We discuss possible avenues to study fission dynamics starting from a time-dependent mean-field approach. Previous attempts to study fission dynamics using the time-dependent Hartree-Fock (TDHF) theory are analyzed. We argue that different initial conditions may be needed to describe fission dynamics depending on the specifics of the fission phenomenon and propose various approaches towards this goal. In particular, we provide preliminary calculations for studying fission following a heavy-ion reaction using TDHF with a density contraint. Regarding prompt muon-induced fission, we also suggest a new approach for combining the time-evolution of the muonic wave function with a microscopic treatment of fission dynamics via TDHF.
We evaluate the performance of modern nuclear energy density functionals for predicting inner and outer fission barrier heights and energies of fission isomers of even-even actinides. For isomer energies and outer barrier heights, we find that the self-consistent theory at the HFB level is capable of providing quantitative agreement with empirical data.
Our understanding of nuclear fission, a fundamental nuclear decay, is still incomplete due to the complexity of the process. In this paper, we describe a study of spontaneous fission using the symmetry-unrestricted nuclear density functional theory. Our results show that the observed bimodal fission can be explained in terms of pathways in multidimensional collective space corresponding to different geometries of fission products. We also predict a new phenomenon of trimodal spontaneous fission for some rutherfordium, seaborgium, and hassium isotopes.
The time dependent equations of motion for the pair breaking effect were corroborated with a condition that fixes dynamically the number of particles on the two fission fragment. The single particle level scheme was calculated with the Woods-Saxon superasymmetric two center shell model. This model provides a continuous variation of the energies from one nucleus up to two separated fragments. The dissipated energy resorts from the time dependent pairing equations. A peculiar phenomenon was observed experimentally in cold fission: the odd partition yields are favored over the even ones. This odd…
Selected aspects of the description of neutron-induced fission in 240Pu in the framework of the nuclear energy density functional theory at finite temperature are presented. In particular, we discuss aspects pertaining to the choice of thermodynamic state variables, the evolution of fission barriers as function of the incident neutron energy, and the temperatures of the fission fragments.
Recent progresses in the description of the latter stage of nuclear fission are reported. Dynamical effects during the descent of the potential towards scission and in the formation of the fission fragments are studied with the time-dependent Hartree-Fock approach with dynamical pairing correlations at the BCS level. In particular, this approach is used to compute the final kinetic energy of the fission fragments. Comparison with experimental data on the fission of 258Fm are made.
The concept of fission barrier - a parameter which enters in quantitative estimates of various observables related to nuclear fission - is presented from the point of view of theory based on the picture of nuclear deformation and energy dependent on it. We describe the macroscopic-microscopic method of calculating energy landscapes which is simpler than the selfconsistent mean field approach, and, due to its two-component nature, seems to be easier to adjust to experimental data. We present some models and methods used to find the fission saddles. For the purpose of illustration, we present re…
Modelling fission with microscopic input: excitation-energy-dependent fission paths for neutron-induced reactions
Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles（布鲁塞尔天文与天体物理研究所，布鲁塞尔自由大学）；Brussels Laboratory of the Universe - BLU-ULB（宇宙布鲁塞尔实验室）
In this letter, we outline a methodology to calculate microscopically mass and charge distributions of spontaneous fission yields. We combine the multi-dimensional minimization of collective action for fission with stochastic Langevin dynamics to track the relevant fission paths from the ground-state configuration up to scission. The nuclear potential energy and collective inertia governing the tunneling motion are obtained with nuclear density functional theory in the collective space of shape deformations and pairing. We obtain a quantitative agreement with experimental data and find that bo…
A static microscopic study of potential-energy surfaces within the Skyrme-Hartree-Fock-plus-BCS model is carried out for the 256Fm and 258Fm isotopes with the goal of deducing some properties of spontaneous fission. The calculated fission modes are found to be in agreement with the experimentaly observed asymmetric-to-symmetric transition in the fragment-mass distributions and with the high- and low-total-kinetic-energy modes experimentally observed in 258Fm. Most of the results are similar to those obtained in macroscopic-microscopic models as well as in recent Hartree-Fock-Bogolyubov calcula…
Fission-fragment properties have been calculated for thermal neutron-induced fission on a $^{239}\textrm{Pu}$ target, using constrained Hartree-Fock-Bogoliubov calculations with a finite-range effective interaction. A quantitative criterion based on the interaction energy between the nascent fragments is introduced to define the scission configurations. The validity of this criterion is benchmarked against experimental measurements of the kinetic energies and of multiplicities of neutrons emitted by the fragments.
Two systematic sources of error in most current microscopic evaluations of fission-barrier heights are studied. They are concerned with an approximate treatment of the Coulomb exchange terms (known as the Slater approximation) in the self-consistent mean fields and the projection on good parity states (e.g., of positive parity for the spontaneous fission of an even-even nucleus) of left-right reflection asymmetric intrinsic solutions (e.g., around the second barrier). Approximate or unprojected solutions are shown to lead each to an underestimation of the barrier heights by a few hundred keV.
Following the formation of an excited muonic atom, inner shell transitions may proceed without photon emission by inverse internal conversion, i.e. the muonic excitation energy is transferred to the nucleus. In actinides, the 2p -> 1s and the 3d -> 1s muonic transitions result in excitation of the nuclear giant dipole and giant quadrupole resonances, respectively, which act as doorway states for fission. The nuclear excitation energy is typically 6.5 - 10 MeV. Because the muon lifetime is long compared to the timescale of prompt nuclear fission, the motion of the muon in the Coulomb field of t…
The problem of an apparent inconsistency between the fission rates derived on the basis of Bohr-Wheeler's transition-state method and Kramers' dynamical model of nuclear fission, first pointed out by Strutinsky in 1973, is revisited. The study is based on studying the features of individual trajectories on the fission path.
The temperature dependence of the liquid-drop fission barrier is considered, the critical temperature for the liquid-gas phase transition in nuclear matter being a parameter. Experimental and calculated data on the fission probability are compared for highly excited $^{188}$Os. The calculations have been made in the framework of the statistical model. It is concluded that the critical temperature for the nuclear liquid--gas phase transition is higher than 16 MeV.
We investigate microscopically the tunneling dynamics in spontaneous fission of atomic nuclei. To this end, we employ a schematic solvable model with a pairing-plus-quadrupole interaction. The spontaneous decay of a system is simulated by introducing a small imaginary part to the energy of a fission doorway state. We show that the many-body Hamiltonian can be reduced to an effective 2$\times$2 Hamiltonian, from which one can derive a simple approximate formula for the decay width. We particularly investigate the applicability of the adiabatic approximation, which has often been used in the lit…
The well established macroscopic-microscopic (mac-mic) description of nuclear fission enables the prediction of fission fragment yields for a broad range of fissioning systems. In this work, we present several key enhancements to this approach. We improve upon the microscopic sector of nuclear potential energy surfaces by magnifying the resolution of the Lipkin-Nogami equations and strengthening the Strutinsky procedure, thus reducing spurious effects from the continuum. We further present a novel deterministic method for calculating fission dynamics under the assumption of strongly damped nuc…
The microscopic description of nuclear fission still remains a topic of intense basic research. Un- derstanding nuclear fission, apart from a theoretical point of view, is of practical importance for energy production and the transmutation of nuclear waste. In nuclear astrophysics, fission sets the upper limit to the nucleosynthesis of heavy elements via the r-process. In this work we initiated a systematic study of intermediate energy proton-induced fission using the Constrained Molecu- lar Dynamics (CoMD) code. The CoMD code implements an effective interaction with a nuclear matter compressi…
A systematic study of 160 heavy and super-heavy nuclei is performed in the Hartree-Fock-Bogoliubov approach with the finite range and density dependent Gogny force with the D1S parameter set. We show calculations in several approximations: with axially symmetric and reflexion symmetric wave functions, with axially symmetric and non-reflexion symmetric wave functions and finally some representative examples with triaxial wave functions are also discussed. Relevant properties of the ground state and along the fission path are thoroughly analyzed. Fission barriers, Q$_α$-factors and lifetimes wit…
We investigate the cold fission of 252Cf within the two center shell model to compute the potential energy surface. The fission yields are estimated by using the semiclassical penetration approach. It turns out that the inner cold valley of the total potential energy is strongly connected with Z=50 magic number. The agreement with experimental values is very much improved only by considering mass and charge asymmetry degrees of freedom. Thus, indeed cold fission of 252Cf is a Sn-like radioactivity, related the other two "magic radioactivities", namely alpha-decay and heavy-cluster decay, calle…
We describe the fissioning dynamics of Pu240 from a configuration in the proximity of the outer fission barrier to full scission and the formation of the fragments within an implementation of density functional theory extended to superfluid systems and real-time dynamics. The fission fragments emerge with properties similar to those determined experimentally, while the fission dynamics appears to be quite complex, with many excited shape and pairing modes. The evolution is found to be much slower than previously expected, and the ultimate role of the collective inertia is found to be negligibl…
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 …
The microscopic approach of fission rates and neutron emission rates in compound nuclei have been applied to $^{258}$No and $^{286}$Cn. The microscopic framework is based on the finite-temperature Skyrme-Hartree-Fock+BCS calculations, in which the fission barriers and mass parameters are self-consistently temperature dependent. The fission rates from low to high temperatures can be obtained based on the imaginary free energy method. The neutron emission rates are obtained with neutron gases at surfaces. Finally the survival probabilities of superheavy nuclei can be calculated microscopically. …
The transformation of an atomic nucleus into two excited fission fragments is modeled as a strongly damped evolution of the nuclear shape, until scission occurs at a small critical neck radius, at which point the mass, charge, and shape of each fragment are extracted. The available excitation energy then is divided statistically on the basis of the microscopic level densities. This approach takes account of the important (and energy-dependent) finite-size effects. After the fragments have been fully accelerated and their shapes have relaxed to their equilibrium form, they undergo sequential ne…
The current status of the application of covariant density functional theory to microscopic description of nuclear fission with main emphasis on superheavy nuclei (SHN) is reviewed. The softness of SHN in the triaxial plane leads to an emergence of several competing fission pathes in the region of the inner fission barrier in some of these nuclei. The outer fission barriers of SHN are considerably affected both by triaxiality and octupole deformation.
To describe of dynamics of ternary fission of $^{252}$Cf an equation of motion of the tri-nuclear system is calculated. The fission of the $^{70}$Ni+$^{50}$Ca+$^{132}$Sn channel was chosen as one of the more probable channels of true ternary fission of $^{252}$Cf. The collinearity of ternary fission has been checked by analyzing results of the equation of motion. The results show that if initially all nuclei are placed collinearly (potential energy of this position is the smallest) and the component of the middle fragment's initial velocity which is perpendicular to this line, is zero then ter…
This article reviews how nuclear fission is described within nuclear density functional theory. In spontaneous fission, half-lives are the main observables and quantum tunnelling the essential concept, while in induced fission the focus is on fragment properties and explicitly time-dependent approaches are needed. The cornerstone of the current microscopic theory of fission is the energy density functional formalism. Its basic tenets, including tools such as the HFB theory, effective two-body effective nuclear potentials, finite-temperature extensions and beyond mean-field corrections, are pre…
Mean field calculations, based on the D1S, D1N and D1M parametrizations of the Gogny energy density functional, have been carried out to obtain the potential energy surfaces relevant to fission in several Ra isotopes with the neutron number 144 $\le$ N $\le$ 176. Inner and outer barrier heights as well as first and second isomer excitation energies are given. The existence of a well developed third minimum along the fission paths of Ra nuclei, is analyzed in terms of the energetics of the "fragments" defining such elongated configuration. The masses and charges of the fission fragments are stu…
Recent developments in theoretical modeling and in computational power have allowed us to make significant progress on a goal not achieved yet in nuclear theory: a fully microscopic theory of nuclear fission. The complete microscopic description remains a computationally demanding task, but the information that can be provided by current calculations can be extremely useful to guide and constrain phenomenological approaches. First, a truly microscopic framework that can describe the real-time dynamics of the fissioning system can justify or rule out assumptions and approximations incompatible …
Static and dynamic aspects of the fission process of $^{226}$Th are analyzed in a self-consistent framework based on relativistic energy density functionals. Constrained relativistic mean-field (RMF) calculations in the collective space of axially symmetric quadrupole and octupole deformations, based on the energy density functional PC-PK1 and a $δ$-force pairing, are performed to determine the potential energy surface of the fissioning nucleus, the scission line, the single-nucleon wave functions, energies and occupation probabilities, as functions of deformation parameters. Induced fission d…
With a help of the selfconsistent Hartree-Fock-Bogoliubov (HFB) approach with the D1S effective Gogny interaction and the Generator Coordinate Method (GCM) we incorporate the transverse collective vibrations to the one-dimensional model of the fission-barrier penetrability based on the traditional WKB method. The average fission barrier corresponding to the least-energy path in the two-dimensional potential energy landscape as function of quadrupole and octupole degrees of freedom is modified by the influence of the transverse collective vibrations along the nuclear path to fission. The set of…
Theoretical methods, interpretations and conclusions on the fission dynamics in a recent publication of H. Hofmann and F. A. Ivanyuk on the mean first passage time are critically considered.
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 Hauser-Feshbach Fission Fragment Decay (HF$^3$D) model is extended to calculate the prompt fission neutron spectrum (PFNS) for the thermal neutron induced fission on $^{235}$U, where the evaporated neutrons from all possible fission fragment pairs are aggregated. By studying model parameter sensitivities on the calculated PFNS, as well as non-statistical behavior of low-lying discrete level spin distribution, we conclude that discrepancies between the aggregation calculation and the experimental PFNS seen at higher neutron emission energies can be attributed to both the primary fission fra…
Fission fragments' charge and mass distribution is an important input to applications ranging from basic science to energy production or nuclear non-proliferation. In simulations of nucleosynthesis or calculations of superheavy elements, these quantities must be computed from models, as they are needed in nuclei where no experimental information is available. Until now, standard techniques to estimate these distributions were not capable of accounting for fine-structure effects, such as the odd-even staggering of the charge distributions. In this work, we combine a fully-microscopic collective…
The fission process is a fascinating phenomenon in which the atomic nucleus, a compact self-bound mesoscopic system, undergoes a spontaneous or induced quantum transition into two or more fragments. A predictive, accurate and precise description of nuclear fission, rooted in a fundamental quantum many-body theory, is one of the biggest challenges in science. Current approaches assume adiabatic motion of the system with internal degrees of freedom at thermal equilibrium. With parameters adjusted to data, such modelling works well in describing fission lifetimes, fragment mass distributions, or …
In the paper, it is studied the influence of Landau-Zener transitions between nuclear many-body states on the dissipative properties of nuclear large--amplitude collective motion. Within the cranking-like approach, we describe the time evolution of a nuclear many-body system as the self-consistent motion in a space of intrinsic excitations and in a space of a single collective (deformation) parameter. By that we measure how the spectral statistics of the nuclear energy levels affects the fission rate at quite large initial temperatures of heavy nuclei.
The K-matrix formalism is applied to a schematic model for nuclear fission. The purpose is to explore the dependence of observables on the assumptions made about the configuration space and nucleon interaction in the Hamiltonian of the fissile nucleus. As expected, branching ratios in induced fission are found to depend sensitively on the character of the residual interaction, whether it is pairing in form or taken from a random ensemble. On the other hand, the branching ratio is not much affected by the presence of additional configurations that do not introduce new fission paths.
Accounting for viscous damping within Fokker-Planck equations led to various improvements in the understanding and analysis of nuclear fission of heavy nuclei. Analytical expressions for the fission time are typically provided by Kramers' theory, which improves on the Bohr-Wheeler estimate by including the time-scale related to many-particle dissipative processes along the deformation coordinate. However, Kramers' formula breaks down for sufficiently high excitation energies where Kramers' assumption of a large barrier no longer holds. In the regime $T>1$ MeV, Kramers' theory should be replace…
The pre-scission particle multiplicities suggest a lifetime of 10 20 s for the nuclear fission to occur which is in contrast to the fission lifetime 10 18 s as predicted by atomic probe. This long standing ambiguity, arising due to the orders of magnitude differences among the fission lifetime measured from the nuclear and atomic probes, has been addressed within a dynamical model which includes the contributions from the nuclear shell effects. We show that, at lower excitation energies, these two probes decouples as the fissioning system survives for a long time without any particle evaporati…
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…
A theoretical model of one-body nuclear friction which was developed earlier, namely the chaos-weighted wall formula, is applied to a dynamical description of compound nuclear decay in the framework of the Langevin equation coupled with statistical evaporation of light particles and photons. We have used both the usual wall formula friction and its chaos-weighted version in the Langevin equation to calculate the fission probability and prescission neutron multiplicity for the compound nuclei $^{178}$W, $^{188}$Pt, $^{200}$Pb, $^{213}$Fr, $^{224}$Th, and $^{251}$Es. We have also obtained the co…
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…
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…
We solve the time-dependent Dirac equation for a muon which is initially bound to a fissioning actinide nucleus. The computations are carried out on a 3-D cartesian lattice utilizing the Basis-Spline collocation method. The muon dynamics is sensitive to the nuclear energy dissipation between the outer fission barrier and the scission point. From a comparison with experimental data we find a dissipated energy of about 10 MeV and a fission time delay due to friction of order $2 \times 10^{-21}$ s.
Within a density matrix approach for nuclear many--body system, it is derived non--Markovian Langevin equations of motion for nuclear collective parameters, where memory effects are defined by memory time. The developed stochastic approach is applied to study both the nuclear descent from fission barrier to a scission point and thermal diffusive overcoming of the barrier. The present paper is partly a review of our results obtained earlier and contains new results on the non--Markovian generalization of Kramers' theory of escape rate and on time features of the collective dynamics in the prese…
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…
For 75 years the theory of nuclear fission has been based on the existence of a collective coordinate associated with the nuclear shape, an assumption required by the Bohr-Wheeler formula as well as by the R-matrix theory of fission. We show that it is also possible to formulate the theory without the help of collective coordinates. In the new formulation, fission is facilitated by individual states in the barrier region rather than channels over the barrier. In a certain limit the theory reduces to a formula closely related to the formula for electronic conductance through resonant tunneling …
