Fusion of ${}^{\text{40}}\text{Ca}+{}^{\text{40}}\text{Ca}$ and other $\text{Ca}+\text{Ca}$ systems near and below the barrier

The fusion excitation function of ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ has been measured from well above the Coulomb barrier, down to low energies where the cross section is as small as $\simeq$20 $\mu$b, and the astrophysical $S$ factor possibly reaches a maximum vs energy. The results of coupled-channels calculations using the $\mathrm{M}3\mathrm{Y}+\mathrm{repulsion}$ potential are presented. A detailed comparison is made with recently published data on the fusion of ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ and of ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$, including the excitation functions, their low-energy slopes and the barrier distributions. The presence of the fusion hindrance phenomenon in all cases is pointed out, as well as the influence of the strong octupole excitation in ${}^{40}$Ca and of nucleon transfer channels with positive $Q$ values in ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$.

Figure 1

Fusion excitation function of ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ measured in the present work and in the previous experiment by Aljuwair et al. [1]. CC calculations (lines) are discussed in the text.

Figure 2

(a) Barrier distribution derived from the present data for ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ (solid circles). (b) Logarithmic derivative (slope) $L\left(E\right)=\text{d}\left[\text{ln}\right(E\sigma \left)\right]/\text{d}E$ of the excitation function, obtained using finite steps $\Delta E=1$ MeV. The line marked $L_{CS}$ is the slope expected for a constant $S$ factor. Here and in the right panel, the data obtained from the previous measurements [1] are plotted as square symbols. (c) The astrophysical $S$ factor. The data are compared to the CC calculations described in the text, which are based on the Woods-Saxon (WS), the $\mathrm{M}3\mathrm{Y}+\mathrm{repulsion}$ (Sol-1, Sol-2) potentials and the associated no-coupling limit using the $\mathrm{M}3\mathrm{Y}+\mathrm{repulsion}$ potential Sol-1.

Figure 3

Entrance channel potentials for ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$. The ordinate $U\left(r\right)+V_{\text{C}}\left(r\right)$ is the sum of the nuclear and Coulomb potentials, respectively. The two solutions Sol-1 and Sol-2 are based on the $\mathrm{M}3\mathrm{Y}+\mathrm{repulsion}$ potential. The entrance channel potentials for the Woods-Saxon (WS) and the pure M3Y potentials (without repulsion) are also shown. The horizontal line “${}^{80}$Zr” marks the ground-state energy of the compound nucleus ${}^{80}$Zr at 14.2 MeV.

Figure 4

Fusion excitation functions of $\mathrm{Ca}+\mathrm{Ca}$ systems are compared to the CC calculations that provide the best fits to the data. The inset shows the low-energy cross sections, where the energy scale is normalized to the barrier resulting from the WS potential used in this article [23].

Figure 5

Linear plot of the fusion cross sections for the three systems vs energy. Here the errors include the systematic uncertainty on the absolute cross-section scale ($\pm$7$\%$).

Figure 6

The calculated fusion cross sections for ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ obtained in the no-coupling limit (dotted line), using the correct soft ${}^{40}$Ca excitation spectrum (red, full line) and the stiff ${}^{48}$Ca spectrum (green, dash-dotted line), are compared to the data. The measured [11] and calculated [14] fusion cross sections for ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ are also shown (blue, continuous line).

Figure 7

(a) Fusion BDs of ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$, ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$, and ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$. Three arrows are drawn at the energy thresholds of hindrance (see text). (b) Logarithmic derivatives of the three systems. The lines corresponding to $L_{CS}$ are reported for reference. The arrows and the $L_{CS}$ lines are drawn with continuous, dash-dotted, and dotted lines for ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$, ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$, and ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$, respectively.