Hindrance in the fusion of ${}^{48}\mathrm{Ca}$$+$${}^{48}\mathrm{Ca}$

The coupled-channels technique is applied to analyze recent fusion data for ${}^{48}\mathrm{Ca}$$+$${}^{48}\mathrm{Ca}$. The calculations include the excitations of the low-lying $2^{+}$, $3^{-}$, and $5^{-}$ states in projectile and target, and the influence of mutual excitations as well as the two-phonon quadrupole excitations is also investigated. The ion-ion potential is obtained by double-folding the nuclear densities of the reacting nuclei with the M3Y$+$repulsion effective interaction but a standard Woods-Saxon potential is also applied. The data exhibit a strong hindrance at low energy compared to calculations that are based on a standard Woods-Saxon potential but they can be reproduced quite well by applying the M3Y$+$repulsion potential with an adjusted radius of the nuclear density. The influence of the polarization of high-lying states on the extracted radius is discussed.

Figure 1

Entrance channel potentials for ${}^{48}\mathrm{Ca}$$+$${}^{48}\mathrm{Ca}$ obtained from the Woods-Saxon (WS, with $R_0=8.562$ fm) and the pure M3Y (M3Y-1) potentials. The two upper curves are the shallow M3Y$+$repulsion potentials determined in Sec. 5. The energy of the ground state of the compound nucleus ${}^{96}\mathrm{Zr}$ is also indicated.

Figure 2

Fusion cross sections for ${}^{48}\mathrm{Ca}$$+$${}^{48}\mathrm{Ca}$. The error bars reflect the statistical uncertainties. The curves are coupled-channels calculations with 24 channels that use Woods-Saxon (WS) potentials with two different radii.

Figure 3

Ratios of the measured and calculated cross sections shown in Fig. 2. The error bars on the open diamonds were determined by the statistical uncertainties and a 7$\%$ systematic error.

Figure 4

Results of the ${\chi }^2$ analysis of the ${}^{48}\mathrm{Ca}$$+$${}^{48}\mathrm{Ca}$ fusion data [5]. ${\chi }^2/N$, minimized with respect to the radius $R$, is shown as a function of the diffuseness parameter $a_r$. The solid curve (24 ch) is for calculations that include 24 channels; the dashed curve (9 ch) is for 9 channels.

Figure 5

Fusion cross sections for ${}^{48}\mathrm{Ca}$$+$${}^{48}\mathrm{Ca}$. The upper two curves are coupled-channels calculations with 24 channels that are based on the Woods-Saxon (WS) and the M3Y$+$repulsion (M3Y$+$rep-1 and M3Y$+$rep-2) potentials discussed in the text. The lowest dashed curve is the no-coupling limit based on the same M3Y$+$repulsion potential.

Figure 6

$S$ factors for the fusion cross sections shown in Fig. 5. Also shown is the $S$ factor obtained in calculations with the M3Y$+$rep-1 potential. All calculations include 24 channels, except the no-coupling limit, which has only 1 channel.

Figure 7

The logarithmic derivative of the energy-weighted cross sections shown in Fig. 5. Also shown is the result of coupled-channels calculations based on the M3Y$+$rep-1 potential. The top curve is the constant $S$ factor limit, $L_{\mathit{CS}}(E_{\mathrm{c}.\mathrm{m}.})$.