Density variation effects in $\alpha + {}^{208}\mathrm{Pb}$ and ${}^{16}\mathrm{O} + {}^{208}\mathrm{Pb}$ fusion reactions

We propose a hybrid approach between the sudden model and the adiabatic model to explain the fusion hindrance in heavy-ion fusion reactions at deep subbarrier energies. Due to the strong Pauli blocking effects at the density overlap region, the $\alpha$ particle approaching the target ${}^{208}\mathrm{Pb}$ becomes enlarged and its density distribution gets changed. By introducing energy dependence into this density variation effect, the potentials at different colliding energies are diverse. With the decrease of colliding energies, the significantly enhanced density variation effects make the nuclear potential more attractive and result in a deeper pocket in the inner part. A reasonable description of the experimental fusion cross sections is achieved for the $\alpha + {}^{208}\mathrm{Pb}$ reaction. On account of highlighting the influence of density variation effects on the fusion process at deep subbarrier energies, we further investigate the density variation effects in the ${}^{16}\mathrm{O} + {}^{208}\mathrm{Pb}$ fusion reaction based on the $\alpha$-cluster structures in ${}^{16}\mathrm{O}$ nuclei. The fusion hindrance at deep subbarrier energies is described well by considering the density variation effects. In addition, an astrophysical $S$ factor between the sudden model and adiabatic model is predicted at the energies below the experimental data.

Figure 1

The width parameter $\beta$ and corresponding central density ${\rho }_0$ of $\alpha$ particle vs the distance between the center of mass of two nuclei at energies of 23.5 MeV (dotted line) and 15.6 MeV (dashed line) for $\alpha + {}^{208}\mathrm{Pb}$ fusion reaction. The solid lines denote the results without considering DVE.

Figure 2

The experimental cross sections for $\alpha + {}^{208}\mathrm{Pb}$ fusion reaction [44] compared with the results calculated by M3Y $+$ Pauli potential (solid line) and M3Y $+$ Pauli $+$ DVE potential (dashed line). The insert shows the M3Y $+$ Pauli potential (solid line) and M3Y $+$ Pauli $+$ DVE potentials at energies of 23.5 MeV (dotted line) and 15.6 MeV (dashed line).

Figure 3

The density variation of ${}^{16}\mathrm{O}$ nuclei and $\alpha$ cluster approaching the target ${}^{208}\mathrm{Pb}$ at colliding energy of 75.65 MeV. The solid, dashed, and dotted lines denote the density distributions of ${}^{208}\mathrm{Pb}, {}^{16}\mathrm{O}$, and $\alpha$ cluster, respectively. Note that the density of $\alpha$ cluster has been multiplied by 0.25. The numbers in box show the root-mean-square radii of corresponding nuclei at specified distance.

Figure 4

(a) The comparison of the Coulomb potential $V_C$, Pauli blocking potential $V_P$, and nuclear potential $V_N$ between M3Y $+$ Pauli potential (solid line) and M3Y $+$ Pauli $+$ DVE potential (dashed line) at Coulomb barrier energy $E=V_b=95.65$ MeV. (b) The comparison between M3Y $+$ Pauli potential (solid line) and M3Y $+$ Pauli $+$ DVE potential (dashed line) at energy of 109.52, 95.65, and 65.85 MeV. The dotted line denotes M3Y $+$ repulsive potential obtained from the sudden model [16].

Figure 5

The experimental fusion cross sections for ${}^{16}\mathrm{O} + {}^{208}\mathrm{Pb}$ [10, 48] fusion system compared with the CC calculations obtained by M3Y $+$ Pauli potential (dashed line) and M3Y $+$ Pauli $+$ DVE potential (dotted line). The solid line denotes the no-coupling results of standard M3Y potential. The insert shows the comparison of $S$ factors between experimental data and the results calculated by adiabatic model (dotted line), sudden model (dashed line), and M3Y $+$ Pauli $+$ DVE approach (solid line).