Examination of the different roles of neutron transfer in the sub-barrier fusion reactions ${}^{32}\mathrm{S}+{}^{94,96}\mathrm{Zr}$ and ${}^{40}\mathrm{Ca}+{}^{94,96}\mathrm{Zr}$

The sub-barrier capture (fusion) reactions ${}^{32}\mathrm{S}+{}^{90,94,96}\mathrm{Zr}$, ${}^{36}\mathrm{S}+{}^{90,96}\mathrm{Zr}$, ${}^{40}\mathrm{Ca}+{}^{90,94,96}\mathrm{Zr}$, and ${}^{48}\mathrm{Ca}+{}^{90,96}\mathrm{Zr}$ with positive and negative $Q$ values for neutron transfer are studied with the quantum diffusion approach and the universal fusion function representation. For these systems, the $s$-wave capture probabilities are extracted from the experimental excitation functions and are also analyzed. Different effects of the positive $Q_{xn}$-value neutron transfer in the fusion enhancement are revealed in the relatively close reactions ${}^{32}\mathrm{S}+{}^{94,96}\mathrm{Zr}$ and ${}^{40}\mathrm{Ca}+{}^{94,96}\mathrm{Zr}$.

Figure 1

The experimental reduced fusion excitation functions $\frac{2E_{\mathrm{c}.\mathrm{m}.}}{\hslash {\omega }_b{R}_b^2}{\sigma }_{\mathrm{cap}}\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ (symbols) [5, 6, 7, 18, 20, 34, 35, 36] and the universal fusion function $F_0=ln\{1+exp[2\pi (E_{\mathrm{c}.\mathrm{m}.}-V_b)/\left(\hslash {\omega }_b\right)]\}$ (solid line) vs $\frac{E_{\mathrm{c}.\mathrm{m}.}-V_b}{\hslash {\omega }_b}$ for the reactions indicated.

Figure 2

The extracted $s$-wave capture probabilities for the reactions indicated by employing Eq. (3) (symbols connected by lines). The experimental capture (fusion) excitation functions used are from Refs. [5, 6, 7, 18, 20].

Figure 3

The extracted $s$-wave capture probabilities for the reactions indicated by employing Eq. (3) (symbols connected by lines). The experimental capture (fusion) excitation functions used are from Refs. [5, 18, 34, 35].

Figure 4

The calculated capture cross sections vs $E_{\mathrm{c}.\mathrm{m}.}$ for the reactions (a) ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ (solid line), ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$ (dotted line), and ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ (dashed line) and (b) ${}^{48}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ (solid line) and ${}^{48}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ (dashed line). The error bars are shown if they are larger than the size of the symbols. The experimental data (symbols) are from Refs. [5, 6, 7, 34].

Figure 5

The calculated capture cross sections vs $E_{\mathrm{c}.\mathrm{m}.}$ for the reactions (b) ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ (solid line), ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$ (dotted line), and ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ (dashed line) and (a) ${}^{32}\mathrm{S}+{}^{96}\mathrm{Zr}$ (solid line), ${}^{32}\mathrm{S}+{}^{94}\mathrm{Zr}$ (dotted line), and ${}^{32}\mathrm{S}+{}^{90}\mathrm{Zr}$ (dashed line). For the reactions ${}^{32}\mathrm{S}$,${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ and ${}^{32}\mathrm{S}$,${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$, the capture cross sections calculated without taking into consideration the neutron transfer are shown by dash-dot-dotted and dash-dotted [it is matched with the solid line in part (a)] lines, respectively.

Figure 6

The calculated capture cross sections vs $E_{\mathrm{c}.\mathrm{m}.}$ for the reactions (a) ${}^{32}\mathrm{S}+{}^{96}\mathrm{Zr}$ (solid line), ${}^{32}\mathrm{S}+{}^{94}\mathrm{Zr}$ (dotted line), and ${}^{32}\mathrm{S}+{}^{90}\mathrm{Zr}$ (dashed line) and (b) ${}^{36}\mathrm{S}+{}^{96}\mathrm{Zr}$ (solid line) and ${}^{36}\mathrm{S}+{}^{90}\mathrm{Zr}$ (dashed line). The error bars are shown if they are larger than the size of the symbols. The experimental data (symbols) are from Refs. [18, 20, 35].

Figure 7

The calculated (lines) and experimental (symbols) fusion barrier distributions $D$ [Eq. (5)] for the ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ reaction. The solid (dashed) (the energy increment is 1.75 MeV) and dotted (dash-dotted) (the energy increment is 4.9 MeV) lines show the values of $D$ calculated with (without) taking neutron transfer into consideration. The solid (the energy increment is 4.9 MeV) and open (the energy increment is 1.75 MeV) circles are extracted from the experimental excitation function [5].