Statistical model calculation of the compound nuclear fission timescale in fusion-fission reactions

In the decay of an excited heavy nucleus, both the fission or evaporation residue (ER) cross sections and the fission time are determined by the competition between the various evaporation channels and fission. Statistical model (SM) calculations for several heavy-ion-induced fusion-fission reactions are presented which reproduce the fission or ER excitation functions. The corresponding calculated fission times are found to depend upon the compound nuclei and their excitation energies. The multichance fission probabilities and the corresponding fission times are also obtained, and they are shown to depend sensitively on the shell correction energies and the neutron separation energies of the compound nuclei involved. The experimental neutron multiplicities are reproduced by adjusting the saddle-to-scission transition time interval. An additional delay time for compound nuclear formation is found necessary to fit the experimental neutron multiplicities for a system with higher entrance channel mass symmetry.

Figure 1

The fission cross sections (${\sigma }_{\mathrm{fiss}}$) for the reactions ${}^{12}\mathrm{C}+{}^{194}\mathrm{Pt}$ and ${}^{12}\mathrm{C}+{}^{198}\mathrm{Pt}$ forming the compound nuclei ${}^{206}\mathrm{Po}$ and ${}^{210}\mathrm{Po}$, respectively. Statistical model results with $\beta =0$ and best-fit values are shown. The experimental points are from Ref. [40].

Figure 2

The evaporation residue cross sections (${\sigma }_{ER}$) for the reactions ${}^{16}\mathrm{O}+{}^{194}\mathrm{Pt}$ and ${}^{18}\mathrm{O}+{}^{194}\mathrm{Pt}$ forming the compound nuclei ${}^{210}\mathrm{Rn}$ and ${}^{212}\mathrm{Rn}$, respectively. Statistical model results with $\beta =0$ and best-fit values are shown. The experimental points are from Refs. [41, 42].

Figure 3

The reduced evaporation residue cross sections ${\tilde{\sigma }}_{ER}={\sigma }_{ER}/\pi$ $\lambda -{}^2$ where $\lambda -$ is the de Broglie wavelength in the entrance channel, for the reactions ${}^{12}\mathrm{C}+{}^{204}\mathrm{Pb}$ and ${}^{19}\mathrm{F}+{}^{197}\mathrm{Au}$, both forming the compound nucleus ${}^{216}\mathrm{Ra}$. Statistical model results with $\beta =0$ and 2.2 ${\mathrm{zs}}^{-1}$ are shown. The experimental points are from Ref. [44].

Figure 4

The prescission neutron multiplicities ($n_{\mathrm{pre}}$) for the reactions ${}^{12}\mathrm{C}+{}^{194}\mathrm{Pt}$ and ${}^{12}\mathrm{C}+{}^{198}\mathrm{Pt},$ forming the compound nuclei ${}^{206}\mathrm{Po}$ and ${}^{210}\text{Po},$ respectively. Statistical model results for presaddle neutrons obtained with $\beta =0$ and best-fit $\beta$ values from fission excitation functions are shown (lines a and b, respectively). Lines c and d show $n_{\mathrm{pre}}$ values which include saddle-to-scission contributions obtained with $\beta =\beta \left(\text{best-fit}\right)$ and ${\beta }_{ss}=10{\mathrm{zs}}^{-1}$, respectively. $n_{\mathrm{pre}}$ values calculated with $\beta \left(\text{best-fit}\right)$ and ${\tau }_{\mathrm{ssc}}=65\mathrm{zs}$ are given by the line e. The experimental points are from Ref. [46].

Figure 5

The prescission neutron multiplicities ($n_{\mathrm{pre}}$) for the reactions ${}^{16}\mathrm{O}+{}^{194}\mathrm{Pt}$ and ${}^{18}\mathrm{O}+{}^{194}\mathrm{Pt},$ forming the compound nuclei ${}^{210}\mathrm{Rn}$ and ${}^{212}\mathrm{Rn}$, respectively. Statistical model results for presaddle neutrons obtained with $\beta =0$ and best-fit $\beta$ values from ER excitation functions are shown (lines a and b, respectively). Lines c and d shows $n_{\mathrm{pre}}$ values which include saddle-to-scission contributions obtained with $\beta \left(best\text{−}fit\right)$ and ${\beta }_{ss}=10{\mathrm{zs}}^{-1}$, respectively. $n_{\mathrm{pre}}$ values calculated with $\beta \left(best\text{−}fit\right)$ and ${\tau }_{\mathrm{ssc}}=65$ zs are given by the line e. The experimental points are from Ref. [51].

Figure 6

The prescission neutron multiplicities ($n_{\mathrm{pre}}$) for the reactions ${}^{12}\mathrm{C}+{}^{204}\mathrm{Pb}$ and ${}^{19}F+{}^{197}\mathrm{Au}$, both forming the compound nucleus ${}^{216}\mathrm{Ra}$. Statistical model results for presaddle neutrons obtained with $\beta =0$ and best-fit $\beta$ values from ER excitation functions are shown (lines a and b, respectively). Lines c and d show $n_{\mathrm{pre}}$ values, which include saddle-to-scission contributions obtained with $\beta \left(best\text{−}fit\right)$ and ${\beta }_{ss}=10{\mathrm{zs}}^{-1}$, respectively. $n_{\mathrm{pre}}$ values calculated with $\beta \left(best\text{−}fit\right)$ and ${\tau }_{\mathrm{ssc}}=65\mathrm{zs}$ are given by line e. The experimental points are from Ref. [52].

Figure 7

Prescission neutron multiplicity ($n_{\mathrm{pre}}$) obtained with ${\tau }_{\mathrm{form}}$. The presaddle ($n_{\mathrm{pre}-\mathrm{sad}}$) and saddle-to-scission ($n_{\mathrm{ssc}}$) contributions to $n_{\mathrm{pre}}$ are also shown.

Figure 8

Effect of formation time delay (${\tau }_{\mathrm{form}}$) on reduced ER cross section.

Figure 9

Fission-to-neutron width ratios for the compound nuclei ${}^{210}\mathrm{Po}$, ${}^{212}\text{Rn,}$ and ${}^{216}\mathrm{Ra}$ at CN spin $\ell$ of $20\hslash$. The width ratios for ${}^{206}\mathrm{Po}$ and ${}^{210}\mathrm{Rn}$ are higher than ${}^{210}\mathrm{Po}$ and ${}^{212}\text{Rn,}$ respectively, and are not shown here.

Figure 10

Saddle-time distribution of the compound nucleus ${}^{216}\mathrm{Ra}$ populated in the ${}^{12}\mathrm{C}+{}^{204}\mathrm{Pb}$ reaction. The average saddle time (259 zs) is marked by the arrow.

Figure 11

The excitation energy dependence of the average saddle time (${\tau }_{\mathrm{saddle}}$) of ${}^{206}\mathrm{Po}$ and ${}^{210}\mathrm{Po}$ compound nuclei obtained with $\beta =0$ and also with the $\beta$ that best fits the respective fission excitation functions.

Figure 12

The excitation energy dependence of the average saddle time (${\tau }_{\mathrm{saddle}}$) of ${}^{210}\mathrm{Rn}$ and ${}^{212}\mathrm{Rn}$ compound nuclei obtained with $\beta =0$ and also with the $\beta$ that best fits the respective ER excitation functions.

Figure 13

The excitation energy dependence of the average saddle time (${\tau }_{\mathrm{saddle}}$) of the compound nucleus ${}^{216}\mathrm{Ra}$ populated through the ${}^{12}\mathrm{C}+{}^{204}\mathrm{Pb}$ and ${}^{19}\mathrm{F}+{}^{197}\mathrm{Au}$ reactions obtained with $\beta =0$ and 2.2 ${\mathrm{zs}}^{-1}$.

Figure 14

Fission-to-neutron width ratios for the compound nuclei ${}^{206}\mathrm{Po}$ and ${}^{210}\mathrm{Po}$ for CN spin $\ell$ of $20\hslash$.

Figure 15

The excitation energy dependence of the relative probability of various chance fission events for the compound nuclei ${}^{206}\mathrm{Po}$ and ${}^{210}\mathrm{Po}$.

Figure 16

The chance number dependence of the relative probability of various chance fission events for the compound nuclei ${}^{206}\mathrm{Po}$ and ${}^{210}\mathrm{Po}$.

Figure 17

The excitation energy dependence of saddle time of various chance fission events for the compound nuclei ${}^{206}\mathrm{Po}$ and ${}^{210}\mathrm{Po}$.