Structure of ${}^{72,74}\mathrm{Ge}$ nuclei probed with a combined analysis of heavy-ion fusion reactions and Coulomb excitation

We study the fusion reactions of the ${}^{72}\mathrm{Ge}+{}^{72}\mathrm{Ge}$ and ${}^{74}\mathrm{Ge}+{}^{74}\mathrm{Ge}$ systems in terms of the full order coupled-channels formalism. We obtain the fusion excitation function as well as the fusion barrier distribution of these reactions using the coupling matrix suggested from the recent Coulomb excitation experiments. We compare those results with the results obtained by the coupling matrix based on the pure vibrational and rotational models. The barrier distribution for the ${}^{74}\mathrm{Ge}+{}^{74}\mathrm{Ge}$ reaction obtained with the experimental coupling matrix is in good agreement with that obtained with the vibrational model, in contrast to the rotational model. On the other hand, the calculations for ${}^{72}\mathrm{Ge}+{}^{72}\mathrm{Ge}$ system show that the fusion barrier distribution obtained with the experimental coupling matrix significantly differs from those obtained with vibrational and rotational models. Our study indicates that the shape of ${}^{74}\mathrm{Ge}$ is closer to spherical than to deformed, while ${}^{72}\mathrm{Ge}$ has a shape admixture in its ground state, which can be described by neither the vibrational nor the rotational models, as suggested by the Coulomb excitation experiments. This finding will resolve the debates concerning the structure of these nuclei.

Figure 1

Comparison of (a) the excitation function of the fusion cross section and (b) the fusion barrier distribution for the ${}^{74}\mathrm{Ge}+{}^{74}\mathrm{Ge}$ reactions obtained with three different two-level calculations. The solid line is obtained with the experimental coupling matrix (“TS” stands for “transitional structure”), while the dotted and the dashed lines are with the pure vibrational and rotational models, respectively. The dot-dashed line is for the one dimensional calculation. Experimental data are taken from Ref. [29].

Figure 2

Comparison of (a) the excitation function of the fusion cross section and (b) the fusion barrier distribution for the ${}^{74}\mathrm{Ge}+{}^{74}\mathrm{Ge}$ reactions calculated in different models. The solid line is obtained by using the full five-dimensional experimental coupling matrix given in Eq. (4). The dotted line is obtained for the pure vibrational model including up to the double phonon states, while the dashed line for the rotational model including the coupling up to $4^{+}$ member of the ground state rotational band. Experimental data are taken from Ref. [29].

Figure 3

The same as Fig. 2 but when the coupling up to the triple quadrupole phonon states in the vibrational model and the coupling up to the $6^{+}$ member of the ground state rotational band in the rotational coupling model are taken into account. Experimental data are taken from Ref. [29].

Figure 4

Effect of octupole excitation on (a) the fusion excitation function and (b) the fusion barrier distribution for the reaction between two ${}^{74}\mathrm{Ge}$ nuclei. The solid line is the same as Fig. 2. The result of calculations which include the octupole vibration together with the experimental coupling of Eq. (4) is denoted by the dashed line and that with the pure double quadrupole couplings by the dashed line. The dotted-dashed line is the same as in Fig. 1. Experimental data are taken from Ref. [29].

Figure 5

Comparison of (a) the fusion cross section and (b) the fusion barrier distribution obtained with different models for ${}^{72}\mathrm{Ge}+{}^{72}\mathrm{Ge}$ reactions. The dotted and dashed lines are obtained by the two-channel calculations with the vibrational and rotational models, respectively. The solid line is obtained by assuming the coupling matrix given in Eq. (11) (“SA” stands for “shape admixture”). The dot-dashed line is the result of one dimensional calculation.

Figure 6

The same as Fig. 5, but when the vibrational model is extended to include up to two phonon states, and the rotational coupling model up to the $4^{+}$ member of the ground states rotational band. The solid line is the same as that in Fig. 5.

Figure 7

The same as Fig. 6, but when the coupling up to the triple quadrupole phonon states for the vibrational model and the coupling up to the 6${}^{+}$ member of the ground state rotational band for the rotational coupling model are taken into account.