A formalism for semi-adiabatic cranking mass parameters is presented. For the fission process of 234U, the time-dependent pairing equations of motion were used to calculate the excitation energy and to extract values of the cranking inertia. A fission barrier is determined by minimizing the action trajectory in a five dimensional configuration space spanned by elongation, necking, deformations of fragments and mass-asymmetry. The deformation energy is computed in the the frame of the microscopic-macroscopic model. The two center shell model with Woods-Saxon potentials is used in this context. Values of the inertia for excited fissioning systems are reported. A dependence between the cranking mass parameters and the intrinsic excitation energy is evidenced.
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.
Collective mass tensors derived in the cranking approximation to the adiabatic time-dependent Hartree-Fock-Bogoliubov (ATDHFB) method are employed in a study of induced fission dynamics. Together with a collective potential determined in deformation-constrained self-consistent mean-field calculations based on nuclear energy density functionals, the mass tensors specify the collective Hamiltonian that governs the time evolution of the nuclear wave function from an initial state at equilibrium deformation, up to scission and the formation of fission fragments. In an illustrative calculation of l…
The fission-fragments mass-yield of 236U is obtained by an approximate solution of the eigenvalue problem of the collective Hamiltonian that describes the dynamics of the fission process whose degrees of freedom are: the fission (elongation), the neck and the mass-asymmetry mode. The macroscopic-microscopic method is used to evaluate the potential energy surface. The macroscopic energy part is calculated using the liquid drop model and the microscopic corrections are obtained using the Woods-Saxon single-particle levels. The four dimensional modified Cassini ovals shape parametrization is used…
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 …
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…
The fission processes of thermal excited nuclei are conventionally studied by statistical models which rely on inputs of phenomenological level densities and potential barriers. Therefore the microscopic descriptions of spontaneous fission and induced fission are very desirable for a unified understanding of various fission processes. We propose to study the fission rates, at both low and high temperatures, with microscopically calculated temperature-dependent fission barriers and collective mass parameters. The fission barriers are calculated by the finite-temperature Skyrme-Hartree-Fock+BCS …
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 dynamics of induced fission of $^{226}$Th is investigated in a theoretical framework based on the finite-temperature time-dependent generator coordinate method (TDGCM) in the Gaussian overlap approximation (GOA). The thermodynamical collective potential and inertia tensor at temperatures in the interval $T=0 - 1.25$ MeV are calculated using the self-consistent multidimensionally constrained relativistic mean field (MDC-RMF) model, based on the energy density functional DD-PC1. Pairing correlations are treated in the BCS approximation with a separable pairing force of finite range. Constrai…
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 …
Low-energy positive and negative parity collective states in the equilibrium minimum, and the dynamics of induced fission of actinide nuclei are investigated in a unified theoretical framework based on the generator coordinate method (GCM) with the Gaussian overlap approximation (GOA). The collective potential and inertia tensor, both at zero and finite temperature, are computed using the self-consistent multidimensionally constrained relativistic mean field (MDC-RMF) model, based on the energy density functional DD-PC1. Pairing correlations are treated in the BCS approximation with a separabl…
Collective inertial mass coefficients with respect to translational, relative, and rotational motions are microscopically calculated, along the collective reaction path self-consistently determined, based on the adiabatic self-consistent collective coordinate (ASCC) method. The impact of the time-odd component of the mean-field potential on the inertial masses are investigated. The results are compared with those calculated with the cranking formulae. The inertial masses based on the ASCC method reproduce the exact total nuclear mass for the translational motion as well as the exact reduced ma…
In this article, we derive a quantal self-consistent time-reversal invariant parameter-free cranking model for isoscalar monopole excitation coupled to intrinsic motion in even-even light nuclei. The model uses a wavefunction that is a product of monopole and intrinsic wavefunctions and a constrained variational method to derive, from a many-particle Schrodinger equation, a pair of coupled self-consistent cranking-type Schrodinger equations for the monopole and intrinsic systems. The monopole co-ordinate used is the trace of the quadrupole tensor and hence describes the overall deformation of …
Towards the microscopic theoretical description for large amplitude collective dynamics, we calculate the coefficients of inertial masses for low-energy nuclear reactions. Under the scheme of energy density functional, we apply the adiabatic self-consistent collective coordinate (ASCC) method, as well as the Inglis' cranking formula to calculate the inertias for the translational and the relative motions, in addition to those for the rotational motion. Taking the scattering between two $α$ particles as an example, we investigate the impact of the time-odd components of the mean-field potential…
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 …
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…
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…
High-spin rotational bands in rare-earth Er ($Z=68$), Tm ($Z=69$) and Yb ($Z=70$) isotopes are investigated by three different nuclear models. These are (i) the cranked relativistic Hartree-Bogoliubov (CRHB) approach with approximate particle number projection by means of the Lipkin-Nogami (LN) method, (ii) the cranking covariant density functional theory (CDFT) with pairing correlations treated by a shell-model-like approach (SLAP) or the so called particle-number conserving (PNC) method, and (iii) cranked shell model (CSM) based on the Nilsson potential with pairing correlations treated by t…
The two stages dynamical stochastic model developed earlier for description of fusion-fission reactions is applied to the calculation of mass- and energy-distributions of fission fragments of platinum isotopes in reaction ${\rm ^{36}Ar + ^{142}Nd \to ^{178-x}Pt + xn}$. The first stage of this model is the calculation of the approaching of projectile nucleus to the target nucleus. On the second stage of the model, the evolution of the system formed after the touching of the projectile and target nuclei is considered. The evolution of the system on both stages is described by three-dimensional L…
We have decomposed to symmetric and asymmetric modes the mass-TKE fission fragment distributions calculated by 4-dimensional Langevin approach and observed how the dominant fission mode and symmetric mode change as functions of $Z^2/\sqrt[3]{A}$ of the fissioning system in the actinides and trans-actinide region. As a result, we found that the symmetric mode makes a sudden transition from super-long to super short fission mode around $^{254}$Es. The dominant fission modes on the other hand, are persistently asymmetric except for $^{258}$Fm, $^{259}$Fm and $^{260}$Md when the dominant fission m…
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-…
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…
The parametrization D1M of the Gogny energy density functional is used to study fission in the odd-mass Uranium and Plutonium isotopes with A=233,\ldots,249 within the framework of the Hartree-Fock-Bogoliubov (HFB) Equal Filling Approximation (EFA). Ground state quantum numbers and deformations, pairing energies, one-neutron separation energies, barrier heights and fission isomer excitation energies are given. Fission paths, collective masses and zero point rotational and vibrational quantum corrections are used to compute the systematic of the spontaneous fission half-lives t$_{SF}$, the mass…
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…
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…
We report here the signature of bi-modal fission, one asymmetric and the other symmetric, in Uranium nuclei in the mass range A = 230 to 236. The finding is unexpected and striking and is based on a model independent analysis of experimental mass distributions (cumulative yields) at various excitations from about 23 to 66 MeV in the alpha induced fission of 232Th. It has been found that the observed asymmetry in the mass distributions and the unusually narrow peak in the symmetry region, can both be explained in a consistent manner if one assumes: a) multi-chance fission, b) bi-modal fission a…
Post-neutron emission fission product mass distributions are calculated by using pre-neutron emission fission product yields (FPYs) and neutron multiplicity. A semi-empirical model is used to calculate the pre-neutron FPY, first. Then the neutron multiplicity for each fission fragment mass is used to convert the pre-neutron FPY to the post-neutron FPY. In doing so, assumptions are made for the probability for a pre-emission fission fragment with a mass number $A^*$ to decay to a post-emission fragment with a mass number $A$. The resulting post-neutron FPYs are compared with the data available.…
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 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 time-dependent generator coordinate method with the gaussian overlap approximation (TDGCM+GOA) formalism is applied to describe the fission of $^{252}$Cf. We perform analysis of fission from the initial states laying in the energetic range from the ground state to the state located 4 MeV above the fission barrier. The fission fragment mass distributions, obtained for different parity, energy of levels and types of mixed states, are calculated and compared with experimental data. The impact of the total time of wave packet propagation on the final results is studied as well. The weak depend…
The Hauser-Feshbach statistical model is applied to the de-excitation of primary fission fragments using input mass yields calculated with macroscopic-microscopic models of the potential energy surface. We test the sensitivity of the prompt fission observables to the input mass yields for two important reactions, $^{235}$U$(n_\mathrm{th},f)$ and $^{239}$Pu$(n_\mathrm{th},f)$, for which good experimental data exist. General traits of the mass yields, such as the location of the peaks and their widths, can impact both the prompt neutron and $γ$-ray multiplicities, as well as their spectra. Speci…
A comparative study of fission of actinides specially $^{238}$U, by proton and bremsstrahlung photon is performed. Relative mass distribution of $^{238}$U fission fragments have been explored theoretically for both proton and photon induced fission. The integrated yield along with charge distribution of the products are calculated to find out the neutron richness in comparison to the nuclei produced by r-process in nucleosynthesis. Some r-process nuclei in intermediate mass range for symmetric fission mode are found to be produced almost two order of magnitude more for proton induced fission t…
Constrained mean-field calculations, based on the Gogny-D1M energy density functional, have been carried out to describe fission in the isotopes $^{250-260}$No. The even-even isotopes have been considered within the standard Hartree-Fock-Bogoliobov (HFB) framework while for the odd-mass ones the Equal Filling Approximation (HFB-EFA) has been employed. Ground state quantum numbers and deformations, pairing energies, one-neutron separation energies, inner and outer barrier heights as well as fission isomer excitation energies are given. Fission paths, collective masses and zero-point quantum vib…
Semi-empirical mass formula of the atomic nucleus describe binding energies of the nuclei. In the simple form of this formula, there are five terms related to the properties of the nuclear structure. The coefficients in each terms can be determined by various approach such as fitting on experimental binding energy values. In this study, the surface energy coefficient in the formula which is a correction on total binding energy has been obtained by a method that is not previously described in the literature. The experimental fission barrier energies of nuclei have been used for this task. Accor…
We evaluate the impact of using sets of fission yields given by the GEF code for spontaneous (sf), neutron-induced ((n,f)), and beta-delayed (betadf) fission processes which take into account the approximate initial excitation energy of the fissioning compound nucleus. We further explore energy-dependent fission dynamics in the r process by considering the sensitivity of our results to the treatment of the energy sharing and de-excitation of the fission fragments using the FREYA code. We show that the asymmetric-to-symmetric yield trends predicted by GEF can reproduce the high-mass edge of the…
