Comprehensive analysis of fusion data well above the barrier

We report on the comprehensive systematics of nearly 400 fusion-evaporation and/or fusion-fission cross-section data for a very large variety of systems over an energy range $\sim 3A$ to $155A$ MeV. Scaled by the reaction cross section and expressed as a function of the center-of-mass energy per nucleon, the fusion cross section displays a universal behavior. Within experimental errors, this behavior does not depend on system mass, mass asymmetry, or system isospin. The deduced homographic functional dependence for complete and summed complete and incomplete fusion excitation functions is derived from basic strong absorption model formulas for reaction cross sections and allows us to draw the main properties of these functions. The limiting energy for the complete fusion and the main characteristics (onset, maximum, and extinction) of the incomplete fusion excitation functions are determined. The complete fusion reaction process disappears around 6.5 MeV/nucleon and the incomplete one disappears at about 13 MeV/nucleon in the center-of-mass frame. The regularity in fusion data is particularly obvious for the evaporation-residue subset of the data ensemble. Adding the fusion-fission data component does not alter the general data trend but somewhat obscures it owing to the larger uncertainty and/or possible normalization problems.

Figure 1

Raw fusion cross sections ${\sigma }_{\mathrm{fus}}$ plotted as a function of incident energy per nucleon in the laboratory reference frame $E_{\mathrm{lab}}/\text{nucleon}=E_{\mathrm{in}}$. The inventoried systems are distinguished among them by symbols and levels of gray (a color code online). The same symbols and the levels of gray (color code) are used throughout this paper. References to all ${\sigma }_{\mathrm{fus}}$ data are given in Table 1.

Figure 2

Statistics of 382 fusion cross-section data points (CF and CF + IF) as a function of (a) system mass $A_{\mathrm{tot}}$, (b) mass asymmetry $\mu$, (c) system neutron-to-proton ratio $N_{\mathrm{tot}}/Z_{\mathrm{tot}}$, and (d) reaction incident energy $E_{\mathrm{in}}$, while (e) shows the correlation between $A_{\mathrm{tot}}$ and $\mu$. The size of squares in the scatter plot (e) is proportional to the number of data points within the crossed bins.

Figure 3

Ratios of reaction cross sections calculated with the Bass [3], Kox [104] (a), and Shen [105] (b) formulas vs that of Tripathi [24] at energies of fusion data of Table 1. Inset (c) shows the projection of the Shen-Tripathi ratio onto the ordinate axis taking either all points (hollow histogram) or those of energy $E_{\mathrm{c}.\mathrm{m}.}/A_{\mathrm{tot}}>1.06$ MeV/nucleon (filled histogram). Meaning of the symbols and levels of gray (colors online) are listed in Fig. 1.

Figure 4

Histograms of the ratios of experimental and model reaction cross sections using (a) Bass [3], (b) Kox [104], (c) Shen [105], and (d) Tripathi [24] formulas, respectively. Hollow histograms refer to all the used experimental reaction data and those filled to the data fulfilling the condition $E_{\mathrm{c}.\mathrm{m}.}/A_{\mathrm{tot}}>1.06$ MeV/nucleon.

Figure 5

Normalized fusion cross sections ${\sigma }_{\mathrm{fus}}^N$ plotted as a function of the laboratory energy per nucleon $E_{\mathrm{in}}$. To better resolve the stacked points, panels (c) and (b) repeat the data subsets for $A_{\mathrm{tot}}\le 44$ and $46\le A_{\mathrm{tot}}\le 59$, respectively. The meaning of the symbols and levels of gray (colors online) may be inferred from Fig. 1.

Figure 6

Same as Fig. 5 but for mass symmetric systems.

Figure 7

Normalized fusion cross sections ${\sigma }_{\mathrm{fus}}^N$ plotted as a function of the center-of-mass energy per nucleon $E_{\mathrm{c}.\mathrm{m}.}/A_{\mathrm{tot}}$. The meaning of the symbols and levels of gray (colors online) may be inferred from Fig. 1. A light-gray band for $E_{\mathrm{c}.\mathrm{m}.}/A_{\mathrm{tot}}<1.06$ MeV/nucleon delimits a portion of the data set which is of no particular relevance in the present study because we focus on fusion region III and fusion disappearance. For the discussion on the groups of data labeled by $①$ and $②$, see text.

Figure 8

${\sigma }_{\mathrm{fus}}^N$ plotted as a function of $E_{\mathrm{c}.\mathrm{m}.}/A_{\mathrm{tot}}$ for those data points which survived the iterative fitting procedure. The curve is due to a fit with the homographic function (4) restricted to 256 data points and 78 reaction systems lying outside the light-gray band. (For the explanation of the light-gray band see the caption of Fig. 7 and text.) The background band around the best fit curve is due to the errors on the fit parameters provided by the used code [111]. The histogram in the inset shows the distribution of relative errors of experimental fusion cross sections $\Delta {\sigma }_{\mathrm{fus}}^N$ when $\Delta {\sigma }_{\mathrm{fus}}$ are normalized by the Tripathi reaction cross section ${\sigma }_{\mathrm{reac}}^T$.

Figure 9

${\sigma }_{\mathrm{fus}}^N$ yields as a function of $E_{\mathrm{c}.\mathrm{m}.}/A_{\mathrm{tot}}$ of those fusion experiments in which no fission contribution has been reported. These 48 reactions and 238 data points are labeled by the same symbols and levels of gray (colors online) as in Fig. 1. Data points of the group $①$ in Fig. 7 are drawn by thinner lines and lighter colors. The full (red) curve is due to the best fit with the homographic function (4) to the subset of 181 evaporation-residue data points satisfying the condition $E_{\mathrm{c}.\mathrm{m}.}/A_{\mathrm{tot}}>1.0$ MeV/nucleon, i.e., to the points lying outside the light-gray band. (For the explanation of the light-gray band, see the caption to Fig. 7 and the text.) The thin dashed black curve is the same kind of fit to the full set of CF + IF data fulfilling the above energy requirement, i.e., the fit curve of Fig. 8. Those 126 data points which survived the rejection through the iterative fitting procedure are made visible by thin dark gray circles drawn over them.

Figure 10

Normalized complete fusion cross section ${\sigma }_{\mathrm{fus}}^N$ as a function of $E_{\mathrm{c}.\mathrm{m}.}/A_{\mathrm{tot}}$ (main panel) and as a function of the inverse of the same energy (inset) from those measurements which have reported data on both complete and incomplete fusion. The full (violet) curve in the inset and the main panel is due to the best fit of the complete fusion data by the homographic function (4) in the falling (third) region of fusion cross section. The (dark green) dash-dotted curve in inset is due to the same kind of fit to the stagnating (second) region of fusion cross section. The background band around the best fit curve in the main panel and in the inset is due to the errors on the fit parameters provided by the code used [111]. The (red) dashed curve is from Fig. 8 and it represents the best-fit homographic function of the normalized fusion cross sections of Fig. 7. The (blue) dotted curve is the difference of both. Those data points which survived the rejection through the iterative fitting procedure are made visible by the thin dark-gray circles drawn over them.