Fission yield model for astrophysical use based on the four-dimensional Langevin model

. Nuclear fission is expected to play an essential role in the r-process nucleosynthesis via the fission recycling process. However, the impact of the fission recycling process on the r-process is still unclear due to the ambiguity of the fission yields for nuclei far from the beta-stability line and superheavy nuclei. Our four-dimensional Langevin model can reproduce experimental nuclear fission properties, such as fission fragment mass distributions and total kinetic energies in a systematic manner including their sudden changes due to shell structure. Then we developed a phenomenological yield model for nuclei Z=92 to 122, ranging from neutron deficient to very neutron-rich ones, by fitting the fission fragment mass yield obtained by the Langevin model with five Gaussians. We use the charge distributions for each isobar based on a parametrization which we have evaluated based on thousands of experimental data for actinides, to obtain the fission yields in the form of (Z, N) distributions. What is unique here is the sudden change of the mass distribution from a 2-peak to 1-peak structure for the region of 256 Fm, and the appearance of a 5-peak structure for superheavy nuclei predicted by the 4D Langevin calculation.


Introduction
Nuclear fission is a fundamental phenomenon. More than 80 years have passed since nuclear fission was found. Nevertheless its detailed mechanism is still unknown. Especially, we do not have concrete information on fissions of superheavy nuclei at very low temperature below a few MeV. Such information is important to investigate the origin of heavy elements, i.e., r-process nucleosynthesis. Fission yields of superheavy nuclei contribute to the r-process nucleosynthesis via fission recycling. In the fission recycling scenario, the fission fragments synthesized by nuclear fission of a heavy nucleus will be the seed nuclei of the following r-process path. Since the time scale of the r-process is much longer than that of nuclear fission, several fission recycling paths can take place during the r-process nucleosynthesis.
The network calculation of r-process nucleosynthesis requires various nuclear inputs such as neutron capture rates, nuclear masses, nuclear structures, beta-decay rates, delayed neutron emissions, fission barrier height, and fission fragment yields etc. Among nuclear inputs for the r-process calculations, the uncertainty of fission yields is still opaque because the fission product yields, especially the yields of superheavy nuclei, are very model-dependent. To improve such a situation, we need reliable predictions for fission yields. To predict the fission yield more precisely, various theoretical approaches have been developed. * Corresponding author: ishizuka.c.aa@m.titech.ac.jp In this decade, multi-messenger astrophysics has opened the door to a new era of the r-process. Multimessenger astrophysics provides information on the site of the r-process. After the epoch making obsesrvation of GW170817, now we know that the r-process will take place in binary neutron star mergers and black hole neutron star mergers. In addition to these, type-II supernovae are still possible site for the r-process because many previous works suggest that we need another site apart from neutron star mergers, neutron star-black hole mergers to explain r-process components in the present solar abundance. In this paper, we focus on a fission fragment yield model for the rprocess.

Model
We have developed a fission yield model for the rprocess based on the four-dimensional Langevin model [1]. This model describes neutron-induced nuclear fission as the time evolution of the shape of the compound nucleus, following equations of motion with friction and random force, i.e., the Langevin equation, where and are four shape coordinates and their conjugate momenta. The quantities, , , and , are initial mass tensor, the friction tensor, and the random force, respectively.The nuclear shape model is the two center model proposed by Maruhn and Greiner [2]. Parameters are elongation, mass asymmetry, and independent deformations of two fragments. We adopted the Werner-Wheeler approximation [3] for the mass tensor, and the Wall and Window formula [4] for the friction tensor. The shell and pairing corrections were calculated by the BCS approximation and the Strutinsky prescription from the single-particle energies obtained with Two-Center Woods-Saxon models. In our Langevin model, effective temperature T* and temperature T are included. T* relates to and via Einstein relation, while T is derived from the inertial energy Eint by Eint=aT 2 . Our 4D model can reproduce two experimental systematics, including anomalies around Fm and Md, without fitting parameters [1,5]. First is the systematics on the averaged fission fragment mass as a function of the mass number of a compound nucleus, and the second is that on the averaged total kinetic energy (TKE) of fission fragments as a function of 2 1/3 ⁄ , where the Z and A are the charge and mass numbers of a compound nucleus. Using the Langevin model, we investigated the fission fragment mass yields of about two hundred nuclei with a charge number from 92 to 122 from the neutron-deficient to the neutron-rich side. According to our previous work [6], nuclear fissions of superheavy nuclei typically show three or four peak structures at low excitation energy near the fission barriers. Such peak structures of fission fragment mass yields can be easily understood by TKE components. For example, in nuclear fission of 294 Og, the main components of fission fragments are super-asymmetric and super-short modes. However, additionally, we can observe standard and super-long modes, though the abundances of such components are very small. Note that the peak position of super-short and that of super-long are the same in the fission mass yields because both consist of symmetric fissions. Then, we need five Gaussians to fit a nuclear mass yield of a superheavy nucleus. Fig.1 Gaussian fitting parameters as a function of neutron number of a compound nucleus.
In Fig.1 the left two panels show schematic views of superposition of four Gaussians for three-peaks' mass yield and five Gaussians for four peaks. The right two panels show three parameters of each Gaussian component, side-left, side-right, center, center-left, center-right and center-base in the case of Oganesson (Og) isotopes. Red, blue and green symbols are parameters a, b, and c of a Gauss function, respectively. The upper one of the right panels in Fig.1 corresponds to the center-based plotted with purple solid line in left two panels. The lower-right panel of Fig.1 corresponds to the center case plotted with red solid line in the upperleft panel. Note that the center Gaussian does not appear in four-peak mass yields, though the Og-isotopes with N=180 and 184 have four-peaks. On the other hand, the center-base Gaussian is the common component in any peak structures.
We found that all three Gaussian parameters show clear linear correlation to the neutron mass number of a compound nucleus, respectively. Such a strong correlation can be seen in all the Gaussians. It suggests that we can interpolate the Gaussian parameters to some nucleus for which we did not perform the Langevin calculations.
Then we constructed our semi-empirical fission fragment yield model Y( , ) combining the fission mass yield model Y( ) based on the superposition of five Gaussians fitted to results of the four-dimensional Langevin calculations, and our empirical charge distribution formula, which we found when we analyzed thousands of experimental fission yields [7]. For the details of the above formula, please see the reference [7]. Thus, we developed a semi-empirical fission yield model Y(Z, A) for nuclides with charge number Z=90-122, A=221-360, which are required in the r-process nucleosynthesis.. In this section, we first show a representative mass and charge distribution of our yield model. Then we discuss the influence of the fission yield model on the rprocess by performing the simple r-process calculation.   Fig.3 shows the fission fragment mass yields of 260 Pu of our model. 260 Pu is known as one of the critical elements of the fission recycling process, especially for the r-process in binary neutron star mergers. We found that the two peaks in our model are much sharper than the other models. We fitted fission yields simulated by the four-dimensional Langevin model with five Gaussians. Obtained Gaussian parameter (a, b, c) shows the strong linear correlation between neutron number among isotopes, respectively, as shown in Fig.1. That is true in Pu isotopes. For Pu isotopes, we simulated nuclear fissions of compound nuclei  Pu with the Langevin model. In those Pu isotopes, 238-262 Pu resulted in asymmetric fissions, and the others were symmetric fissions. Gaussian parameter (a, b, c) for 238-262 Pu aligns on a straight line as a function of neutron numbers contained in isotopes, respectively. Then we calculated the 260 Pu in Fig.3 using three Gaussian parameters (a, b, c) interpolated by neighborhood isotopes. Note that we simulated the nuclear fission of the compound nucleus 240 Pu (n+ 239 Pu), which is one of the most examined fissioning nuclei, and fitted Gaussian parameters are also on those straight lines. Our Langevin result of 240 Pu reproduces standard nuclear libraries well, and the peak positions (parameter b shown with blue squares) and the widths (parameter c shown in green triangles) continuously change among Pu isotopes. Therefore, our peaks of 260 Pu are considered more reasonable, though the peak positions of fission fragments are very modeldependent and different from ours. Using such a yield model, we evaluated the influence of the r-process nucleosynthesis with the simple model proposed by Shinya Wanajo in 2018 [8] which can provide a nuclear abundance synthesized by single rprocess path. In Wanajo's model, the temperature profile is based on the EOS proposed by Timmes-Swesty in 2000. The density profile ρ( ) is the typical trajectory of Hayashi et al. (2022) [9]. ρ( )= 0 ((t+t0)/t0) -3 , (4) where 0 = 4e11 g/cc (n-drip density), and t0 = 1.56e-3 sec. The entropy time evolves as follows, s0 = 1 kB /nucleon, si+1 = si + dsi (5) ds = dq/T, where i is the calculation step. The electron fraction is Ye=0.05 which will be realized in black hole-neutron star mergers. Fig.4 shows the abundance of a single r-process path at the electron fraction Ye=0.05. The Red line is the abundance pattern with the present model, while the black dots are the solar abundance. For comparison, the results with other fission yields are also shown. The blue, pink, green lines are GEF, Kodama-Takahashi, Kodama-Takahashi (symmetric version), respectively.

Results and Discussion
Note that we should focus on the results at larger mass numbers than A=130, the main source of the solar abundance pattern below A=130 is not the r-process. The difference in the fission yield pattern directly affects the abundance pattern. In the rare-earth peak, the GEF model looks better around 170, but our model agrees well around 155 to 160, i.e., the third peak of the r-process elements which is generally difficult to reproduce. In our model, the abundant superasymmetric components of super-heavy nuclear fission may improve the agreement to the third peak.
In this paper, we investigate how our yield model works in the r-process nucleosynthesis. To that aim, we fixed the electron ratio at very low Ye=0.05 because we can clearly see the contribution of the fission model to the r-process abundance under such a low Ye environment. When we perform the same calculation with higher Ye expected during neutron-neutron star mergers, we cannot see so large difference from Kodama-Takahashi. We believe that our model based on the four-dimensional Langevin results is reliable enough to perform the quantitative discussion. We plan to apply our model to the full calculation of the r-process nucleosynthesis.

Summary
To summarize, we performed the four-dimensional Langevin calculations for two hundred nuclei. After that, we fitted the obtained mass yields by five Gaussians. We found that these Gaussian parameters have clear systematics against the neutron number of the isotopes. Combining the fission mass yield model based on that systematics with our semi-empirical charge distribution formula, we constructed a fission yield model Y(Z, A) for nuclides with charge number Z=90-122, A=221-360. Then we showed the mass yield of 260 Pu as a representative case of the main contributors to the rprocess. Our model gives sharper peaks and broader tails than the other models. For further discussion, we will compare our model with others in more nuclei with mass numbers 260 to 280, which are the main contributors to the r-process via fission recycling.
We also performed a simple calculation to examine the influence to the r-process nucleosynthesis assuming the black hole-neutron star merger As a result, we found that our model can reproduce the third peak of the r-process well. For more detailed study, we will perform the full calculation of the r-process nucleosynthesis.