Determination of the compound nucleus survival probability $P_{\mathrm{surv}}$ for various “hot” fusion reactions based on the dynamical cluster-decay model

After a successful attempt to define and determine recently the compound nucleus (CN) fusion/ formation probability $P_{\text{CN}}$ within the dynamical cluster-decay model (DCM), we introduce and estimate here for the first time the survival probability $P_{\text{surv}}$ of CN against fission, again within the DCM. Calculated as the dynamical fragmentation process, $P_{\text{surv}}$ is defined as the ratio of the evaporation residue (ER) cross section ${\sigma }_{\text{ER}}$ and the sum of ${\sigma }_{\text{ER}}$ and fusion-fission (ff) cross section ${\sigma }_{\text{ff}}$, the CN formation cross section ${\sigma }_{\text{CN}}$, where each contributing fragmentation cross section is determined in terms of its formation and barrier penetration probabilities $P_0$ and $P$. In DCM, the deformations up to hexadecapole and “compact” orientations for both in-plane (coplanar) and out-of-plane (noncoplanar) configurations are allowed. Some 16 “hot” fusion reactions, forming a CN of mass number $A_{\text{CN}}\sim 100$ to superheavy nuclei, are analyzed for various different nuclear interaction potentials, and the variation of $P_{\text{surv}}$ on CN excitation energy $E^{*}$, fissility parameter $\chi$, CN mass $A_{\text{CN}}$, and Coulomb parameter $Z_1{Z}_2$ is investigated. Interesting results are that three groups, namely, weakly fissioning, radioactive, and strongly fissioning superheavy nuclei, are identified with $P_{\text{surv}}$, respectively, $\sim 1,\sim {10}^{-6}$, and $\sim {10}^{-10}$. For the weakly fissioning group ($100<A_{\text{CN}}\lesssim 200$), independent of the interaction potential, different isotopes and for coplanar or noncoplanar collisions, $P_{\text{surv}}$ decreases from one to zero as $E^{*}$ increases, whereas, independent of entrance channel effects, the same is surprisingly the reverse for the radioactive group ($A_{\text{CN}}\sim 200–250$), i.e., $P_{\text{surv}}$ increases with the increase of $E^{*}$. This is shown to be so due to the different relative magnitudes of ${\sigma }_{\text{ER}}$ and ${\sigma }_{\text{ff}}$ and their variations with $E^{*}$ in the two cases. For the superheavy nuclei also $P_{\text{surv}}$ is a decreasing function of $E^{*}$. Furthermore, of particular interest are the cases of ${}^{105}{\mathrm{Ag}}^{*}$, isotopes of ${\mathrm{Pt}}^{*}$, and ${}^{213,215,217}{\mathrm{Fr}}^{*}$ nuclei — for ${}^{105}{\mathrm{Ag}}^{*}$, whereas the $P_{\text{CN}}$ belongs to the strongly fissioning superheavy group, $P_{\text{surv}}$ belongs to weakly fissioning nuclei; for ${\mathrm{Pt}}^{*}$ isotopes, the inverse of all the compound systems studied, both $P_{\text{CN}}$ and $P_{\text{surv}}$ decrease with the increase of $E^{*}$; for ${}^{213,215,217}{\mathrm{Fr}}^{*}$ nuclei, though fissility $\chi$ is nearly the same, $P_{\text{surv}}$ for ${}^{213,217}{\mathrm{Fr}}^{*}$ is of the same order as for weakly fissioning nuclei, but that for ${}^{215}{\mathrm{Fr}}^{*}$ is of the order of radioactive nuclei. Apparently, further calculations are called for.

Figure 1

The DCM-calculated $P_{\text{surv}}$ as a function of CN excitation energy $E^{*}$, using for nuclear proximity potential, the Blocki et al. [40] pocket formula, (a) for coplanar ($\Phi =0^{\circ }$) case of different CN, compared to the noncoplanar ($\Phi \ne 0^{\circ }$) case of ${}^{164}{\mathrm{Yb}}^{*}$, (b) for different isotopes ${}^{176,182,188,196}{\mathrm{Pt}}^{*}$ and ${}^{213,217}{\mathrm{Fr}}^{*}$, and (c) $\Phi =0^{\circ }$ case of SEDF, using SIII, SSk, and GSkI forces in ${}^{196}{\mathrm{Pt}}^{*}$ [20] and ${}^{164}{\mathrm{Yb}}^{*}$ [19].

Figure 2

Same as for Fig. 1 ($\Phi =0^{\circ }$ case), but for superheavy nuclei ${}^{286}{\mathrm{Cn}}^{*}$ and ${}^{292}{\mathrm{Fl}}^{*}$.

Figure 3

Same as Fig. 1 ($\Phi =0^{\circ }$ case), but for (a) ${}^{246}{\mathrm{Bk}}^{*}$ and (b) ${}^{215}{\mathrm{Fr}}^{*}$.

Figure 4

The DCM-calculated ${\sigma }_{\text{ER}}$ and ${\sigma }_{\text{ff}}$ plotted as a function of $E^{*}$ for the radioactive ${}^{246}{\mathrm{Bk}}^{*}$ and the weakly fissioning ${}^{202}{\mathrm{Pb}}^{*}$ nuclei, compared to the experimental data. For ${}^{246}{\mathrm{Bk}}^{*}$, the experimental data are only for ${\sigma }_{\text{fission}}$ ($\equiv {\sigma }_{\text{ff}}\equiv {\sigma }_{\text{fusion}}$), and ${\sigma }_{\text{ER}}$ is for the DCM-predicted numbers. $P_{\mathrm{surv}}$ is also plotted in each case.

Figure 5

Variation of $P_{\text{surv}}$ with the fissility parameter $\chi$ for all the reactions under consideration.

Figure 6

Variation of $P_{\text{surv}}$ with the compound nucleus mass number $A_{\text{CN}}$ for all the reactions under consideration.

Figure 7

Variation of $P_{\text{surv}}$ with product $Z_1{Z}_2$ of target and projectile charge numbers.