Fusion cross sections and barrier distributions for ${}^{16}\mathrm{O}+{}^{70,72,73,74,76}\mathrm{Ge}$ and ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ reactions at energies near and below the Coulomb barrier

In this paper, we have investigated the sensitivity of fusion cross section and the barrier distribution data with respect to the choice of the potential parameters. To ascertain this, the fusion dynamics of ${}^{16}\mathrm{O}+{}^{70,72,73,74,76}\mathrm{Ge}$ and ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ reactions at near and sub-barrier energies have been examined. In the present study, we have used the one-dimensional Wong formula, symmetric-asymmetric Gaussian barrier distribution (SAGBD) model, and coupled channel approach. For theoretical estimations, the Woods-Saxon form of nuclear potential has been adopted, and it has been found that both the fusion cross section and barrier distribution are quite sensitive to the choice of potential parameters in the entire range of incident energies. Calculations based on the Wong formula are significantly smaller than the sub-barrier fusion data, which has clearly suggested that the couplings to the nuclear structure's degrees of freedom of the colliding nuclei are needed to reproduce the experimental data. On the other hand, in the SAGBD model calculations the multidimensional character of nucleus-nucleus potential has been incorporated and it has reasonably recovered both the fusion cross-section data and the barrier distribution data of ${}^{16}\mathrm{O}+{}^{70,72,73,74,76}\mathrm{Ge}$ and ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ reactions. In the SAGBD model, channel coupling parameter $\lambda$ has been determined from barrier distribution and hence $\lambda$ is related to the various channel coupling effects which have arisen because of the nuclear structure of the participating nuclei. The increase in $\lambda$ with an increase in target isotopic mass has clearly indicated that there are shape transition effects in Ge isotopes, which change their shape from spherical symmetry to prolate deformed shape as one moves from ${}^{70}\mathrm{Ge}$ to ${}^{76}\mathrm{Ge}$. Similarly, the percentage reduction of effective fusion barrier $V_{\mathit{CBRED}}$ with respect to uncoupled Coulomb barrier has also shown an increasing trend with the increase in target isotopic mass. In order to confirm the shape transition effects, the coupled channel calculations have been performed by using the code CCFULL. The present coupled channel analysis has clearly pointed out the importance of nuclear structure degrees of freedom of participating nuclei in the sub-barrier region. Similar results have also been evident from SAGBD analysis. Hence the model predictions incorporate the contributions from vibrational excitations and/or static deformation in the fusion of ${}^{16}\mathrm{O}+{}^{70,72,73,74,76}\mathrm{Ge}$ reactions. For the ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ reaction, the vibrational excitations along with the neutron transfer channel have led to larger sub-barrier fusion enhancement and their influences have been automatically included in the SAGBD calculations. Furthermore, ${}^{16}\mathrm{O}+{}^{76}\mathrm{Ge}$ and ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ reactions have formed the same compound nucleus ${}^{92}\mathrm{Zr}$ via different entrance channels. Their comparative study has indicated that the effect of the neutron transfer channel has dominated over the entrance channel mass asymmetry effects. In addition, the minimum value of ${\chi }^2$ that has been obtained for the SAGBD calculations is consistent with that of the coupled channel calculations.

Figure 1

(a) The variation of the Coulomb barrier $\left(V_{CB}\right)$ as a function of radial separation between projectile and target for the ${}^{16}\mathrm{O}+{}^{70,72,73,74,76}\mathrm{Ge}$ and ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ reactions, and (b) amplified version of (a) near barrier positions.

Figure 2

The variations of the Coulomb $\left(V_C\right)$ and nuclear $\left(V_N\right)$ potentials along with the Coulomb barrier $\left(V_{CB}\right)$ as a function of radial separation between projectile and target for (a) ${}^{16}\mathrm{O}+{}^{70}\mathrm{Ge}$, (b) ${}^{16}\mathrm{O}+{}^{72}\mathrm{Ge}$, (c) ${}^{16}\mathrm{O}+{}^{73}\mathrm{Ge}$, (d) ${}^{16}\mathrm{O}+{}^{74}\mathrm{Ge}$, (e) ${}^{16}\mathrm{O}+{}^{76}\mathrm{Ge}$, and (f) ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ systems.

Figure 3

The fusion cross section (a) and barrier distribution (b) for ${}^{16}\mathrm{O}+{}^{70}\mathrm{Ge}$ reaction as a function of energy in center-of-mass frame “$E_{c.m.}$” for different values of range $\left(r_0\right)$ parameter. The fusion cross section and barrier distribution obtained by using the SAGBD model are compared with experimental data taken from Ref. [13].

Figure 4

The fusion cross section (a) and barrier distribution (b) for ${}^{16}\mathrm{O}+{}^{70}\mathrm{Ge}$ reaction as a function $E_{c.m.}$ for different values of diffuseness $\left(a_0\right)$ parameter. The fusion cross section and barrier distribution obtained by using the SAGBD model are compared with the experimental data taken from Ref. [13].

Figure 5

The fusion cross section of ${}^{16}\mathrm{O}+{}^{70}\mathrm{Ge}$ (a) and ${}^{16}\mathrm{O}+{}^{72}\mathrm{Ge}$ (b) reactions as a function of $E_{c.m.}$ obtained by using Wong formula and SAGBD model. These results are compared with the experimental data taken from Ref. [13].

Figure 6

The barrier distribution (BD) for ${}^{16}\mathrm{O}+{}^{70}\mathrm{Ge}$ (a), and ${}^{16}\mathrm{O}+{}^{72}\mathrm{Ge}$ reactions as a function of $E_{c.m.}$. The discrete symbols denote the barrier distribution data [13] obtained by using double differentiation method [17]. For both cases, the solid line denotes the outcomes of the present model.

Figure 7

Same as Fig. 5 but for ${}^{16}\mathrm{O}+{}^{73}\mathrm{Ge}$ (a) and ${}^{16}\mathrm{O}+{}^{74}\mathrm{Ge}$ (b) reactions. The results are compared with experimental data taken from Ref. [13].

Figure 8

Same as in Fig. 6 but for ${}^{16}\mathrm{O}+{}^{73}\mathrm{Ge}$ (a) and ${}^{16}\mathrm{O}+{}^{74}\mathrm{Ge}$ (b) reactions.

Figure 9

Same as Fig. 5 but for ${}^{16}\mathrm{O}+{}^{76}\mathrm{Ge}$ (a) and ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ (b) reactions. The results are compared with experimental data taken from Ref. [23].

Figure 10

Same as Fig. 6 but for the ${}^{16}\mathrm{O}+{}^{76}\mathrm{Ge}$ (a) and ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ (b) reactions. The results are compared with the experimental data taken from Ref. [23].

Figure 11

The fusion excitation functions of ${}^{16}\mathrm{O}+{}^{70,72,73}\mathrm{Ge}$ reactions obtained by using the coupled channel method. The calculated fusion cross sections are compared with the available experimental data taken from Ref. [13].

Figure 12

Same as Fig. 11 but for ${}^{16}\mathrm{O}+{}^{74,76}\mathrm{Ge}$ and ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ reactions. The calculated fusion cross sections are compared with the available experimental data taken from Refs. [13, 23].

Figure 13

Fusion cross-section data for ${}^{16}\mathrm{O}+{}^{70,72,73,74,76}\mathrm{Ge}$ systems (a) and ${}^{16}\mathrm{O}+{}^{76}\mathrm{Ge}$ and ${}^{18}\mathrm{O}+{}^{74}\mathrm{Ge}$ systems (b) are compared in reduced scale. The experimental data of chosen reactions are taken from Refs. [13, 23].