Extended optical model analyses of elastic scattering and fusion cross section data for the ${}^7\mathrm{Li}+{}^{208}\mathrm{Pb}$ system at near-Coulomb-barrier energies using a folding potential

Simultaneous ${\chi }^2$ analyses previously made for elastic scattering and fusion cross section data for the ${}^6\mathrm{Li}+{}^{208}\mathrm{Pb}$ system are extended to the ${}^7\mathrm{Li}+{}^{208}\mathrm{Pb}$ system at near-Coulomb-barrier energies based on the extended optical model approach, in which the polarization potential is decomposed into direct reaction (DR) and fusion parts. Use is made of the double folding potential as a bare potential. It is found that the experimental elastic scattering and fusion data are well reproduced without introducing any normalization factor for the double folding potential and that both the DR and fusion parts of the polarization potential determined from the ${\chi }^2$ analyses satisfy separately the dispersion relation. Further, we find that the real part of the fusion portion of the polarization potential is attractive while that of the DR part is repulsive except at energies far below the Coulomb barrier energy. A comparison is made of the present results with those obtained from the coupled discretized continuum channels calculations and a previous study based on the conventional optical model with a double folding potential. We also compare the present results for the ${}^7\mathrm{Li}+{}^{208}\mathrm{Pb}$ system with the analysis previously made for the ${}^6\mathrm{Li}+{}^{208}\mathrm{Pb}$ system.

Figure 1

$P_E$ values for (a) the ${}^7\mathrm{Li}+{}^{208}\mathrm{Pb}$ system and (b) the ${}^6\mathrm{Li}+{}^{208}\mathrm{Pb}$ system.

Figure 2

The Stelson plot of $S_i=\sqrt{E{\sigma }_i}$ for DR ($i=D$, open circles) and fusion ($i=F$, filled circles) cross sections. Use is made of the semi-experimental DR cross section for $S_D$, while the experimental fusion cross section is employed for $S_F$. The intercepts of the straight lines with the energy axis give us the threshold energies $E_{0,D}^{\mathrm{semi}\text{−}\exp }=19.3$ MeV and $E_{0,F}^{\exp }=26.5$ MeV.

Figure 3

The strength parameters $V_i$ (upper panel) and $W_i$ (lower panel) for $i=D$ and $F$ as functions of $E_{\mathrm{c}.\mathrm{m}.}$. The open and filled circles are the strength parameters for $i=D$ and $F$, respectively. The dotted and solid lines in the lower panel denote $W_D$ and $W_F$ from Eqs. (16) and (17), respectively, while the dotted and solid curves in the upper panel represent $V_D$ and $V_F$ calculated by using the dispersion relation of Eq. (8) with $W_i$ given by Eqs. (16) and (17). The potential values and the corresponding reference energies used in Eq. (8) are such that $V_F$ ($E_s=29.9$ MeV) $=2.2$ MeV and $V_D$ ($E_s=29.3$ MeV) $=-0.03$ MeV, respectively.

Figure 4

Ratios of the elastic scattering cross sections to the Rutherford cross section calculated with our final dispersive optical potential are shown in comparison with the experimental data. The data are taken from Ref. [2].

Figure 5

DR and fusion cross sections calculated with our final dispersive optical potentials are shown in comparison with the experimental data. ${\sigma }_D^{\mathrm{semi}\text{−}\exp }$ denoted by the open circles are obtained as described in Sec. 2. The fusion data are from Refs. [11] and [12].

Figure 6

The values of $W_F(r,E),W_D(r,E)$, and the sum $W_{\mathrm{tot}}(r,E)=W_F(r,E)+W_D(r,E)$ as functions of $E$ calculated by using Eqs. (6), (7), (16), and (17) at the strong absorption radius, $r=R_{\mathrm{sa}}=12.4$ fm, for all energies.

Figure 7

The values of the total imaginary potential $W(r,E)$ at $r=R_{\mathrm{sa}}=12.4$ fm deduced in the present ${\chi }^2$ analyses and those obtained in Ref. [2]. The values from Ref. [2] are multiplied by factors 1.23 and 1.11 for the ${}^7\mathrm{Li}+{}^{208}\mathrm{Pb}$ and ${}^6\mathrm{Li}+{}^{208}\mathrm{Pb}$ systems, respectively.