Indications of a shallow potential in ${}^{48}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ fusion reactions

Fusion data for ${}^{48}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ are analyzed by coupled-channels calculations. Puzzling features of a previous analysis are eliminated by applying a potential that has a shallow pocket in the entrance channel. Thus the observed $S$ factor for fusion, which develops a maximum at low energy, can be reproduced fairly well. The high-energy data can also be accounted for but that requires the use of a weak, short-ranged imaginary potential that absorbs the incoming flux near the location of the minimum of the potential pocket. Predictions of the fusion hindrance in other $\mathrm{Ca}+\mathrm{Zr}$ systems are made and are compared with the systematics that has been developed previously.

Figure 1

The $\mathrm{M}3\mathrm{Y}\hspace{0.5em}+\hspace{0.5em}\mathrm{repulsion}$ potential (red solid curve) is compared to the Woods-Saxon potential (middle dashed curve) and to the pure $\mathrm{M}3\mathrm{Y}\hspace{0.5em}+\hspace{0.5em}$exchange potential (lowest black dashed curve). The diamonds show the potential with large diffuseness that was used in Ref. [6]. The energy of the compound nucleus ${}^{144}\mathrm{Nd}$ and the energy band where the $S$ factor for fusion has a maximum (≈88 MeV) are also indicated.

Figure 2

Measured fusion cross sections for ${}^{48}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ [6, 7] are compared to coupled-channels calculations based on the Woods-Saxon (WS, green dashed curve) and $\mathrm{M}3\mathrm{Y}+\mathrm{repulsion}$ (red solid curve) potentials. The lowest dashed (black) curve is the no-coupling limit for the WS potential.

Figure 3

Barrier distributions obtained from the fusion cross sections shown in Fig. 2 using $\Delta E=2$ MeV (open circles for $\Delta E=1$ MeV). The red solid curve was derived from the $\mathrm{M}3\mathrm{Y}+\mathrm{repulsion}$ calculation show in Fig. 2. The blue dashed curve was obtained from a calculation that includes up to five phonons of the octupole excitation in ${}^{96}\mathrm{Zr}$.

Figure 4

(a) The measured $S$ factors for the fusion of ${}^{48}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ [6, 7] are compared to the two coupled-channels calculations shown in Fig. 2. The value of ${\eta }_0$ was set to ${\eta }_0=74.58$. The blue dashed curve is the prediction of Eq. (8) and the parameters of Table . The logarithmic derivatives, Eq. (5), are shown in panel (b). The top short-dashed curve is the constant $S$ factor limit, Eq. (6); the long-dashed curve is the no-coupling limit for the WS potential.

Figure 5

Linear plot of the fusion cross sections for ${}^{48}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ [6]. The upper dashed curve is based on the WS potential with diffuseness $a=0.677$ fm and the structure input from Ref. [6]. The next (blue) dashed curve is based on the same potential but stucture input is from Table . The red solid and the lower dot-dashed curves are the coupled-channels calculations, based on the $\mathrm{M}3\mathrm{Y}+\mathrm{repulsion}$ potential and the structure input of Table , with and without the effect of the short-range imaginary potential $\mathit{iW}(r)$, respectively.

Figure 6

Logarithmic derivative (a) and $S$ factors (b) for the fusion of ${}^{40}\mathrm{Ca}+{}^{90,94,96}\mathrm{Zr}$ [14, 15] and ${}^{48}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ [6]. The curves in panel (a) are fits to the measured $L(E)$ using the parametrization Eq. (7). The upper, nearly horizontal, dashed lines show the constant $S$ factor limit, Eq. (6). The curves in panel (b) are the $S$ factors predicted by Eq. (8) and the parameters in Table . The parameters $E_0$ and ${\eta }_0$ used in the plot are $E_0=0,1,3,0$ MeV and ${\eta }_0=68.59,70.87,71.71,73.98$ for the systems ${}^{40}\mathrm{Ca}+{}^{90,94,96}\mathrm{Zr}$ and ${}^{48}\mathrm{Ca}+{}^{90}\mathrm{Zr}$, respectively.

Figure 7

The energy $E_s$ for $\mathrm{Ca}+\mathrm{Zr}$ and $\mathrm{Ni}+\mathrm{Ni}$ systems. The solid curve is the reference value, Eq. (9). The open symbols are extrapolated values. The solid symbols are for systems where a maximum in the $S$ factor has been observed.