Focused on the generation and evolution of vast complementary pairs of the primary fission fragments at scission moment, Dinuclear and Statistical Model (DSM) is proposed. (1) It is assumed that the fissile nucleus elongates along a symmetric coaxis until it breaks into two primary fission fragments. (2) Every complementary pair of the primary fission fragments is approximatively described as two ellipsoids with large deformation at scission moment. (3) The kinetic energy in every complementary pair of the primary fragments is mainly provided by Coulomb repulsion, which is explicitly expressed through strict six-dimensional integrals. (4) Only three phenomenological coefficients are obtained to globally describe the quadrupole deformation parameters of arbitrary primary fragments both for $^{235}$U($n_{th}, f$) and $^{239}$Pu($n_{th}, f$) reactions, on the basis of the common characteristics of the measured data, such as mass and charge distributions, kinetic energy distributions. In the framework of DSM, the explicit average total kinetic energy distribution $\overline{TKE}(A)$ and the average kinetic energy distribution $\overline{KE}(A)$ are consistently represented. The theoretical results in this paper agree well with the experimental data. Furthermore, this model is expected as the reliable approach to generally evaluate the corresponding observebles for thermal neutron-induced fission of actinides.
A simplified, though realistic, model describing two receding and accelerating fission fragments, due to their mutual Coulomb repulsion, shows that fission fragments share excitation energy well after they ceased to exchange nucleons. This mechanism leads to a lower total kinetic energy of the fission fragments, particularly if the pygmy resonances in the fission fragments are excited. Even though the emphasis here is on fission, similar arguments apply to fragments in heavy-ion reactions.
We present the first fully unrestricted microscopic calculations of the primary fission fragment intrinsic spins and of the fission fragments' relative orbital angular momentum for $^{236}$U$^*$, $^{240}$Pu$^*$, and $^{252}$Cf using the time-dependent density functional theory framework. Within this microscopic approach, free of restrictions and unchecked assumptions and which incorporates the relevant physical observables for describing fission, we evaluate the triple distribution of the fission fragment intrinsic spins and of their fission fragments' relative orbital angular momentum and sho…
We propose an efficient approach to describe the fission-fragment charge yields for actinides based on the driving potential of fissioning system. Considering the properties of primary fission fragments at their ground states, the driving potential, which represents the potential energies of the system around scission configuration and closely relates to the yields of fragments, can be unambiguously and quickly obtained from the Skyrme energy-density functional together with the Weizsaecker-Skyrme mass model. The fission-fragment charge distributions for thermal-neutroninduced fission and spon…
Fissioning nuclei and fission fragments, nuclear fragments emerging from energetic collisions, or nuclei probed with various external fields can emit one or more pre-equilibrium neutrons, protons, and potentially other heavier nuclear fragments. I describe a formalism which can be used to evaluate the pre-equilibrium neutron emission probabilities and the excitation energies of the remnant fragments.
In the present paper, we explore the idea of isospin conservation in new situations and contexts based on the directions provided by our earlier works. We present the results of our calculations for the relative yields of neutron-rich fission fragments emitted in fast neutron-induced fission, 238U (n, fission) reaction by using the concept of the conservation of isospin and compare with the experimental data. Our results successfully reproduced the gross features of partition wise fission fragments distribution of 238U (n, fission). This confirms that in all kinds of fission, isospin remains p…
On the occasion of the $75^{th}$ anniversary of the fission phenomenon, we present a surprisingly simple result which highlights the important role of isospin and its conservation in neutron rich fission fragments. We have analysed the fission fragment mass distribution from two recent heavy-ion reactions $^{238}$U($^{18}$O,f) and $^{208}$Pb($^{18}$O,f) as well as a thermal neutron fission reaction $^{245}$Cm(n$^{th}$,f). We find that the conservation of the total isospin explains the overall trend in the observed relative yields of fragment masses in each fission pair partition. The isospin v…
The CGMF code implements the Hauser-Feshbach statistical nuclear reaction model to follow the de-excitation of fission fragments by successive emissions of prompt neutrons and $γ$ rays. The Monte Carlo technique is used to facilitate the analysis of complex distributions and correlations among the prompt fission observables. Starting from initial configurations for the fission fragments in mass, charge, kinetic energy, excitation energy, spin, and parity, $Y(A,Z,KE,U,J,π)$, CGMF samples neutron and $γ$-ray probability distributions at each stage of the decay process, conserving energy, spin an…
Experimental data on angular distributions of gamma rays emitted from binary and ternary spontaneous fission of $^{252}{\rm Cf}$ are analyzed. Their difference indicates that the alignment of fragments is higher in ternary fission than in binary one. The consequences of possible relation between the mechanism of ternary fission and the excitation of collective modes during the saddle-to-scission stage are discussed.
The even-odd effect in fission is explained by a model based on statistical mechanics. It reveals that the variation of the even-odd effect with the mass of the fissioning nucleus and the increase towards asymmetric splits is due to the important statistical weight of configurations where the light fission fragment populates the ground state of an even-even nucleus. This implies that entropy drives excitation energy and unpaired nucleons predominantly to the heavy fragment. Therefore, the even-odd effect is an additional signature of the recently discovered energy-sorting process in fission.
In the frame of proposed statistical method the calculation of fission-fragment yields mass and charge spectra for pre- actinide nuclei on example of natPb isotopes had been conducted.
We propose event by event velocity fluctuations of nuclear fission fragments as an additional interesting observable that gives access to the nuclear temperature in an independent way from spectral measurements and relates the diffusion and friction coefficients for the relative fragment coordinate in Kramer-like models (in which some aspects of fission can be understood as the diffusion of a collective variable through a potential barrier). We point out that neutron emission by the heavy fragments can be treated in effective theory if corrections to the velocity distribution are needed.
The characteristics of the odd-even effect in fission-fragment Z distributions are compared to a model based on statistical mechanics. Special care is taken for using a consistent description for the influence of pairing correlations on the nuclear level density. The variation of the odd-even effect with the mass of the fissioning nucleus and with fission asymmetry is explained by the important statistical weight of configurations where the light nascent fission fragment populates the lowest energy state of an even-even nucleus. This implies that entropy drives excitation energy and unpaired n…
The strong quadrupolar component of the Coulomb field between newly formed fission fragments can affect the internal energy of the fragments and their angular momentum. Previous estimates of these effects gave contradictory conclusions about their magnitude. Here we calculate them by solving the time-dependent Hamiltonian equation for the daughter $^{100}$Zr produced in the fission reaction n + $^{235}$U $\rightarrow$ $^{136}$Te + $^{100}$Zr. The Hamiltonian was constructed from the projected angular momentum eigenstates of an aligned deformed mean-field configuration. For typical initial cond…
The possibilities of the proposed statistical approach are shown in the task of investigating the post-scission transformation of mass and charge yields of fission fragments with considering the emission of different length of chains of elementary particles as the beta particles and neutrons fission. Using as the example the 232 Th isotope, which is located on the border between pre-actinide and actinides nuclei, the changes in the topology of fission fragments yields, the role of emission of nuclear particles in the competition of symmetric and asymmetric modes in fission processes are invest…
Consideration of memory of spin and parity in the fissioning compound nucleus by applying the Hauser-Feshbach fission fragment decay model to photonuclear reactions
Static self-consistent methods usually allow to determine the most probable fission fragments mass asymmetry. We have applied random neck rupture mechanism to the nuclei in the configuration at the end of fission paths. Fission fragment mass distributions have been deduced from the pre-scission nuclear density distribution obtained from the self-consistent calculations. Potential energy surfaces as well as nuclear shapes have been calculated in the fully microscopic theory, namely the constrained Hartree-Fock-Bogolubov model with the effective Gogny D1S density-dependent interaction. The metho…
A consistent framework for treating the energy dependence of fission-fragment neutron multiplicities is presented. The shape evolution of the compound nucleus towards scission is treated in the strong damping limit using the Metropolis walk method. The available excitation energy at scission is then divided statistically between the two fragments using microscopic level densities. Deformation energies, which contribute to the excitation energy when the fragments relax to their ground-state shapes, are also computed. From the total fragment excitation energies, the number of emitted neutrons is…
During nuclear fission, a heavy nucleus splits into two rotating fragments. The associated angular momentum is large, yet the mechanism of its generation and its dependence on the mass of fragments remain poorly understood. In this Letter, we provide the first microscopic calculations of angular momentum distributions in fission fragments for a wide range of fragment masses. For the benchmark case of $^{239}$Pu($n_{\text{th}}$,f), we find that the angular momentum of the fragments is largely determined by the nuclear shell structure and deformation, and that the heavy fragments therefore typic…
The origin of the asymmetry in the fragment mass distribution of low-energy nuclear fission is considered from the semiclassical point of view. Using the semiclassical periodic-orbit theory, one can define and quantify the shell effect associated with spatially localized nascent-fragment (prefragment) part of the potential. We investigate the roles of prefragments in the deformed shell effect using a simple cavity potential model but with realistic shape degrees of freedom for describing the fission processes. The results suggest that the prefragment magic numbers play essential roles in deter…
The structure effects of the fission fragments on their yields are studied within the statical theory with the inputs, like, excitation energies and level density parameters for the fission fragments at a given temperature calculated using the temperature dependent relativistic mean field formalism (TRMF). For the comparison, the results are also obtained using the finite range droplet model. At temperatures $T =1-2$ MeV, the structural effects of the fission fragments influence their yields. It is also seen that at $T = $ 3 MeV, the fragments become spherical and the fragments distribution pe…
Fission-fragment mass and total-kinetic-energy (TKE) distributions following fission of even-even nuclides in the region $74 \leq Z \leq 126$ and $92 \leq N \leq 230$, comprising 896 nuclides have been calculated using the Brownian shape-motion method. The emphasis is the region of superheavy nuclei. To show compatibility with earlier results the calculations are extended to include earlier studied regions. An island of asymmetric fission is obtained in the superheavy region, $106\leq Z\leq114$ and $162\leq N\leq 176$, where the heavy fragment is found to be close to $^{208}$Pb and the light f…
We calculate neutron multiplicities from fission fragments with specified mass numbers for events having a specified total fragment kinetic energy. The shape evolution from the initial compound nucleus to the scission configurations is obtained with the Metropolis walk method on the five-dimensional potential-energy landscape, calculated with the macroscopic-microscopic method for the three-quadratic-surface shape family. Shape-dependent microscopic level densities are used to guide the random walk, to partition the intrinsic excitation energy between the two proto-fragments at scission, and t…
The fission-fragment mass and total kinetic energy (TKE) distributions are evaluated in a quantum mechanical framework using elongation, mass asymmetry, neck degree of freedom as the relevant collective parameters in the Fourier shape parametrization recently developed by us. The potential energy surfaces (PES) are calculated within the macroscopic-microscopic model based on the Lublin-Strasbourg Drop (LSD), the Yukawa-folded (YF) single-particle potential and a monopole pairing force. The PES are presented and analysed in detail for even-even Plutonium isotopes with $A=236 -246$. They reveal …
Nuclear fission produces fragments whose spins are coupled to the relative angular motion via angular momentum conservation. It is shown how ensembles of such spins can readily be obtained by either direct microcanonical sampling or by sampling of the associated normal modes of rotation. The resulting distribution of the spin-spin opening angle is illustrated in various three- and two- dimensional scenarios and it is demonstrated how recent mutually conflicting model calculations can be well reproduced with different assumptions about the scission geometry.
Eighty years after its experimental discovery, a microscopic description of induced nuclear fission based solely on the interactions between neutrons and protons and quantum many-body methods still poses formidable challenges. The goal of this paper is to contribute to the development of a predictive microscopic framework for the accurate calculation of static properties of fission fragments for hot fission and thermal or slow neutrons. To this end, we focus on the 239Pu(n,f) reaction and employ nuclear density functional theory with Skyrme energy densities. Potential energy surfaces are compu…
The recent advent of experimental techniques in which the dynamical characteristics of fission fragments are determined more accurately, prompted us to investigate the angular momentum acquired by fragments in a model which describes the cold(neutronless) fission of $^{252}$Cf as the decay of a giant nuclear molecule. The molecular configuration is a consequence of the interplay between the attractive nuclear part and the repulsive Coulomb+nuclear forces. The basic ideea of the present approach is to separate the radial(fission) modes describing the decay of the molecule from the modes associa…
We present the results of our calculations for the relative yields of neutron-rich fission fragments emitted in $^{208}$Pb ($^{18}$O, fission) reaction by using the concept of the conservation of isospin and compare with the experimental data. We take into account a range of isospin values allowed by the isospin algebra and assume that the fission fragments are formed in Isobaric Analog States. We also take into account the neutron multiplicity data for various neutron-emission channels in each partition, and use them to obtain the weight factors in calculating the yields. We then calculate th…
Neutron star (NS) merger ejecta offer a viable site for the production of heavy r-process elements with nuclear mass numbers A >140. The crucial role of fission recycling is responsible for the robustness of this site against many astrophysical uncertainties. Here, we introduce new improvements to our scission-point model, called SPY, to derive the fission fragment distribution for all neutron-rich fissioning nuclei of relevance in r-process calculations. These improvements include a phenomenological modification of the scission distance and a smoothing procedure of the distribution. Such corr…
An electron localization measure was originally introduced to characterize chemical bond structures in molecules. Recently, a nucleon localization based on Hartree-Fock densities has been introduced to investigate $α$-cluster structures in light nuclei. Compared to the local nucleonic densities, the nucleon localization function has been shown to be an excellent indicator of shell effects and cluster correlations. Using the spatial nucleon localization measure, we investigate the emergence of fragments in fissioning heavy nuclei. To illustrate basic concepts of nucleon localization, we employ …
Fission properties of the actinide nuclei are deduced from theoretical analysis. We investigate potential energy surfaces and fission barriers and predict the fission fragment mass-yields of actinide isotopes. The results are compared with experimental data where available. The calculations were performed in the macroscopic-microscopic approximation with the Lublin-Strasbourg Drop (LSD) for the macroscopic part and the microscopic energy corrections were evaluated in the Yukawa-folded potential. The Fourier nuclear shape parametrization is used to describe the nuclear shape, including the non-…
We have, recently, shown that the general trends of partition-wise fission fragment mass distribution in heavy ion (HI) induced compound nuclear (CN) fission of heavy nuclei can be reproduced reasonably well by using the concept of isospin conservation, hence providing a direct evidence of isospin conservation in neutron-rich systems [1, 2, 3, 4]. In this paper, we test the concept of isospin conservation to reproduce the fission fragment mass distribution emerging from thermal neutron-induced CN fission reaction, 245Cm(nth, f). As earlier, we use Kelson's conjectures [5] to assign isospin to …
The constrained Hartree-Fock-Bogoliubov method is used with the Gogny interaction D1S to calculate potential energy surfaces of fissioning nuclei ${}^{226}$Th and ${}^{256,258,260}$Fm up to very large deformations. The constraints employed are the mass quadrupole and octupole moments. In this subspace of collective coordinates, many scission configurations are identified ranging from symmetric to highly asymmetric fragmentations. Corresponding fragment properties at scission are derived yielding fragment deformations, deformation energies, energy partitioning, neutron binding energies at sciss…
The goal of this paper is to better understand the structure of fission fragment distributions by investigating the competition between the static structure of the collective manifold and stochastic dynamics. In particular, we study the characteristics of the tails of yield distributions, which correspond to very asymmetric fission. We use the stochastic Langevin framework to simulate the nuclear evolution after the system tunnels through the multi-dimensional potential barrier. For a representative sample of different initial configurations along the outer turning-point line, we define effect…
Accurate knowledge of fission fragment yields is an essential ingredient of numerous applications ranging from the formation of elements in the r-process to fuel cycle optimization for nuclear energy. The need for a predictive theory applicable where no data is available is an incentive to develop a fully microscopic approach to fission dynamics. In this work, we calculate the pre-neutron emission charge and mass distributions of the fission fragments formed in the neutron-induced fission of 239Pu using a microscopic method based on nuclear energy density functional (EDF) method, where large a…
We analyzed the total kinetic energy (TKE) of fission fragments with three-dimensional Langevin calculations for a series of actinides and Fm isotopes at various excitation energies. This allowed us to establish systematic trends of TKE with $Z^2/A^{1/3}$ of the fissioning system and as a function of excitation energy. In the mass-energy distributions of fission fragments we see the contributions from the standard, super-long, and super-short (in the case of $^{258}$Fm) fission modes. For the fission fragments mass distribution of $^{258}$Fm we obtained a single peak mass distribution. The dec…
A recent analysis of experimental data [J. Wilson $et. al$, Nature $\mathbf 590$, 566 (2021)] found that the angular momenta of nuclear fission fragments are uncorrelated. Based on this finding, the authors concluded that the spins are therefore determined only $after$ scission has occurred. We show here that the nucleon-exchange mechanism, as implemented in the well-established event-by-event fission model $\mathtt{FREYA}$, while agitating collective rotational modes in which the two spins are highly correlated, nevertheless leads to fragment spins that are largely uncorrelated. This fact inv…
The intrinsic excitation energy of fission fragments is dynamically evaluated in terms of the time dependent pairing equations. These equations are corroborated with two conditions. One of them fixes the number of particles and the another separates the pairing active spaces associated to the two fragments in the vicinity of the scission configuration. The fission path is obtained in the frame of the macroscopic-microscopic model. The single particle level schemes are obtained within the two center Woods-Saxon shell model. It is shown that the available intrinsic dissipated energy is not share…
The population of rotational states in the ground-state band of neutron-rich fragments emitted in the spontaneous fission of $^{252}$Cf is described within a time-dependent quantum model similar to the one used for Coulomb excitation. The initial population probability of the states included in the selected basis is calculated according to the bending model at scission. Subsequently these initial amplitudes are feeding the coupled dynamical equations describing the population of rotational states in both fragments during the tunneling and post-barrier (pure Coulomb) motion. As application we c…
Fusion-fission dynamics is investigated with a special emphasis on fusion reactions at low energy for which shell effects and pairing correlations can play a crucial role leading in particular to multi-modal fission. To follow the dynamical evolution of an excited and rotating nucleus we solve a 2-dimensional Langevin equation taking explicitly light-particle evaporation into account. The confrontation theory-experiment is demonstrated to give interesting information on the model presented, its qualities as well as its shortcomings.
The cold(neutronless) fission of $^{252}$Cf is studied in the frame of a molecular model in which the scission configuration is described by two aligned fragments interacting by means of Coulomb (+ nuclear) forces. The study is carried out for different distances between the fragments tips and excitation energies. For a given deformation, the fragment's total energy is computed via the constrained Hartree-Fock + BCS formalism. The total excitation energy present in the fragments is supposed to contribute only to the fragments deformation and the asymptotic value of the kinetic energy is equate…
In current simulations of fission, the number of protons and neutrons in a given fission fragment is almost always obtained by integrating the total density of particles in the sector of space that contains the fragment. Because of the antisymmetry of the many-body wave function of the whole nucleus, this procedure systematically gives noninteger numbers of particles in the fragments. We introduce a novel sampling method to estimate rigorously the probability of finding $Z$ protons and $N$ neutrons in a fission fragment without resorting to projectors, which can sometimes give unwieldy results…
Theoretical description of nuclear fission remains one of the major challenges of quantum many-body dynamics. The slow, mostly adiabatic motion through the fission barrier is followed by a fast, non-adiabatic descent of the potential between the fragments. The latter stage is essentially unexplored. However, it is crucial as it generates most of the excitation energy in the fragments. The superfluid dynamics in the latter stage of fission is obtained with the time-dependent Hartree-Fock theory including BCS dynamical pairing correlations. The fission modes of the 258Fm nucleus are studied. The…
Following upon our earlier paper [1] containing some initial results, we present detailed discussion and complete results in this paper, which provide the first direct evidence for the validity of isospin as a nearly good quantum number in neutron-rich systems. The evidence comes from the reproduction of the general features of the partition-wise relative yields of neutron-rich fission fragments produced in two heavy-ion induced fusion fission reactions, namely $^{208}$Pb ($^{18}$O, f) and $^{238}$U ($^{18}$O, f), by using the concept of isospin conservation. To fix the isospin values and use …
The collective dynamics of low energy fission in 238U is described within a time-dependent formalism based on the Gaussian Overlap Approximation of the time-dependent Generator Coordinate Method. The intrinsic deformed configurations of the nucleus are determined from the self-consistent Hartree-Fock-Bogoliubov procedure employing the effective force D1S with constraints on the quadrupole and octupole moments. Fragment kinetic energy and mass distributions are calculated and compared with experimental evaluations. The effect of the collective dynamics along the fission paths and the influence …
Microscopic Description of Rotational Nuclear Fission Elucidates Fragment Spin Generation and Scission Mechanism
Hangzhou International Innovation Institute of Beihang University（北京航空航天大学杭州国际创新学院）；School of Physics, and State Key Laboratory of Nuclear Physics and Technology, Peking University（北京大学物理学院及核物理与核技术国家重点实验室）；Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences（中国科学院现代物理研究所南方核科学理论中心）
Energy-Resolved Spin and Parity Distributions of Fission Fragments
University of Zagreb（萨格勒布大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Université de Toulouse（图卢兹大学）
Although nuclear fission can be understood qualitatively as an evolution of the nuclear shape, a quantitative description has proven to be very elusive. In particular, until now, there exists no model with demonstrated predictive power for the fission fragment mass yields. Exploiting the expected strongly damped character of nuclear dynamics, we treat the nuclear shape evolution in analogy with Brownian motion and perform random walks on five-dimensional fission potential-energy surfaces which were calculated previously and are the most comprehensive available. Test applications give good repr…
