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 literature. With typical value of the parameters, we find that the adiabatic approximation works within a factor of around 5 for the decay width.
The event-by-event fission model FREYA is extended to spontaneous fission of actinides and a variety of neutron observables are studied for spontaneous fission and fission induced by thermal neutrons with a view towards possible applications for detection of special nuclear materials.
A simple phenomenological model, based on the Świ{\c a}tecki idea for evaluation of the spontaneous fission half-lives is proposed. The model contains only one adjustable parameter fixed to the data for even-even nuclei and two additional hindrance factors to the life-times, which give the effect of odd particles. A good agreement with the experimental data for all fissioning nuclei is achieved.
The spontaneous fission lifetime of 264Fm has been studied within nuclear density functional theory by minimizing the collective action integral for fission in a two-dimensional quadrupole collective space representing elongation and triaxiality. The collective potential and inertia tensor are obtained self-consistently using the Skyrme energy density functional and density-dependent pairing interaction. The resulting spontaneous fission lifetimes are compared with the static result obtained with the minimum-energy pathway. We show that fission pathways strongly depend on assumptions underlyin…
Spontaneous fission and alpha decay are the main decay modes for superheavy nuclei. The superheavy nuclei which have small alpha decay half-life compared to spontaneous fission half-life will survive fission and can be detected in the laboratory through alpha decay. We have studied the alpha decay half-life and spontaneous half-life of some superheavy elements in the atomic range Z = 100-130. Spontaneous fission half-lives of superheavy nuclei have been calculated using the phenomenological formula and the alpha decay half-lives using Viola-Seaborg-Sobiczewski formula (Sobiczewski et al. 1989)…
We examine the fission barriers, mass parameters and spontaneous fission half lives of fermium isotopes within the framework of macroscopic-microscopic model with a $δ$-pairing interaction. Four different macroscopic models are applied and studied. The results are compared to experimental data and to the corresponding monopole pairing ones. The half lives obtained in the $δ$-pairing model are comparable with experimental data. The state-dependence of pairing has an important effect on the calculated fission half-lives.
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…
We systematically calculate the spontaneous fission half-lives for heavy and superheavy nuclei between U and Fl isotopes. The spontaneous fission process is studied within the semi-empirical WKB approximation. The potential barrier is obtained using a generalized liquid drop model, taking into account the nuclear proximity, the mass asymmetry, the phenomenological pairing correction, and the microscopic shell correction. Macroscopic inertial-mass function has been employed for the calculation of the fission half-life. The results reproduce rather well the experimental data. Relatively long hal…
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…
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…
This is a short review of methods and results of calculations of fission barriers and fission half-lives of even-even superheavy nuclei. An approvable agreement of the following approaches is shown and discussed: The macroscopic-microscopic approach based on the stratagem of the shell correction to the liquid drop model and a vantage point of microscopic energy density functionals of Skyrme and Gogny type selfconsistently calculated within Hartree-Fock-Bogoliubov method. Mass parameters are calculated in the Hartree-Fock-Bogoliubov cranking approximation. A short part of the paper is devoted t…
Spontaneous-fission half-lives of the heaviest even-even nuclei are evaluated and compared with their alpha-decay mode. Calculations of spontaneous-fission half-lives are performed in the dynamical way with potential energy obtained by the macroscopic-microscopic method and the inertia tensor obtained by the cranking approximation. The alpha-decay half-lives are calculated for the same region of nuclei by use of the Viola and Seaborg formula. The ground-state properties such as mean square radii and electric quadrupole moments are also studied. From the analysis of spontaneous-fission half-lif…
Recent microscopic studies, based on the theoretical framework of nuclear energy density functionals, have analyzed dynamic (least action) and static (minimum energy) fission paths, and it has been shown that in addition to the important role played by nonaxial and/or octupole collective degrees of freedom, fission paths crucially depend on the approximations adopted in calculating the collective inertia. The dynamics of spontaneous fission of $^{264}$Fm and $^{250}$Fm is explored. The fission paths, action integrals and the corresponding half-lives predicted by the functionals PC-PK1 and DD-P…
It is shown, that the Gamow-like model with only one adjustable parameter - radius constant is able to reproduce well the alpha-decay half-lives for all even-even nuclei with the proton number larger than 50. The systematics for odd-A and odd-odd isotopes can be also well described when ones introduces an additional hindrance factor. Similar model based on the W. J. Swiatecki idea from 1955 is developed to reproduce the spontaneous fission half-lives of transactinide nuclei. The achieved accuracy of reproduction of the data is better than that obtained in more advanced theories.
Collective inertia is strongly influenced at the level crossing at which quantum system changes diabatically its microscopic configuration. Pairing correlations tend to make the large-amplitude nuclear collective motion more adiabatic by reducing the effect of those configuration changes. Competition between pairing and level crossing is thus expected to have a profound impact on spontaneous fission lifetimes. To elucidate the role of nucleonic pairing on spontaneous fission, we study the dynamic fission trajectories of $^{264}$Fm and $^{240}$Pu using the state-of-the-art self-consistent frame…
Using the idea of the instanton approach to quantum tunneling we try to obtain a method of calculating spontaneous fission rates for nuclei with the odd number of neutrons or protons. This problem has its origin in the failure of the adiabatic cranking approximation which serves as the basis in calculations of fission probabilities. Selfconsistent instanton equations, with and without pairing, are reviewed and then simplified to non-selfconsistent versions with phenomenological single-particle potential and seniority pairing interaction. Solutions of instanton-like equations without pairing an…
We study the role of different collective degrees of freedom in spontaneous fission process of even--even Fm isotopes. To find a proper collective space we examine a reach collection of nuclear shape parameters $\{β_λ\}$, with $λ$=2,3,4,5,6 and 8; as well as the paring degrees of freedom i.e. proton $Δ_{p}$ and neutron $Δ_{n}$ pairing gaps. On the basis of the multidimensional dynamic--programming method (MDPM) the optimal collective space $\{β_{2}, β_{4}, β_{6}, Δ_{p}, Δ_{n}\}$ for Fm isotopes was found.
Fortran subroutines have been written to simulate the production of fission neutrons from the spontaneous fission of 252Cf and 240Pu, and from the thermal neutron induced fission of 239Pu and 235U. The names of these four subroutines are getnv252, getnv240, getnv239, and getnv235, respectively. These subroutines reproduce measured first, second, and third moments of the neutron multiplicity distributions, measured neutron-fission correlation data for the spontaneous fission of 252Cf, and measured neutron-neutron correlation data for both the spontaneous fission of 252Cf and the thermal neutron…
Studies of fission dynamics, based on nuclear energy density functionals, have shown that the coupling between shape and pairing degrees of freedom has a pronounced effect on the nonperturbative collective inertia and, therefore, on dynamic (least-action) spontaneous fission paths and half-lives. Collective potentials and nonperturbative cranking collective inertia tensors are calculated using the multidimensionally-constrained relativistic mean-field (MDC-RMF) model. Pairing correlations are treated in the BCS approximation using a separable pairing force of finite range. Pairing fluctuations…
In this paper the essential theoretical predictions for the nuclear muonic radioactivity are presented by using a special fission-like model similar with that used in description of the pionic emission during fission. Hence, a fission-like model for the muonic radioactivity takes into account the essential degree of freedom of the system: muon-fissility, muon-fission barrier height, etc. Using this model it was shown that most of the SHE-nuclei lie in the region where the muonic fissility parameters attain their limiting value X=1. Hence, the SHE-region is characterized by the absence of a cla…
The systematic of the spontaneous fission half-lives for the nuclei $^{242-262}$Fm and $^{250-260}$No is analyzed, within a least action scheme, with the parametrization D1M of the Gogny energy density functional. The properties of the dynamic (least action) fission paths are analyzed and compared to those of the static (minimal energy) ones. The constrained Hartree-Fock-Bogoliubov approximation is used to compute deformed mean-field configurations, zero-point quantum corrections and collective inertias. It is shown that a cumbersome full variational search of the least action fission path, wi…
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.
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）
We investigate the dynamics of spontaneous fission in a configuration-interaction (CI) approach. In that formalism the decay rate is governed by an effective interaction coupling the ground-state configuration and a fission doorway configuration, with the interaction strength determined by inverting a high-dimensioned CI Hamiltonian matrix that may have a block-tridiagonal structure. It is shown that the decay rate decreases exponentially with the number of blocks at a rate determined by the largest eigenvalue of a matrix in the block space for Hamiltonians with identical off-diagonal blocks. …
Background: The complete event fission simulation code FREYA is used to study correlations in fission. To make the best possible simulations, FREYA one must find the optimized values of the five physics-based parameters which characterize the events produced by FREYA. So far this has been done only empirically or by brute force computational techniques. Purpose: We seek to develop a method of parameter optimization for FREYA that would include uncertainty quantification and an analysis of the correlations between the parameters. Methods: We focus on spontaneous fission as a simpler test case o…
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…
Exact solution of adiabatic time dependent Hartree-Fock-Bogoliubov inertia for fission
Technische Universität Darmstadt（达姆施塔特工业大学）；GSI Helmholtzzentrum für Schwerionenforschung（GSI亥姆霍兹重离子研究中心）；Helmholtz Forschungsakademie Hessen für FAIR（黑森FAIR亥姆霍兹研究院）；Instituto de Estructura de la Materia, CSIC（CSIC物质结构研究所）；Universidad Autónoma de Madrid（马德里自治大学）；Universidad Politécnica de Madrid（马德里理工大学）
The physics and the technical algorithms of the statistical de-excitation code ABLA07 are documented. The new developments of ABLA07 have been guided by the empirical knowledge obtained in a recent experimental campaign on the nuclide distributions measured at GSI, Darmstadt. Besides distinct signatures of very asymmetric binary splits, lighter systems show clear features of multifragmentation, while heavy systems reveal the influence of dynamics and microscopic structure on the fission process. ABLA07 includes elaborate but efficient descriptions of all these processes, with one set of the mo…
We explore the properties of the Barcelona Catania Paris Madrid (BCPM) energy density functional concerning fission dynamics. Potential energy surfaces as well as collective inertias relevant in the fission process are computed for several nuclei where experimental data exists. Inner and outer barrier heights as well as fission isomer excitation energies are reproduced quite well in all the cases. The spontaneous fission half lives $t_{\textrm{\textrm{SF}}}$ are also computed using the standard semiclassical approach and the results are compared with the experimental data. A reasonable agreeme…
Within MCNP6, users can use multiplicity distributions for spontaneous and neutron-induced fission from an evaluation performed by Lestone (LA-UR-05-0288). This evaluation assumes the multiplicity distributions are Gaussian and adjusts the standard deviations to reproduce measured 2nd and 3rd factorial moments. In MCNPX, users have the option of using multiplicity distributions for spontaneous fission using an evaluation of the available experimental data by Santi and Miller (LA-UR-07-6229). The spontaneous fission multiplicity distributions and corresponding factorial moments from these two e…
The decimal logarithm of spontaneous fission half-life of the superheavy nucleus $^{286}$Fl experimentally determined is $\log_{10} T_f^{exp} (s) = -0.632$. We present a method to calculate the half-life based on the cranking inertia and the deformation energy, functions of two independent surface coordinates, using the best asymmetric two center shell model. In the first stage we study the statics. At a given mass asymmetry up to about $η=0.5$ the potential barrier has a two hump shape, but for larger $η$ it has only one hump. The touching point deformation energy versus mass asymmetry shows …
In a previous article published in Phys. Rev. C 94 (2016) 014309 we have shown for the first time that the best dynamical trajectory during the deformation toward fission of the superheavy nucleus $^{286}$Fl is a linearly increasing radius of the light fragment, $R_2$. This macroscopic-microscopic result reminds us about the $α$ or cluster preformation at the nuclear surface, assumed already in 1928, and proved microscopically many times. This time we give more detailed arguments for the neighboring nucleus $^{282}$Cn. Also similar figures are presented for heavy nuclei $^{240}$Pu and $^{252}$…
Superheavy nuclei produced until now are decaying mainly by $α$ emission and spontaneous fission. Calculated $α$ decay half-lives are in agreement with experimental data within one order of magnitude. The discrepancy between theory and experiment can be as high as ten orders of magnitude for spontaneous fission. We analyze a way to improve the accuracy by using the action integral based on cranking inertia and a potential barrier computed by the macroscopic-microscopic method with a two-center shell model. Illustrations are given for $^{282}$Cn which has a measured fission half-life.
Using the microscopic-macroscopic model based on the deformed Woods-Saxon single-particle potential and the Yukawa-plus-exponential macroscopic energy we calculated static fission barriers $B_{f}$ for 1305 heavy and superheavy nuclei $98\leq Z \leq 126$, including even - even, odd - even, even - odd and odd - odd systems. For odd and odd-odd nuclei, adiabatic potential energy surfaces were calculated by a minimization over configurations with one blocked neutron or/and proton on a level from the 10-th below to the 10-th above the Fermi level. The parameters of the model that have been fixed pr…
The yields of light elements ($Z=1,2$) obtained from spontaneous ternary fission of $^{252}$Cf are treated within a nonequilibrium approach, and the contribution of unstable nuclei and excited bound states is taken into account. These light cluster yields may be used to probe dense matter, and to infer in-medium corrections. Continuum correlations are calculated from scattering phase shifts using the Beth-Uhlenbeck formula, and the effect of medium modification is estimated. The relevant distribution is reconstructed from the measured yields of isotopes. This describes the state of the nucleon…
We present a description of nuclear spontaneous fission, and generally of quantum tunneling, in terms of instantons - periodic imaginary-time solutions to time-dependent mean-field equations - that allows for a comparison with more familiar and used generator coordinate (GCM) and adiabatic time-dependent Hartree-Fock (ATDHF) methods. It is shown that the action functional whose value for the instanton is the quasiclassical estimate of the decay exponent fulfils the minimum principle when additional constraints are imposed on trial fission paths. In analogy with mechanics, these are conditions …
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…
Dynamical selection of fragment shell effects in spontaneous fission of $^{240}$Pu, $^{232}$Th, and $^{264}$Fm
Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University（中山大学中法核工程与技术学院）；Key Laboratory of Nuclear Data, China Institute of Atomic Energy（中国原子能科学研究院核数据重点实验室）
Background: Microscopic description of spontaneous fission is one of the most challenging subjects in nuclear physics. It is necessary to evaluate the collective potential and the collective inertia along a fission path for a description of quantum tunneling in spontaneous or low-energy fission. In past studies of the fission dynamics based on nuclear energy density functional (EDF) theory, the collective inertia has been evaluated with the cranking approximation, which neglects dynamical residual effects. Purpose: The purpose is to provide a reliable and efficient method to include dynamical …
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 …
Some time ago the possibility of classical (without Gamow tunneling) universal description of radioactive nuclei decay was demonstrated. Such possibility is basis on the classical interpretation of Bohmian Psi-field reality in Bohmian-Chetaev mechanics and the hypothesis for the presence of dissipative forces, generated from the Gryzinski translational precession of the charged particles spin, in Langevin- Kramers diffusion mechanism. In this paper is present an unified model of proton, alpha decay, cluster radioactivity and spontaneous fission half-life as explicit function which depends on t…
We show that the total kinetic energy (TKE) of nuclei after the spontaneous fission of $^{258}$Fm can be well reproduced using simple assumptions on the quantum collective phase-space explored by the nucleus after passing the fission barrier. Assuming energy conservation and phase-space exploration according to the stochastic mean-field approach, a set of initial densities is generated. Each density is then evolved in time using the nuclear time-dependent density-functional theory. This approach goes beyond mean-field by allowing spontaneous symmetry breaking as well as a wider dynamical phase…
In this paper new predictions for the spontaneous pion emission accompanied by fission are presented for all nuclei with Z between 100 and 108. The bimodal fission as an indirect experimental evidence for the pionic radioactivity is demonstrated. The experimental tests of this important connection are suggested.
Spontaneous-fission properties of 256Fm, 258Fm, and 260Fm isotopes are studied within the Skyrme-Hartree-Fock+BCS framework. In the particle-hole channel we take the Skyrme SkM* effective force, while in the particle-particle channel we employ the seniority pairing interaction. Three static fission paths for all investigated heavy fermium isotopes are found. The analysis of these fission modes allows to describe observed asymmetric fission of 256Fm, as well as bimodal fission of 258Fm and symmetric fission in 260Fm.
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…
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）
The nuclear cluster radioactivities have been studied theoretically in the framework of a microscopic superasymmetric fission model (MSAFM). The nuclear interaction potentials required for binary cold fission processes are calculated by folding in the density distribution functions of the two fragments with a realistic effective interaction. The microscopic nuclear potential thus obtained has been used to calculate the action integral within the WKB approximation. The calculated half lives of the present MSAFM calculations are found to be in good agreement over a wide range of observed experim…
We introduce a general variational framework to address the tunneling of hot Fermi systems. We use the representation of the trace of the imaginary time $τ=it$ propagator as a functional integral type of a sum over complete sets of states at intermediate propagation slices. We assume that these states are $τ$-dependent and generated by an arbitrary trial Hamiltonian $H_0(τ)$. We then use the convexity inequality to derive $H_0(τ)$ controlled variational bound for a trial action functional. This functional has a general structure consisting of two parts - statistically weighted quantum penetrab…
