Dissipation strength of the tilting degree of freedom in fusion-fission reactions

The four-dimensional Langevin model was applied to calculate a wide set of experimental observables for compound nuclei, formed in heavy-ion fusion-fission reactions. A modified one-body mechanism for nuclear dissipation with a reduction coefficient ks of the contribution from a ”wall” formula was used for shapes parameters. Different possibilities of deformation-dependent dissipation coefficient for the K coordinate (γK) were investigated. Presented results demonstrate that the influence of the ks and γK parameters on the calculated quantities can be selectively probed. It was found that it is possible to describe experimental data with the deformation-dependent γK coefficient. One of the possibility is to use large values of γK 0.2 (MeV zs)−1/2 for compact shapes featuring no neck and small values of γK 0.0077 (MeV zs)−1/2 for elongated shapes. Fission still is one of the most interesting and challenging topics in nuclear physics providing a perfect opportunity to investigate the large scale evolution of initial compound nucleus into fission products. During the past two decades stochastic approach based on multidimensional Langevin equations has been extensively and rather successfully used to elucidate many problems of collective nuclear dynamics in fusion-fission reactions at high excitation energies [1, 2]. A reasonable choice of collective degrees of freedom for modeling shape evolution and considering particle evaporation allow modeling the complex interplay between static and dynamical effects in fission and succeeding in explaining a wide range of experimental data [3, 4]. 201 , 0 0 EPJ Web of Conferences DOI: 10.1051/ conf/201611 0 0 epj

Fission still is one of the most interesting and challenging topics in nuclear physics providing a perfect opportunity to investigate the large scale evolution of initial compound nucleus into fission products.During the past two decades stochastic approach based on multidimensional Langevin equations has been extensively and rather successfully used to elucidate many problems of collective nuclear dynamics in fusion-fission reactions at high excitation energies [1,2].A reasonable choice of collective degrees of freedom for modeling shape evolution and considering particle evaporation allow modeling the complex interplay between static and dynamical effects in fission and succeeding in explaining a wide range of experimental data [3,4].The significance of orientation degree of freedom (K coordinate), which is the projection of the total angular momentum onto the symmetry axis of fissioning nucleus, was demonstrated and an overdamped Langevin equation for K coordinate were introduced [5,6].
In the present study we used recently developed 4D dynamical model for the description of fissioning nucleus shape evolution based on a stochastic approach [1,2,4,7,8].The detailed description of the model could be found in Refs.[2,4] and here we give only a short summary.Three collective shape coordinates and the orientation degree of freedom (K coordinate) were considered dynamically from the ground state deformation to the scission into fission fragments.A modified one-body mechanism of nuclear dissipation [9,10] was employed to determine the dissipative part of the driving forces with reduction coefficient from the "wall" formula k s .The value k s = 1.0 corresponds to the "wall" and "wall-plus-window" formulas, whereas values 0.2 < k s < 0.5 allow to reproduce different features of the experimental fission fragment MED and particle multiplicities in multidimensional Langevin calculations [2,8,11].
In Refs.[12][13][14] it was argued that chaos-theory related ideas [14] can be used to calculate the value of the reduction coefficient k s as a function of deformation of the fissioning nucleus.The applications of this approach to calculate the coefficient k s (q) for studying different fission characteristics were rather successful, and have shown that such calculations yield almost the same results as those using the constant k s coefficient from the interval 0.25 < k s < 0.5 [6,12,15].
We use three different options to model the deformation dependence of γ K (q) coefficient as described in Ref. [16].As the first one we used , for shapes without a neck; γ neck K (q), for shapes with a neck, (1) where γ const K is a variable parameter independent of nuclear deformation.After appearance of the neck in nuclear shape the γ const K value is joined smoothly with γ neck K (q), which was obtained in Refs.[5,16].The third one is given by equation for shapes without a neck; γ neck K (q), for shapes with a neck, (2) where γ cshape K (q) is the extrapolation of γ neck K (q) to the mononuclear shapes, featuring no neck.The detailed description of these three prescriptions could be found in Ref. [16].
In the present calculations we investigate the influence of k s and deformation-dependent γ K (q) parameters on the fusion-fission (σ fis ), evaporation residues (σ ER ) cross sections, and anisotropy of fission fragment angular distribution in 4D dynamical model.We performed calculations for the reactions 16 O + 188 Os → 204 Po (E lab = 84, 89, 94, and 99 MeV) [17][18][19] and 16 O + 184 Pt → 200 Pb (E lab = 91.6,102.5, and 107.9 MeV) [20][21][22][23].The calculations were performed at different values of k s and γ K .In figures we present the options which allow good reproduction of experimental data.In 201 fig. 1 one can see that k s = 0.25 or k s (q) obtained from chaos-weighted wall formula allows well reproduce the fission cross section for the fission of 204 Po.The calculated fission cross section does not depend neither on the γ K option used nor on the absolute γ K values, so we did not show it in fig. 1 in order not to complicate figure.Our investigations [16] show that only the anisotropy of fission fragment angular distribution is dependent on both k s and γ K values, as it was found in previous investigation with heavy compound nuclei [6].The calculated anisotropy of fission fragment angular distribution is presented in fig 2. One can see that at both k s values (k s = 0.5 and k s (q)) it is possible to reproduce the experimental data on anisotropy at constant γ K = 0.077 (MeV zs) −1/2 or with the deformation-dependent γ K coefficient: to use large values of γ K 0.2 (MeV zs) −1/2 for compact shapes featuring no neck and small values of γ K (q) for dinuclear shapes.The 4D calculations with γ (3) K (q) option underestimate the experimental anisotropy of fission fragment angular distribution.
The comparison of theoretical 4D calculations with experimental data for the 200 Pb is presented in fig. 3.In figure 3(a) the calculated fusion-fission (σ fis ) and evaporation residues (σ ER ) cross sections are compared with the experimental data at different excitation energies and k s values.We found that γ K does not influence the cross sections, so we did not present in fig.3(a) the results obtained with different γ K values.One can see that 4D calculations reproduce worse the cross sections for the 200 Pb compound nucleus than for the 204 Po compound nucleus.The same is true also for the calculated results for the anisotropy of fission fragment angular distribution presented in fig.3(b).In the present calculations we needed to increase the k s and/or γ K values, which generate large values of the anisotropy.However, even very large values of γ K (the option γ K (q) with the large values of γ const K = 0.4 -0.6 (MeV zs) −1/2 for compact shapes predicts the anisotropy about 15% less than experimental data.However, we should mention that there is significant difference in experimental data on fusion cross section for the 200 Pb compound nucleus between old [23] and recent [20] experimental data.Thus, possible revision of experimental data [23]   between theoretical and experimental data.
Concluding we can state that it was found that in the 4D Langevin calculations it is possible to describe experimental data with the deformationdependent γ K coefficient.One of the possibility is to use large values of γ K 0.2 (MeV zs) −1/2 for compact shapes featuring no neck and small values of γ K 0.0077 (MeV zs) −1/2 for elongated shapes.Using such a different value of γ K for deformations before saddle and along the saddleto-scission descent tends the dynamics of the K coordinate to the prediction of the transition state model at saddle point.
This study was partially supported by the Russian Foundation for Basic Research, Research Project No. 13-02-00168 (Russia).

Figure 1 :
Figure 1: (Color online) The fusionfission cross section for 204 Po.The open squares are experimental data.The filled symbols are results of 4D calculations: circles with k s =0.25, triangles with k s (q), and diamonds k s =1.

( 2 )
K (q) with the large values of γ const K = 0.4 -0.6 (MeV zs) −1/2 for compact shapes) and large k s values does not allow to reproduce the experimental values of the anisotropy of fission fragment angular distribution.Moreover, the anisotropy reaches the saturation at γ const K = 0.4 (MeV zs) −1/2 , and does not grow up after the further increase of γ const K independently on k s value used.The results of present 4D calculations with k s =1.0 and γ(2)

Figure 3 :
Figure 3: (Color online) The σ fis and σ ER (a) and the anisotropy of fission fragment angular distribution (b) for 200 Pb at different k s values and γ K options.The panel (b) demonstrates results for the E lab = 107.9MeV.The open symbols are experimental data.The hatched area at panel (b) determines experimental data with error bars.The filled symbols are results of 4D calculations: circles with k s =0.25; triangles with k s (q); and diamonds with k s =1.0.