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 analyzed the experimental data of the pre-scission neutron multiplicity in connection with fission fragments in the reaction $^{58}$Ni+$^{208}$Pb at the incident energy corresponding to the excitation energy of compound nucleus $E^{*}$=185.9 MeV, which was performed by DéMoN group. The relation between the pre-scission neutron multiplicity and each reaction process having different reaction time is investigated. In order to study the fusion-fission process accompanied by neutron emission, the fluctuation-dissipation model combined with a statistical model is employed. It is found that the f…
In the present work the formal definition of the scission point - the maximal elongation at which the nucleus splits into two fragments - is given. The shape and the deformation energy at the scission point are calculated within the macroscopic-microscopic model. Three minima in the scission point deformation energy are found corresponding to the "standard", "supershort" and "superlong" fission modes. The contribution of each fission mode to the mass distribution of the fission fragments and total kinetic energy is discussed and compared with the experimental results. On the example of the fis…
An outstanding problem in the theory of nuclear fission is understanding the Hamiltonian dynamics at the scission point. Here we apply the Generator Coordinate Method to calculate decay widths for pre-scission configurations into the two-fragment continuum. Transitions that are allowed under diabatic dynamics can have widths up to several MeV. For non-diabatic decays through the pairing interaction, typical widths to a specific final state channel are 2-3 orders of magnitude smaller. The nucleus U-236 is taken as a representative example in the calculations.
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）
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…
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…
A static scission configuration in cold ternary fission has been considered in the framework of two mean field approaches. The virial theorems has been suggested to investigate correlations in the phase space, starting from a kinetic equation. The inverse mean field method is applied to solve single-particle Schredinger equation, instead of constrained selfconsistent Hartree-Fock equations. It is shown, that it is possible to simulate one-dimensional three-center system via inverse scattering method in the approximation of reflectless single-particle potentials
In this note we provide answers to criticisms made the appendix D of the article [Phys. Rev. C100, 034615 (2019)] against to the theory developed recently and known a "Stochastic Mean-Field" (SMF) approach and its applications. Briefly, we list our replies and leave the judgment to the readers.
We reconsider the importance of statistical fluctuations for fission dynamics beyond the saddle in the light of recent evaluations of transport coefficients for average motion. The size of these fluctuations are estimated by means of the Kramers-Ingold solution for the inverted oscillator, which allows for an inclusion of quantum effects.
Within the two-center shell model parameterization we have defined the optimal shape which fissioning nuclei attain just before the scission and calculated the total deformation energy (liquid drop part plus the shell correction) as function of the mass asymmetry and elongation at the scission point. The three minima corresponding to mass symmetric and two mass asymmetric peaks in the mass distribution of fission fragments are found in the deformation energy at the scission point. The calculated deformation energy is used in quasi-static approximation for the estimation of the total kinetic an…
Survival of Pairing Correlations and Shell Effects at Scission in Finite-Temperature Nuclear Fission: Implications for Odd-Even Staggering
National Centre for Nuclear Research（国家核研究中心）；China Institute of Atomic Energy, Nuclear Data Center（中国原子能科学研究院核数据中心）；Department of Theoretical Physics, Maria Curie-Skłodowska University（玛丽亚·居里-斯克沃多夫斯卡大学理论物理系）；Department of Physics, Tsinghua University（清华大学物理系）
The fusion-fission process accompanied by neutron emission is studied in the superheavy-mass region on the basis of the fluctuation-dissipation model combined with a statistical model. The calculation of the trajectory or the shape evolution in the deformation space of the nucleus with neutron emission is performed. Each process (quasi-fission, fusion-fission, and deep quasi-fission processes) has a characteristic travelling time from the point of contact of colliding nuclei to the scission point. These dynamical aspects of the whole process are discussed in terms of the pre-scission neutron m…
A point-by-point answer to the comment authored by S. Ayik and D. Lacroix is presented. At this point in time this text is not aimed at being submitted to Phys. Rev. C or any other journal, unless the authors of the comment choose to follow such an avenue. I also suggest a possible formulation of a stochastic mean field approach free of the difficulties in the stochastic mean field model due to S. Ayik.
Hartree-Fock-Bogoliubov calculations of hot fission in $^{240}\textrm{Pu}$ have been performed with a newly-implemented code that uses the D1S finite-range effective interaction. The hot-scission line is identified in the quadrupole-octupole-moment coordinate space. Fission-fragment shapes are extracted from the calculations. A benchmark calculation for $^{226}\textrm{Th}$ is obtained and compared to results in the literature. In addition, technical aspects of the use of HFB calculations for fission studies are examined in detail. In particular, the identification of scission configurations, t…
An outstanding problem in the theory of nuclear fission is to understand the Hamiltonian dynamics at the scission point. In this work the fissioning nucleus is modeled in self-consistent mean-field theory as a set of Generator Coordinate (GCM) configurations passing through the scission point. In contrast to previous methods, the configurations are constructed in the Hartree-Fock approximation with axially symmetric mean fields and using the K-partition numbers as additional constraints. The goal of this work is to find paths through the scission point where the overlaps between neighboring co…
The calculations of the potential energy surface are essential in the theoretical description of the fission process. In the constrained self-consistent approach, the smooth evolution of nuclear shape is described from the ground state until a very elongated one with a narrow neck. In all microscopic calculations, the rupture of the neck at scission is associated with a substantial change of nuclear matter density distribution and rapid energy decrease. In this paper, we show that there is no discontinuity of the potential energy surface at scission when multi-constrained calculations are appl…
We propose a framework to calculate the dynamics at the scission point of nuclear fission, based as far as possible on a discrete representation of orthogonal many-body configurations. Assuming axially symmetric scission shapes, we use the $K$ orbital quantum number to build a basis of wave functions. Pre-scission configurations are stable under mean-field dynamics while post-scission configurations evolve to separated fragments. In this first exploratory study, we analyze a typical fission trajectory through to scission in terms of these configurations. We find that there is a major rearrange…
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…
Calculations are presented for the time evolution of $^{240}$Pu from the proximity of the outer saddle point until the fission fragments are well separated, using the time-dependent density functional theory extended to superfluid systems. We have tested three families of nuclear energy density functionals and found that all functionals exhibit a similar dynamics: the collective motion is highly dissipative and with little trace of inertial dynamics, due to the one-body dissipation mechanism alone. This finding justifies the validity of using the overdamped collective motion approach and to so…
A new statistical method for systematizing the nuclei fission fragments, investigation of their mass and charge spectra and neutron fission parameters has been suggested. In proposed method, the mass and charge yields of nucleons are determined from the equilibrium conditions of the nuclear fragments post-fission ensemble. Here we consider protons and neutrons of different fragments as statistically non-equivalent. The abilities of the proposed statistical method for explanation of the U, Pa, Th, Ac and Ra post-fission fragments ordering and neutron emission function calculation have been demo…
The scission of a nucleus into two fragments is at present the least understood part of the fission process, though the most important for the formation of the observables. To investigate the potential energy landscape at the largest possible deformations, i.e. at the scission point (line, hypersurface), the Strutinsky's optimal shape approach is applied. For the accurate description of the mass-asymmetric nuclear shape at the scission point, it turned out necessary to construct an interpolation between the two sets of constraints for the elongation and mass asymmetry which are applied success…
The isotopic-yield distributions and kinematic properties of fragments produced in transfer-induced fission of 240Pu and fusion-induced fission of 250Cf, with 9 MeV and 45 MeV of excitation energy respectively, were measured in inverse kinematics with the spectrometer VAMOS. The kinematic properties of identified fission fragments allow to derive properties of the scission configuration such as the distance between fragments, the total kinetic energy, the neutron multiplicity, the total excitation energy, and, for the first time, the proton- and neutron-number sharing during the emergence of t…
At a certain finite neck radius during the descent of a fissioning nucleus from the saddle to the scission point, the attractive nuclear forces can no more withstand the repulsive Coulomb forces producing the neck rupture and the sudden absorption of the neck stubs by the fragments. At that moment, the neutrons, although still characterized by their pre-scission wave functions, find themselves in the newly created potential of their interaction with the separated fragments. Their wave functions become wave packets with components in the continuum. The probability to populate such states gives …
In the exothermic process of fission decay, an atomic nucleus splits into two or more independent fragments. Several aspects of nuclear fission are not properly understood, in particular the formation of the neck between the nascent fragments, and the subsequent mechanism of scission into two or more independent fragments. Using an implementation of time-dependent density functional theory, based on a relativistic energy density functional and including pairing correlations, we analyze the final phase of the process of induced fission of $^{240}$Pu, and show that the time-scale of neck formati…
The simultaneous measurement of the isotopic fission-fragment yields and fission-fragment velocities of $^{239}$U has been performed for the first time. The $^{239}$U fissioning system was produced in one-neutron transfer reactions between a $^{238}$U beam at 5.88 MeV/nucleon and a $^{9}$Be target. The combination of inverse kinematics at low energy and the use of the VAMOS++ spectrometer at the GANIL facility allows the isotopic identification of the full fission-fragment distribution and their velocity in the reference frame of the fissioning system. The proton and neutron content of the fra…
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 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…
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…
A statistical model description of heavy ion induced fusion-fission reactions is presented where shell effects, collective enhancement of level density, tilting away effect of compound nuclear spin and dissipation are included. It is shown that the inclusion of all these effects provides a consistent picture of fission where fission hindrance is required to explain the experimental values of both pre-scission neutron multiplicities and evaporation residue cross-sections in contrast to some of the earlier works where a fission hindrance is required for pre-scission neutrons but a fission enhanc…
The strong deformation present immediately after scission has consequences for the angular momentum population of the fragments as well as the angular distribution of their decay radiation. We find that the usual spin-cutoff parameterization describes very well the angular momentum distribution associated with the deformation of the fragments at the scission point. Depending on the deformation, its contribution can comparable to the thermal contribution to the angular momentum of the newly formed fragments. The $M$-distribution of the angular momentum is highly polarized and gives rise to larg…
Energy dependence of fission observables is a key issue for wide nuclear applications. We studied real-time fission dynamics from low-energy to high excitations in the compound nucleus $^{240}$Pu with the time-dependent Hartree-Fock+BCS approach. It is shown that the evolution time of the later phase of fission towards scission is considerably lengthened at finite temperature. As the role of dynamical pairing is vanishing at high excitations, the random transition between single-particle levels around the Fermi surface to mimic thermal fluctuations is indispensable to drive fission. The obtain…
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…
Influence of the Exit Channel in $^{235}$U(n,f) and $^{239}$Pu(n,f) Reactions in Time-Dependent Density Functional Theory
Facility for Rare Isotope Beams, Michigan State University（稀有同位素束设施，密歇根州立大学）；Department of Physics and Astronomy, Michigan State University（物理与天文学系，密歇根州立大学）；Theoretical Division, Los Alamos National Laboratory（理论部，洛斯阿拉莫斯国家实验室）；Cyclotron Institute, Texas A&M University（回旋加速器研究所，德克萨斯农工大学）
Ternary fission of actinides probes the state of the nucleus at scission. Light clusters are produced in space and time very close to the scission point. Within the nonequilibrium statistical operator method, a generalized Gibbs distribution is constructed from the information given by the observed yields of isotopes. Using this relevant statistical operator, yields are calculated taking excited states and continuum correlations into account, in accordance with the virial expansion of the equation of state. Clusters with mass number $A \le 10$ are well described using the nonequilibrium genera…
Microscopic description of the fission process including intrinsic excitations. Part I: 240Pu adiabatic and asymmetric fission path within the Schrodinger Collective Intrinsic Model
CEA, DAM, DIF（法国原子能委员会）；Université Paris-Saclay（巴黎萨克雷大学）；CEA, DES, IRESNE, DER, SPRC, LEPh（法国原子能委员会）；Center for Computational Simulation, Universidad Politécnica de Madrid（马德里理工大学计算模拟中心）；Departamento de Física Teórica and CIAFF, Universidad Autónoma de Madrid（马德里自治大学理论物理系及CIAFF）；Université Paris-Saclay, CNRS/IN2P3, IJCLab（巴黎萨克雷大学）；Nuclear Science Division, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室核科学部）
The mass and total kinetic energy distributions of the fission fragments in the fission of even-even isotopes of superheavy elements from Hs (Z=108) to Og (Z=118) are estimated using a pre-scission point model. We restrict to nuclei for which spontaneous fission has been experimentally observed. The potential energy surfaces are calculated with Strutinsky's shell correction procedure. The parametrization of the nuclear shapes is based on Cassini ovals. For the just before scission configuration we fix $α$=0.98, what corresponds to $r_{neck}\approx 2$ fm, and take into account another four defo…
Although the overall time-scale for nuclear fission is long, suggesting a slow process, rapid shape evolution occurs in its later stages near scission. Theoretical prediction of the fission fragments and their characteristics are often based on the assumption that the internal degrees of freedom are equilibrated along the fission path. However, this adiabatic approximation may break down near scission. This is studied for the symmetric fission of $^{258,264}$Fm. The non-adiabatic evolution is computed using the time-dependent Hartree-Fock method, starting from an adiabatic configuration where …
The scission neutron kinetic energy spectrum is calculated for $^{236}U$ in the frame of the dynamical scission model. The bi-dimensional time dependent Schrödinger equation with time dependent potential is used to propagate each neutron wave function during the scission process which is supposed to last $1\times 10^{-22}$ sec. At the end, we separate the unbound parts and continue to propagate them as long as possible (in this case $50\times 10^{-22}$ sec) in the frozen fragments approximation. At several time intervals, the Fourier transforms of these wave packets are calculated in order to …
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 concept of collinear spontaneous true ternary fission of 252Cf is subject to critical analysis. The conclusion is that the collinear flight of the fragments turns out to be a natural and most probable mode. The collinearity arises in the model on the prescission stage as a result of the account of the principles of the collective Bohr's model. It is partly destroyed at the post-scission stage of spreading of the fragments due to their Coulomb interaction, with the allowance for the spin effects arising at the moment of scission. The final angular distribution of the fragments is calculated…
Kinetic energy of fission fragments within a dynamical model
Graduate School of Science and Engineering, Kindai University（近畿大学理工学研究科）；Advanced Science Research Center, Japan Atomic Energy Agency（日本原子能研究开发机构先进科学研究中心）
When two fragments are created in a fission decay, any torque due to nuclear and Coulomb interaction can change the fragment's angular momentum. This article explores the character and magnitude of the angular momentum as a function of the initial conditions around the scission point using the time-dependent Hartree-Fock theory. To understand the torque acting on the fragments, the Frozen Hartree-Fock method is also used to determine the collective potential at scission. Two $^{240}$Pu fission channel ( $^{132}$Sn+$^{108}$Ru and $^{144}$Ba+$^{96}$Sr ) are studied. These two channels cover diff…
A model is developed to calculate the total kinetic energy (TKE) distribution of fission fragments in the framework of the time-dependent generator coordinate method (TDGCM), extended to include dissipation effects in the description of induced fission dynamics. Starting from an expression for the dissipative term in the GCM Hamiltonian that determines the time evolution of a statistical collective wave function, derived in the first part of this work, the integrated flux of the probability current through the scission hyper-surface can be computed at different temperatures. The kinetic energy…
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.
A new, rapidly convergent Fourier over spheroid parametrization is developed to describe the shape of a fissioning nucleus: its elongation, non-axiality and left-right asymmetry and neck formation. The 4D Potential Energy Surfaces (PES) of even-even actinide nuclei are evaluated within the macro-micro model. The Langevin trajectories generated on such PESs allow for obtaining the fission fragments' mass and kinetic energy yields. The charge equilibration at the scission configuration and the post-scission neutron emission are also discussed.
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…
Background: Recent experiments on beta-delayed fission in the mercury-lead region and the discovery of asym- metric fission in 180 Hg [1] have stimulated theoretical interest in the mechanism of fission in heavy nuclei. Purpose: We study fission modes and fusion valleys in 180 Hg and 198 Hg to reveal the role of shell effects in pre-scission region and explain the experimentally observed fragment mass asymmetry and its variation with A. Methods: We use the self-consistent nuclear density functional theory employing Skyrme and Gogny energy density functionals. Results: The potential energy surf…
Probability distribution of observables from a Bogoliubov vacuum projected onto good particle number: application to scission configurations of an actinide
