Four-dimensional Langevin dynamics of heavy-ion-induced fission

A four-dimensional dynamical model based on Langevin equations was developed and applied to calculate a wide set of experimental observables for the reactions ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}\to {}^{224}\mathrm{Th}$ and ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}\to {}^{248}\mathrm{Cf}$ over a wide range of excitation energy. The fusion-fission and evaporation residue cross sections, fission fragment mass-energy distribution parameters, prescission neutron multiplicities, and anisotropy of angular distribution of fission fragments could be reasonably reproduced using a modified one-body mechanism for nuclear friction with a reduction coefficient of the contribution from a wall formula $k_s\simeq 0.25$ and a dissipation coefficient for the orientation degree of freedom ($K$ coordinate) ${\gamma }_K\simeq$ 0.077 ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$. Inclusion of the $K$ coordinate into calculation of potential energy changes the stiffness of the nucleus with respect to mass asymmetry coordinate for the values of $K\ne 0$ and results in a shift of the Businaro-Gallone point towards larger $Z^2/A$ values. The experimental data on the fission fragment mass-energy distribution parameters together with mean prescission neutron multiplicity for heavy fissioning nuclei are reproduced through the four-dimensional Langevin calculations more accurately than through three-dimensional calculations.

Figure 1

The Helmholtz free energy along the mean fission trajectory for the ${}^{224}\mathrm{Th}$ compound nucleus as the function of the elongation parameter $q_1$, and corresponding fission barriers ($B_f$) for different combinations of $L$ and $K$ values.

Figure 2

The Helmholtz free energy for the compound nucleus ${}^{224}\mathrm{Th}$ in the collective coordinates $q_1$ and $K$ at $T=1.5$ MeV. The parameters $h$ and $\alpha$ are fixed and equal to zero. The numbers at the contour lines indicate the Helmholtz free energy values in MeV. The dashed curve shows the dependence of saddle-point deformations on $K$.

Figure 3

Quadrupole moments of the saddle-point configurations ${\mathcal{Q}}_{\mathrm{sd}}$ (in barns) as a function of the parameter $Z^2/A$ for beta-stable nuclei at nuclear temperature $T_{\mathrm{sd}}=1.5$ MeV. The solid curve corresponds to the polynomial approximation of the calculated data for a nonrotating nucleus ($L=0$). Dash and dash-dotted curves are polynomial approximations of the results obtained for $L=40$ with $K=0$ and $K=L=40$, respectively.

Figure 4

The stiffness of the free energy landscape for the ${}^{224}\mathrm{Th}$ compound nucleus with respect to the mass asymmetry coordinate $\eta$ along the mean fission trajectory as a function of the elongation parameter $q_1$. Different combinations of $L$ and $K$ are considered as indicated.

Figure 5

The stiffness of the beta-stable nuclei with respect to mass asymmetry coordinate at the saddle (black symbols and curves) and scission (red symbols and curves) points, $d^{2}F/d{\eta }^2$ (in MeV), for nuclei along the beta-stability line at temperature $T=1.5$ MeV. The solid curves correspond to the polynomial approximation for the calculated data for the non-rotating nucleus ($L=0$). Dashed and dash-dotted curves are polynomial approximations for the $L=40$ and $K=0$ and $K=L=40$ cases respectively. The arrows with the numbers indicate the corresponding Businaro-Gallone point for each pair of $L$ and $K$.

Figure 6

The fusion-fission ${\sigma }_{\mathit{FF}}$ (a) and evaporation residues ${\sigma }_{\mathit{ER}}$ (b) cross sections for the compound nucleus ${}^{224}\mathrm{Th}$ as a function of center-of-mass energy. The open symbols are experimental data: Ref. [67] (triangles), Ref. [72] (circles), Ref. [64] (diamonds), and Ref. [63] (inverted triangles). The filled symbols are the calculated results with $k_s=0.25$ and ${\gamma }_K=0.077$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ (circles), with $k_s=0.25$ and ${\gamma }_K=0.06$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ (diamonds), and with $k_s=0.5$ and ${\gamma }_K=0.077$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ (triangles).

Figure 7

The prescission neutron multiplicity for the compound nucleus ${}^{224}\mathrm{Th}$ as a function of center-of-mass energy. The open symbols are experimental data from Ref. [71]. The filled symbols are the calculated results with different values of $k_s$ and ${\gamma }_K$, which are marked with the same symbols as in Fig. 6.

Figure 8

The ${\sigma }_M^2$ (a) and ${\sigma }_{E_K}^2$ (b) for the compound nucleus ${}^{224}\mathrm{Th}$ as a function of center-of-mass energy. The open symbols are experimental data from Ref. [70]. The filled symbols are the calculated results with different values of $k_s$ and ${\gamma }_K$, as in Fig. 6.

Figure 9

The prescission neutron multiplicity (a), ${\sigma }_M^2$ (b), and ${\sigma }_{E_K}^2$ (c) for the compound nucleus ${}^{248}\mathrm{Cf}$ as a function of center-of-mass energy. The open symbols are experimental data from Refs. [73, 74]. The filled symbols are the calculated results with $k_s=0.25$ and ${\gamma }_K=0.077$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ (circles), with $k_s=0.25$ and ${\gamma }_K=0.308$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ (diamonds), and with $k_s=0.5$ and ${\gamma }_K=0.077$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ (triangles).

Figure 10

The fission fragment angular distributions for the compound nucleus ${}^{248}\mathrm{Cf}$ at $E_{\mathrm{lab}}=120$ (a), 140 (b), and 160 (c) MeV. The open symbols are experimental data from Ref. [72] and the solid circles are results of 4D calculations with $k_s=0.25$ and ${\gamma }_K=0.077$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$.

Figure 11

The anisotropy of the fission fragment angular distribution as a function of the center-of-mass energy for the reaction ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}\to {}^{248}\mathrm{Cf}$. The calculated points are connected by lines to guide the eye. The open symbols are the experimental data: Ref. [72] (circles), Ref. [75] (squares). The filled symbols are the calculated results with different values of $k_s$ and ${\gamma }_K$, which are marked in the same order as in Fig. 9. The dotted and dashed curves present the results predicted by the SCTS and SPTS models, respectively.

Figure 12

The anisotropy of the fission fragment angular distribution as a function of the center-of-mass energy for the reaction ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}\to {}^{224}\mathrm{Th}$. The calculated points are connected by lines to guide the eye. The open symbols are the experimental data [72, 103]. The filled symbols are the calculated results with different values of $k_s$ and ${\gamma }_K$, as in Fig. 6. The dotted and dashed curves present the results predicted by the SCTS and SPTS models, respectively.

Figure 13

The anisotropy of the angular distribution as a function of the fission fragment mass for the reaction ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}\to {}^{248}\mathrm{Cf}$. The open squares are experimental data at $E_{\mathrm{lab}}=96$ MeV from Ref. [76], and the circles are calculated results at $E_{\mathrm{lab}}=95$ MeV with $k_s=0.25$ and ${\gamma }_K=0.077$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$.

Figure 14

The relaxation time of $K$ coordinate ${\tau }_K$ as a function of the elongation parameter for the nucleus ${}^{224}\mathrm{Th}$ obtained with Eq. (21). The solid curve is calculated at ${\gamma }_K=0.077$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ and $I=40$, the dashed curve at ${\gamma }_K=0.077$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ and $I=60$, and the dotted curve at ${\gamma }_K=0.06$ ${\left(\mathrm{MeV}\mathrm{zs}\right)}^{-1/2}$ and $I=60$.