Systematic study of the Woods-Saxon potential parametersbetween heavy-ions
本站小编 Free考研考试/2022-01-01
Lin Gan1, , Zhi-Hong Li2,3, , Hui-Bin Sun4, , Shi-Peng Hu4, , Er-Tao Li4, , Jian Zhong4, , 1.Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China 2.China Institute of Atomic Energy, Beijing 102413, China 3.School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 101408, China 4.College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China Received Date:2020-06-30 Available Online:2021-05-15 Abstract:Experimental elastic scattering angular distributions of 11B, 12C, and 16O + heavy-ions are used to study the Woods-Saxon potential parameters. Best fitted values of the diffuseness parameters are found for each system, and a linear relationship is expressed between the diffuseness parameters and $A_1^{1/3}+A_2^{1/3}$ . The correlation of the potential depth and radius parameters with $A_1^{1/3}+A_2^{1/3}$ is also revealed within the limitations of the diffuseness parameter formula. Because the incident energies of most of the analyzed reactions are below or around the Coulomb barrier, the energy dispersion relation between the real and imaginary potentials is considered in order to investigate the ratio between the imaginary and real potential well depths, resulting in an expression of $W/V$ . Within the limitation of the volume integrals calculated with the S$\tilde{a}$o Paulo potential, parameterized formulas for the depth and radius parameters are obtained. The systematic Woods-Saxon potential parameters derived in the 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.
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II.EXTRACTION OF THE WOODS-SAXON POTENTIAL PARAMETERSThe Woods-Saxon model potential was proposed by Woods and Saxon to approximate the shape of the nuclear component of nucleus-nucleus interactions [17]. Although it is a phenomenological approximation, reactions of different incident energies and different projectile-target combinations can be well reproduced with the six parameters used in the calculations [7]. The six-parameter Woods-Saxon potential contains a real component and an imaginary component and is expressed as
where V and W are potential depth parameters for the real and imaginary potentials, respectively; $ R_i $ = $ r_i(A_1^{1/3}+ $$ A_2^{1/3}) $, in which $ i = V, \;W $, denotes the real and imaginary radius parameters; $ a_V $ and $ a_W $ respectively refer to the real and imaginary diffuseness; and $ A_{1} $ and $ A_{2} $ are the mass numbers of the projectile and target nuclei. The Coulomb potential is always expressed as
where $ R_{\rm C} = r_{\rm C}(A_1^{1/3}+A_2^{1/3}) $ is the radius parameter. It has been proved that theoretical angular distributions are not sensitive to changes in the Coulomb radius [18]. Therefore, a fixed $ r_{\rm C} $ equal to 1.0 fm is used throughout the following processes. In the present work, 27 sets of elastic scattering angular distribution data are adopted to extract the Woods-Saxon potential parameters. The targets range from 12C to 209Bi, and the incident energies range from 25 to 420 MeV. Most of the data are from the National Nuclear Data Center, while the angular distributions of 12C elastic scattering from 90Zr, 91Zr, 96Zr , and 116Sn were measured at the China Institute of Atomic Energy (CIAE), Beijing. Because the optical potential parameters have large ambiguities, several different sets of parameter combinations can reproduce the experimental data well [19]. It has been confirmed that when fitting two or more optical potential parameters at the same time, we easily fall into a minimum trap. In order to study the intrinsic relationship between these parameters, we define more than 1 million sets of parameter combinations of W, $ R_V $, $ R_W $, $ a_V $ , and $ a_W $ in advance and then use each of them to fit V for all the data. W varies in the range of 10 - 300 MeV by steps of 2 MeV. The geometric parameters of the real and imaginary components are set to the same value, i.e., $ r_V $ = $ r_W $, and vary between 0.5 and 1.5 fm in increments of 0.01 fm. Finally, $ a_V $ = $ a_W $ vary between 0.3 and 1.0 fm in increments of 0.01 fm. The analysis process can be divided into four steps: 1) Use all the parameter combinations to fit each elastic scattering dataset with the nuclear reaction code PTOLEMY [20]. In this way, the output V and corresponding $ \chi^2 $ are obtained for every calculation. 2) Analyze the correlation of $ \chi^2 $ with the depth, radius, and diffuseness parameters. We find that, for each system, there exists a suitable value of the diffuseness parameter that gives the minimum $ \chi^2 $. As a typical result, Figure 1 shows $ \chi^2 $ versus the diffuseness parameter for 12C + 90Zr elastic scattering. However, the depth and radius parameters cannot be extracted in the same way as the diffuseness parameter because there are no best fit values for these parameters. We list the extracted diffuseness for each interaction system in Table 1. The analysis of the diffuseness uncertainties employs the $ \chi^2 $ envelope method proposed in Ref. [21]. A linear formula for the diffuseness parameter a versus $ A_1^{1/3}+A_2^{1/3} $ is summarized in Eq. (3) and shown in Fig. 2; the uncertainties of the slope and intercept are 0.011 and 0.063 fm, respectively. The slope is negative, indicating that as the mass of the system increases, the diffuseness parameter tends to decrease. Figure1.$\chi^2$ versus the diffuseness parameter for 12C + 90Zr at 66 MeV. Each dot represents a parameter set. It is found that a = 0.59 fm corresponds to the minimum $\chi^2$.
Reaction
$E_{\rm 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
Table1.The most suitable diffuseness parameters extracted in the analysis.
Figure2. (color online) The diffusion parameter as a function of $A_1^{1/3}+A_2^{1/3}$.
3) When the diffuseness parameters are fixed as in Eq. (3), we find a correlation between the depth and radius parameters with the real and imaginary parts, respectively. As an example, $ \chi^2 $ versus the depth parameters of 12C + 90Zr elastic scattering at 66 MeV is shown in Fig. 3. It is obvious that the curve for V is much steeper than that for W, which indicates that $ \chi^2 $ is more sensitive to V than to W. Each curve corresponds to a specific radius parameter value, and the correlations of the depth and radius parameter combinations of best fit are as follows: Figure3. (color online) $\chi^2$ versus V (a) and W (b) for 12C + 90Zr at 66 MeV. Each curve corresponds to a different radius parameter; the valleys give almost the same $\chi^2$. The radius parameters for each curve from left to right are 1.07 to 1.03 fm, by steps of 0.01 fm. Observe that the curves for V (a) are much steeper than those for W (b).
The values of $ const1 $ and $ const2 $ are listed in Table 2. The errors stem from the uncertainties of the diffuseness parameters. Since the radius parameter and diffuseness parameter are the same for the real and imaginary components, the ratio of W to V is equal to the ratio of $ const2 $ to $ const1 $.
Reaction
$E_{\rm lab}{\rm{ /MeV}}$
${{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
Table2.Volume integrals (Vol) calculated with SPP, $const1$ , and $const2$ in Eq. (4); the real depth (V) and radius (R) parameters are determined by the Volume integrals and $const1$.
For most of the nuclear reactions analyzed here, because the incident energies are around or below the Coulomb barrier, the energy dispersion relation between the real and imaginary potentials [22-25] cannot be ignored. Due to the scarcity of energy points around the Coulomb barrier for each system analyzed in the present work, we cannot conduct a detailed study of the energy dispersion relation. We instead propose a rough method to analyze the relationship between the real and imaginary parts near the Coulomb barrier. As the Coulomb barrier is proportional to $ Z_1Z_2/ (A_1^{1/3}+A_2^{1/3}) $, we adopt ${{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 to 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 thus determine 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 $. Figure4. (color online) Value of $W/V$ versus ${{E}}_{\rm{lab}}\times(A_1^{1/3}+A_2^{1/3})/Z_1Z_2$. The dots represent the value for each system. $W/V$ decreases sharply when ${{E}}_{\rm{lab}}\times(A_1^{1/3}+A_2^{1/3})/Z_1Z_2$$<$ 10 and remains relatively stable when ${{E}}_{\rm{lab}}\times(A_1^{1/3}+A_2^{1/3})/Z_1Z_2$$\geqslant$ 10. The red line is a simple linear fit to describe the behavior of $W/V$.
4) The volume integral is considered to be a reliable physical quantity for describing the total strength of a potential [26, 27]; it is expressed as
The SPP can describe the realistic potential successfully [12, 28], and it is adopted to calculate the volume integrals for every system in this work. The depth and radius parameters are determined by requiring that they satisfy Eq. (4) [with a determined by Eq. (3)] and their corresponding $ J_V $-values are the same as those given by the SPP systematics. The calculated volume integrals and depth and radius parameters are shown in Table 2 and in Figs. 5, 6, and 7, respectively. Figure5. Volume integrals versus $(A_1^{1/3}+A_2^{1/3})$ (a) and incident energy (b). The volume integrals of the different systems are similar, and they decrease slightly as the incident energy increases.
Figure6. (color online) Depth of the real part determined by $const1$ and the volume integral. The values show system independence.
Figure7. (color online) Radius parameters determined by $const1$ and the volume integral. There is a strong linear relationship between the radius parameter and $(A_1^{1/3}+A_2^{1/3})$.
In Fig. 5, it is evident that the volume integrals of the various systems are relatively close and that they decrease slightly as the incident energy increases. The depth of the real part of each system exhibits system independence and an insignificant energy dependence. The variations in the real depth induced by the incident energy are less than 10 MeV for all systems, and the theoretical angular distributions rarely change. An average value of 268.7 $ \pm $ 24.1 MeV is thus adopted for the depth, as shown in Fig. 6. The analysis of the radius parameters is shown in Fig. 7, where we find that the linear expression with $ A_1^{1/3}+A_2^{1/3} $ describes the trend of the radius parameter very well. The expressions of the depth and radius parameters are shown in Eq. (6). Using the expressions for the optical potential parameters in Eq. (3) and Eq. (6), $ const1 $ and $ const2 $ can be calculated for all systems, the results of which are plotted in Fig. 8. Figure8. (color online) Values of $const1$ (a) and $const2$ (b) in Eq. (4) versus $A_1^{1/3}+A_2^{1/3}$ for each reaction. The values of $const1$ and $const2$ increase sharply as $A_1^{1/3}+A_2^{1/3}$ increases. The dots with error bars represent the data, and the red triangles denote the results calculated by adopting the formulas in Eqs. (3) and (6). The theoretical points are connected by the red curves to describe the trend of $const1$ and $const2.$