Empirical nucleus-nucleus potential deduced from fusion excitation functions

Existing data on near-barrier fusion excitation functions for 48 medium and heavy nucleus-nucleus systems have been analyzed using a simple “diffused-barrier formula” derived assuming the Gaussian shape of the barrier height distributions. The obtained mean values of the barrier height have been used then for determination of the parameters of the empirical nucleus-nucleus potential, assumed to have Woods-Saxon shape. The mean barrier heights calculated with this potential are reproduced with an accuracy of about $1\text{MeV}$, while other frequently used potentials, i.e., the proximity potential and the Akyüz-Winther potential, considerably overpredict the experimental values, especially for heavy systems. In order to predict fusion excitation functions with the diffused-barrier formula, we propose a simple method of theoretical prediction of the second parameter of the barrier distribution, its width. The proposed formula accounts for the quantum effect of sub-barrier tunneling, static quadrupole deformations, and collective surface vibrations of the colliding nuclei. With the theoretical knowledge of the mean height and width of the barrier distributions, one can predict cross sections for overcoming the barrier (“sticking” or “capture”) in reactions of very heavy systems used to produce superheavy nuclei.

Figure 1

Fusion excitation functions (top) and the deduced barrier distributions (bottom) for the ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ [12] and ${}^{34}\mathrm{S}+{}^{168}\mathrm{Er}$ [18] reactions. The barrier distributions determined with the standard “point difference method” and the “polynomial fit method” are shown with full and open circles, respectively. The Gaussian-barrier distributions obtained by fitting the “diffused-barrier formula,” Eq. (6), to the fusion excitation functions are shown by solid lines. After Ref. 20.

Figure 2

The barrier distributions $d^2\left(E{\sigma }_{fus}\right)∕d{E}^2$ determined with the standard “point difference method” (full circles) and the “polynomial fit method” (open circles) for fusion reactions of ${}^{16}\mathrm{O}$ ions with ${}^{144}\mathrm{Sm}$, ${}^{154}\mathrm{Sm}$, ${}^{184}\mathrm{W}$, and ${}^{208}\mathrm{Pb}$ target nuclei. (Data taken from Refs. 10, 14.) The Gaussian-barrier distributions obtained by fitting the “diffused-barrier formula,” Eq. (6), are shown by solid lines. After Ref. 20.

Figure 3

Fusion excitation function for the ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ reaction, measured by Timmers et al. [12] (full circles), compared with the least-square fit of the “diffused-barrier formula,” Eq. (6), shown by solid line. Dashed lines illustrate sensitivity of the calculated curve to the variation of the mean barrier $B_0$ increased by $1\%$ and the width $w$ increased by $10\%$.

Figure 4

Fusion excitation functions measured for the ${}^{16}\mathrm{O}+{}^{144,154}\mathrm{Sm}$ [10], ${}^{16}\mathrm{O}+{}^{186}\mathrm{W}$, and ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$ [14] reactions (full circles) compared with predictions (solid lines) of the “diffused-barrier formula,” Eq. (6), for values of $B_0$, $w$, and $R_{\sigma }$ obtained with the least-square method.

Figure 5

Fusion excitation functions measured for the ${}^{48}\mathrm{Ca}+{}^{40,48}\mathrm{Ca}$ [19], ${}^{58}\mathrm{Ni}+{}^{60}\mathrm{Ni}$ [8], and ${}^{36}\mathrm{S}+{}^{110}\mathrm{Pd}$ [9] reactions (full circles) compared with predictions (solid lines) of the “diffused-barrier formula,” Eq. (6), for values of $B_0$, $w$, and $R_{\sigma }$ obtained with the least-square method.

Figure 6

Fusion excitation functions measured for the ${}^{40}\mathrm{Ca}+{}^{90,96}\mathrm{Zr}$ [12], ${}^{40}\mathrm{Ca}+{}^{192}\mathrm{Os}$, and ${}^{40}\mathrm{Ca}+{}^{194}\mathrm{Pt}$ [11] reactions (full circles) compared with predictions (solid lines) of the “diffused-barrier formula,” Eq. (6), for values of $B_0$, $w$ and $R_{\sigma }$ obtained with the least-square method.

Figure 7

Contours of constant value of ${\chi }^2$ in the procedure of fitting the complete set of experimental fusion barriers $B_0\left(\text{exp}\right)$ with those calculated with the Woods-Saxon nuclear potential, Eqs. (8, 11). The best fit $\left({\chi }_{min}^2\right)$ was found for $r_0=1.18\text{fm}$ and $a=0.675\text{fm}$. The contours correspond to ${\chi }^2$ increased by $10\%$ and $50\%$ relative to ${\chi }_{min}^2$, respectively.

Figure 8

Comparison of experimental barrier heights $B_0$ with theoretical predictions for the Akyüz-Winther potential [24], proximity potential [25], and the “empirical potential,” Eqs. (8, 11), with parameters $r_0$ and $a$ given by Eq. (15).

Figure 9

Comparison of experimental values of the width $w$ of the fusion barrier distributions, deduced for 48 reactions (fifth column in Table ) with those calculated with Eq. (27).