Coupled-channels description of multinucleon transfer and fusion reactions at energies near and far below the Coulomb barrier

We investigate heavy-ion multinucleon transfer reactions using the coupled-channels formalism. We first use the semiclassical approximation and show that a direct coupling between the entrance and the pair transfer channels improves a fit to the experimental one- and two-neutron transfer cross sections for the ${}^{40}\text{Ca}+{}^{96}\text{Zr}$ and ${}^{60}\text{Ni}+{}^{116}\text{Sn}$ systems. We then discuss the validity of the perturbative approach and highlight the effect of high-order terms. The effect of absorption is also investigated for energies around the Coulomb barrier. Finally, we use a quantal coupled-channels approach to achieve a simultaneous description of the fusion cross sections and the transfer probabilities for the ${}^{40}\text{Ca}+{}^{96}\text{Zr}$ reaction. We find a significant effect of the couplings to the collective excited states on the transfer probabilities around the Coulomb barrier.

Figure 1

The ground-state energies for the one-neutron $\left(1n\right)$ and the two-neutron $\left(2n\right)$ transfer processes for the ${}^{40}\text{Ca}+{}^{96}\text{Zr}$ and ${}^{60}\text{Ni}+{}^{116}\text{Sn}$ systems. The dashed lines denote the optimum $Q$ value for each transfer channel.

Figure 2

The transfer probabilities for the ${}^{40}\text{Ca}+{}^{96}\text{Zr}$ (the upper panel) and ${}^{60}\text{Ni}+{}^{116}\text{Sn}$ (the lower panel) reactions as a function of the distance of the closest approach, $D$, for the Rutherford trajectory. The experimental data, taken form Refs. [23, 24], for the one-neutron (the crosses) and the two-neutron (the squares) transfers are compared to transfer probabilities for the ground state to the ground-state transitions obtained with the pure sequential scheme (the thin solid line for the $1n$ transfer and the dotted line for the $2n$ transfer) and with the direct transfer scheme (the thick solid line for the $1n$ transfer and the dashed line for the $2n$ transfer).

Figure 3

Same as Fig. 2, but with the optimum $Q$ values (see the text).

Figure 4

The one-neutron transfer probability as a function of the distance of the closest approach for the ${}^{40}\text{Ca}+{}^{96}\text{Zr}$ reaction. The sequential two-neutron scheme with the optimum $Q$ value is employed. The dashed line, the triangles, and the dotted line are the results of the time-dependent perturbation theory with the first, the second, and the third orders, respectively. Those results are compared to the exact solution (the solid line) and to the experimental data (the crosses). The coupling parameters used here are $a_1=1.55$ fm, ${\beta }_1=26.4$ MeV fm, $a_2=1.286$ fm, and ${\beta }_2=150.4$ MeV fm.

Figure 5

The two-neutron transfer probability for the ${}^{40}\text{Ca}+{}^{96}\text{Zr}$ system as a function of the distance of the closest approach, $D$. The direct two-neutron transfer scheme with the optimum $Q$ values is employed. The results of the time-dependent perturbation theory up to the first and the second orders are shown by the dotted and the solid lines, respectively. The result of the second-order calculation neglecting the direct two-neutron transfer couplings is also shown by the dashed line (see text). The experimental data are taken from Ref. [23].

Figure 6

The transfer probabilities as a function of the distance of the closest approach, $D$, calculated by including the effect of absorption of the classical trajectories with the imaginary potential. The direct two-neutron transfer scheme with the optimum $Q$ value is employed. The solid lines denote the results for the one-neutron transfer process, while the dashed lines are for the two-neutron transfer process. The experimental data are taken from Refs. [23, 24].

Figure 7

The $s$-wave survival probability for each channel as a function of the center of mass energy. It is obtained by solving the time-dependent Schrödinger equation with a single channel including the imaginary potential. The crosses and the squares show a fit with the complementary error function given by Eq. (8).

Figure 8

A comparison between the transfer probabilities obtained with the semiclassical approximation (the crosses and the circles) and those with the quantal coupled-channels calculations (the solid and the dashed lines) for the ${}^{40}\text{Ca}+{}^{96}\text{Zr}$ and ${}^{60}\text{Ni}+{}^{116}\text{Sn}$ systems.

Figure 9

The one-neutron transfer (the upper panel) and the two-neutron transfer (the lower panel) probabilities as a function of the distance of the closest approach for the ${}^{60}\text{Ni}+{}^{116}\text{Sn}$ reaction. The solid and the dotted lines are obtained with the nuclear potential, which yields the barrier heights of $B=166$ and 179.9 MeV, respectively.

Figure 10

The three eigenbarriers obtained by diagonalizing the intrinsic Hamiltonian matrix, Eq. (3), at each internuclear separation, $r$.

Figure 11

(a) The fusion cross sections for the ${}^{40}\text{Ca}+{}^{96}\text{Zr}$ reaction. The dashed line shows the results of the coupled-channels calculations including only the collective excitations in the colliding nuclei, while the solid line shows those with both the inelastic and the multineutron transfer channels. (b) The corresponding fusion barrier distribution. The experimental data are taken from Refs. [7, 8]. (c) The probabilities for the multineutron transfer reactions obtained with (the solid lines) and without (the dashed lines) taking into account the inelastic excitations. The experimental data are taken from Ref. [23].