in Eq. (4) versus $A_1^{1/3}+A_2^{1/3}$ of each reaction. It is obvious that the value of $const1$ and $const2$ are increasing sharply as $A_1^{1/3}+A_2^{1/3}$ increasing. The dots with error bar represent the data, and the red triangles are the calculated results by adopting the formulas in Eq. (3) and (6). The theoretical points were connected with red curve to describe the trend of $const1$ and $const2$

      $ \begin{aligned}[b] V =& (268.7 \pm 24.1) \ \ {({\rm{MeV}})}\\ W/V =& \left\{ \begin{array}{l} 0.0416\times EDRP+0.4124 \ \ \ EDRP<10; \\ 0.8284\ \ \ \ \ \ EDRP\geqslant10 \\ \end{array}\right.\\ R_V =& R_W = (1.772\pm 0.039)\times (A_1^{1/3}+A_2^{1/3})\\ &\ \ \ \ \ \ \ \ \ \ \ \ -(4.881\pm 0.256)\ \ {\rm{(fm)}}\\ EDRP =& E_{\rm{Lab}}\times(A_1^{1/3}+A_2^{1/3})/Z_1Z_2 \end{aligned} $

    • The systematics of Woods-Saxon parameters extracted from present work were checked with various elastic scattering data The systematics derived by Wang et al [13, 14] and Xu et al [15] were also employed to calculate angular distributions for comparison. The results can be seen in Fig. 9-11.

      Figure 9.  (color online) Comparison of the differential cross sections from experimental data and theoretical calculations. The solid curves are calculated with the systematics summarized in this work. The results of Xu et al [15] (dashed line) and Wang et al [13, 14] (short dashed line) are also shown in figure. Different data sets are offset by factors of 104. The experimental data are from Refs. [29-33].

      Figure 10.  (color online) Same as Fig. 9 but with different targets. The experimental data are from Refs. [34-42].

      Figure 11.  (color online) Elastic scattering angular distributions of 16O + heavy-ions. The experimental data are from Refs. [43-49].

      Figure 12.  (color online) Elastic scattering angular distributions of 16O + heavy-ions in panel (a) and (b), and 40Ca+90,96Zr and 28Si+27Al in panel (c). The experimental data are from Refs. [50-53].

      One can found that, the current systematic potential parameters can reproduce the various experimental data very well, not only the elastic scattering induced by 11B, 12C and 16O, but also some other systems, such as 28Si, 32S and 40Ca + nucleus. For most of the systems, our results can give similar description as the ones by Xu et al. [15] and Wang et al. [13, 14]. For some scattering systems, especially for relatively light systems, such as 12C + 12C at 158.8 MeV and 360 MeV (Fig. 9), 12C + 13C at 127.2 MeV (Fig. 9), and 11B + 58Ni at 25 MeV (Fig. 13), the present systematic results are even better.

      Figure 13.  (color online) Elastic scattering angular distributions of 11B+58Ni at different incident energies in panel (a), and 32S+58Ni at different incident energies in panel (b). Different data sets are offset by factors of 102. The experimental data are from Refs. [54, 55].

    • The elastic scattering angular distributions of 11B, 12C and 16O + heavy-ions at incident energies between 2.3 and 35 A MeV were fitted with six-parameter Woods-Saxon potential. We obtained $ \chi^{2} $ for each calculation to find the best-fit parameter combinations. The diffuseness parameter was determined firstly, and the relationship with $ A_1^{1/3}+A_2^{1/3} $ was concluded. Using the formula of diffuseness parameter, the ratio of imaginary depth and real depth was obtained, and the correlations between depths and radius parameters were discovered. With the limitation of volume integrals calculated with SPP, the values of depths and radius parameters of various systems were determined. The real depths show system independence, and ignorable energy dependence, so the average value was adopted for all systems. The values of $ W/V $ show strong energy dependence when the incident energy is below and around the Coulomb barrier. The radius parameters have a very obvious linear relationship with $ A_1^{1/3}+A_2^{1/3} $, and the expression was fitted. The systematic potential parameters summarized in present work can considerably reproduce various interaction systems not only induced by 11B, 12C and 16O, but also some elastic scattering induced by other heavy ions, such as 28Si, 32S and 40Ca + nucleus.

      However, the present work is far from being satisfactory. For some heavy system, such as 40Ca+96Zr at incident energies of 152 MeV [52], our result is not quite satisfactory. More detailed researches are needed to analyze the relevant parameters, such as isospin.

      It is difficult to directly extract the optical potential between some heavy ions, especially those between unstable nuclei. The systematic study of optical potential parameters in these situations is particularly important. Although there have been some studies on the nuclear-nuclear interaction potential between heavy ions, these works have more or less system limitations, that is, some systems can be reproduced well, and some can not. Therefore, research in this area is still needed.

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