Nuclear fission is a fundamental phenomenon that brings us enormous energy in the form of nuclear energy. There is abundant data on the fission of some significant nuclei, such as neutron-induced fission on ²³⁵U or ²³⁸Pu for the safe and efficient use of nuclear energy. However, it is still difficult to predict or measure observables such as the distribution of fission fragment yield or the total kinetic energies (TKEs) and their correlations accurately enough even today [1–3]. For example, radioactivities of the heavy actinide nuclei often make it difficult to measure fission-related quantities with good accuracy. From the theoretical side, it has been known that average mass numbers of the heavy and light fragments and TKE has an anomalous trend superposed on a systematical one.

Information on nuclear fission is essential not only for the development of nuclear energy but also in nuclear astrophysics, such as the r-process (rapid neutron capture process) nucleosynthesis. The concept of r-process with fission recycling has a long history after Burbidge-Burbidge proposed it in 1957 [4]. Since then, the r-process sites have been hot topics in astrophysics. A few astronomical phenomena, such as type-II supernovae and neutron star mergers, have been enthusiastically studied as their actual site. In 2017, this situation on the r-process drastically changed. The gravitational wave from GW170817 [5] elucidated that one of the r-process sites was neutron star mergers. Now the r-process is believed to be the origin of heavy elements during neutron star (NS) mergers. Under the extremely high neutron flux produced by NS-NS or NS-BH (black hole) mergers, the r-process synthesizes various superheavy nuclei (SHN). Then the fission fragments from SHNs are considered to be seed nuclei of the subsequent r-process path [6, 7]. Several fission recyclings have been expected to take place in the r-process. It opened a new era for the r-process and has required nuclear physicists to access more information on the nuclear fission of superheavy elements because that brings a significant ambiguity among nuclear inputs of the r-process simulations.

In the r-process environment, fission yields provide information [8] on which nuclei will be the dominant seed nuclei of the r-process. TKEs are also essential to estimate the number of neutrons emitted by fission recycling and how much energy is released as a local source of heating. Some use a microscopic approach as density functional theory [9] to predict these, while we use the Langevin model of the macroscopic-microscopic approaches here.

The Langevin approach [10] is successfully applied in various branches of theoretical physics and chemistry for many years. In nuclear physics, this approach is used to describe fission or fusion processes at the excitations well above the fission barrier [11–21]. The approach describes quite well the mass distributions and kinetic energies of fission fragments, the multiplicities of emitted neutrons, and other observable of fission or fusion processes.

Our Langevin model can simultaneously reproduce fission yields and TKEs well without parameter adjustments [18, 20, 22]. Thus we can provide reliable fission properties of superheavy elements (SHEs) at low excitation energies [23, 24] where the experiments are challenging.

This paper is organized as follows. In Section 2, we briefly introduce our four-dimensional Langevin model used to predict features of the SHN fission and Brosa model [25], which is used to analyze the TKEs. In Section 3, we show fission fragment mass distributions (FFMDs) and the total kinetics energies (TKEs) released in the nuclear fission of actinides to SHEs. We also display the fragment deformations of these nuclei. Based on those results, we found the systematics in the FFMDs and TKEs. We summarize our results in Sec.4.

This section reports the fission simulations of important actinides, more heavy nuclei for which the experimental information is available, and selected SHEs that demonstrate the role of the shell effects in the fission yields. The calculations were done with the same set of parameters for all nuclei. Figure 2 displays the fission fragment mass yields at the excitation energies slightly above the fission barrier. Note, that the reference data of ²³⁶U to ²⁵²Cm are from the JENDL/FPY-2011 [34] and those reference data are for post-neutron emission. The peaks of mass disstribution are not symmetric with respect to A/2 and are shifted a little bit to the left. Our distribution are calculated for the pre-neutron emission and are symmetric with respect to A/2. The reference data of ^256−259 Fm, ²⁶⁰Md, and ²⁵⁹Lr are the experimental data of spontaneous fission [35–37]. These data and our calculations are both for pre-neutron emission. In this case the position of peaks is perfectly reproduced.

On average, our calculations reproduce quite well the peak width and position for nuclear fission of ²³⁶U to ²⁵⁹Lr, including Fm-isotopes, as a compound system. The transition from mass-asymmetric to mass-symmetric distribution between ²⁵⁷Fm and ²⁵⁸Fm is also qualitatively reproduced, though the transition is not as sharp as in experiments on spontaneous fission [37]. The similar mild transition from asymmetric to symmetric yield between ²⁵⁶Fm and ²⁵⁸Fm was obtained also in a resent work [38] within a Metropolis method for strongly damped fission dynamics. The transition from asymmetric to symmetric yield in Fermium isotopes was qualitatively reproduced also in [39] within the time-dependent generator-coordinate formalism and in [40] within the three-dimensional Langevin approach. The mildness of transition could be related to the fact that the calculations were done at the finite excitation energies, which partly suppresses the shell effects.

In the region of light super-heavies, the fragment yields are mass symmetric, like in the case of ²⁷⁴Hs. For heavier SHEs the strongly mass asymmetric peaks appear in mass distributions at A _H ≈ 208. For the nuclei from ²⁹⁶Lv to ³⁰⁶122, the symmetric and strongly mass asymmetric peaks are approximately of the same magnitude. In the case of ³⁰²120 and ³⁰⁶122, the symmetric peak is split into two components. In ³⁰²120 and ³⁰⁶122, the mean fragment mass of the lighter middle peak becomes almost constant at A _F = 144 as in the case of the middle peaks of ²⁷⁴Hs to ²⁹⁶Lv and the heavier peaks in actinides. On the other hand, the heavier middle peaks in ³⁰²120 and ³⁰⁶122 clearly deviate from A _F = 144 as shown in Figure 3.

The heavy fragment with A _H ≈ 208 was found recently [21] in Langevin calculations with the so-called Fourier shape parametrization. The ²⁰⁸Pb was obtained also as the main fission fragment of ²⁹⁴Og in [41] assuming the cluster radioactivity as an important decay mode in super-heavy nuclei.

In experimental work of Itkis group [42] the mass symmetric fission of Hassium isotopes was found in reactions ²²Ne + ²⁴⁹Cf →²⁷¹Hs and ²⁶Mg + ²⁴⁸Cm →²⁷⁴Hs and the mass asymmetric fission with the main fragment close to ²⁰⁸Pb in reactions ³⁶S + ²³⁸U →²⁷¹Hs and ⁵⁸Fe + ²⁰⁸Pb →²⁶⁶Hs. At the same time, the later experimental results of Itkis [43] show ¹³²Sn as the main fragment of mass asymmetric fission of super-heavy nuclei.

The reason of the difference may be the fact that the calculations and experiment are done not for the same process. The calculations are carried out for the fission of a completely equilibrated compound nucleus. Whereas in Itkis experiments, the super-heavy system is created in the fusion-fission reactions between heavy ions. The re-separation of ions can happen at each stage of the evolution of the combined system from the touching configuration till the separation of fission fragments. The main part of fission fragments comes from the quasi-fission events. It is not clear how well the quasi-fission events are separated in the experiment from the true fission events. This uncertainty could be the reason for disagreement between experimental and calculated results. The question, what is the main fission fragment of fission of super-heavy nuclei,¹³²Sn or ²⁰⁸Pb, still has to be clarified both from the experimental and theoretical sides.

To summarize the fission mass yields, we plot the averaged peak positions in Figure 3. Filled circles, open squares, filled squares, filled triangles, and open triangles denote the averaged mass of the heaviest peak A _H , the lightest peak A _L , the middle peak A _M (three peak FFMD case), the lighter middle peak A M L and the heavier middle peak A M H , respectively. Open circles with error bars are the measured averaged peaks [43] but at high E _x compared to our results at E _x = 7 MeV. The experimental values lie along the extrapolation line of the averaged mass systematics of actinides. In our calculations, the shell structures of A _F = 132–144 govern fission fragments with the heaviest peak position in actinides and the middle peaks (A _M and A M L ) in SHEs. However, the heaviest and lightest peaks in SHEs behave quite differently than in actinides. Our calculations suggest the strong influence of the double shell closure of ¹³²Sn and the deformed shell at Z _H = 54 in actinides as experiments, while the dominant shell changes to the double-shell closure of ²⁰⁸Pb.

To investigate the Brosa-like fission modes defined in the previous section, we plot the TKEs as a function of fission fragment mass number in Figure 4. We define the total kinetic energy of fission fragments by the sum of the pre-scission kinetic energy and the energy of Coulomb repulsion of fragments just before the scission in approximation of point charges, T K E = K E pre + e 2 Z L Z H / D . (4) Here KE _pre is the kinetic energy of relative motion of future fragments in fission direction, and D is the distance between the centers of mass of parts nucleus to the left and to the right from the neck.

It is seen from Figure 4 that the fission modes continuously change from the standard mode to the coexisting standard mode and supershort mode in actinides from ²³⁶U to ²⁵⁹Lr. SHEs clearly have a different mode component from that in actinides. We find the super asymmetric mode, which commonly appears in SHEs, in ²⁷⁴Hs, while the super asymmetric components in the fission yield of ²⁷⁴Hs are too small to see in Figure 2. The other fission modes in SHEs seem to be in common with those in actinides. We also find that the dominant mode gradually changes from the super short to super asymmetric from ²⁷⁴Hs to ²⁹²Fl.

In Figure 5, we display the quadrupole moments Q ₂₀ to elucidate which nuclear shell structure affects nuclear fissions in actinides and SHEs.

We calculate the quadrupole (Q ₂₀) and octupole (Q ₃₀) momentum of each fragment shape just before scission, as follows. First, we define the volume density as ρ i r = A i 1 r ⊂ V i V i , 1 r ⊂ V i = 1 : r ⊂ V i 0 : r ⊈ V i . (5) where r is the distance from the center of each fragment, and V _i is the volume of each fragment.

We calculate the Q ₂₀ and Q ₃₀ using the sharp-cut approximation of the fission fragment just before scission. Q 20 i = 5 16 π ∫ ρ i r 2 z 2 − x 2 − y 2 d 3 r , Q 30 i = 7 16 π ∫ ρ i r 2 z 2 − 3 z x 2 + y 2 d 3 r . (6)

Actinides from ²³⁶U to ²⁵⁹Lr show a similar trend of the Q ₂₀. We found that Q ₂₀ localize around (A _F , Z _F ) = (132, 50) with Q ₂₀ = 0, and spreads to (A _F , Z _F ) = (140, 52–56) where both of the Q ₂₀ and Q ₃₀ develop well at the ground state. Such a deformed shape is compatible with a fragment shape at scission. We also find the common structure in Q ₂₀ of SHEs from ²⁷⁴Hs to ³⁰⁶122. In SHEs, the double-closed shell of ²⁰⁸Pb becomes dominant as the mass number increases instead of ¹³²Sn, though we can observe both influences on Q ₂₀. In the nuclear fission of SHEs, spherical fragments with Q ₂₀ = 0 appear at A _F ≃ 208.

In the same manner, as Figures 2, 4, 5, the octupole moment Q ₃₀ of major actinides and SHEs are illustrated in Figure 6. In contrast to the strongly mass-dependent Q ₂₀, the octupole moments Q ₃₀ are almost constant as Q ₃₀ ≃ 2 b^3/2 for all nuclei picked up in Figure 6. Those Q ₃₀ values are consistent with experiments [44] and recent results [45] with time-dependent density functional theory, which describes quantum dynamics well. We also find that the Q ₃₀s of ²⁹⁴Og, ³⁰²120, and ³⁰⁶122 become larger than that of Uranium. Such strong octupole deformations in these nuclei result in fission fragments with A _F = 140 to 160. Thus, fission fragment shapes strongly depend on the mass number of a fissioning nucleus.

Figure 7 summarizes the Brosa-like fission modes shown in Figure 4. The lower panel shows the fission modes observed in our results of nuclear fissions at E _x = 7 MeV. The upper panel corresponds to a similar plot as the lower panel but for the fission fragments at Ex = 7, 10, and 30 MeV in the case of SHEs. Reference values are taken from the previous works by Brosa (with open squares and open diamonds) [25] and Itkis et al. (with open and filled triangles) [43]. The other symbols are all our results. For the details of those symbols, see the legend in Figure 7.

We carefully included experimental data of Itkis et al. [43]. For ²⁷⁴Hs case, we took a higher symmetric component of ²⁷⁴Hs, which was produced via ²⁶Mg + ²⁴⁸Cm as a black filled-triangle because the lower symmetric component corresponds to a liquid-drop-like state. The averaged TKE of the asymmetric component of ²⁷⁴Hs corresponds to the maximum yields. In their experiments, the symmetric components are very sparse for the nuclear fission of ²⁸⁶Cn at E _lab = 232 MeV and ³⁰²120 at E _lab = 328 MeV. So we only plot the asymmetric components with three filled triangles for each nucleus. In Itkis’s measurements, the TKE can be fitted with three Gaussian components, i.e., lower, Viola-like (middle), and higher components. Among those mean asymmetric-TKEs, Viola-like and higher components agreed well with our calculations.

We marked each dominant mode in centered black dots. The standard mode distributes along the solid red line in all actinides. The symmetric modes change from superlong to supershort at Es, though these two modes are on each mode’s expected lines. Extending our focus from the actinides to SHEs, we find that actinides and SHEs have common systematics for the standard, the superlong, and supershort modes. In addition to them, the super asymmetric mode merges. In Figures 7A,B, we also investigated the changes in the TKE systematics accompanied by the change in Ex. The systematics for E _x = 7 MeV is almost the same as that for E _x = 10 MeV, though the TKEs become lower at E _x = 10 MeV. At E _x = 30 MeV, all modes except for the symmetric mode suddenly disappear, i.e., we cannot see any asymmetric component in SHEs at E _x = 30 MeV. In order to clarify what happens at E _x = 30 MeV, we plot the fission fragment mass distribution and TKE(A) in Figure 8 for ²⁹⁶Lv as a representative case.

As shown in Figure 7B, we found that the asymmetric mode does not appear in the TKE systematics at E _x = 30 MeV. To understand it, we plot the FFMD and the TKE distributions in the case of ²⁹⁶Lv in Figure 8. Compared to the FFMDs at low E _x shown in Figure 2, FFMD at the high E _x of Figure 8 spreads over a broader mass range. The TKEs of these fission fragments show that the high excitation energy washes out the shell structure, and the system becomes a liquid-drop-like state.

At the end of this section, we focus on the influence of the shell effects on nuclear fission in SHEs. Figure 9 is the shell energy color map on the nuclear chart with the positions of sub and main peaks of FFMDs. The shell energy was taken from the KTUY mass formula [46] consisting of a gross term, an even-odd term, and a shell term. Their shell energy was obtained with spherical single-particle potentials. They treated deformed nuclei as a superposition of spherical nuclei. The resulting KTUY mass formula agrees with experimental masses very well within about 500 keV. In the case of Z = 120, these isotopes have four peak structures in the FFMDs, as seen in Figure 3. In Figure 9, pairs of filled triangles correspond to the main peaks, while those of open triangles are the sub-peak positions. We plot solid lines based on the assumption that the ratio of proton to neutron density is unchanged (UCD). In SHEs, the double shell closure of ²⁰⁸Pb governs nuclear fissions instead of ¹³²Sn in actinides, as discussed in Figure 5. The UCD lines of two Z = 120 isotopes, ^302,308 120, come very close to ²⁰⁸Pb. The shell correction energy there is much smaller than in the other regions. It makes fission fragments there very stable. However, this situation drastically changes when the isotopes become more neutron-rich. In ^322–338 120, the positions of the main and sub-peaks approach each other. That is because the UCD lines of those nuclei are no longer close to ²⁰⁸Pb.

Nuclear fission is still a challenging research topic even today. The number of prompt neutrons emitted from fission fragments is essential information to operate nuclear reactors and develop advanced reactors. To improve the prediction of prompt neutrons [47], we need an integrated understanding of the fission phenomenon based on nuclear physics. Especially, precise prediction of the TKE is crucial because it turns into the excitation energy of a fission fragment that emits prompt neutrons. In this paper, we systematically simulated nuclear fissions in actinides and SHEs. The fission model we used here is the four-dimensional Langevin model. Our previous works found that it can reproduce quite well representative characteristics of nuclear fission in actinides.

The main focus of the current paper is demonstrating the results of a systematic survey of fission properties from actinides to SHEs. We showed the fission yield FPY(A), the total kinetic energy TKE(A), the quadrupole moment Q ₂₀(A), and the octupole moment Q ₃₀(A) of each fragment produced by nuclear fission of ²³⁶U to ³⁰⁶122 as a compound system.

As a result, our Q ₃₀ values are consistent with those of the microscopic model and previous experiments. We also found that the double closed shell of ¹³²Sn and deformed shell at Z = 54 affect the nuclear fission of actinides and SHEs, though those impacts become weaker in SHEs. Instead, the double-closed shell of ²⁰⁸Pb is dominant in SHEs. Based on the above surveys, we derived two systematics in the averaged peak position of fission yield and the averaged TKE of each component of the Brosa-like model. This two systematics help evaluate the vital fission fragments and interpolate/extrapolate the averaged TKE. For the fission recycling in the r-process, we need to elucidate the fission properties of more neutron-rich SHEs than the nuclei we picked up in this paper. We have started expanding the survey for those nuclei and will report it elsewhere.