Comparative analysis of the fusion reactions ${}^{48}\mathrm{Ti}+{}^{58}\mathrm{Fe}$ and ${}^{58}\mathrm{Ni}+{}^{54}\mathrm{Fe}$

The experimental fusion excitation functions of the reactions ${}^{48}\mathrm{Ti}+{}^{58}\mathrm{Fe}$ and ${}^{58}\mathrm{Ni}+{}^{54}\mathrm{Fe}$, measured down to the low sub-barrier energies, are described within the quantum diffusion approach and the universal fusion function representation. For these systems, the $s$-wave capture probabilities are extracted from the experimental excitation functions and are also analyzed. An enhancement of the sub-barrier fusion cross section observed in the ${}^{48}\mathrm{Ti}+{}^{58}\mathrm{Fe}$ reaction in comparison to the relatively close system ${}^{58}\mathrm{Ni}+{}^{54}\mathrm{Fe}$ is explained.

Figure 1

The experimental reduced fusion excitation functions $\frac{2E_{\mathrm{c}.\mathrm{m}.}}{\hslash {\omega }_b{R}_b^2}{\sigma }_{\mathrm{cap}}\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ [7, 8] (symbols) and the universal fusion function $F_0=ln(1+exp[2\pi (E_{\mathrm{c}.\mathrm{m}.}-V_b)/\left(\hslash {\omega }_b\right)])$ (solid line) vs $\frac{E_{\mathrm{c}.\mathrm{m}.}-V_b}{\hslash {\omega }_b}$ for the reactions indicated. The experimental data have uncertainty bars because of the uncertainties of experimental ${\sigma }_{\mathrm{cap}}\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ and $E_{\mathrm{c}.\mathrm{m}.}$.

Figure 2

The extracted (symbols) and calculated (lines) $s$-wave capture probabilities versus $E_{\mathrm{c}.\mathrm{m}.}$ for the reactions indicated. The experimental $P_{\mathrm{cap}}$ is obtained as the incremental ratio for the successive pairs of experimental points. The used experimental capture (fusion) excitation functions are from Refs. [7, 8]. The experimental data have uncertainty bars because of the uncertainties of experimental ${\sigma }_{\mathrm{cap}}\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ and $E_{\mathrm{c}.\mathrm{m}.}$.

Figure 3

The extracted (symbols connected by lines) $s$-wave capture probabilities versus $E_{\mathrm{c}.\mathrm{m}.}-V_b$ for the reactions indicated. The experimental $P_{\mathrm{cap}}$ is obtained as the incremental ratio for the successive pairs of experimental points. The used experimental capture (fusion) excitation functions are from Refs. [7, 8].

Figure 4

Calculated fusion (capture) cross sections vs $E_{\mathrm{c}.\mathrm{m}.}$ (solid lines) for the indicated reactions. The experimental data (symbols) are from Refs. [7, 8]. The experimental data have uncertainty bars because of the uncertainties of experimental ${\sigma }_{\mathrm{cap}}\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ and $E_{\mathrm{c}.\mathrm{m}.}$.

Figure 5

The calculated (lines) and experimental (symbols [7, 8]) astrophysical $S$ factor vs $E_{\mathrm{c}.\mathrm{m}.}$ for the indicated reactions. The experimental data have uncertainty bars because of the uncertainties of experimental ${\sigma }_{\mathrm{fus}}\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ and $E_{\mathrm{c}.\mathrm{m}.}$.

Figure 6

The calculated (lines) and experimental (symbols [7, 8]) logarithmic derivatives of the excitation functions for the indicated reactions. The experimental $L$ factor is obtained as the incremental ratio for successive pairs of experimental energy points of the of the excitation function. In the calculations, $△E_{\mathrm{c}.\mathrm{m}.}=0.4$ MeV (dashed lines) and $△E_{\mathrm{c}.\mathrm{m}.}=0.8$ MeV (solid lines) are used. The experimental data have uncertainty bars because of the uncertainties of experimental ${\sigma }_{\mathrm{cap}}\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ and $E_{\mathrm{c}.\mathrm{m}.}$.

Figure 7

The calculated (lines) and experimental (symbols [7, 8]) fusion barrier distributions $BD$ [Eq. (10)] for the indicated reactions. For the calculated $BD$, $△E_{\mathrm{c}.\mathrm{m}.}=0.4$ MeV (dashed lines) and $△E_{\mathrm{c}.\mathrm{m}.}=1.0$ MeV (solid lines) are used. For the extraction of the experimental $BD$, the energy step is $△E_{\mathrm{c}.\mathrm{m}.}\approx 1$ MeV. The experimental data have uncertainty bars because of the uncertainties of experimental ${\sigma }_{\mathrm{cap}}\left(E_{\mathrm{c}.\mathrm{m}.}\right)$ and $E_{\mathrm{c}.\mathrm{m}.}$.