Semimicroscopic modeling of heavy-ion fusion reactions with multireference covariant density functional theory

We describe low-lying collective excitations of atomic nuclei with the multireference covariant density functional theory and combine them with coupled-channels calculations for heavy-ion fusion reactions at energies around the Coulomb barrier. To this end, we use the calculated transition strengths among several collective states as inputs to the coupled-channels calculations. This approach provides a natural way to describe anharmonic multiphonon excitations, as well as a deviation of rotational excitations from a simple rigid rotor. We apply this method to subbarrier fusion reactions of ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni},{}^{58}\mathrm{Ni}+{}^{60}\mathrm{Ni}$, and ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ systems. We find that the effect of anharmonicity tends to smear the fusion barrier distributions, better reproducing the experimental data compared to the calculations in the harmonic oscillator limit.

Figure 1

(a) The total energy for the angular-momentum-projected states with $J=0,2,4$, and 6 for ${}^{58}\mathrm{Ni}$ as a function of the intrinsic mass quadrupole deformation $\beta$ of the mean-field states. The particle-number projection has also been implemented. Those curves are obtained with the projected CDFT method with PC-PK1 interaction [37]. The low-lying collective levels, obtained with the configuration mixing calculation, are also plotted at their average deformation $\overline{\beta }$. (b) The collective wave functions given by Eq. (23) for the states indicated in the figure as a function of deformation parameter $\beta$.

Figure 2

Comparison of the experimental and the calculated low-lying energy spectra of ${}^{58}\mathrm{Ni}$. The experimental data are taken from Refs. [38, 39], while the calculated spectrum is obtained with the PC-PK1 force. The $E2$ transition strengths are given in units of $e^2{\mathrm{fm}}^4$.

Figure 3

Same as Fig. 2, but for ${}^{60}\mathrm{Ni}$.

Figure 4

The fusion cross sections (a) and the fusion barrier distributions (b) for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ system. Here the fusion barrier distribution is defined as $D_{\mathrm{fus}}\left(E\right)=d^2\left(E{\sigma }_{\mathrm{fus}}\right)/d{E}^2$. The dashed line is the result of the coupled-channels calculations, including the double quadrupole-phonon excitations in each ${}^{58}\mathrm{Ni}$ nucleus in the harmonic oscillator limit, while the solid line is obtained by including the $0_1^{+},2_1^{+},0_3^{+},2_2^{+}$, and $4_1^{+}$ states with the coupling strengths shown in Table 1. The dotted line denotes the result in the absence of the channel couplings. The experimental data are taken from Ref. [44] for the fusion cross sections and from Ref. [6] for the fusion barrier distribution.

Figure 5

Comparison of the fusion barrier distribution for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ reaction obtained with several methods. (a) Results obtained with the calculated transition strengths and the excitation energies for the two-phonon couplings in ${}^{58}\mathrm{Ni}$. The solid and dashed lines show the results with the PC-PK1 and PC-F1 interactions, respectively. Panel (b) is obtained by using the experimental values for the excitation energies and also by multiplying a constant factor to the transition strengths so that the $B\left(E2\right)$ value from the first $2^{+}$ state to the ground state coincides with the experimental value.

Figure 6

The fusion barrier distributions for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ reaction obtained with several coupling schemes. In all the panels, the solid line denotes the result of the MR-CDFT calculation shown in Fig. 4. In (a), the dashed line is obtained by setting the energy of the two-phonon triplet to be exactly twice the energy of the first $2^{+}$ state. In (b), the dashed line is obtained by setting all the diagonal couplings to be zero, while in (c) it is obtained by setting the coupling strength to be zero for the transition from the second $2^{+}$ state to the third $0^{+}$ state.

Figure 7

The fusion cross sections (a) and the fusion barrier distributions (b) for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ system obtained with the MR-CDFT method. The solid line is the same as that in Fig. 4, while the dashed line is obtained by inverting the sign of the quadrupole moment of the first $2^{+}$ state. The experimental data are taken from Refs. [6, 44].

Figure 8

A comparison of the coupled-channels calculations for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ system with (the dashed line) and without (the solid line) the couplings to the second $0^{+}$ state. Panels (a) and (b) show the fusion cross sections and the fusion barrier distributions, respectively. The experimental data are taken from Refs. [6, 44].

Figure 9

Same as Fig. 4, but for the ${}^{58}\mathrm{Ni}+{}^{60}\mathrm{Ni}$ system. The experimental data are taken from Ref. [6].

Figure 10

Same as Fig. 4, but for the ${}^{40}\mathrm{Ca}+{}^{58}\mathrm{Ni}$ system. The experimental data are taken from Ref. [48].