New results in low-energy fusion of ${}^{40}\mathbf{Ca}+{}^{90,92}\mathbf{Zr}$

Background: Near- and sub-barrier fusion of various Ca + Zr isotopic combinations have been widely investigated. A recent analysis of ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ data has highlighted the importance of couplings to multiphonon excitations and to both neutron and proton transfer channels. Analogous studies of ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ tend to exclude any role of transfer couplings. However, the lowest measured cross section for this system is rather high ($840\mu \mathrm{b}$). A rather complete data set is available for ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$, while no measurement of ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ fusion has been performed in the past.

Purpose: Our aim is to measure the full excitation function of ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ near the barrier and to extend downward the existing data on ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$, in order to estimate the transfer couplings that should be used in coupled-channels calculations of the fusion of these two systems and of ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$.

Methods: ${}^{40}\mathrm{Ca}$ beams from the XTU Tandem accelerator of INFN–Laboratori Nazionali di Legnaro were used, bombarding thin metallic ${}^{90}\mathrm{Zr}$ ($50\mu \mathrm{g}/{\mathrm{cm}}^2$) and ${{}^{92}\mathrm{ZrO}}_2$ targets (same thickness) enriched to $99.36\%$ and $98.06\%$ in masses 90 and 92, respectively. An electrostatic beam deflector allowed the detection of fusion evaporation residues (ER) at very forward angles, and angular distributions of ER were measured.

Results: The excitation function of ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ has been measured down to the level of $\simeq 60\mu \mathrm{b}$. Coupled-channels (CC) calculations using a standard Woods-Saxon (WS) potential and following the line of a previous analysis of ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ fusion data give a good account of the new data, as well as of the existing data for ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$. The previous excitation function of ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ has been extended down to $40\mu \mathrm{b}$.

Conclusions: Transfer couplings play an important role in explaining the fusion data for ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ and ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$. The strength of the pair-transfer coupling is deduced by applying a simple recipe based on the value obtained for ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$. The logarithmic slopes and the $S$ factors for fusion are reproduced fairly well for all three systems by the CC calculations, and there are no indications of a fusion hindrance at the lowest energies. In contrast, the new data for ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ indicate the onset of a fusion hindrance at the lowest energies.

Figure 1

Excitation functions of ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ (a) and ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ (b) are compared to the CC calculations described in the text. The solid diamonds are the new data, whereas the open circles in (a) are the ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ data by Timmers et al. [5].

Figure 2

Measured cross sections for the fusion of ${}^{40}\mathrm{Ca}+{}^{90,96}\mathrm{Zr}$ [4, 5], ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$ [11], and the new data for ${}^{40}\mathrm{Ca}+{}^{90,92}\mathrm{Zr}$ (solid diamonds) are compared to CC calculations in a logarithmic (a) and a linear (b) plot, respectively. The parameters of the calculations are indicated with a (*) in Table 3.

Figure 3

Logarithmic slopes of the excitation function of ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ (a) and ${}^{40}\mathrm{Ca}+{}^{92}\mathrm{Zr}$ (b) are compared to CC calculations. The solid (red) points in (a) are the results of the new measurements for ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$.

Figure 4

The logarithmic slopes (a) and the astrophysical $S$ factors (b) for the fusion of the four ${}^{40}\mathrm{Ca}+{}^A\mathrm{Zr}$ systems ($A=90$, 92, 94, and 96) are compared to the results of CC calculations. The results were obtained from the measured and calculated cross sections shown in Fig. 2.