Improved Wong and classical approximations, and reduction of fusion data

We present an improved version of the Wong formula for heavy-ion fusion, where the parameters of the parabolic approximation of the Coulomb barrier are replaced by parameters of the $l$-dependent potential at an effective partial wave. A pocket formula for this $l$ dependence is given. This version reproduces the fusion cross sections of quantum-mechanical calculations very well, even when the original Wong formula is invalid. The same procedure is used to derive an improved expression for the classical fusion cross section, which is very accurate at above-barrier energies. Based on this classical expression, we propose a new method to reduce fusion data in this energy range. This method is used to perform a comparative study of complete fusion suppression in collisions of weakly bound projectiles. This study indicates that the suppression of complete fusion is essentially due to the action of nuclear breakup couplings.

Figure 1

Comparison of BPM fusion cross sections, ${\sigma }_{\mathrm{BPM}}$, with fusion cross sections of QM calculations with the SPP and short-range absorption, ${\sigma }_{\mathrm{F}}^{1\mathrm{ch}}$, for three systems in different mass regions. For comparison, we also show cross sections of QM calculations with the long-range imaginary potential of Eq. (7), ${\sigma }_{\mathrm{R}}^{1\mathrm{ch}}$.

Figure 2

The Wong fusion function and its asymptotic limit.

Figure 3

The Wong fusion cross section in comparison with the cross sections predicted by the BPM. See the text for details.

Figure 4

The Coulomb barriers for the ${}^7\mathrm{Li}+{}^{27}\mathrm{Al}, {}^7\mathrm{Bi}+{}^{209}\mathrm{Al}$, and ${}^{24}\mathrm{Mg}+{}^{138}\mathrm{Ba}$ systems, in comparison with the corresponding parabolic fits.

Figure 5

The BPM fusion cross sections for the ${}^7\mathrm{Li}+{}^{27}\mathrm{Al}, {}^7\mathrm{Li}+{}^{209}\mathrm{Bi}$, and ${}^{24}\mathrm{Mg}+{}^{138}\mathrm{Ba}$ systems, in comparison to those obtained by the improved versions of the Wong formula, ${\sigma }_{\mathrm{W}}^{\mathrm{g}}$ and ${\overline{\sigma }}_{\mathrm{W}}$. The improved classical cross sections, ${\overline{\sigma }}_{\mathrm{cl}}$, are also shown.

Figure 6

The function $f_{\mathrm{R}}\left(y\right)$ of Eq. (53), for the ${}^7\mathrm{Li}+{}^{27}\mathrm{Al}, {}^7\mathrm{Li}+{}^{209}\mathrm{Bi}$, and ${}^{24}\mathrm{Mg}+{}^{138}\mathrm{Ba}$ systems, plotted versus the dimensionless energy variable, $y$.

Figure 7

Reduced BPM cross sections for the ${}^7\mathrm{Li}+{}^{27}\mathrm{Al}, {}^7\mathrm{Li}+{}^{209}\mathrm{Bi}$, and ${}^{24}\mathrm{Mg}+{}^{138}\mathrm{Ba}$ systems, reduced by the two fusion function procedures discussed in the text.

Figure 8

Reduced BPM cross sections for the same systems as the previous figure, but the CFF method now carries out the reductions.

Figure 9

Experimental CFF for a few tightly bound systems with negligible channel coupling effects.

Figure 10

Experimental CFF for weakly bound projectiles with different targets: (a) collisions of ${}^6\mathrm{Li}$, (b) collisions of ${}^7\mathrm{Li}$, (c) collisions of ${}^9\mathrm{Be}$, (d) collisions of ${}^6\mathrm{He}$. See the text for details.

Figure 11

The ratio $\overline{\mathcal{H}}\left(y\right)$ for (a) tightly bound and (b) weakly bound systems.