Effects of transfer channels on near- and sub-barrier fusion of ${}^{\mathbf{32}}$S $+$ ${}^{\mathbf{48}}$Ca

The fusion excitation function of ${}^{32}$S + ${}^{48}$Ca has been experimentally studied in a wide energy range, from above the Coulomb barrier down to cross sections in the sub-$\mu$b region. The measurements were done at INFN–Laboratori Nazionali di Legnaro, using the ${}^{32}$S beam from the XTU Tandem accelerator. The excitation function has a smooth behavior below the barrier, and no evident hindrance character shows up in the measured energy region. The fusion barrier distribution has a peculiar shape with two distinct peaks of similar height, lower and higher than the Akyüz-Winther barrier. Coupled-channels calculations using the M3Y + repulsion potential are presented for this system and for ${}^{36}$S + ${}^{48}$Ca. The results of these calculations give a good account of the data, and indicate the influence of one- and two-nucleon transfer channels with positive $Q$ values, which are only open for ${}^{32}$S + ${}^{48}$Ca.

Figure 1

Angular distributions measured at two different energies near the Coulomb barrier together with Gaussian fits. Only statistical errors are reported, not exceeding the symbol size in most cases.

Figure 2

Fusion excitation function (full symbols, left scale) and extracted barrier distribution (open symbols, right scale) for ${}^{32}$S + ${}^{48}$Ca. Plotted errors are purely statistical.

Figure 3

Logarithmic slope and the $L_{\mathrm{CS}}$ limit (solid line); see text.

Figure 4

Top panel (a): The M3Y+repulsion entrance channel potentials (Sol-1, Sol-2) for the two stable solutions to the fusion of ${}^{36}$S + ${}^{48}$Ca are compared to the Woods-Saxon (WS) potential described in the text, and to the pure M3Y+exchange potential. Bottom panel (b): The corresponding potentials used for ${}^{32}$S + ${}^{48}$Ca. The ground-state energies of the two compound nuclei ${}^{84,80}$Kr are indicated by the solid horizontal lines.

Figure 5

In the top panel (a) fusion cross sections for ${}^{36}$S + ${}^{48}$Ca [12] are compared to CC calculations (Ch-17) based on a WS potential, and on the M3Y+repulsion potential Sol-2. The no-coupling limit Ch-1 is also shown. The lower panel (b) reports the ratio of the measured and calculated cross sections, using the two potentials. The energy of the Coulomb barrier is indicated.

Figure 6

Fusion cross sections for ${}^{32}$S + ${}^{48}$Ca are compared to CC calculations based on the M3Y+repulsion potential. The calculation Ch-14 includes the surface excitations, Ch-28 takes into account, additionally, the one-proton transfer, and Ch-42 includes couplings to the one-proton and pair-transfer. Ch-1 is the no-coupling limit, and the result of the Ch-42 calculation using the WS potential is also shown. The predicted cross sections for one-proton (1P) and two-nucleon transfer (2N) are reported. The lower panel (b) shows the ratio of the measured and calculated cross sections.

Figure 7

Logarithmic (a) and linear (b) plots of the fusion cross sections for ${}^{36}$S + ${}^{48}$Ca [12] (open symbols) and ${}^{32}$S + ${}^{48}$Ca (full symbols), compared to each other and to the full CC calculations. The calculation for ${}^{32}$S + ${}^{48}$Ca without transfer coupling (“no transf”) is also shown.

Figure 8

Logarithmic derivatives $L\left(E\right)$ of the (energy-weighted) cross sections shown in Fig. 7. The constant $S$ factor limit $L_{\mathrm{CS}}$ is also indicated.

Figure 9

First derivative of the energy-weighted fusion cross sections shown in Fig. 7.

Figure 10

Barrier distributions $B\left(E\right)$ obtained as the second derivative of the energy-weighted cross sections shown in Fig. 7. The experimental distributions were obtained with the $\Delta E$ shown in the figures; the calculated distributions were obtained with $\Delta E$ = 1.5 MeV.