Applicability of the Wong formula for fusion cross sections from light to heavy systems

We discuss the applicability of the Wong formula for fusion cross sections in a single-channel problem. To this end, we carry out a systematic study and compare the approximate fusion cross sections with the exact results in a wide mass region of reaction systems. We show that the deviation of the approximate results from the exact cross sections is large for light systems, even though the Wong formula provides a reasonable approximation for heavy systems. We also discuss the energy dependence of the deviation, and show that for a given projectile nucleus the critical energy, at which the deviation exceeds 5% of the exact cross sections, increases as a function of the mass number of the target nucleus.

Figure 1

Comparisons between the exact (solid line) and approximate (dashed line) fusion cross sections for four selected systems. The approximate solutions are obtained with the Wong formula. The Akyüz-Winther potential is employed for the internuclear potential.

Figure 2

The deviation from the exact results, defined by Eq. (12), of the fusion cross sections obtained with the Wong formula. This quantity is plotted as a function of the mass number of the target nucleus for five projectile nuclei indicated in the figure.

Figure 3

The deviation of the Wong formula given by Eq. (13) for ${}^6\mathrm{Li}$ induced fusion reactions as a function of the incident energy relative to the barrier height for each system. The target nuclei are ${}^{238}\mathrm{U}$, ${}^{152}\mathrm{Sm}$, ${}^{74}\mathrm{Ge}$, ${}^{40}\mathrm{Ar}$, ${}^{28}\mathrm{Si}$, ${}^{20}\mathrm{Ne}$, and ${}^{16}\mathrm{O}$, as shown in the figure in the decreasing order at $E-V_B=4$ MeV.

Figure 4

The critical energy, $E_{\mathrm{crit}}$, at which the deviation defined by Eq. (13) exceeds 5%, measured relative to the barrier height for each system as a function of the mass number of the target nucleus.