Compound nucleus formation probability $P_{CN}$ determined within the dynamical cluster-decay model for various “hot” fusion reactions

The compound nucleus (CN) fusion/formation probability $P_{CN}$ is defined and its detailed variations with the CN excitation energy $E^{*}$, center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}$, fissility parameter $\chi$, CN mass number $A_{CN}$, and Coulomb interaction parameter $Z_1{Z}_2$ are studied for the first time within the dynamical cluster-decay model (DCM). The model is a nonstatistical description of the decay of a CN to all possible processes. The (total) fusion cross section ${\sigma }_{\mathrm{fusion}}$ is the sum of the CN and noncompound nucleus (nCN) decay cross sections, each calculated as the dynamical fragmentation process. The CN cross section ${\sigma }_{CN}$ is constituted of evaporation residues and fusion-fission, including intermediate-mass fragments, each calculated for all contributing decay fragments ($A_1$, $A_2$) in terms of their formation and barrier penetration probabilities $P_0$ and $P$. The nCN cross section ${\sigma }_{\mathit{nCN}}$ is determined as the quasi-fission (qf) process, where $P_0=1$ and P is calculated for the entrance-channel nuclei. The DCM, with effects of deformations and orientations of nuclei included in it, is used to study the $P_{CN}$ for about a dozen “hot” fusion reactions forming a CN of mass number $A\sim 100$ to superheavy nuclei and for various different nuclear interaction potentials. Interesting results are that $P_{CN}=1$ for complete fusion, but $P_{CN}<1$ or $P_{CN}\ll 1$ due to the nCN contribution, depending strongly on different parameters of the entrance-channel reaction but found to be independent of the nuclear interaction potentials used.

Figure 1

Schematic of the formation/decay path used to calculate the compound nucleus and noncompound nucleus decay cross sections.

Figure 2

Schematic of two equal or unequal axially symmetric deformed, oriented nuclei, lying in the same plane (azimuthal angle $\Phi =0^{\circ }$) for various ${\theta }_1$ and ${\theta }_2$ values in the range 0 to ${180}^{\circ }$. The ${\theta }_i$ are measured anticlockwise from the colliding axis, and angle ${\alpha }_i$, clockwise from the symmetry axis.

Figure 3

Same as Fig. 2, but for noncoplanar ($\Phi \ne 0^{\circ }$) nuclei. For details, see Ref. [24].

Figure 4

DCM-calculated $P_{CN}$ as a function of the compound nucleus excitation energy $E^{*}$, using the Blocki et al. [38] pocket formula for the nuclear proximity potential. Reactions are shown in the legend.

Figure 5

Same as Fig. 4, but for its variation with the center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}$.

Figure 6

Variation of $P_{CN}$ with $E^{*}$, showing a comparison of the use of different nuclear inteaction potentials (pocket formula of Blocki et al. versus potentials based on SEDF using SIII and GSkI forces) for $\Phi =0$ and for two ${}^{64}\mathrm{Ni}$-based compound systems, ${}^{164}{\mathrm{Yb}}^{*}$ and ${}^{186}{\mathrm{Pt}}^{*}$. The case of $\Phi \ne 0$ is also added for ${}^{164}{\mathrm{Yb}}^{*}$.

Figure 7

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

Figure 8

Variation of $P_{CN}$ with compound nucleus mass number $A_{CN}$.

Figure 9

Variation of $P_{CN}$ with target-projectile charge numbers product $Z_1{Z}_2$.