Dynamic approach to description of entrance channel effects in angular distributions of fission fragments

Background: It is well known that the anomalous behavior of angular anisotropies of fission fragments at sub- and near-barrier energies is associated with a memory of conditions in the entrance channel of the heavy-ion reactions, particularly, deformations and spins of colliding nuclei that determine the initial distributions for the components of the total angular momentum over the symmetry axis of the fissioning system and the beam axis.

Purpose: We develop a new dynamic approach, which allows the description of the memory effects in the fission fragment angular distributions and provides new information on fusion and fission dynamics.

Methods: The approach is based on the dynamic model of the fission fragment angular distributions which takes into account stochastic aspects of nuclear fission and thermal fluctuations for the tilting mode that is characterized by the projection of the total angular momentum onto the symmetry axis of the fissioning system. Another base of our approach is the quantum mechanical method to calculate the initial distributions over the components of the total angular momentum of the nuclear system immediately following complete fusion.

Results: A method is suggested for calculating the initial distributions of the total angular momentum projection onto the symmetry axis for the nuclear systems formed in the reactions of complete fusion of deformed nuclei with spins. The angular distributions of fission fragments for the ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th},{}^{12}\mathrm{C}+{}^{235,236,238}\mathrm{U}$, and ${}^{13}\mathrm{C}+{}^{235}\mathrm{U}$ reactions have been analyzed within the dynamic approach over a range of sub- and above-barrier energies. The analysis allowed us to determine the relaxation time for the tilting mode and the fraction of fission events occurring in times not larger than the relaxation time for the tilting mode.

Conclusions: It is shown that the memory effects play an important role in the formation of the angular distributions of fission fragments for the reactions induced by heavy ions. The approach developed for analysis of the effects is a suitable tool to get insight into the complete fusion-fission dynamics, in particular, to investigate the mechanism of the complete fusion and fission time scale.

Figure 1

Schematic diagram of the summation of angular momenta in the complete fusion reaction between a light nucleus and a heavy target nucleus having a nonzero spin in the ground state (left). The total angular momentum of the nuclear system produced in a subbarrier complete fusion for the most probable orientation of the target nuclei with $s_t=0\hslash$ (middle) and $s_t\ne 0\hslash$ (right).

Figure 2

Fission cross sections as functions of $E_{\mathrm{c}.\mathrm{m}.}$ for the ${}^{12}\mathrm{C}+{}^{236}\mathrm{U}$ (a), ${}^{12}\mathrm{C}+{}^{235}\mathrm{U}$ (b), ${}^{12}\mathrm{C}+{}^{238}\mathrm{U}$ (c), ${}^{13}\mathrm{C}+{}^{235}\mathrm{U}$ (d), and ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}$ (e) reactions. Experimental data are represented by points: $▴$ are taken from Ref. [31], $\diamond$ from Ref. [32], $\circ$ from Ref. [14], $\square$ from Ref. [33], $\blacksquare$ from Ref. [34], $•$ from Ref. [4], and $⧫$ from Ref. [35]. The curves are the model calculations (see text).

Figure 3

Initial distributions over $K$ at the subbarrier energy $E_{\mathrm{c}.\mathrm{m}.}=55$ MeV. The calculations are presented for ${}^{247}\mathrm{Cf}$ produced with $J=17/2\hslash$ in the ${}^{12}\mathrm{C}+{}^{235}\mathrm{U}$ reaction $\left(▴\right)$ and for ${}^{248}\mathrm{Cf}$ produced with $J=9\hslash$ in the ${}^{12}\mathrm{C}+{}^{236}\mathrm{U}$ reaction $(•)$.

Figure 4

Initial distributions over $K$ for ${}^{247}\mathrm{Cf}$ produced with $J=17/2\hslash$ in the ${}^{12}\mathrm{C}+{}^{235}\mathrm{U}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=55\left(\blacksquare \right)$, 60 ($▴$), 65 ($•$), and 70 ($⧫$) MeV.

Figure 5

Initial distributions over $K$ for ${}^{248}\mathrm{Cf}$ produced with $J=8\hslash$ in the ${}^{12}\mathrm{C}+{}^{236}\mathrm{U}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=55\left(\blacksquare \right)$, 60 ($▴$), 65 ($•$), and 70 ($⧫$) MeV.

Figure 6

Initial distributions over $M$ for ${}^{247}\mathrm{Cf}$ produced in the ${}^{12}\mathrm{C}+{}^{235}\mathrm{U}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=55\left(\blacksquare \right)$, 60 ($▴$), 65 ($•$), and 70 ($⧫$) MeV.

Figure 7

Multiplicity of prescission neutrons (${\nu }_{\mathrm{pre}}$). Experimental data for the ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}$ reaction are taken from Ref. [46]. The calculation results are presented for the reaction leading to production of the ${}^{248}\mathrm{Cf}$ compound nuclei: the solid line is for the ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}$ reaction, the dashed line for the ${}^{12}\mathrm{C}+{}^{236}\mathrm{U}$ reaction, and the dashed-dotted line for the ${}^{13}\mathrm{C}+{}^{235}\mathrm{U}$ reaction.

Figure 8

Comparison of ADA calculation with different ${\tau }_K=10\times {10}^{-21}$ s ($\square$), $20\times {10}^{-21}(\circ )$, and $30\times {10}^{-21}$ s ($▵$) and the experimental data for the ${}^{12}\mathrm{C}+{}^{236}\mathrm{U}$ reaction ($•$, Refs. [14, 15]; $\blacksquare$, Ref. [32]).

Figure 9

Comparison of the experimental ADA and the calculations performed with ${\tau }_K=20\times {10}^{-21}$ s. For the ${}^{12}\mathrm{C}+{}^{236}\mathrm{U}$ reaction the experimental points are marked as in Fig. 8, the calculation results are presented by $\circ$. For the ${}^{12}\mathrm{C}+{}^{235}\mathrm{U}$ reaction, the experimental points [14, 15] are marked as $▴$, and the calculation results are presented by $\square$.

Figure 10

The same as in Fig. 9. For the ${}^{12}\mathrm{C}+{}^{238}\mathrm{U}$ reaction the experimental points [14, 15] are marked as $•$, and the calculation results are marked as $\circ$. For the ${}^{13}\mathrm{C}+{}^{235}\mathrm{U}$ reaction, the experimental points [34] are marked as $▴$, and the calculation results are marked as $▵$.

Figure 11

The same as in Fig. 9 for the ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}$ reaction. Experimental points are marked as follows: $•$, from Ref. [4]; $\blacksquare$, from Ref. [32]; $⧫$, from Ref. [9]; and $▴$, from Ref. [47]. The calculation results are presented as $\circ$.

Figure 12

The fraction of fission events occurring in a time not exceeding ${\tau }_K$ in the total yield of fission fragments for the ${}^{12}\mathrm{C}+{}^{236}\mathrm{U}(\circ ),{}^{12}\mathrm{C}+{}^{235}\mathrm{U}(\diamond ),{}^{12}\mathrm{C}+{}^{238}\mathrm{U}(\square ),{}^{13}\mathrm{C}+{}^{235}\mathrm{U}\left(▵\right)$, and ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}(▿)$ reactions. For the ${}^{12}\mathrm{C}+{}^{235}\mathrm{U}$ reaction the calculation results are not presented in the range $E_{\mathrm{c}.\mathrm{m}.}=58–84$ MeV due to the overlap with the results for the ${}^{12}\mathrm{C}+{}^{236}\mathrm{U}$ and ${}^{13}\mathrm{C}+{}^{235}\mathrm{U}$ reactions.