Nuclear Fission Dynamics: Past, Present, Needs, and Future

NT@UW-19-18,LA-UR-19-32211

Aurel Bulgac,1 Shi Jin,1 and Ionel Stetcu2 1Department of Physics, University of Washington, Seattle, Washington 98195–1560, USA 2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: March 23, 2020) Significant progress in the understanding of the fission process within a microscopic framework has been recently reported. Even though the complete description of this important nuclear reaction remains a computationally demanding task, recent developments in theoretical modeling and computational power have brought current microscopic simulations to the point where they can provide guidance and constraints to phenomenological models, without making recourse to parameters. An accurate treatment compatible with our understanding of the inter-nucleon interactions should be able to describe the real-time dynamics of the fissioning system and could justify or rule out assumptions and approximations incompatible with the underlying universally accepted quantum- mechanical framework. Of particular importance are applications to observables that cannot be directly measured in experimental setups (such as the angular momentum distribution of the fission fragments, or the excitation energy sharing between the fission fragments, or fission of nuclei formed during the r-process), and their dependence of the excitation energy in the fissioning system. Even if accurate predictions are not within reach, being able to extract the trends with increasing excitation energy is important in various applications. The most advanced microscopic simulations of the fission process do not support the widely used assumption of adiabaticity of the large amplitude collective motion in fission, in particular for trajectories from the outer saddle towards the scission configuration. Hence, the collective potential energy surface and inertia tensor, which are the essential elements of many simplified microscopic theoretical approaches, become irrelevant. In reality, the dynamics of the fissioning system is slower than in the case of pure adiabatic motion by a factor of three to four times and is strongly overdamped. The fission fragment properties are defined only after the full separation, while in most of the current approaches no full separation can be achieved, which increases the uncertainties in describing fission-related observables in such methods.

I. THE PAST gammas can be described quite accurately using statistical methods [12, 13]; iii) and that the non-relativistic In a matter of days after Hahn and Strassmann [1] com- Schrödinger equation should be adequate as well, as no municated their yet unpublished results to Lise Meitner, genuine relativistic effects, such a retardation, are ex- she and her nephew Otto Frisch [2] understood that an pected to play any noticeable role in fission dynamics. unexpected and qualitatively new type of nuclear reac- Whether a fissioning nucleus undergoes either sponta- tion has been put in evidence and they dubbed it nuclear neous fission or induced fission the time it takes the nu- fission, in analogy to cell divisions in biology. Until that cleus to evolve from its ground state shape until outside moment in time nuclear fission was considered a totally the barrier or past the outer saddle is very long in case of −15 unthinkable process [3,4], “as excluded by the small pen- neutron induced fission ≈ O(10 ) sec. in comparison etrability of the Coulomb barrier [5], indicated by the with the time the nucleus slides downhill until scission, −20 Gamov’s theory of alpha-decay” [2]. Meitner and Frisch which is estimated to be O(10 ) sec. Therefore, the [2] also gave the correct physical interpretation of the saddle-to-scission stage of nuclear shape evolution is the nuclear fission mechanism. They understood that Bohr’s fastest and arguably the most non-equilibrium stage of compound nucleus [6] is formed after the absorption of a the nucleus dynamics from the moment a neutron has neutron, which eventually slowly evolves in shape, while been captured and the stage which plays a crucial role arXiv:1912.00287v2 [nucl-th] 19 Mar 2020 the volume remains constant, and that the competition in in determining the FFs properties. The many intri- between the surface energy of a nucleus and its Coulomb cacies of the fission process, the multitude of aspects, energy leads to the eventual scission. Meitner and Frisch which required a deep theoretical understanding, ulti- [2] also correctly estimated the total energy released in mately rooted in the quantum nature of this phenomenon this process to be about 200 MeV. A few months later defied the efforts of many generations of theorists, and a Bohr and Wheeler [7] filled in all the technical details huge plethora of mostly phenomenological models have and the long road to developing a microscopic theory of been put forward, often based on contradictory assump- nuclear fission ensued. In the years since, a few more tions between models. The extensive range of assump- crucial theoretical results have been firmly established: tions, range from adiabatic evolution on top of which i) the defining role of quantum shell effects [8,9] and in one adds (relatively weak) dissipation and fluctuations, particular the special role played by the pairing type of to strongly overdamped motion, when the role of the col- nucleon-nucleon interaction in shape evolution [10, 11]; lective inertia becomes irrelevant, to full statistical equi- ii) the fact that the subsequent emission of neutrons and librium near the scission configuration, and to mixing quantum and classical descriptions. 2

Microscopically inspired models are typically based on the (ill suited choice of words, as the our analysis shows) adiabatic approximation, which is often conflated with slow motion. The class of adiabatic transformations, during which only mechanical work is performed and no heat transfer or entropy production occurs, are only a sub- class of slow motion or quasistatic processes. Theorists believed that the nuclear shape evolution until the moment of scission was so slow that individual nucleons had a sufficient time to adapt to avoided single-particle level crossings [14] and the entire nucleus would follow the lowest “molecular term,” using the Born-Oppenheimer chem- ical terminology [15], see Fig.1. Following this assumption at first the generator coordinate method (GCM) has been introduced by Hill and Wheeler [14] and Griffin and Wheeler [16] and later on a related alternative approach, the adiabatic time-dependent Hartree-Fock (AT- DHF) method [17–20]. GCM is still one of the most popular tools still in use in the microscopic theories of fission [20–27]. The GCM and ATDHF method have been shown to be basically equivalent [28], when GCM is defined with complex generator coordinates [29]. As however Goeke and Reinhard [28] succinctly state: “Usually Figure 1. The schematic evolution of the single-particle the |qi is obtained by an educated guess using the precon- nucleons levels (upper panel) and of the total nuclear energy ceived knowledge of the process.” (Typically |qi stands (lower panel) as a function of deformation parameter q [10, for a generalized Slater determinant, aka Hartree-Fock- 31, 32]. The thick line represents the Fermi level and the Bogoliubov many-nucleon wave function.) Even though up/down arrows depict the Cooper pairs of nucleons on the many efforts have been dedicated to find a better way to Fermi level only, in time-reversed orbits (m, −m). choose the collective or generator coordinates the methods proved to be too difficult to implement in practice and the quality of the decoupling between collective and posite. If a railroad car is released on top of a hill, it intrinsic degrees of freedom either not very good or diffi- will convert basically all the gravitational potential en- cult to assess [30]. An exact separation between collec- ergy difference when reaching the bottom of the hill into tive and intrinsic degrees of freedom (DoF) it is equiva- kinetic energy. Thus only pure mechanical work on the lent to an adiabatic evolution of the set of collective DoF. intrinsic DoF of the railroad car will occur with essen- Then the collective DoF would always follow the lowest tially no heat transfer or entropy production. This is an “molecular orbital” and only work would be performed adiabatic process. However, if one were instead to block on the intrinsic DoF, and thus with no heat transfer, and the wheels of the railroad car, the friction will slow down the intrinsic system would remain “cold” during the entire the car and almost the entire gravitational potential en- evolution. ergy difference will be converted into heat, the wheels will get red hot, thus increasing the “intrinsic energy” of the car, and the speed of the car at the bottom of the II. THE PRESENT hill would be rather small. The only practical theoretical framework to con- Unfortunately until recently this crucial aspect of large sider in the treatment of the dynamics of large nu- amplitude collective nuclear dynamics, whether the large clei is the (Time-Dependent) Density Function Theory amplitude collective motion in fission is indeed adiabatic ((TD)DFT), which has been formulated a long time was never tested and, as our results unequivocally show, ago [33–38]. One of the main difficulties consist in con- the adiabatic assumption is strongly violated. Surpris- structing the energy density functional, for which no ingly at first glance, we have recently shown that the rigorous recipes exist, DFT and TDDFT arrive at the evolution of the nuclear shape is in reality significantly mathematical conclusion that the stationary or time- slower that the adiabatic assumption would predict. One dependent solution of the many-body Schrödinger equa- would naively expect that if the motion is even slower tion is in a one-to-one-to-one correspondence with the then at an avoided level crossing, see Fig.1, the proba- number density, an (arbitrary) applied one-body external bility that the system would follow the lower “molecular potential, and that the number density can be obtained term” is even greater and thus the adiabatic assumption by solving the much simpler DFT or TDDFT equations. would be even more likely be valid. An analogy with a There is a continual debate in nuclear physics that DFT classical system can help and demonstrate just the op- is not applicable to nuclei, which are self-bound isolated 3 systems. At the same time however, no one would ar- hσσ0 (r, t), and the pairing field ∆(r, t) are functionals gue that DFT cannot describe neutrons and protons in of various neutron and proton densities, which are com- the neutron star crust and deeper into the star. Neu- puted from the qpwfs, see Ref. [41] for technical details. trons are delocalized in the neutron star crust and below No proton-neutron pairing is assumed in the present it. In the rod, slab, and tube phases in the neutron star study, and the pairing field is singlet in character. A TD- crust and below both protons and neutrons are delocal- SLDA extension to a more complex pairing mechanisms ized and in this respect they are similar to electrons in is straightforward. solids. One can imagine that in the future one might While a nuclear system evolves in time one can produce a nuclear trap using some kind of γ-lasers, simi- uniquely separate the energy into collective kinetic larly to what nowadays experiments are made with cold energy and intrinsic energy contributions [32] us- atoms. Until then one can mentally imagine that one ing the nuclear energy density functional (NEDF) can put an isolated nucleus in a spherical infinite square E (τ(r, t), n(r, t), ...), in a similar manner as in hydrody- well potential with a radius about 3...5x the nuclear ra- namics: dius (or even a harmonic potential) and compare the re- Z mn(r, t)v2(r, t) sults of such a DFT treatment of the nucleus with the Etot= Ecoll(t) + Eint(t) ≡ dr widely accepted DFT alter ego, the NEDF approach, a 2 Z tool of choice in theoretical nuclear calculations. The + dr E τ(r, t) − n(r, t)m2v2(r, t), n(r, t), ... . (2) results of the these two approaches are numerically in- distinguishable under these conditions, and therefore the Above n , t is the number density, τ , t is the kinetic debate alluded above is merely pedantic. One deficiency (r ) (r ) density energy, and , t mn , t , t are linear of a pure DFT approach is that the number density alone p(r ) = (r )v(r ) momentum and local collective/hydrodynamic velocity cannot disentangle between a normal and a superfluid densities, and ellipses stand for various other densities. system, and one needs an order parameter, as one does , t /n , t is the position of the center of the local in the case of magnetization. The practical local density p(r ) (r ) Fermi sphere in momentum space. The first term in approximation (LDA) [34], which is the local formula- Eq. (2) is the collective/hydrodynamic energy flow E tion of DFT, has to be augmented with the anomalous coll and the second term is the intrinsic energy E in the density [39, 40], and it was dubbed the superfluid LDA int local rest frame. For the sake of simplicity we have sup- (SLDA). In Refs. [39, 40] one can find detailed reviews pressed the spin and isospin DoF, even though they are of the developments, verification, and validation of TD- included in all the actual calculations. The collective SLDA for a variety of physical systems, ranging from cold energy E t is not vanishing only in the presence of atoms, nuclei, and to neutron star crust. coll( ) currents and vanishes exactly for stationary states. The A (TD)DFT framework for nuclear structure and dy- inertia tensor in E t in the case of irrotational col- namics should satisfy several requirements (in this or- coll( ) lective motion is fully equivalent to the Werner-Wheeler der of importance): i) the DFT and the Schrödinger de- inertial tensor [21]. The intrinsic energy E t is description of observables should be identical, as both in int( ) termined only by the fermionic matter distribution. The ultimate instance rely on the same inter particle inter- qualitative new result established in Ref. [32] and also actions; ii) both DFT and Schrödinger equations should illustrated here in Fig.2 is that in an fully unrestricted describe correctly Nature, thus we need accurate inter- TDSLDA the collective flow energy is almost negligible actions between nucleons; iii) the numerical implemen- until scission, in total discrepancy with what one would tation of the (TD)DFT should faithfully reproduce the have naively expected if the adiabatic assumptions would theory. At present we definitely do not have acceptable be satisfied. answers to the requirements i) and ii) and rely instead to Our simulations point to an unexpectedly small E a significant amount of phenomenology. The numerical coll from saddle-to-scission, corresponding to a collective implementation of DFT without any drastic physical re- speed v /c ≈ . ··· . , significantly smaller than strictions became possible only relatively recently, with coll 0 002 0 004 the Fermi velocity v /c ≈ . , see Fig.2. Since in TD- the advent of supercomputers [39, 40]. F 0 25 SLDA one simulates the one-body dynamics exactly, it The TDSLDA is formulated in terms of Bogoliubov is natural to discuss adiabaticity at the mean-field level. quasi-particle wave functions (qpwfs). The evolution of The transition rate between sp states is suppressed if the qpwfs is governed by the equations: the time to cross an avoided level-crossing configuration   satisfies the restriction ∆t ¯h/∆ ≈ 400 fm/c, where uk↑ h↑↑ h↑↓ 0 ∆ uk↑ ∂ h h − ∆ = 1/ρsp(F ) is the average sp energy level spacing uk↓  ↓↑ ↓↓ ∆ 0  uk↓ i¯h   =  − ∗ −h∗ −h∗    , (1) at the Fermi level. Since on the way from saddle-to- ∂t vk↑  0 ∆ ↑↑ ↑↓ vk↑ 3 v ∆∗ 0 −h∗ −h∗ v scission the time required is 1 ... 3 × 10 fm/c and sev- k↓ ↓↑ ↓↓ k↓ eral dozen of avoided level crossings occur [31, 42], this where we have suppressed the spatial r and time coordi- condition is clearly violated. Somewhat surprisingly, the nate t, and k labels the qpwfs (including the isospin) adiabatic assumption is also violated even in the case of u , t , v , t , with σ ↑, ↓ the z-projection of SLy4 NEDF, see Ref. [43] and Fig.2, when the saddle-to- [ kσ(r ) kσ(r )] = 4 the nucleon spin. The single-particle (sp) Hamiltonian scission time is O(10 ) fm/c as well. The collective mo- 4

2.5 Sly4 2.0 Sly4 (larger pairing) SeaLL1-1 1.5 SeaLL1-2

[MeV] SkM∗ 1.0 ﬂow 0.5 E 0.0 0 100 200 300 400 500 600 700 800 t [fm/c]

Figure 2. The collective flow energy evaluated for NEDFs [43] realistic pairing SLy4 (dash-dot line), enhanced pairing SLy4* (dash line), and for SkM*(dotted and dash-dot lines with error bars), and SeaLL1 (solid and dashed lines with errors bars) Figure 3. In nuclei the level density increases with the sets [32]. The error bars illustrate the size of the variations excitation energy quite fast, practically exponentially at en- due to different initial conditions in case of various SeaLL1- ergies of the order of the neutron separation energy, when √ 1,2 and SkM*-1,2 NEDFs used. In the case of realistic pairing ρ(E∗) ∝ exp 2aE∗ [54, 55], and it reaches values of NEDF Sly4 (larger pairing) the time has been scaled by a 5 −1 factor of 1/10. O(10 ) MeV and various potential energy surfaces, corresponding to different “molecular terms” display a large number of avoided level crossings, see Ref. [56]. tion is thus expected to be strongly overdamped. From saddle-to-scission the nucleus behaves as a very viscous fluid, the role of collective inertia is strongly suppressed, populated during the evolution, see Fig.3. The sep- and the trajectories follow predominantly the direction aration between such potential energy surfaces reaches of the steepest descent with the terminal velocity deter- values of order O(10) keV or even less, and only if the mined by the balance between the friction and the driving system traverses a level crossing in a time much longer conservative forces, see Fig.2. than ≈ ¯h/10 keV ≈ 20, 000 fm/c or longer the nucleus will remain on the lowest potential energy surface. On This result serves as the first microscopic justification 3 for the assumption of the overdamped Brownian mo- the other hand the saddle-to-scission time is O(10 ) fm/c tion model [44–49] and partially to the scission-point and this is why the adiabatic assumption is in the final model [50–53]. In both these phenomenological models analysis strongly violated. it is assumed that the preformed FFs are in statistical Why do pairing correlations play an important role equilibrium and that the collective energy flow is either in fission dynamics? In a static nuclear configuration vanishing or very small. The main difference is that in each single-particle level is doubly-degenerate, due to the the scission-point model there is no mechanism to en- Kramers degeneracy. At the level crossing both nucleons sure that all equilibrium scission configurations could be on the highest occupied level (homo - highest occupied reached dynamically, while the nucleus evolves from the molecular orbital) would have to transition simultane- saddle-to-scission. Moreover, the relaxed FF properties ously to the lowest down-sloping level (lumo - lowest un- are defined only after the FFs become sufficiently well occupied molecular orbital) to ensure that the local mo- separated, see below. It is equally unexpected that in mentum distribution remains approximately spherical, as the case of enhanced pairing, when the pairing conden- otherwise it would acquire an oblate shape [10, 11, 14], sates retain their long-range order throughout the entire while the shape of the nucleus becomes more prolate. saddle-to-scission evolution, the collective dynamics re- Nucleon-nucleon interactions at low momentum transfer mains strongly overdamped. can be modeled with a reasonable accuracy with a zero- While evolving from saddle-to-scission a nucleus en- range δ-interaction, which favors the transitions between counters a large number of avoided level crossings and pairs of time-reversed orbitals, exactly as the Kramers instead of following the lowest potential energy surface, degenerate orbitals, see Fig.1. The up-sloping levels are characterized by larger projections of the angular mo- as would happen in an adiabatic evolution, many transi- 1/3 tions to higher excited levels occur. Similarly to what is mentum on the fission axis, |m| ≈ kF r0A , and these levels should be depopulated, since in a FF the largest known for decades in chemistry [57–59], one should con- 1/3 sider not only the lowest potential energy surface, when angular momenta are smaller, ≈ kF r0(A/2) . While initially the system could be found on the lowest with evolving from one level crossing to the next, the entire unit probability, but all “molecular terms,” which become evolution is likely rather well reproduced by a simple 5

1.00 30 SeaLL1-1 SeaLL1-2 0.75 /2 25 SkM∗-1 SkM∗-2 0.50 0 20

0.25 /2 15 [b] 20 0.00 10 Q 4 5

3 /2 0

2 0 16 18 20 22 24 26 28 30 1 /2 distance [fm]

0 2 SeaLL1-1 SeaLL1-2

SkM∗-1 Figure 4. These are some selected snapshots of the neu- 1 SkM∗-2

tron/proton magnitudes (left) and phases (right) of pairing ] 2 gaps (upper half/lower/half of each frame) in the case of re- / 3 0 alistic pairing strengths (upper panels) and enhanced pair- [b ing strength (lower panels) [32], obtained in the case of SLy4

30 1 NEDF. − Q

2 − one-body dynamics, as each single-particle level occupa- 3 tion probability changes little. What one-body dynamics − lacks is the contributions arising from the Boltzmann col- 16 18 20 22 24 26 28 30 lision integral. However, at each level crossing the two- distance [fm] correlated nucleon pairs will undergo a collision, and at low energies transitions between pairs of time-reversed Figure 5. The evolution of the quadrupole and octupole orbitals expected to dominate the collision rate. One moments of the FFs as a functions of their separation after should take with a grain of salt this simplistic picture of scission. [32]. Different lines correspond to different initial “collisions” and jumps between sp levels, as nothing hap- conditions and different NEDFs, see Ref. [32] more details. pens instantaneously or at one point in space in quantum The solid/dashed lines are for the light/heavy FFs respec- mechanics. In the presence of a Bose-Einstein condensate tively. of nucleon Cooper pairs the nucleus has a superfluid com- ponent and pair transfers are enhanced due to the Bose enhancement factor. The dynamics of the nuclear sys- TDGCM [24, 66], and scission-point [50–53] models the tems then approaches the evolution of classical inviscid calculation of the FFs yields consider only a very limited (no viscosity) or perfect fluid. An illustration of this be- range of nuclear shapes. In particular in such simulations havior was exemplified in Fig. 4 in Ref. [40] and in Fig.4. one never introduces the octupole FF moments. Our When the magnitude of the pairing field was artificially results, as well recent analysis by Scamps and Simenel increased from a realistic value to a value 3...4x larger [67], clearly demonstrate that the FFs emerge at scission the evolution time from saddle-to-scission decreased by octupole deformed and also with a significantly larger a factor of ≈ 10 and at the same time the long range quadrupole deformation than the relaxed values. More- coherence of the pairing field across the entire nucleus over, even after scission the FFs there is a significant survived. For realistic values of pairing strengths during Coulomb interaction between them, which leads to the the descent from the saddle-to-scission both proton and excitation of both low energy and giant resonances in neutron pairing fields fluctuate both in space and time, FF [68, 69]. This interaction enables additional exci- long range order basically vanishes, but quite often it is tation energy exchange between the FFs after scission, revived. and it also affects their total kinetic energy, a behav- Another important aspect which emerged from our ior also seen in our simulation, but yet not documented. TDSLDA fission simulations [32, 43], which is of signifi- In statistical scission-point models there is no dynam- cant importance in the implementation of various phe- ics, and only the competition between FFs configurations nomenological models, concerns the shapes of FFs at at the scission point are considered [50–53], a model to scission and when they reach their relaxed shapes, see which our results lend partial support. However, the only Fig.5. This behavior is apparently confirmed indirectly shapes considered are quadrupole deformation of the re- by experiments. In Langevin or Fokker-Planck [60–65], laxed FFs, which clearly is not what our dynamical sim- 6 ulation demonstrate. 5 Our simulations put in evidence another very impor- En = 0.50 MeV (CGMF+TDSLDA) tant aspect, the mechanism of the excitation energy shar- En = 5.55 MeV (CGMF+TDSLDA) 4 E = 0.50 MeV ing between the FFs. As a rule the heavy FF emerges n 235 En = 5.55 MeV U(n,f) in the end cooler than the light FF, even though they n o have been in contact for quite a long time before scission. i 3 s s Moreover, when increasing the initial energy excitation of i f / )

the ﬁssioning nucleus we have established that only the A

( 2 heavy FF becomes hotter and that is reflected in the average neutron multiplicity number of emitted, and results which is in apparent agreement with experimental find- 1 ings, see Fig.6. Experimentally it is extremely difficult Preliminary to infer the excitation energies of the FFs, which are a 0 crucial input in various statistical codes [70–72]. Bertsch 80 90 100 110 120 130 140 150 160 et al. [73] argue that the FFs spin distribution, which de- A (amu) termines the prompt gamma angular distribution, can be used to infer information of the excitation energy shar- Figure 6. We compare here the average neutron multiplicity ing between FFs. Randrup et al. [72] point to a pro- ν¯(A) emitted by FFs using a CGMF simulation [71], which nounced anti-correlation between ν¯(A) and mean total assumes an En dependence for the energy sharing extracted kinetic energy TKE¯ , which can be used reduce uncertain- using the the excitation energy sharing between the FFs in our ties in data analysis. Schmidt and Jurado [74, 75] sug- calculation with NEDF SeaLL1, as a function of the equivalent incident neutron energy in 235U(n,f) reaction along with gested a phenomenological model, the “energy-sorting” available experimental data [77]. Note that in this figure the mechanism based on the empirical constant tempera- parametrization was based on 240Pu calculations, while 236U ture parametrization of the nuclear level densities due to calculations are in progress. Gilbert and Cameron [76]. In this “energy-sorting” model the FFs before scission have different temperatures, with a lower temperature of the heavy fragment, which would We have presented rather simple arguments that most generate an energy flow from the light/hotter to the likely such an accurate representation does not exist in heavy/cooler fragment. Our simulations demonstrate general, particularly in fission large amplitude collective however that the near scission the two FFs have prop- motion [32]. A rather simple estimate of the possible erties quite different (shape, excitation energy, pairing number of different shapes, and therefore of independent correlations) from the properties of relaxed fragments. terms in Eq. (3), shows that it is basically infinitely small It is therefore problematic to relate the properties of ex- in comparison with the number of possible terms in an cited isolated nuclei with the properties of FFs in contact exact multi-configurational Slater determinant represen- before rupture. tation of Ψexact. Can we check whether GCM is a good As we have mentioned above, the number of collec- approximation for the saddle-to-scission evolution? Now tive DoF and their character is a long standing problem we definitely can and we have the answer, and since in microscopic inspired theoretical and phenomenological models. The choice of collective DoF and their character Etotal = Ecoll(t)+Eint(t) ≈ Eint(q, T ) ≈ V (q, T ) ≈ const. is guided by the authors’ intuition, their computational (5) and other abilities, educated guess, imagination, and/or where T is the temperature of the intrinsic system, we available resources [28]. It was never proven or demon- know that the collective motion is strongly overdamped, strated that a GCM representation of the nuclear wave that there is an irreversible energy flow from the collec- function is ever accurate. The wave function of a many- tive/shape DoF to the intrinsic DoF, and that during the fermion system in TDGCM is constructed according to saddle-to-scission evolution the temperature of the intrin- the following prescription sic system increases, as does its (entanglement) entropy as well. We should also remember, that the evolution of Z the fissioning nucleus from saddle-to-scission is a truly ΨGCM(x, t) = dqf(q, t)Φ(x|q), (3) non-equilibrium one, and the notion of a “slowly evolving” nuclear temperature T might be questionable. This where Φ(x|q) are (generalized) Slater determinants de- suggest that in phenomenological calculations a more pending on nucleon spatial coordinates, spin, and isospin physically motivated choice for the potential energy sur- x = (x1, . . . , xA), xk = (rk, σk, τk) and parameterized face would be one in which with increasing deforma- by the collective coordinates q = (q1, . . . , qn), and where tion one would increase the “temperature T ,” so as to f(q, t) is the collective wave function. There is no crite- keep the “collective potential” energy V (q, T ) essentially rion or small parameter which controls the accuracy of equal to the initial excitation energy of the compound such a representation, that is nucleus. Basically this is the prescription that Randrup et al. [48, 49] have implemented lately. − |h | i| . (4) 1 ΨGCM Ψexact There were indirect indication in microscopic ap- 7

III. WHAT ARE THE NEEDS?

Nuclear fission is a complex process, in which a heavy nucleus evolves from a compact shape to a configuration in which two or more fragments are produced, and is ac- companied by emission of prompt neutrons and gamma rays (and, eventually, electrons and antineutrinos). Most of the energy is released in form of kinetic energy of the fragments, while prompt neutrons emitted before beta decays play the main role in applications like energy production. The dynamics of the nuclear system from the formation of compound nucleus until after the acceleration of FFs and prompt particle emission is too rapid to be experimentally resolved. On the other hand, the time scale of the weak interaction, which governs the decay of FFs in experiments towards stability ranges from sec- onds to minutes, and thus the dynamics involving beta decay towards stability, including delayed neutron and gamma emissions, can be decoupled from the initial more rapid part that is only governed by the strong interactions. Consequently, the fragment properties directly Figure 7. In the upper panel we show a typical scission influence the properties of prompt fission and gamma configuration when unlimited type of fluctuations are allowed rays, which have been the subject of comprehensive, al- as compared to a typical TDSLDA scission configuration, in beit not exhaustive, experimental investigations over the which axial symmetry is assumed in this case for 258Fm [78]. years. Thus, at Los Alamos National Laboratory, experimental campaigns have investigated the prompt fission neutron spectra for neutron-induced fission of ma- proaches that the number of collective DoF can vary jor actinides (235,238U, 239Pu) for a large range of in- along the fission path. In the overwhelming majority of cident neutron energies using the ChiNu experimental fission studies the only deformation of the nucleus close to setup [84, 85]. On the other hand, experiments that the ground state is the axial symmetric quadrupole mo- measure the average neutron multiplicity as a function ment Q , in spite of the fact that the Bohr-Mottelson 20 of pre-fission neutron mass [86–91], which can be used five-dimensional collective Hamiltonian is one tool of to guide energy sharing in fission fragments, are more choice to describe the low energy structure of open-shell scarce, and for a limited number of reactions (usually nuclei [55, 79, 80]. Ryssens et al. [81] have shown in a spontaneous fission or neutron-induced fission with ther- beyond mean field calculation of the 1-dimensional fis- mal neutrons). The surrogate reaction technique is used sion path the nucleus 240Pu is axially deformed around by Lawrence Livermore National Lab and Texas A&M the ground state configuration, it becomes triaxially de- to measure the 239Pu and 241Pu prompt fission neutron formed when it reaches the fission isomer region, and multiplicity (average and distribution) as a function of before it reaches the outer fission barrier it breaks the equivalent incident neutron energy [92, 93]. Complemen- axial symmetry as well. We have recently developed a tary, significant resources have been devoted toward mea- framework into which one can include fluctuations and suring the prompt fission gamma rays produced in the dissipation in a quantum approach [78], which allows for decay of fission fragments, after the neutron emission, shape fluctuations of any kind. What we observed that at Los Alamos, using the DANCE calorimeter [94–97], by allowing a bending mode to become active, see Fig.7, and worldwide employing high-resolution detectors [98– the mass and TKE distributions acquire shapes in quite 102]. Such measurements complement existing measure- nice agreement with observations. It was discussed quite ments of prompt-gamma rays by Verbinski et al. [103], some time ago that such bending modes might be respon- and many others [104–106]. Furthermore, data on to- sible for the angular distributions of the FFs [82] and this tal gamma production can be useful in evaluating the type of distributions can be also extracted from TDSLDA prompt fission gamma properties, if no other data ex- calculations in the near future [73, 83]. Phenomenological ist, fission dominates, and the other gamma-producing or even TDGCM approaches do not consider so far such channels can be modeled with reasonable accuracy [107]. bending modes, which are definitely physically relevant In addition to detecting and measuring the products of for a large number of observables. decay of FFs, a concerted effort has been directed towards the direct measurement of the FF distributions. While simpler, the “2E” measurements [108–111] have the short- coming that additional information is required to identify 8 the mass of the fragment, hence the poor 4-6 amu reso- atics far from stability, as prompt neutron and gamma lution. Reducing such large uncertainties was the main observables are sensitive to ingredients like level densi- argument for the construction of the SPIDER “2E-2v” ex- ties, optical models, or gamma strength functions. perimental setup [112, 113] at the Los Alamos Neutron Several models have been proposed to describe the Science Center, which should start taking data in the shape dynamics, and to some extent many of them are near future. In this setup the two fragment kinetic ener- able to reproduce experimental quantities like the pre- gies and velocities are measured simultaneously and can neutron emission mass distributions, irrespective of the achieve one mass unit resolution, as also demonstrated approximations involved. But, as noted above, the di- by similar setups, COSI FAN TUTTE at ILL, France rect measurement of the FFs before neutron emission and VERDI at JRC-Geel, Belgium. The Sofia experi- is not possible. Therefore, even before one goes into ment [114] at GSI has produced an extensive amount of details regarding the validity of the approximations in- experimental data with accuracies in mass and charge volved in theoretical models, one needs to consider the less than one mass unit, but suffer from the fact that a corrections involved in the analysis of the experimen- large range of excitation energies that cannot be disen- tal data. The post-neutron emission FF mass distribu- tangled is produced in Coulomb excitations. LLNL and tions are obtained using information on average prompt LANL activation measurements at TUNL have provided neutron multiplicity as a function of the fragment mass, invaluable data on the incident energy dependence of cu- ν¯(A), either from measurements, where available, or from mulative fission product yields [93, 115], and ongoing ef- phenomenological models. Given the scarcity of the data, forts are directed toward testing the Bohr hypothesis of and the fact that these phenomenological models have independence between entrance and outgoing channels been built in a systematics based on little data, the un- in a compound reaction by comparing fission yields in certainties arising from ν¯(A) could be considerable. An- the 239Pu(n,f) and 240Pu(γ,f) reactions [116] that lead to other assumption is that no neutrons are emitted during the same fissioning system, at least in mass and charge. scission or at neck rupture. It is conceivable that the At CARIBOU, two projects are underway: (i) measure- number of neutrons emitted during the scission dynam- ment of fission product properties (isomeric yield ratios, ics is not necessarily large, but how large is large? If gamma-ray decay branching ratios, and β-delayed neu- the fraction of scission neutrons is significant, as some tron emission properties) and (ii) improvements of the phenomenological models predict, citing the experimen- antineutrino spectrum simulations by performing mea- tal prompt fission spectrum as evidence to support the surements of the β-decay data using GAMASPHERE claim [120–125], what is the impact on the experimen- [117]. Other experiments related to the properties of FFs tal analysis? While these models are not universally ac- have concentrated on measuring the average total kinetic cepted, a microscopic approach should be able to answer energy (TKE) of the fission fragments at the LANSCE such a question, and the assumption of small numbers of WNR facility for energy from a few hundred keVs to 200 neutrons emitted during scission should not be implicitly MeV [111, 113]. The High Rigidity Spectrometer [118] included in the model. Moreover, it would be rather im- at FRIB [119] will allow to explore the fissioning of very possible to asses the fraction of scission neutrons if the neutron-rich nuclei, study their shell structure, the pres- model does not follow the dynamics of the process until ence of superheavy, the limits of stability and the equa- full separation. tion of state of neutron-rich nuclear matter, the neutron All the codes that model the prompt neutron and skins, and shed light on the nature of the r-process, so gamma emissions have been built in the assumption that far an unknown territory. the neutron emission proceeds after the full acceleration Treating the evolution of a heavy nucleus from a com- of the fission fragments [71, 126–130]. This assumption pact configuration until the start of beta decays in the can have significant consequences, as at full acceleration FFs, including prompt neutron and gamma emissions, is the neutrons are maximally boosted. If the emission oc- a complicated task, computationally unfeasible within a curs instead during the acceleration, then the prompt unified microscopic approach. Within this reality, one neutron spectrum in the lab frame can noticeably be al- has to consider a mixture of approaches, in which the tered. initial part of the fission process is treated within a mi- In addition to validating assumptions in phenomeno- croscopic framework that can inform more phenomenological codes, a reliable theoretical model should provide logical treatments that model the emission of prompt information on observables that cannot be measured, but neutrons and gamma rays, and whose results can be di- are essential in modeling the neutron and gamma emis- rectly compared against a large set of experimental data. sion. Thus, while the total excitation energy available And the input from microscopic models does not nec- in the fragments can be determined from the Q-value essarily need to be restricted to the fission process. The and TKE, no additional constrains on how this energy systematics of several physical quantities used in the phe- is shared between the FFs are available. As the most nomenological models of neutron and gamma emission is efficient way to lower the energy in FFs is via neutron based on data for stable nuclei. Since the FFs are nuclei emission, ν¯(A) can be used to parametrize the excitation far from stability, it is important to investigate within energy sharing. However, such data are scarcely avail- more microscopic models the validity of various system- able, usually for a limited number of spontaneous and 9 neutron-induced fission with thermal neutrons, and thus sight into the real quantum dynamics, and it revealed ex- guidance from microscopic models on how the excitation tremely valuable information into nuclear processes and energy is shared with increasing incident neutron energy quantities, which are either not easy or impossible to ob- is required not only for minor actinides, but even for tain in laboratory or observations: FFs excitation ener- induced fission reaction of major actinides 235,238U and gies and angular momenta distributions prior to neutron 239Pu isotopes. and gamma emission, element formation in astrophysi- Gamma emission competes with the neutron emission cal environments, as well as other nuclear reactions in when the excitation energy in the nucleus is around the a parameter free approach. In particular, the excitation neutron separation energy. The spin of the fragments, energy sharing mechanism between FFs and its evolu- which cannot be directly measured experimentally, de- tion with the initial excitation energy of the compound termines the exact strength on the neutron-gamma com- nucleus was not accessible until now within a dynamic petition. Only indirect information can be extracted re- approach. Fluctuations, which are essential in order to garding the spin distribution of the FFs from properties reproduce mass and charge yields for example, can be of prompt gamma rays, like average multiplicity, prompt now incorporated into a pure quantum framework [78]. gamma fission spectra [131], or isomeric ratios [132]. The The quality of the agreement with experimental data is measurements of gamma rays are also scarce with increas- surprisingly good, especially taking into account the fact ing the incident neutron energy. Experimentally, one ob- that no attempt was made to reproduce any fission data. serves an increase of the total prompt fission gamma en- Basically all phenomenological NEDFs satisfy the most ergy released in fission [133], which has been interpreted important requirements to describe the gross properties as an increase of the average spin with the incident en- of nuclear fission: saturation, realistic surface tension and ergy. However, only little experimental data exist and symmetry energy, Coulomb energy, realistic pairing and no microscopic model has been able to reproduce such shell corrections energy. Nevertheless, the quality of ex- trends. isting NEDFs needs improving. One can make a strong The main message of this section is that experimentally argument that we have now a clear path from more phe- one cannot isolate and measure properties of post-scission nomenology and adjusted parameters to more fundamen- FFs, and any full model will have to include simulations tal theory and increased predictive power [137]. of prompt neutron and gamma emission, in order to com- Perhaps the most important aspect we have observed pare and validate against experimental data. Given that in all our simulations is the strong violation of the adia- many quantities used in the modeling of prompt neutron batic assumption in fission large amplitude collective mo- and gamma emission are taken from data systematics, tion. Basically since the 1950’s the adiabatic assumption which in general is available only for nuclei close to sta- was the main simplification included in all microscopic bility, the reliability of the theoretical model near scission frameworks, GCM, ATDHF and a large majority of phe- and beyond, until full separation, is very important. Such nomenological models as well, such as the Langevin and models should allow for a full separation of the fragments, the Fokker-Planck equations, where one needs to intro- and employ approximations that are validated and under duce a potential energy surface and an inertia tensor in control, as to reduce any uncertainties regarding the FF the space of the collective variables. If the collective mo- properties. tion is overdamped, the inertia tensor becomes irrelevant, and moreover, considering only the lowest potential energy surface is physically unacceptable, see Figs.3 and IV. WHAT LESSONS HAVE WE LEARNED SO 2 and the corresponding discussion in sectionII. As we FAR AND WHAT IS THE MOST LIKELY PATH have established in Ref. [78] the fluctuations, or equiva- TO THE FUTURE? lently the role of two-body collisions, does not affect this conclusion. While pairing is not the engine driving the fission dy- A somewhat unexpected result was the character of namics, pairing provides the essential lubricant, with- the energy sharing mechanism between the fission frag- out which the evolution may arrive rather quickly to a ments, the fact that the heavy fragment is cooler than the screeching halt [134–136]. So far we have not considered light fragment and it has less excitation energy [32, 43]. whether proton-neutron pairing might have a role in fis- And this conclusion is not a result of the fact that the sion. It is very unlikely that a condensate of proton- heavy FF is closer to the double magic 132Sn nucleus. neutron pairs exists in heavy nuclei, but as we have In the original experiment of Hahn and Strassmann [1] learned from our simulations, proton-neutron transitions the heavy fragment had a charge closer to Z ≈ 52 − 56, with L=0 between single-particle orbitals could be im- a fact recognized also by Meitner and Frisch [2, 138], portant as the neutron-neutron and proton-proton tran- and explained by Scamps and Simenel [67], this is due to sitions with (S,L,T)=(0,0,1) at low excitation energies, a stabilization of the octupole deformation in FFs, also even in the absence of a condensate of such pairs, see the observed in our simulations [32, 43]. With increasing ex- discussion concerning Figs.1 and4. citation energy of the fissioning nucleus the heavy FF TDSLDA framework for fission dynamics, while it does appears to be the only one who absorb the increase and not incorporate fluctuations, has provided a lot of in- emits more neutrons, see Fig.6 and the accompanying 10 discussion. The character of the excitation energy shar- ACKNOWLEDGMENTS ing mechanism has major consequences on the predicted spectrum of emitted neutrons and gammas. We thank many people with whom we had discussion Another important outcome was the clear indications over the years and for their input: I. Abdurrahman, G.F. that many more collective DoF appear to be relevant Bertsch, P. Magierski, K.J. Roche, N. Schunck. We thank in fission than have ever been considered in either phe- G.F. Bertsch and J. Randrup who made a number of nomenological or microscopically inspired models such as suggestions on our manuscript, which has been released GCM and ATDHF frameworks. At scission, both FFs are as a preprint [140]. octupolly deformed. Moreover, the FFs attain their re- The work of AB and SJ was supported by U.S. Depart- laxed shapes only after the separation between them is ment of Energy, Office of Science, Grant No. DE-FG02- ≈ 5...6 fm. No phenomenological or microscopically in- 97ER41014 and in part by NNSA cooperative agreement spired approach on the market follow the FFs to such de-na0003841. The work of IS was supported by the US large separations. As we stressed in sectionIII the spec- Department of Energy through the Los Alamos National trum of emitted neutrons is affected by the number of Laboratory. Los Alamos National Laboratory is oper- neutrons emitted before the full FFs acceleration. An- ated by Triad National Security, LLC, for the National other DoF, which appears to be relevant as well is the Nuclear Security Administration of U.S. Department of bending mode, see Fg.7 and Ref. [78], the inclusion of Energy (Contract No. 89233218CNA000001). The TD- which likely is going to influence the angular momentum SLDA calculations have been performed at the OLCF distributions of the FFs [82]. Summit and Titan, and CSCS Piz Daint, and for gener- We have pointed to several directions into which phe- ating initial configurations for direct input into the TD- nomenological models and theoretical models such as SLDA code at OLCF Titan and Summit and NERSC GCM and ATDHF would have to be altered, in order Edison. This research used resources of the Oak Ridge to describe nuclear fission in a manner more consistent Leadership Computing Facility, which is a U.S. DOE Of- with theoretical expectations inferred from unrestricted fice of Science User Facility supported under Contract quantum mechanical simulations, see sectionIII. We did No. DE- AC05-00OR22725 and of the National Energy not cover or mention all phenomenological models on the Research Scientific computing Center, which is supported market and not all microscopically inspired theoretical by the Office of Science of the U.S. Department of En- frameworks, as this is not a review of such approaches, ergy under Contract No. DE-AC02-05CH11231. We ac- for which we recommend Refs. [129, 139]. knowledge PRACE for awarding us access to resource Piz Daint based at the Swiss National Supercomputing Centre (CSCS), decision No. 2018194657. This work is supported by "High Performance Computing Infrastruc- ture" in Japan, Project ID: hp180048. A series of simulations were carried out on the Tsubame 3.0 supercomputer at Tokyo Institute of Technology. This research used resources provided by the Los Alamos National Laboratory Institutional Computing Program.

[1] O. Hahn and F. Strassmann, “Über den Nachweis und [8] V.M. Strutinsky, “Shell effects in nuclear masses and das Verhalten der bei der Bestrahlung des Urans mit- deformation energies,” Nucl. Phys. A 95, 420 (1967). tels Neutronen entstehenden Erdalkalimetalle,” Natur- [9] M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. wissenschaften 27, 11 (1939). Strutinsky, and C. Y. Wong, “Funny Hills: The Shell- [2] L. Meitner and O. R. Frisch, “Disintegration of Uranium Correction Approach to Nuclear Shell Effects and Its by Neutrons: a New Type of Nuclear Reaction,” Nature Applications to the Fission Process,” Rev. Mod. Phys. 143, 239 (1939). 44, 320 (1972). [3] R. Stuewer, “An act of creation: The Meitner–Frisch in- [10] G. Bertsch, “The nuclear density of states in the space terpretation of nuclear fission,” (2010), http://edition- of nuclear shapes,” Phys. Lett. B 95, 157 (1980). open-access.de/proceedings/5/11/index.html. [11] G. F. Bertsch and A. Bulgac, “Comment on “Sponta- [4] J. M. Pearson, “On the belated discovery of fission,” neous Fission: A Kinetic Approach”,” Phys. Rev. Lett. Physics Today 68 (6), 40 (2015). 79, 3539 (1997). [5] E. Fermi, E. Amaldi, O. d’Agostino, F. Rasetti, and [12] V. Weisskopf, “Statistics and Nuclear Reactions,” Phys. E. Segre, “Artificial radioactivity produced by neutron Rev. 52, 295 (1937). bombardment,” Proc. Roy. Soc. A 146, 249 (1934). [13] W. Hauser and H. Feshbach, “The inelastic scattering [6] N. Bohr, “Neutron Capture and Nuclear Constitution,” of neutrons,” Phys. Rev. 87, 366 (1952). Nature 137, 344 and 351 (1936). [14] D. L. Hill and J. A. Wheeler, “Nuclear Constitution and [7] N. Bohr and J. A. Wheeler, “The Mechanism of Nuclear the Interpretation of Fission Phenomena,” Phys. Rev. Fission,” Phys. Rev. 56, 426 (1939). 89, 1102 (1953). 11

[15] M. Born and J.R. Oppenheimer, “Zur Quantentheorie [37] M. A. L. Marques, C. A. Ullrich, F. Nogueira, A. Rubio, der Molekeln,” Annalen der Physik 389, 457 (1927). K. Burke, and E. K. U. Gross, eds., Time-Dependent [16] J. J. Griffin and J. A. Wheeler, “Collective Motions in Density Functional Theory, Lecture Notes in Physics, Nuclei by the Method of Generator Coordinates,” Phys. Vol. 706 (Springer-Verlag, Berlin, 2006). Rev. 108, 311 (1957). [38] M. A. L. Marques, N. T. Maitra, F. M. S. Nogueira, [17] M. Baranger, “Microscopic view of nuclear collective E. K. U. Gross, and A. Rubio, eds., Fundamentals properties,” J. Phys. 33, C5 (1972). of Time-Dependent Density Functional Theory, Lecture [18] M. Baranger and M. Vénéroni, “An adiabatic time- Notes in Physics, Vol. 837 (Springer, Heidelberg, 2012). dependent Hartree-Fock theory of collective motion in [39] A. Bulgac, “Time-Dependent Density Functional The- finite systems,” Ann. Phys. 114, 123 (1978). ory and the Real-Time Dynamics of Fermi Superfluids,” [19] F. Villars, “Adiabatic time-dependent Hartree-Fock the- Ann. Rev. Nucl. and Part. Sci. 63, 97 (2013). ory in nuclear physics,” Nucl. Phys. A 285, 269 (1978). [40] A. Bulgac, “Time-Dependent Density Functional The- [20] P. Ring and P. Schuck, The Nuclear Many-Body Prob- ory for Fermionic Superfluids: from Cold Atomic gases, lem, 1st ed., Theoretical and Mathematical Physics Se- to Nuclei and Neutron Star Crust,” Physica Status So- ries No. 17 (Springer-Verlag, Berlin Heidelberg New lidi B 2019, 1800592 (2019). York, 2004). [41] S. Jin, A. Bulgac, K. Roche, and G. Wlazłowski, [21] J. K. Krappe and K. Pomorski, Theory of Nuclear Fis- “Coordinate-space solver for superfluid many- sion (Springer Heidelberg, 2012). fermion systems with the shifted conjugate-orthogonal [22] N. Schunck and L. M. Robledo, “Microscopic theory of conjugate-gradient method,” Phys. Rev. C 95, 044302 nuclear fission: a review,” Rep. Prog. Phys. 79, 116301 (2017). (2016). [42] G. F. Bertsch, W. Younes, and L. M. Robledo, “Scission [23] H. Goutte, J. F. Berger, P. Casoli, and D. Gogny, “Mi- dynamics with k partitions,” Phys. Rev. C 97, 064619 croscopic approach of fission dynamics applied to frag- (2018). ment kinetic energy and mass distributions in 238U,” [43] A. Bulgac, P. Magierski, K. J. Roche, and I. Stetcu, “In- Phys. Rev. C 71, 024316 (2005). duced Fission of 240Pu within a Real-Time Microscopic [24] D. Regnier, N. Dubray, N. Schunck, and M. Ver- Framework,” Phys. Rev. Lett. 116, 122504 (2016). rière, “Fission fragment charge and mass distributions [44] H. A. Weidenmüller and J.-S. Zhang, “Nuclear fission in 239Pu(n, f) in the adiabatic nuclear energy density viewed as a diffusion process: Case of very large fric- functional theory,” Phys. Rev. C 93, 054611 (2016). tion,” Phys. Rev. C 29, 879 (1984). [25] A. Zdeb, A. Dobrowolski, and M. Warda, “Fission dy- [45] J. Randrup and P. Möller, “Brownian Shape Motion namics of 252Cf,” Phys. Rev. C 95, 054608 (2017). on Five-Dimensional Potential-Energy Surfaces:Nuclear [26] N. Schunck, editor, Energy Density Functional Methods Fission-Fragment Mass Distributions,” Phys. Rev. Lett. for Atomic Nuclei (IOP Publishing, Bristol, UK, 2019). 106, 132503 (2011). [27] W. Younes, D. M. Gogny, and J. F. Berger, A Micro- [46] J. Randrup, P. Möller, and A. J. Sierk, “Fission- scopic Theory of Fission Based on the Generator Co- fragment mass distributions from strongly damped ordinate Method, Lectures Notes in Physics, Vol. 950 shape evolution,” Phys. Rev. C 84, 034613 (2011). (Springer, Berlin, 2019). [47] J. Randrup and P. Möller, “Energy dependence [28] K. Goeke and P.-G. Reinhard, “The generator- of fission-fragment mass distributions from strongly coordinate-method with conjugate parameters and the damped shape evolution,” Phys. Rev. C 88, 064606 unification of microscopic theories for large amplitude (2013). collective motion,” Ann. Phys. 124, 249 (1980). [48] D. E. Ward, B. G. Carlsson, T. Døssing, P. Möller, [29] R.E. Peierls and D.J. Thouless, “Variational approach J. Randrup, and S. Åberg, “Nuclear shape evolution to collective motion,” Nucl. Phys. 38, 154 (1962). based on microscopic level densities,” Phys. Rev. C 95, [30] G.D. Dang, A. Klein, and N.R. Walet, “Self-consistent 024618 (2017). theory of large-amplitude collective motion: applica- [49] M. Albertsson, B.G. Carlsson, T. Døssing, P. Møller, tions to approximate quantization of nonseparable sys- J. Randrup, and S. Åberg, “Excitation energy partition tems and to nuclear physics,” Phys. Rep. 335, 93 (2000). in fission,” Phys. Lett. B 803, 135276 (2020). [31] F. Barranco, G.F. Bertsch, R.A. Broglia, and [50] B.D. Wilkins and E.P. Steinberg, “Semi-empirical in- E. Vigezzi, “Large-amplitude motion in superfluid Fermi terpretation of nuclear fission based on deformed-shell droplets,” Nucl. Data Sheets Phys. A 512, 253 (1990). effects,” Phys. Lett. B 42, 141 (1972). [32] A. Bulgac, S. Jin, K. J. Roche, N. Schunck, and [51] B. D. Wilkins, E. P. Steinberg, and R. R. Chas- I. Stetcu, “Fission dynamics of 240Pu from saddle to man, “Scission-point model of nuclear fission based on scission and beyond,” Phys. Rev. C 100, 034615 (2019). deformed-shell effects,” Phys. Rev. C 14, 1832 (1976). [33] P. Hohenberg and W. Kohn, “Inhomogeneous Electron [52] J.-F. Lemaître, S. Panebianco, J.-L. Sida, S. Hilaire, Gas,” Phys. Rev. 136, B864–B871 (1964). and S. Heinrich, “New statistical scission-point model [34] W. Kohn and L. J. Sham, “Self-consistent equations in- to predict fission fragment observables,” Phys. Rev. C cluding exchange and correlation effects,” Phys. Rev. 92, 034617 (2015). 140, A1133–A1138 (1965). [53] J.-F. Lemaître, S. Goriely, S. Hilaire, and J.-L. Sida, [35] W. Kohn, “Nobel Lecture: Electronic structure of “Fully microscopic scission-point model to predict fis- matter—wave functions and density functionals,” Rev. sion fragment observables,” Phys. Rev. C 99, 034612 Mod. Phys. 71, 1253–1266 (1999). (2019). [36] R. M. Dreizler and E. K. U. Gross, Density Func- [54] H. A. Bethe, “An attempt to calculate the number of tional Theory: An Approach to the Quantum Many– energy levels of a heavy nucleus,” Phys. Rev. 50, 332– Body Problem (Springer-Verlag, Berlin, 1990). 341 (1936). 12

[55] A. Bohr and B.R. Mottelson, Nuclear Structure (Ben- of fission fragments,” Phys. Rev. C 83, 061601 (2011). jamin Inc., New York, 1969). [76] A. Gilbert and A. G. W. Cameron, “A composite [56] A. Bulgac, G. Do Dang, and D. Kusnezov, “Stochas- nuclear-level density formula with shell correction,” tic aspects of large amplitude collective motion,” Phys. Can. J. Phys. 43, 1446 (1965). Rep. 264, 67 (1996). [77] R. Müller, A. A. Naqvi, F. Käppeler, and F. Dick- [57] J. C. Tully and R. K. Preston, “Trajectory Surface Hop- mann, “Fragment velocities, energies, and masses from ping Approach to Nonadiabatic Molecular Collisions: fast neutron induced fission of 235U,” Phys. Rev. C 29, + The reaction of H with D2,” J. Chem. Phys. 55, 562 885 (1984). (1971). [78] A. Bulgac, S. Jin, and I. Stetcu, “Unitary evolution [58] J. C. Tully, “Molecular dynamics with electronic transi- with fluctuations and dissipation,” Phys. Rev. C 100, tions,” J. Chem. Phys. 93, 1061 (1990). 014615 (2019). [59] S. Hammes-Schiffer and J. C. Tully, “Proton transfer [79] G. F. Bertsch, M. Girod, S. Hilaire, J.-P. Delaroche, in solution: Molecular dynamics with quantum transi- H. Goutte, and S. Péru, “Systematics of the first 2+ tions,” J. Chem. Phys. 101, 4657 (1994). excitation with the gogny interaction,” Phys. Rev. Lett. [60] P. Grangé, J.-Q. Li, and H. A. Weidenmüller, “Induced 99, 032502 (2007). nuclear fission viewed as a diffusion process: Tran- [80] J. P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hi- sients,” Phys. Rev. C 27, 2063 (1983). laire, S. Péru, N. Pillet, and G. F. Bertsch, “Structure [61] P. Fröbrich and I.I. Gontchar, “Langevin description of of even-even nuclei using a mapped collective hamilto- fusion, deep-inelastic collisions and heavy-ion-induced nian and the d1s gogny interaction,” Phys. Rev. C 81, fission,” Phys. Rep. 292, 131 (1998). 014303 (2010). [62] A. J. Sierk, “Langevin model of low-energy fission,” [81] W. Ryssens, P.-H. Heenen, and M. Bender, “Numerical Phys. Rev. C 96, 034603 (2017). accuracy of mean-field calculations in coordinate space,” [63] C. Ishizuka, M. D. Usang, F. A. Ivanyuk, J. A. Maruhn, Phys. Rev. C 92, 064318 (2015). K. Nishio, and S. Chiba, “Four-dimensional Langevin [82] L. G. Moretto, G. F. Peqaslee, and G. J. Wozniak, approach to low-energy nuclear fission of 236U,” Phys. “Angular-Momentum-Bearing Modes in Fission,” Nucl. Rev. C 96, 064616 (2017). Phys. A 502, 453c (1989). [64] J. Sadhukhan, W. Nazarewicz, and N. Schunck, “Micro- [83] A. Bulgac, “Projection of Good Quantum Numbers for scopic modeling of mass and charge distributions in the Reaction Fragments,” Phys. Rev. C 100, 034612 (2019). spontaneous fission of 240Pu,” Phys. Rev. C 93, 011304 [84] R.C. Haight, C.Y. Wu, H.Y. Lee, T.N. Taddeucci, B.A. (2016). Perdue, J.M. O’Donnell, N. Fotiades, M. Devlin, J.L. [65] J. Sadhukhan, C. Zhang, W. Nazarewicz, and Ullmann, T.A. Bredeweg, M. Jandel, R.O. Nelson, S.A. N. Schunck, “Formation and distribution of fragments Wender, D. Neudecker, M.E. Rising, S. Mosby, S. Sjue, in the spontaneous fission of 240Pu,” Phys. Rev. C 96, M.C. White, B. Bucher, and R. Henderson, “The 061301 (2017). LANL/LLNL Prompt Fission Neutron Spectrum Pro- [66] D. Regnier, N. Dubray, and N. Schunck, “From gram at LANSCE and Approach to Uncertainties,” Nu- asymmetric to symmetric fission in the fermium iso- clear Data Sheets 123, 130 – 134 (2015), special Issue on topes within the time-dependent generator-coordinate- International Workshop on Nuclear Data Covariances method formalism,” Phys. Rev. C 99, 024611 (2019). April 28 - May 1, 2014, Santa Fe, New Mexico, USA [67] G. Scamps and C. Simenel, “Impact of pear-shaped fis- http://t2.lanl.gov/cw2014. sion fragments on mass-asymmetric fission in actinides,” [85] Kelly, Keegan J., Devlin, Matthew, Gomez, Jaime A., Nature 564, 382 (2018). O´Donnell, John M., Taddeucci, Terry N., Haight, [68] M.G. Mustafa, H.W. Schmitt, and U. Mosel, “Dipole Robert C., Lee, Hye Young, Mosby, Shea M., Foti- excitations in fission fragments,” Nucl. Phys. A 178, 9 ades, Nikolaos, Neudecker, Denise, Talou, Patrick, Ris- (1971). ing, Michael E., White, Morgan C., Solomon, Clell J., [69] C. Simenel and A. S. Umar, “Formation and dynamics Wu, Ching-Yen, Bucher, Brian, Buckner, Matthew Q., of fission fragments,” Phys. Rev. C 89, 031601 (2014). and Henderson, Roger A., “Measurements of the prompt [70] J. Randrup and R. Vogt, “Calculation of fission observ- fission neutron spectrum at lansce: The chi-nu experi- ables through event-by-event simulation,” Phys. Rev. C ment,” EPJ Web Conf. 193, 03003 (2018). 80, 024601 (2009). [86] E. E. Maslin, A. L. Rodgers, and W. G. F. Core, [71] B. Becker, P. Talou, T. Kawano, Y. Danon, and “Prompt neutron emission from u235 fission fragments,” I. Stetcu, “Monte Carlo Hauser-Feshbach predictions of Phys. Rev. 164, 1520–1527 (1967). prompt fission gamma rays: Application to n+235U, [87] K. Nishio, Y. Nakagome, H. Yamamoto, and I. Kimura, n+239Pu, and 252Cf (sf),” Phys. Rev. C 87, 014617 “Multiplicity and energy of neutrons from 235u(nth,f) (2013). fission fragments,” Nuclear Physics A 632, 540 – 558 [72] J. Randrup, P. Talou, and R. Vogt, “Sensitivity of neu- (1998). tron observables to the model input in simulations of [88] C. Tsuchiya, Y. Nakagome, H. Yamana, H. Moriyama, 252Cf(sf),” Phys. Rev. C 99, 054619 (2019). K. Nishio, I. Kanno, K. Shin, and I. Kimura, “Simul- [73] G. F. Bertsch, T. Kawano, and L. M. Robledo, “Angu- taneous Measurement of Prompt Neutrons and Fission lar momentum of fission fragments,” Phys. Rev. C 99, Fragments for 239Pu(nth,f),” J. Nucl. Sci. Tech. 37, 941 034603 (2019). (2000). [74] K.-H. Schmidt and B. Jurado, “Entropy driven exci- [89] O. A. Batenkov, G. A. Boykov, F.-J. Hambsch, J. H. tation energy sorting in superfluid fission dynamics,” Hamilton, V. A. Jakovlev, V. A. Kalinin, A. B. Laptev, Phys. Rev. Lett. 104, 212501 (2010). V. E. Sokolov, and A. S. Vorobyev, “Prompt Neutron [75] K.-H. Schmidt and B. Jurado, “Final excitation energy Emmision in the Neutron-Induced Fission of 239Pu and 13

235U,” AIP Conf. Proc. 769, 1003 (2005). neutron-induced fission of 235U and 238U,” Phys. Rev. [90] A. Göök, F.-J. Hambsch, and M. Vidali, “Prompt C 92, 034618 (2015). neutron multiplicity in correlation with fragments from [101] A. Gatera, T. Belgya, W. Geerts, A. Göök, F.-J. spontaneous fission of 252Cf,” Phys. Rev. C 90, 064611 Hambsch, M. Lebois, B. Maróti, A. Moens, A. Ober- (2014). stedt, S. Oberstedt, F. Postelt, L. Qi, L. Szent- [91] Alf Göök, Franz-Josef Hambsch, Stephan Oberstedt, miklósi, G. Sibbens, D. Vanleeuw, M. Vidali, and and Marzio Vidali, “Prompt neutrons in correlation F. Zeiser, “Prompt-fission γ-ray spectral characteristics 235 239 with fission fragments from U(n, f),” Phys. Rev. C from Pu(nth, f),” Phys. Rev. C 95, 064609 (2017). 98, 044615 (2018). [102] H. Makii, K. Nishio, K. Hirose, R. Orlandi, R. Léguil- [92] O. A. Akindele, B. S. Alan, J. T. Burke, R. J. Casper- lon, T. Ogawa, T. Soldner, U. Köster, A. Pollitt, F.-J. son, R. O. Hughes, J. D. Koglin, K. Kolos, E. B. Nor- Hambsch, I. Tsekhanovich, M. Aïche, S. Czajkowski, man, S. Ota, and A. Saastamoinen, “Expansion of the L. Mathieu, C. M. Petrache, A. Astier, S. Guo, T. Oht- surrogate method to measure the prompt fission neu- suki, S. Sekimoto, K. Takamiya, R. J. W. Frost, and tron multiplicity for 241Pu,” Phys. Rev. C 99, 054601 T. Kawano, “Effects of the nuclear structure of fission (2019). fragments on the high-energy prompt fission γ-ray spec- 235 [93] B. S. Wang, J. T. Harke, O. A. Akindele, R. J. Casper- trum in U(nth, f),” Phys. Rev. C 100, 044610 (2019). son, R. O. Hughes, J. D. Koglin, K. Kolos, E. B. [103] V. V. Verbinski, Hans Weber, and R. E. Sund, “Prompt Norman, S. Ota, and A. Saastamoinen, “Determin- gamma rays from 235U(n, f), 239Pu(n, f), and spon- ing the average prompt-fission-neutron multiplicity for taneous fission of 252Cf,” Phys. Rev. C 7, 1173–1185 0 239Pu(n, f) via a 240Pu(α, α f) surrogate reaction,” (1973). Phys. Rev. C 100, 064609 (2019). [104] R. W. Peelle and F. C. Maienschein, “Spectrum of pho- [94] J. L. Ullmann, E. M. Bond, T. A. Bredeweg, A. Cou- tons emitted in coincidence with fission of 235U by ther- ture, R. C. Haight, M. Jandel, T. Kawano, H. Y. Lee, mal neutrons,” Phys. Rev. C 3, 373–390 (1971). J. M. O’Donnell, A. C. Hayes, I. Stetcu, T. N. Tad- [105] Frances Pleasonton, Robert L. Ferguson, and H. W. deucci, P. Talou, D. J. Vieira, J. B. Wilhelmy, J. A. Schmitt, “Prompt gamma rays emitted in the thermal- Becker, A. Chyzh, J. Gostic, R. Henderson, E. Kwan, neutron-induced fission of 235U,” Phys. Rev. C 6, 1023– and C. Y. Wu, “Prompt γ-ray production in neutron- 1039 (1972). induced fission of 239Pu,” Phys. Rev. C 87, 044607 [106] Frances Pleasonton, “Prompt γ-rays emitted in the (2013). thermal-neutron induced fission of 233U and 239Pu,” [95] A. Chyzh, C. Y. Wu, E. Kwan, R. A. Henderson, J. M. Nucl. Phys. A 213, 413 – 425 (1973). Gostic, T. A. Bredeweg, A. Couture, R. C. Haight, A. C. [107] I. Stetcu, M. B. Chadwick, T. Kawano, P. Talou, Hayes-Sterbenz, M. Jandel, H. Y. Lee, J. M. O’Donnell, R. Capote, and A. Trkov, “Evaluation of the Prompt and J. L. Ullmann, “Systematics of prompt γ-ray emis- Fission Gamma Properties for Neutron Induced Fission sion in fission,” Phys. Rev. C 87, 034620 (2013). of 235;238U and 239Pu,” Nuclear Data Sheets 163, 261 [96] A. Chyzh, C. Y. Wu, E. Kwan, R. A. Henderson, T. A. (2020). Bredeweg, R. C. Haight, A. C. Hayes-Sterbenz, H. Y. [108] Ch. Straede, C. Budtz-Jørgensen, and H.-H. Knitter, Lee, J. M. O’Donnell, and J. L. Ullmann, “Total prompt “235u(n, f) fragment mass-, kinetic energy- and angu- γ-ray emission in fission of 235U, 239,241Pu, and 252Cf,” lar distributions for incident neutron energies between Phys. Rev. C 90, 014602 (2014). thermal and 6 mev,” Nuclear Physics A 462, 85 – 108 [97] M. Jandel, G. Rusev, E.M. Bond, T.A. Bredeweg, M.B. (1987). Chadwick, A. Couture, M.M. Fowler, R.C. Haight, [109] F. Vivès, F.-J. Hambsch, H. Bax, and S. Oberstedt, T. Kawano, A.L. Keksis, S.M. Mosby, J.M. O’Donnell, “Investigation of the fission fragment properties of the R.S. Rundberg, I. Stetcu, P. Talou, J.L. Ullmann, D.J. reaction 238u(n,f) at incident neutron energies up to 5.8 Vieira, J.B. Wilhelmy, M.A. Stoyer, R.J. Haslett, R.A. mev,” Nuclear Physics A 662, 63 – 92 (2000). Henderson, J.A. Becker, and C.Y. Wu, “Prompt fis- [110] E. Birgersson, A. Oberstedt, S. Oberstedt, and F.-J. sion gamma-ray studies at dance,” Physics Procedia 59, Hambsch, “Properties of the reaction 238u at the vi- 101 – 106 (2014), gAMMA-2 Scientific Workshop on the brational resonances,” Nuclear Physics A 817, 1 – 34 Emission of Prompt Gamma-Rays in Fission and Re- (2009). lated Topics. [111] D. L. Duke, F. Tovesson, A. B. Laptev, S. Mosby, F.-J. [98] R. Billnert, F.-J. Hambsch, A. Oberstedt, and S. Ober- Hambsch, T. Bryś, and M. Vidali, “Fission-fragment stedt, “New prompt spectral γ-ray data from the reac- properties in 238U(n, f) between 1 and 30 mev,” Phys. tion 252Cf(sf) and its implication on present evaluated Rev. C 94, 054604 (2016). nuclear data files,” Phys. Rev. C 87, 024601 (2013). [112] F. Tovesson, C. W. Arnold, T. Bredeweg, M. Jan- [99] A. Oberstedt, T. Belgya, R. Billnert, R. Borcea, del, A. B. Laptev, K. Meierbachtol, A. Sierk, White T. Bryś, W. Geerts, A. Göök, F.-J. Hambsch, Z. Kis, M., A. A. Hecht, D. Mader, Blakeley R., U. Greife, T. Martinez, S. Oberstedt, L. Szentmiklosi, K. Takàcs, B. Moore, D. Shields, and L. Snyder, “Spider: A new and M. Vidali, “Improved values for the characteris- instrument for fission yield measurement,” in Proceed- tics of prompt-fission γ-ray spectra from the reaction ings of the Fifth International Conference on ICFN5, 235 U(nth,f),” Phys. Rev. C 87, 051602 (2013). edited by J. H. Hamilton and A. V. Ramayya (World [100] M. Lebois, J. N. Wilson, P. Halipré, A. Oberstedt, Scientific, 2013) p. 361. S. Oberstedt, P. Marini, C. Schmitt, S. J. Rose, S. Siem, [113] K. Meierbachtol, F. Tovesson, D. L. Duke, V. Geppert- M. Fallot, A. Porta, and A.-A. Zakari, “Comparative Kleinrath, B. Manning, R. Meharchand, S. Mosby, and measurement of prompt fission γ-ray emission from fast- D. Shields, “Total kinetic energy release in 239Pu(n, f) post-neutron emission from 0.5 to 50 mev incident neu- 14

tron energy,” Phys. Rev. C 94, 034611 (2016). [126] R. Vogt and J. Randrup, “Event-by-event Modeling of [114] E. Pellereau, J. Taïeb, A. Chatillon, H. Alvarez- Prompt Neutrons and Photons from Neutron-induced Pol, L. Audouin, Y. Ayyad, G. Bélier, J. Benlliure, and Spontaneous Fission with FREYA,” Physics Proce- G. Boutoux, M. Caamaño, E. Casarejos, D. Cortina-Gil, dia 47, 82 (2013). A. Ebran, F. Farget, B. Fernández-Domínguez, T. Gor- [127] O. Litaize, O. Serot, and L. Berge, “Fission modelling binet, L. Grente, A. Heinz, H. Johansson, B. Jurado, with FIFRELIN,” Eur. Phys. Jour. A 51, 177 (2015). A. Keli ć Heil, N. Kurz, B. Laurent, J.-F. Martin, C. No- [128] K.-H. Schmidt, B. Jurado, C. Amouroux, and ciforo, C. Paradela, S. Pietri, J. L. Rodríguez-Sánchez, C. Schmitt, “General description of fission observables: K.-H. Schmidt, H. Simon, L. Tassan-Got, J. Vargas, Gef model code,” Nuclear Data Sheets 131, 107 – 221 B. Voss, and H. Weick, “Accurate isotopic fission yields (2016). of electromagnetically induced fission of 238U measured [129] K.-H. Schmidt and B. Jurado, “Review on the progress in inverse kinematics at relativistic energies,” Phys. Rev. in nuclear fission - experimental methods and theoretical C 95, 054603 (2017). descriptions,” Rep. Prog. Phys. 81, 106301 (2018). [115] M.E. Gooden, C.W. Arnold, J.A. Becker, C. Bha- [130] P. Talou, R. Vogt, J. Randrup, M. E. Rising, S. A. tia, M. Bhike, E.M. Bond, T.A. Bredeweg, B. Fallin, Pozzi, J. Verbeke, M. T. Andrews, S. D. Clarke, P. Jaf- M.M. Fowler, C.R. Howell, J.H. Kelley, Krishichayan, fke, M. Jandel, T. Kawano, M. J. Marcath, K. Meier- R. Macri, G. Rusev, C. Ryan, S.A. Sheets, M.A. Stoyer, bachtol, L. Nakae, G. Rusev, A. Sood, I. Stetcu, and A.P. Tonchev, W. Tornow, D.J. Vieira, and J.B. Wil- C. Walker, “Correlated prompt fission data in transport helmy, “Energy dependence of fission product yields simulations,” Eur. Phys. Jour. A 54, 9 (2018). from 235u, 238u and 239pu for incident neutron ener- [131] I. Stetcu, P. Talou, T. Kawano, and M. Jandel, “Prop- gies between 0.5 and 14.8 mev,” Nuclear Data Sheets erties of prompt-fission γ rays,” Phys. Rev. C 90, 024617 131, 319 – 356 (2016), special Issue on Nuclear Reac- (2014). tion Data. [132] I. Stetcu, P. Talou, T. Kawano, and M. Jandel, “Iso- [116] J. Silano, A. Tonchev, R. Henderson, N. Schunck, mer production ratios and the angular momentum dis- W. Tornow, C. Howell, F. Krishichayan, S. Finch, and tribution of fission fragments,” Phys. Rev. C 88, 044603 M. Gooden, “Comparing fission-product yields from (2013). 240 photon-induced fission of Pu and neutron-induced fis- [133] J. Fréhaut, A. Bertin, and R. Bois, “Mesure de ν¯p pour sion of 239Pu as a test of the bohr hypothesis in nuclear la fission de 232Th, 235U et 237Np induite par des neu- fission,” (2019), subbmited to the Proceedings of the trons d’energie comprise entre 1 et 15 MeV,” in Inter- Santa Fe workshop on fission product yields. national Conference on Nuclear Data for Science and [117] G. Savard, “Fission product yield measurements using Technology, Antwerp, Belgium (Reidel, Dordrech, Hol- 252cf spontaneous fission and neutron-induced fission on land, 1983) p. 78. actinide targets at CARIBU,” (2019), talk at the Inter- [134] P. Goddard, P. Stevenson, and A. Rios, “Fis- national Workshop on Fission Product Yields. sion dynamics within time-dependent Hartree-Fock: [118] “A High Rigidity Spectrometer fro FRIB, Deformation-induced fission,” Phys. Rev. C 92, 054610 hrs.lbl.gov/documents/HRS-WhitePaper122017.pdf,”. (2015). [119] “Facility for Rare Isotopes Beams, www.frib.msu.edu,”. [135] P. Goddard, P. Stevenson, and A. Rios, “Fission dy- [120] N. Carjan and M. Rizea, “Scission neutrons and other namics within time-dependent Hartree-Fock. II. Boost- scission properties as function of mass asymmetry in induced fission,” Phys. Rev. C 93, 014620 (2016). 235 U(nth,f),” Phys. Rev. C 82, 014617 (2010). [136] Y. Tanimura, D. Lacroix, and G. Scamps, “Collec- [121] N. Carjan, F.-J. Hambsch, M. Rizea, and O. Serot, tive aspects deduced from time-dependent microscopic “Partition between the fission fragments of the excita- mean-field with pairing: Application to the fission pro- tion energy and of the neutron multiplicity at scission cess,” Phys. Rev. C 92, 034601 (2015). in low-energy fission,” Phys. Rev. C 85, 044601 (2012). [137] A. Bulgac, S. Jin, P. Magierski, K. J Roche, N. Schunck, [122] M. Rizea and N. Carjan, “Dynamical scission model,” and I. Stetcu, “Nuclear Fission: from more phenomenol- Nucl. Phys. A 909, 50 (2013). ogy and adjusted parameters to more fundamental the- [123] N. Carjan and M. Rizea, “Similarities between calcu- ory and increased predictive power,” EPJ Web of Con- lated scission-neutron properties and experimental data ferences 163, 00007 (2017). on prompt fission neutrons,” Phys. Lett. B 747, 178 [138] L. Meitner and O. R. Frisch, “Products of the Fission (2015). of the Uranium Nucleus,” Nature 143, 471 (1939). [124] R. Capote, N. Carjan, and S. Chiba, “Scission neutrons [139] A. N. Andreyev, K. Nishio, and K.-H. Schmidt, “Nu- for U, Pu, Cm, and Cf isotopes: Relative multiplicities clear fission: a review of experimental advances and phe- calculated in the sudden limit,” Phys. Rev. C 93, 024609 nomenology,” Rep. Prog. Phys. 81, 016301 (2018). (2016). [140] A. Bulgac, S. Jin, and I. Stetcu, “Nuclear Fission Dy- [125] N. Carjan and M. Rizea, “Structures in the energy namics: Past, Present, Needs, and Future,” (2019), distribution of the scission neutrons: Finite neutron- arXiv:1912:00287v1. number effect,” Phys. Rev. C 99, 034613 (2019).