Fusion probability in heavy nuclei

Background: Fusion between two massive nuclei is a very complex process and is characterized by three stages: (a) capture inside the potential barrier, (b) formation of an equilibrated compound nucleus (CN), and (c) statistical decay of the CN leading to a cold evaporation residue (ER) or fission. The second stage is the least understood of the three and is the most crucial in predicting yield of superheavy elements (SHE) formed in complete fusion reactions.

Purpose: A systematic study of average fusion probability, $\langle P_{\mathrm{CN}}\rangle$, is undertaken to obtain a better understanding of its dependence on various reaction parameters. The study may also help to clearly demarcate onset of non-CN fission (NCNF), which causes fusion probability, $P_{\mathrm{CN}}$, to deviate from unity.

Method: ER excitation functions for 52 reactions leading to CN in the mass region 170–220, which are available in the literature, have been compared with statistical model (SM) calculations. Capture cross sections have been obtained from a coupled-channels code. In the SM, shell corrections in both the level density and the fission barrier have been included. $\langle P_{\mathrm{CN}}\rangle$ for these reactions has been extracted by comparing experimental and theoretical ER excitation functions in the energy range $\sim 5$%–35% above the potential barrier, where known effects of nuclear structure are insignificant.

Results: $\langle P_{\mathrm{CN}}\rangle$ has been shown to vary with entrance channel mass asymmetry, $\eta$ (or charge product, $Z_p{Z}_t)$, as well as with fissility of the CN, ${\chi }_{\mathrm{CN}}$. No parameter has been found to be adequate as a single scaling variable to determine $\langle P_{\mathrm{CN}}\rangle$. Approximate boundaries have been obtained from where $\langle P_{\mathrm{CN}}\rangle$ starts deviating from unity.

Conclusions: This study quite clearly reveals the limits of applicability of the SM in interpreting experimental observables from fusion reactions involving two massive nuclei. Deviation of $\langle P_{\mathrm{CN}}\rangle$ from unity marks the beginning of the domain of dynamical models of fusion. Availability of precise ER cross sections over a wider energy range for many more reactions is desired for accurate determination of $\langle P_{\mathrm{CN}}\rangle$ and more insight into the dynamics of fusion in the heavy mass region.

Figure 1

Capture cross sections as a function of $\frac{E_{\text{c.m.}}}{V_B}$ for 19 reactions leading to CN ${}^{170}\mathrm{Hf},{}^{181}\mathrm{Re},{}^{185}\mathrm{Ir},{}^{188}\mathrm{Pt},{}^{194}\mathrm{Hg},{}^{198}\mathrm{Pb},{}^{200}\mathrm{Pb}$, and ${}^{202}\mathrm{Pb}$. The solid line in each panel represents the result of the ccfull calculation. The shaded region denotes range of $E_{\text{c.m.}}$ from 5% to 35% above the $V_B$, the focus of the present work. See text for details.

Figure 2

Capture cross sections as a function of $\frac{E_{\text{c.m.}}}{V_B}$ for 19 reactions leading to CN ${}^{202}\mathrm{Po},{}^{204}\mathrm{Po},{}^{207}\mathrm{At},{}^{210}\mathrm{Po},{}^{210}\mathrm{Rn},{}^{213}\mathrm{Fr},{}^{216}\mathrm{Ra},{}^{218}\mathrm{Ra}$, and ${}^{220}$ Th. The solid line in each panel represents the result of the ccfull calculation. The shaded region denotes range of $E_{\text{c.m.}}$ from 5% to 35% above the $V_B$, the focus of the present work. See text for details.

Figure 3

ER cross sections (or ${\sigma }_{\Sigma xn})$ as a function of $E^{*}$ for 19 reactions leading to CN ${}^{170}\mathrm{Hf},{}^{181}\mathrm{Re},{}^{185}\mathrm{Ir},{}^{188}\mathrm{Pt},{}^{194}\mathrm{Hg},{}^{198}\mathrm{Pb},{}^{200}\mathrm{Pb}$, and ${}^{200}\mathrm{Po}$. The solid line in each panel represents the result of the SM calculation, whereas the dashed line stands for the scaled-down SM calculation to match experimental data. The shaded region denotes the range of $E^{*}$ corresponding to $\frac{E_{\text{c.m.}}}{V_B}$ from 5% to 35% above the $V_B$. See text for details.

Figure 4

ER cross sections (or ${\sigma }_{\Sigma xn})$ as a function of $E^{*}$ for 20 reactions leading to CN ${}^{202}\mathrm{Pb},{}^{202}\mathrm{Po},{}^{204}\mathrm{Po},{}^{207}\mathrm{At},{}^{210}\mathrm{Po},{}^{210}\mathrm{Rn},{}^{212}\mathrm{Rn}$, and ${}^{213}\mathrm{Fr}$. The solid line in each panel represents the result of the SM calculation, whereas the dashed line stands for the scaled-down SM calculation to match experimental data. The shaded region denotes the range of $E^{*}$ corresponding to $\frac{E_{\text{c.m.}}}{V_B}$ from 5% to 35% above the $V_B$. See text for details.

Figure 5

ER cross sections (or ${\sigma }_{\Sigma xn})$ as a function of $E^{*}$ for 13 reactions leading to CN ${}^{216}\mathrm{Ra},{}^{218}\mathrm{Ra}$, and ${}^{220}$ Th. The solid line in each panel represents the result of the SM calculation, whereas the dashed line stands for the scaled-down SM calculation to match experimental data. The shaded region denotes the range of $E^{*}$ corresponding to $\frac{E_{\text{c.m.}}}{V_B}$ from 5% to 35% above the $V_B$. See text for details.

Figure 6

Variation of $\langle P_{\mathrm{CN}}\rangle$ as a function of (a) $Z_p{Z}_t$, (b) $\eta$, (c) ${\chi }_{\mathrm{eff}}$, and (d) ${\chi }_{\mathrm{CN}}$. See text for details.

Figure 7

$\langle P_{\mathrm{CN}}\rangle$ surface plots as a function of (a) $Z_p{Z}_t$ and ${\chi }_{\mathrm{CN}}$, (b) $\eta$ and ${\chi }_{\mathrm{CN}}$.

Figure 8

Projection of $\langle P_{\mathrm{CN}}\rangle$ surface plots (shown in Fig. 7) on the (a) $Z_p{Z}_t$–${\chi }_{\mathrm{CN}}$ and (b) $\eta$–${\chi }_{\mathrm{CN}}$ planes. Variation of $\langle P_{\mathrm{CN}}\rangle$ is shown with a color gradient. The two straight lines denote the “boundaries” between the regions with $\langle P_{\mathrm{CN}}\rangle =1$ and $\langle P_{\mathrm{CN}}\rangle <1$.