Coupled-channels analyses for large-angle quasi-elastic scattering in massive systems

We discuss in detail the coupled-channels approach for large-angle quasi-elastic scattering in massive systems, where many degrees of freedom may be involved in the reaction. We especially investigate the effects of single-, double-, and triple-phonon excitations on the quasi-elastic scattering for ${}^{48}\mathrm{Ti}$, ${}^{54}\mathrm{Cr}$, ${}^{56}\mathrm{Fe}$, ${}^{64}\mathrm{Ni}$, and ${}^{70}\mathrm{Zn}+{}^{208}\mathrm{Pb}$ systems, for which the experimental cross sections have been measured recently. We show that the present coupled-channels calculations well account for the overall width of the experimental barrier distribution for these systems. In particular, it is shown that the calculations taking into account single-quadrupole phonon excitation in ${}^{48}\mathrm{Ti}$ and triple-octupole phonon excitations in ${}^{208}\mathrm{Pb}$ reasonably well reproduce the experimental quasi-elastic cross section and barrier distribution for the ${}^{48}\mathrm{Ti}+{}^{208}\mathrm{Pb}$ reaction. However, ${}^{54}\mathrm{Cr}$, ${}^{56}\mathrm{Fe}$, ${}^{64}\mathrm{Ni}$, and ${}^{70}\mathrm{Zn}+{}^{208}\mathrm{Pb}$ systems seem to require the double-quadrupole phonon excitations in the projectiles to reproduce the experimental data.

Figure 1

Effects of multiphonon excitations on the quasi-elastic scattering cross section (upper panel) and on the quasi-elastic barrier distribution (lower panel) for the ${}^{48}\mathrm{Ti}+{}^{208}\mathrm{Pb}$ system. The dotted line is the result of the coupled-channels calculations, including coupling to the one-quadrupole phonon state in the projectile and the one-octupole phonon states in the target nucleus, whereas the solid line is obtained by including the coupling in addition to the two-octupole phonon state in the target nucleus. The dashed line is the result of double-quadrupole phonon couplings in the projectile and the double-octupole phonon couplings in the target nucleus. The experimental data are taken from Ref. [18].

Figure 2

The quasi-elastic scattering cross sections [(a), (c), (e), and (g)] and the quasi-elastic barrier distributions [(b), (d), (f), and (h)] for the ${}^{54}\mathrm{Cr}$, ${}^{56}\mathrm{Fe}$, ${}^{64}\mathrm{Ni}$, and ${}^{70}\mathrm{Zn}+{}^{208}\mathrm{Pb}$ systems obtained with two coupling schemes as indicated in the insets. The dashed line is obtained by including the one-quadrupole phonon state in the projectile nuclei, whereas the solid line is obtained with the double-phonon couplings. The double-octupole phonon excitations in the target nucleus is included in all the calculations. The experimental data are taken from Ref. [18].

Figure 3

Effects of triple-phonon excitations on the quasi-elastic scattering cross section (upper panel) and on the quasi-elastic barrier distribution (lower panel) for the ${}^{48}\mathrm{Ti}+{}^{208}\mathrm{Pb}$ system. The dashed line is the result of the coupled-channels calculations taking into account the coupling to the one-phonon state in the projectile and the two-phonon states in the target nuclei. The solid line is obtained by including the coupling to the one-phonon state in the projectile and the three-phonon states in the target. The experimental data are taken from Ref. [18].

Figure 4

Effects of triple-phonon excitations on the quasi-elastic cross sections [(a), (c), (e), and (g)] and on the quasi-elastic barrier distributions [(b), (d), (f), and (h)] for the ${}^{54}\mathrm{Cr}$, ${}^{56}\mathrm{Fe}$, ${}^{64}\mathrm{Ni}$, and ${}^{70}\mathrm{Zn}+{}^{208}\mathrm{Pb}$ systems. The dashed line is the same as the solid line in Fig. 2, whereas the dotted line is the results of the calculations taking the coupling to the triple-octupole phonon in the target and the one-quadrupole phonon state in the projectile nucleus into account. The solid line is obtained by including the coupling to the double-quadrupole phonon states in the projectile and the triple-otcupole phonon states in the target nucleus. The experimental data are taken from Ref. [18].

Figure 5

Effect of anharmonic octupole phonon excitations in ${}^{208}\mathrm{Pb}$ on (a) the quasi-elastic cross section and (b) the quasi-elastic barrier distribution for the ${}^{70}\mathrm{Zn}+{}^{208}\mathrm{Pb}$ reaction. The solid line is the coupled-channels calculations obtained by reducing the coupling strengths to multiphonon states, whereas the dashed line denotes the results in the harmonic limit. The experimental data are taken from Ref. [18].

Figure 6

Comparison of the experimental data with the coupled-channels calculations obtained using different values of the surface diffuseness of the nuclear potential for ${}^{64}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ reaction for (a) the quasi-elastic cross section and (b) the quasi-elastic barrier distribution. The solid and the dashed lines are obtained using the surface diffuseness of the nuclear potential $a=0.63$ fm and $a=1.0$ fm, respectively. The dotted line is the results obtained by shifting the barrier height by around $+2.40$ MeV for the calculations using $a=1.0$ fm. Experimental data is taken from Ref. [18].

Figure 7

Comparison of the calculated quasi-elastic (solid line) with fusion (dashed line) barrier distributions for the ${}^{48}\mathrm{Ti}+{}^{208}\mathrm{Pb}$ reaction. The quasi-elastic barrier distribution is plotted as a function of effective energy $E_{\mathrm{eff}}$ defined by Eq. (11) with $\theta ={170}^{°}$, whereas the fusion barrier distribution as a function of the center-of-mass (c.m.) energy $E$. They are normalized to unit area in the energy interval between $E=170$ and 210 MeV. The single-quadrupole phonon excitation in the projectile and the triple-octupole phonon excitation in the target are taken into account.

Figure 8

Same as described in the caption to Fig. 7, but for (a) ${}^{54}\mathrm{Cr}+{}^{208}\mathrm{Pb}$, (b) ${}^{56}\mathrm{Fe}+{}^{208}\mathrm{Pb}$, (c) ${}^{64}\mathrm{Ni}+{}^{208}\mathrm{Pb}$, and (c) ${}^{70}\mathrm{Zn}+{}^{208}\mathrm{Pb}$ systems. The double-quadrupole phonon excitation in the projectile as well as the triple-octupole phonon excitation in the target are taken into account.