Long-range dynamic polarization potentials for ${}^{11}\mathrm{Li}+{}^{208}\mathrm{Pb}$ and ${}^6\mathrm{He}+{}^{208}\mathrm{Pb}$ systems

We investigate the effects of long-range dynamic polarization on elastic cross sections for heavy-ion collisions which comprise the Coulomb dipole excitation (CDE) potential and the long-range nuclear (LRN) potential. To study these effects, we perform a ${\chi }^2$ analysis for the elastic cross sections of ${}^{11}\mathrm{Li}+{}^{208}\mathrm{Pb}$ and ${}^6\mathrm{He}+{}^{208}\mathrm{Pb}$ systems using the long-range dynamic polarization potentials. For the CDE we formulate some empirical models to reproduce the experimental Coulomb dipole strength $B\left(E1\right)$ distribution and apply to the collision of ${}^{11}\mathrm{Li}$ and ${}^6\mathrm{He}$. But the CDE potential turns out to be not enough to explain the experimental data relevant to the nuclei. It leads us to additionally take the LRN potential peculiar to these halo nuclei into account. Our model, which was constructed by adding the CDE and LRN potentials to the conventional short-range nuclear potential corrected by the Coulomb interaction, provides a good description of the experimental data. In particular, the surface type is shown to be more reasonable rather than the volume type in the LRN potential.

Figure 1

Ratios of the elastic scattering cross sections to the Rutherford cross section $P_E$, as a function of the reduced distance $d$ at (a) $E_{\mathrm{lab}}=22$ MeV for ${}^4\mathrm{He}+{}^{208}\mathrm{Pb}$ and ${}^6\mathrm{He}+{}^{208}\mathrm{Pb}$ systems and (b) $E_{\mathrm{c}.\mathrm{m}.}=28.3$ MeV for ${}^9\mathrm{Li}+{}^{208}\mathrm{Pb}$ and ${}^{11}\mathrm{Li}+{}^{208}\mathrm{Pb}$ systems. The data are taken from Ref. [9] for ${}^4\mathrm{He}+{}^{208}\mathrm{Pb}$ and ${}^6\mathrm{He}+{}^{208}\mathrm{Pb}$ systems and from Ref. [13] for ${}^9\mathrm{Li}+{}^{208}\mathrm{Pb}$ and ${}^{11}\mathrm{Li}+{}^{208}\mathrm{Pb}$ systems.

Figure 2

Ratios $P_E$ calculated with the SRN potential for ${}^9\mathrm{Li}+{}^{208}\mathrm{Pb}$. The solid line denotes the calculated ratios $P_E$ obtained by using Eq. (6) and the parameters in Table 3. Red circles represent the experimental data taken from Ref. [13].

Figure 3

$B\left(E1\right)$ distributions for ${}^{11}\mathrm{Li}$ obtained from three models compared with the experimental data (red circles) taken from Ref. [19]. Solid (black), dashed (blue), and dotted (green) lines represent the $B\left(E1\right)$ distributions obtained from M1, M2, and M3, respectively.

Figure 4

Ratios of the elastic scattering cross sections to the Rutherford cross section $P_E$, as a function of the scattering angle ${\theta }_{\mathrm{c}.\mathrm{m}.}$ at $E_{\mathrm{c}.\mathrm{m}.}=23.1$ and 28.3 MeV for the ${}^{11}\mathrm{Li}+{}^{208}\mathrm{Pb}$ system. Solid (black), dashed (blue), and dotted (green) lines are the calculated ratios $P_E$ obtained by using M1, M2, and M3, respectively, for the ${}^{11}\mathrm{Li}+{}^{208}\mathrm{Pb}$ system. Red circles represent the experimental data for ${}^{11}\mathrm{Li}+{}^{208}\mathrm{Pb}$ taken from Ref. [13]. Note that we only consider the SRN potential and the CDE potential in Eq. (7).

Figure 5

Same as in Fig. 4, but including the additional LRN potential in Eq. (14). Note that we only consider M1.

Figure 6

Same as Fig. 4, but for the ${}^6\mathrm{He}+{}^{208}\mathrm{Pb}$ system. The solid (black) line is for the SRN potential and two long-range dynamic polarization potentials including the CDE and LRN potentials; the dashed (blue) line is for only the SRN potential and the CDE potential obtained by using M1. Red circles represent the experimental data for ${}^6\mathrm{He}+{}^{208}\mathrm{Pb}$ taken from Refs. [8, 9].

Figure 7

Same as Fig. 3, but for the $B\left(E1\right)$ distribution of ${}^6\mathrm{He}$ nuclei obtained from M1. Red circles represent the experimental data for ${}^6\mathrm{He}+{}^{208}\mathrm{Pb}$ taken from Ref. [18].

Figure 8

Real and imaginary strength functions, $V_0^{lo}\left(E\right)$ and $W_0^{lo}\left(E\right)$, as a function of the laboratory energy $E_{\mathrm{lab}}$. The solid (black) line in the lower panel denotes $W_0^{lo}\left(E\right)$ from Eq. (15) and the dotted (red) line in the upper panel represents $W_0^{lo}\left(E\right)$ obtained by using the dispersion relation.

Figure 9

Same as Fig. 6, but with all incident laboratory energies considered.