Suppression of the elastic scattering cross section for the ¹⁷Ne + ²⁰⁸Pb system

As depicted in the dotted magenta line in Fig. 2 (a), the experimental $P_{\textrm{el}}$ ratio of the ¹⁷Ne + ²⁰⁸Pb system cannot be reproduced through a potential defined as the sum of $V_{0}(r)$ and $U_{\textrm{F}}(r)$ in Eq.  (2). This indicates that the additional absorption channels are open near the Coulomb barrier energy region. Therefore, the additional DPPs are used to reproduce the experimental results. This study considers inelastic scattering, breakup reactions, and fusion reactions as the additional DPPs.

In principle, to explain the elastic scattering data for the ¹⁷Ne + ²⁰⁸Pb system using these DPPs, various experimental data are required for the input data. However, in the ¹⁷Ne + ²⁰⁸Pb system, there are two sets of experimental data: the elastic scattering and breakup cross-section data. Nevertheless, we can calculate the total fusion cross sections between the incident isotopes and target nucleus through the Barrier penetration model (BPM) hagi99 . The extracted total fusion cross section obtained from the BPM using $V_{0}(r)$ in Eq. (2) is approximately 994.3 mb.

Utilizing the experimental elastic scattering, experimental breakup reaction, and theoretical total fusion cross-section data, we performed the $\chi^{2}$ analysis. As discussed earlier, breakup reactions occur primarily near the surface of the projectile and the target nucleus; therefore, we employed the differential form of the Woods-Saxon potential in the calculations presented in Eq. (6). Consequently, the $\chi^{2}$ fitting could be performed on twelve adjustable parameters ($V^{\textrm{N}}_{\textrm{br}}$, $W^{\textrm{N}}_{\textrm{br}}$, $a^{\textrm{N}}_{\textrm{v, br}}$, $a^{\textrm{N}}_{\textrm{w, br}}$, $r^{\textrm{N}}_{\textrm{v, br}}$, $r^{\textrm{N}}_{\textrm{w, br}}$, $V_{\textrm{F}}$, $W_{\textrm{F}}$, $a_{\textrm{v, F}}$, $a_{\textrm{w, F}}$, $r_{\textrm{v, F}}$, $r_{\textrm{w, F}}$) in $U^{\textrm{N}}_{\textrm{br}}(r)$ and $U_{\textrm{F}}(r)$. However, as specific adjustable parameters were entangled, we reduced the parameters in the $\chi^{2}$ fitting by applying it on eight adjustable parameters ($V^{\textrm{N}}_{\textrm{br}}$, $W^{\textrm{N}}_{\textrm{br}}$, $a^{\textrm{N}}_{\textrm{br}}$, $r^{\textrm{N}}_{\textrm{br}}$, $V_{\textrm{F}}$, $W_{\textrm{F}}$, $a_{\textrm{F}}$, $r_{\textrm{F}}$) by assuming the diffuseness and radius parameters as $a^{\textrm{N}}_{\textrm{V, br}}$ = $a^{\textrm{N}}_{\textrm{w, br}}$ $\equiv$ $a^{\textrm{N}}_{\textrm{br}}$, $a_{\textrm{v, F}}$ = $a_{\textrm{w, F}}$ $\equiv$ $a_{\textrm{F}}$, $r^{\textrm{N}}_{\textrm{v, br}}$ = $r^{\textrm{N}}_{\textrm{w, br}}$ $\equiv$ $r^{\textrm{N}}_{\textrm{br}}$, and $r_{\textrm{v, F}}$ = $r_{\textrm{w, F}}$ $\equiv$ $r_{\textrm{F}}$.

The parameters obtained from $\chi^{2}$ fitting are listed in set (A) of Table 2. Based on these parameters, we plotted the theoretical $P_{\textrm{el}}$ ratio (i.e., the ratio of the elastic scattering cross section to the Rutherford cross section) and the angular distribution of the breakup cross section as the dashed blue line in Fig. 2. Overall, the theoretical $P_{\textrm{el}}$ ratio and the angular distribution of the breakup cross section are consistent with the experimental ratio, except for one or two points. Interestingly, the rainbow peak observed around $\theta_{\textrm{c.m.}}$ = 60^∘ in the dotted magenta line is strongly suppressed owing to the effect of inelastic scattering and breakup reactions. This can be attributed to the ¹⁷Ne nucleus, which exhibits relatively low values for its first excited energy ($E^{1st}_{x}$ = 1.288MeV) and separation energies ($S_{2p}$=0.944 and $S_{p}$ = 1.464 MeV). These similarities are reminiscent of weakly bound ¹¹Li and ¹¹Be nuclei.

To investigate the angular distribution of the breakup cross section, we utilize the following form  kim02 for reference ¹¹1Our previous papers heo22 ; so16 have a typo. The equation (16) in Ref heo22 and Eq. (5) in Ref. so16 is to be replaced by Eq. (8) in the present paper.:

with

The point to note is that the orbital angular momentum ($l$) is treated as an integer value in quantum-mechanical expression. However, according to the equation for $l$ and $\theta_{\textrm{c.m.}}$ kim02 , $l$ $\equiv$ $l_{\theta_{\textrm{c.m.}}}$ = $\frac{ka_{0}}{2}$ $\cot(\frac{\theta_{\textrm{c.m.}}}{2})$ used in the present research, the orbital angular momentum ($l_{\theta_{\textrm{c.m.}}}$) is obtained by continuous scattering angles ($\theta_{\textrm{c.m.}}$), and does not have discrete integer values, but rather has continuous values. To address this issue, we have exploited semi-classical approximation methods customarily used in several semi-classical treatments of the quantum mechanical Rutherford cross section bass80 ; satc80 ; mott65 ; ford59 . That is, since the main contribution to the differential cross sections stems from the partial waves in the vicinity of $l$ $\approx$ $l_{\theta_{\textrm{c.m.}}}$, for a given scattering angle $\theta_{\textrm{c.m.}}$, one can found the nearest $l$ with the integer values and calculate Eq. (9) using the $l$. In order to obtain more reasonable calculations in the present work, however, we have calculated $\sigma_{\textrm{br};l_{\theta_{\textrm{c.m.}}}}$ by taking an interpolation of the results obtained by the two partial wave cross sections ($\sigma_{\textrm{br};l}$ and $\sigma_{\textrm{br};l+1}$) corresponding to the two nearby $l$ and $l+1$ for $l$ $<$ $l_{\theta_{\textrm{c.m.}}}$ $<$ $l+1$ ($l$ is the integer value). Of course, this procedure is most justified when the scattering follows the classical path. Detailed discussion was also done in the previous paper done by one of present authors kim02 .

Furthermore, we evaluated the inelastic, breakup, and fusion cross sections ($\sigma_{\textrm{inel.}}$, $\sigma_{\textrm{br}}$, and $\sigma_{\textrm{F}}$) can be expressed as udag84 ; huss84

These parameters are listed in set (A) of Table 3. Notably, we used $W_{\textrm{i}}(r)$ in Eq.  (10) as $W_{\textrm{inel.}}(r)$ = $W^{\textrm{N}}_{\textrm{inel.}}(r)$ + $W^{\textrm{C}}_{\textrm{inel.}}(r)$, $W_{\textrm{br}}(r)$ = $W^{\textrm{N}}_{\textrm{br}}(r)$ + $W^{\textrm{C}}_{\textrm{br}}(r)$, and $W_{\textrm{F}}(r)$ for $\sigma_{\textrm{inel.}}$, $\sigma_{\textrm{br}}$, and $\sigma_{\textrm{F}}$, respectively.

Based on the calculations performed thus far, we determined the contribution of the elastic scattering, inelastic scattering, breakup, and total fusion cross sections of the ¹⁷Ne + ²⁰⁸Pb system, as shown in set (A) of Table 3. However, owing to the absence of the experimental inelastic scattering cross-section data, the contribution of each component (nuclear and Coulomb) to the inelastic scattering cross section could not be adequately investigated. As inelastic scattering channel can also be caused by the nuclear and Coulomb interactions, we must also consider the inelastic scattering channel by the nuclear interaction. In particular, the absence of the nuclear component of the inelastic scattering interaction suggests that the flux corresponding to this interaction has flowed into other reaction channels. In fact, as shown in set (A) of Table 3, the total fusion cross section obtained in our calculation (1126.8 mb) is approximately 132.5 mb larger than that obtained in the CC calculation (994.3 mb). To isolate the nuclear component of the inelastic scattering interaction, we employ a new interaction. As inelastic scattering primarily occurred near the nucleus surface, the nuclear component of the inelastic scattering potential was generally considered using a differential form of the Woods-Saxon potential, as expressed in Ref. rehm82 :

where $X^{\textrm{N}}_{\textrm{i,~{}inel}}$ = $(r-R^{\textrm{N}}_{\textrm{i,~{}inel}})/a^{\textrm{N}}_{\textrm{i,~{}inel}}$ where $R^{\textrm{N}}_{\textrm{i,~{}inel}}$ = $r^{\textrm{N}}_{\textrm{i,~{}inel}}$ (${A_{1}^{1/3}}$ + ${A_{2}^{1/3}}$).

With the differential form of the Woods-Saxon potential corresponding to the nuclear component of the inelastic scattering potential, we performed the additional $\chi^{2}$ fitting using $U_{\textrm{DPP}}(r)$ = $U^{\textrm{N}}_{\textrm{inel}}(r)$ + $iW^{\textrm{C}}_{\textrm{inel}}(r)$ + $U^{\textrm{N}}_{\textrm{br}}(r)$ + $U^{\textrm{C}}_{\textrm{br}}(r)$ + $U_{\textrm{F}}(r)$ in Eq.  (2). For reference, $W^{\textrm{C}}_{\textrm{inel}}(r)$ and $U^{\textrm{C}}_{\textrm{br}}(r)$ potentials related to the CQE transition did not require adjustable parameters. In principle, we should perform $\chi^{2}$ fitting on eighteen adjustable parameters ($V^{\textrm{N}}_{\textrm{inel}}$, $W^{\textrm{N}}_{\textrm{inel}}$, $a^{\textrm{N}}_{\textrm{v, inel}}$, $a^{\textrm{N}}_{\textrm{w, inel}}$, $r^{\textrm{N}}_{\textrm{v, inel}}$, $r^{\textrm{N}}_{\textrm{w, inel}}$, $V^{\textrm{N}}_{\textrm{br}}$, $W^{\textrm{N}}_{\textrm{br}}$, $a^{\textrm{N}}_{\textrm{v, br}}$, $a^{\textrm{N}}_{\textrm{w, br}}$, $r^{\textrm{N}}_{\textrm{V, br}}$, $r^{\textrm{N}}_{\textrm{w, br}}$, $V_{\textrm{F}}$, $W_{\textrm{F}}$, $a_{\textrm{v, F}}$, $a_{\textrm{w, F}}$, $r_{\textrm{v, F}}$, $r_{\textrm{w, F}}$) in $U^{\textrm{N}}_{\textrm{inel}}(r)$, $U^{\textrm{N}}_{\textrm{br}}(r)$, and $U_{\textrm{F}}(r)$. Because we have already determined the parameters related to the breakup reaction potential that effectively reproduce the experimental breakup cross-section data, we can fix them ($V^{\textrm{N}}_{\textrm{br}}$, $W^{\textrm{N}}_{\textrm{br}}$, $a^{\textrm{N}}_{\textrm{br}}$, $r^{\textrm{N}}_{\textrm{br}}$) obtained through the set (A) of Table 2. To reduce these parameters, we assume that the diffuseness and radius parameters of the inelastic scattering and total fusion potential are $a^{\textrm{N}}_{\textrm{v, inel.}}$ = $a^{\textrm{N}}_{\textrm{w, inel.}}$ $\equiv$ $a^{\textrm{N}}_{\textrm{inel.}}$, $r^{\textrm{N}}_{\textrm{v, inel.}}$ = $r^{\textrm{N}}_{\textrm{w, inel.}}$ $\equiv$ $r^{\textrm{N}}_{\textrm{inel.}}$, $a_{\textrm{v, F}}$ = $a_{\textrm{w, F}}$ $\equiv$ $a_{\textrm{F}}$ and $r_{\textrm{v, F}}$ = $r_{\textrm{w, F}}$ $\equiv$ $r_{\textrm{F}}$. the radius parameters. Thereafter, we perform a second $\chi^{2}$ fitting with eight adjustable parameters ($V^{\textrm{N}}_{\textrm{inel.}}$, $W^{\textrm{N}}_{\textrm{inel.}}$, $a^{\textrm{N}}_{\textrm{inel.}}$, $r^{\textrm{N}}_{\textrm{inel.}}$, $V^{\textrm{N}}_{\textrm{F}}$, $W^{\textrm{N}}_{\textrm{F}}$, $a^{\textrm{N}}_{\textrm{F}}$, $r^{\textrm{N}}_{\textrm{F}}$). The parameters obtained from $\chi^{2}$ fitting are listed in set (B) of Table 2.

Based on parameter set (B) of Table 2, we plotted the theoretical $P_{\textrm{el}}$ ratio and the angular distribution of the breakup cross section emerging from the nuclear component for inelastic scattering interaction, as shown by the solid black line in Fig.  2. In these calculations, the theoretical $P_{\textrm{el}}$ ratio and angular distribution of the breakup cross section were consistent with the experimental data for almost all scattering angles, except for one or two points. As shown in set (B) of Table 3, part of the fusion cross section shifted to inelastic scattering because of the addition of a new nuclear component of the inelastic scattering interaction, as expected.

Finally, we calculated the inelastic scattering, breakup reaction, and fusion cross sections expressed in Eq. (10), which are listed in set (B) of Table 3. The results showed that the inelastic scattering cross sections exhibited a greater contribution through the nuclear interaction than through Coulomb excitation. By contrast, the breakup reaction cross section showed that the contributions of the two interactions were similar. Interestingly, the contribution of the DR cross section (i.e., the sum of the inelastic scattering and breakup reaction cross sections) to the total reaction was approximately half ($\sigma_{\textrm{DR}}$/$\sigma_{\textrm{R}}$ $\approx$ 0.52) of the total reaction cross section, as shown in set (B) of Table 3. This implies that a significant portion of the elastic scattering channel is converted to the DR channel through processes, such as inelastic scattering and breakup reaction.

According to Ref. ovej23 , the authors remarked that the Coulomb excitation interaction due to two low-lying E2 resonance states was not the main contributor to the imaginary part of the optical model potential. Therefore, in this study, we examined the contribution of each channel and determined the main contributor. To this end, we examined the change in the elastic scattering cross section by adding (or switching on) each channel to $V_{0}(r)$ + $U_{\textrm{F}}(r)$ potential and investigate the extent to which the contribution of each channel affects the rainbow peak. We also investigated the contributions of paired channels (two-channel coupling).

Fig. 3 (a) clearly shows that the $P_{\textrm{el}}$ ratio due to the nuclear part of the breakup reaction potential ( solid black line) has the largest suppression. This can also be confirmed by the values of the reaction cross sections listed in Table 3. However, it is unclear whether the contribution of a specific channel causes the rainbow peak to disappear. To clearly observe the contributions of specific channels to the disappearance of the rainbow peak, we paired (or coupled) each channel with its Coulomb, nuclear, inelastic, and breakup reaction components, respectively. Consequently, we obtained Fig. 3 (b), from which we can carefully infer that the largest contribution to the suppression of the rainbow peak is due to the nuclear component[switch on $U^{\textrm{N}}_{\textrm{inel.}}(r)$ + $U^{\textrm{N}}_{\textrm{br}}(r)$; the dash-dotted magenta line]. The contribution of the breakup reaction [switch on $W^{\textrm{C}}_{\textrm{br}}(r)$ + $U^{\textrm{N}}_{\textrm{br}}(r)$; the solid black line] are also larger than that of inelastic scattering [switch on $W^{\textrm{C}}_{\textrm{inel.}}(r)$ + $U^{\textrm{N}}_{\textrm{inel.}}(r)$; short dashed cyan line]. Finally, the contributions of Coulomb excitation interaction due to two low-lying E2 resonance states [switch on $W^{\textrm{C}}_{\textrm{inel.}}(r)$ + $U^{\textrm{C}}_{\textrm{br}}(r)$; dotted green line] are relatively small. Thus, we can confirm that this contribution is not the primary one, as mentioned above ovej23 . Consequently, it can be interpreted that the ¹⁷Ne projectile can be more easily excited or broken up by nuclear interaction than by the Coulomb interaction generated by the target nucleus. Including all the channels, we can obtain significantly better results, as shown in Fig.  2 (a).

Thus far, we have calculated the elastic scattering, inelastic scattering, breakup reaction, and fusion cross sections for the ¹⁷Ne + ²⁰⁸Pb system using the OM. In Ref. ovej23 , the ¹⁵O production cross section was calculated by treating $\frac{5}{2}^{-}$ state [$E_{x}$ = 1.764 MeV] as the breakup channel using the CC method. Here, we attempt to recalculate the same system using the CC method hagi99 for comparison with the analysis in Ref. ovej23 . We examined how the coupling effects of the potentials generated in the OM were realized within the CC framework. To examine the effect of breakup within CC in the CC calculation, the DPP potentials associated with each coupling were excluded from the OM potential. Moreover, the breakup potential $U_{\textrm{br}}$ was replaced with its counterpart in the CC method. It is to confirm the coupling effect of the two resonance states, the $\frac{5}{2}^{-}$ and the $\frac{3}{2}^{-}$ state, in the CC framework, through the potential obtained in our extended optical model.

First, we perform the CC calculations with set (A) of Table 2, which has no nuclear components for the inelastic channel, and the $\frac{5}{2}^{-}$ state [$E_{x}$ = 1.764 MeV] and $\frac{3}{2}^{-}$ state [$E_{x}$ = 2.692 MeV] are treated as the breakup interactions. Note that $\frac{3}{2}^{-}$ state [$E_{x}$ = 2.692 MeV] is added to the CC in Ref. ovej23 . For reference, the nuclear deformation length ($\delta_{n}$) corresponding to the $\frac{5}{2}^{-}$ and $\frac{3}{2}^{-}$ states are $\delta_{1}$ = 2.14 and $\delta_{2}$ = 1.82, respectively.

In Fig. 4, we plotted the theoretical $P_{\textrm{el}}$ ratio and angular distribution of the breakup cross section by the set (A) parameters as the dashed blue line. Because the theoretical angular distribution of the breakup reaction cross section is not significantly large, as shown in Fig. 4 (b), the theoretical $P_{\textrm{el}}$ ratio does not sufficiently suppress the experimental $P_{\textrm{el}}$ data.

Next, we perform the CC calculations using set (B) of Table 2 which considers the additional coupling of the nuclear components for the inelastic channel and the $\frac{5}{2}^{-}$ and $\frac{3}{2}^{-}$ states for the breakup interaction. The theoretical $P_{\textrm{el}}$ ratio and angular distribution of the breakup cross section are plotted as a solid black line in Fig. 4. However, the figure shows that the change in the theoretical $P_{\textrm{el}}$ ratios is also insignificant because there is almost no change in the breakup cross section, similar to the calculation of set (A). This means that, aside from the two low-lying states discussed above, additional channels, such as higher continuum states or transfers, are necessary to describe the breakup cross section. Therefore, in our OM calculations, we inserted a differential form of the Woos-Saxon potential, corresponding to the contribution of the nuclear component of the breakup reaction, to consider the additional channels.

In the CC formalism, considering all channels contributing to the breakup reaction can be challenging. Thus, we simply adjust the nuclear deformation length mentioned above to determine how the theoretical $P_{\textrm{el}}$ ratios and angular distribution of the breakup cross section change, thereby testing the contribution of channels related to nuclear interaction. The contributions of the Coulomb interactions are presented at approximately 40^∘ and show results similar to those of the CC analysis in Ref. ovej23 .

Furthermore, the results were examined by changing the nuclear deformation length two to four times. The results are plotted as dotted magenta, short-dashed green, and short-dotted violet lines in Fig.  4. As the nuclear deformation strength increases, the breakup reaction cross section increases and gradually approaches that of the experimental data. This implies that the channels related to nuclear interaction are crucial in reproducing the experimental data of $P_{\textrm{el}}$ ratios and angular distribution of the breakup cross section.

According to our CC calculations, although an additional excited state [$\frac{3}{2}^{-}$ state ($E_{x}$ = 2.692 MeV)] is added to the calculations performed in Ref. ovej23 , the result could not explain the ¹⁵O production cross section.