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 low-energy induced fission of $^{228}$Th, $^{230}$Th, $^{234}$U, and $^{240}$Pu, we compare the non-perturbative and perturbative cranking ATDHFB mass tensors in the plane of axially-symmetric quadrupole and octupole deformations, as well as the resulting charge yields.
Starting from a quantum theory of dissipation for nuclear collective motion, the time-dependent generator coordinate method (TDGCM) is extended to allow for dissipation effects in the description of induced fission dynamics. The extension is based on a generalization of the GCM generating functions that includes excited states, and the resulting equation of motion in the collective coordinates and excitation energy. With the assumption of a narrow hamiltonian kernel, an expansion in a power series in collective momenta leads to a Schrödinger-like equation that explicitly includes a dissipation…
The role of dynamical pairing in induced fission dynamics is investigated using the time-dependent generator coordinate method in the Gaussian overlap approximation, based on the microscopic framework of nuclear energy density functionals. A calculation of fragment charge yields for induced fission of $^{228}$Th is performed in a three-dimensional space of collective coordinates that, in addition to the axial quadrupole and octupole intrinsic deformations of the nuclear density, also includes an isoscalar pairing degree of freedom. It is shown that the inclusion of dynamical pairing has a pron…
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…
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.
Since its beginnings, fission theory has asumed that low-energy induced fission takes place through transition-state channels at the barrier tops. Neverthess, up to now there is no microscopic theory applicable to those conditions. We suggest that modern reaction theory is suitable for this purpose, and propose a methodology based on a configuration-interaction framework using the Generator Coordinate Method (GCM). Simple reaction-theoretic models are constructed with the Gaussian Overlap Approximation (GOA) to parameterize both the dynamics within the channels and their incoherent couplings t…
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…
Fission transients describe the fission rate as it evolves towards the quasistationary value given by Kramers' formula. The nature of fission transients is dependent on the assumed initial distribution of the compound nuclei along the fission coordinate. Although the standard initial assumption of a near-spherical object leads to a transient suppression of the fission rate (fission delay), a moderate initial fissionlike deformation can reduce the magnitude of this suppression. For still larger initial deformations, transient fission enhancements are possible. Examples of this behavior are illu…
Collective mass tensor derived from the cranking approximation to the adiabatic time-dependent Hartree-Fock-Bogoliubov (ATDHFB) approach is compared with that obtained in the Gaussian Overlap Approximation (GOA) to the generator coordinate method. Illustrative calculations are carried out for one-dimensional quadrupole fission pathways in 256Fm. It is shown that the collective mass exhibits strong variations with the quadrupole collective coordinate. These variations are related to the changes in the intrinsic shell structure. The differences between collective inertia obtained in cranking and…
The dynamics of low-energy induced fission is explored using a consistent microscopic framework that combines the time-dependent generator coordinate method (TDGCM) and time-dependent nuclear density functional theory (TDDFT). While the former presents a fully quantum mechanical approach that describes the entire fission process as an adiabatic evolution of collective degrees of freedom, the latter models the dissipative dynamics of the final stage of fission by propagating the nucleons independently toward scission and beyond. By combining the two methods, based on the same nuclear energy den…
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 …
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…
Spontaneous fission barriers, quadrupole inertia, and zero-point quadrupole-energy corrections are calculated for 252,256,258Fm in the framework of the self-consistent Skyrme-Hartree-Fock+BCS theory. Two ways of computing dynamical inertia are employed: the Gaussian Overlap Approximation to the Generator Coordinate Method and cranking ansatz. The Skyrme results are compared with those of the Gogny-Hartree-Fock-Bogolyubov model.
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…
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…
\noindent\textbf{Background:} While many phenomenological models for nuclear fission have been developed, a microscopic understanding of fission has remained one of the most challenging problems in nuclear physics. \noindent\textbf{Purpose:} We investigate an applicability of the generator coordinate method (GCM) as a microscopic theory for cluster radioactivities of heavy nuclei, which can be regarded as a fission with large mass asymmetry, that is, a phenomenon in between fission and $α$-decays. \noindent\textbf{Methods:} Based on the Gamow theory, we evaluate the preformation probability of…
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 …
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…
Predicting the properties of neutron-rich nuclei far from the valley of stability is one of the major challenges of modern nuclear theory. In heavy and superheavy nuclei, a difference of only a few neutrons is sufficient to change the dominant fission mode. A theoretical approach capable of predicting such rapid transitions for neutron-rich systems would be a valuable tool to better understand r-process nucleosynthesis or the decay of super-heavy elements. In this work, we investigate for the first time the transition from asymmetric to symmetric fission through the calculation of primary fiss…
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 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 …
The non-perturbative method to compute Adiabatic Time Dependent Hartree Fock Bogoliubov (ATDHFB) collective inertias is extended to the Generator Coordinate Method (GCM) including the case of density dependent forces. The two inertias schemes are computed along the fission path of the $^{234}$U and compared with the perturbative results. We find that the non-perturbative schemes predict very similar collective inertias with a much richer structure than the one predicted by perturbative calculations. Moreover, the non-perturbative inertias show an extraordinary similitude with the exact GCM ine…
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…
A brief description of the IntraNuclear cascade, preequilibrium, evaporation, fission, coalescence, and Fermi breakup models used by the last versions of our CEM and LAQGSM event generators is presented, with a focus on our latest development of all these models. The recently developed "S" and "G" versions of our codes, that consider multifragmentation of nuclei formed after the preequilibrium stage of reactions when their excitation energy is above 2A MeV using the Statistical Multifragmentation Model (SMM) code by Botvina et al. ("S" stands for SMM) and the fission-like binary-decay model GE…
The nuclear fission process is a dramatic example of the large-amplitude collective motion in which the nucleus undergoes a series of shape changes before splitting into distinct fragments. This motion can be represented by a pathway in the many-dimensional space of collective coordinates. The collective action along the fission pathway determines the spontaneous fission half-lives as well as mass and charge distributions of fission fragments. We study the performance and precision of various methods to determine the minimum action and minimum-energy fission trajectories in the collective spac…
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…
A description of the IntraNuclear Cascade (INC), preequilibrium, evaporation, fission, coalescence, and Fermi breakup models used by the latest versions of our CEM03.03 and LAQGSM03.03 event generators is presented, with a focus on our most recent developments of these models. The recently developed "S" and "G" versions of our codes, that consider multifragmentation of nuclei formed after the preequilibrium stage of reactions when their excitation energy is above 2A MeV using the Statistical Multifragmentation Model (SMM) code by Botvina et al. ("S" stands for SMM) and the fission-like binary-…
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…
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…
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 …
We apply the generator coordinate method (GCM) to single-$Λ$ hypernuclei in order to discuss the spectra of hypernuclear low-lying states. To this end, we use the same relativistic point-coupling energy functional both for the mean-field and the beyond-mean-field calculations. This relativistic GCM approach provides a unified description of low-lying states in ordinary nuclei and in hypernuclei, and is thus suitable for studying the $Λ$ impurity effect. We carry out an illustrative calculation for the low-lying spectrum of $^{21}_Λ$Ne, in which the interplay between the hypernuclear collective…
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…
We discuss the origin of the narrowness of the single peak at mass-symmetric division in the fragment mass-yield curve for spontaneous fission of $^{258}$Fm. For this purpose, we employ the macroscopic-microscopic model, and calculate a potential-energy curve at the mass-symmetric compact scission configuration, as a function of the fragment mass number, which is obtained from the single-particle wave-function densities. In the calculations, we minimize total energies by varying the deformations of the two fragments, with constraints on the mass quadrupole moment and keeping the neck radius ze…
Cluster radioactivity is described as a very mass asymmetric fission process. The reflection symmetry breaking octupole moment has been used in a mean field HFB theory as leading coordinate instead of the quadrupole moment usually used in standard fission calculations. The procedure has been applied to the study of the ``very mass asymmetric fission barrier'' of several even-even Thorium isotopes. The masses of the emitted clusters as well as the corresponding half-lives have been evaluated on those cases where experimental data exist.
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）
Background: The isospin mixing is an interesting feature of atomic nuclei. It plays a crucial role in astrophysical nuclear reactions. However, it is not straightforward for variational nuclear structure models to describe it. Purpose: We propose a tractable method to describe the isospin mixing within a framework of the generator coordinate method and demonstrate its usability. Method: We generate the basis wave functions by applying the Fermi transition operator to the wave functions of isobars. The superposition of these basis wave functions and variationally obtained wave functions quantit…
We investigate the accuracy of several schemes to calculate ground-state correlation energies using the generator coordinate technique. Our test-bed for the study is the $sd$ interacting boson model, equivalent to a 6-level Lipkin-type model. We find that the simplified projection of a triaxial generator coordinate state using the $S_3$ subgroup of the rotation group is not very accurate in the parameter space of the Hamiltonian of interest. On the other hand, a full rotational projection of an axial generator coordinate state gives remarkable accuracy. We also discuss the validity of the simp…
We present a generator-coordinate method for realistic shell-model Hamiltonians that closely approximates the full shell model calculations of the matrix elements for the neutrinoless double-$β$ decay of $^{124}$Sn, $^{130}$Te, and $^{136}$Xe. We treat not only quadrupole deformations but also the proton-neutron pairing amplitudes as generator coordinates. We validate this method by calculating and comparing spectroscopic quantities with the exact shell model results and experimental data. Our Hamiltonian-based generator-coordinate method produces $0νββ$ matrix elements much closer to the shel…
We propose a consistent prescription for the derivation of the particle number and angular momentum projected QRPA (PQRPA) equation in the Generator Coordinate Method (GCM) framework for calculation of NME's of double-beta decay of axially deformed nuclei. We derive closed formulae for the calculation of excitation energies and wave functions of the intermediate nucleus.
In this work, we derived a transport equation based on a generalized equation of time-dependent generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA). The transport equation is obtained by using quantum-mechanics phase space distributions under a ``quasi-particle" picture and strategy of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hierarchy. The theoretical advantage of this transport equation is that time evolution of $s$-body phase space density distribution is coupled with $s+1$-body phase space density distributions, and thus, non-adiabatic effects and dynamical…
We treat proton-neutron pairing amplitudes, in addition to the nuclear deformation, as generator coordinates in a calculation of the neutrinoless double-beta decay of 76Ge. We work in two oscillator shells, with a Hamiltonian that includes separable terms in the quadrupole, spin-isospin, and pairing (isovector and isoscalar) channels. Our approach allows larger single-particle spaces than the shell model and includes the important physics of the proton-neutron quasiparticle random-phase approximation (QRPA) without instabilities near phase transitions. After comparing the results of a simplifi…
The existing formalism used to compute the operator overlaps necessary to carry out generator coordinate method calculations using a set of Hartree- Fock- Bogoliubov wave functions, is generalized to the case where each of the HFB states are expanded in different arbitrary bases spanning different sub-space of the Hilbert space.
The generator coordinate method (GCM) is an important tool of choice for modeling large-amplitude collective motion in atomic nuclei. The computational complexity of the GCM increases rapidly with the number of collective coordinates. It imposes a strong restriction on the applicability of the method. In this work, we propose a subspace-reduction algorithm that employs optimal statistical ML models as surrogates for exact quantum-number projection calculations for norm and Hamiltonian kernels. The model space of the original GCM is reduced to a subspace relevant for nuclear low energy spectra …
We present a mechanism to efficiently pre-select the number of intrinsic many-body states that are used to define the many-body wave functions within the discrete Generator Coordinate Method (GCM). This procedure, based on the proper definition of a natural basis of orthonormal states, does not require the evaluation of the non-diagonal Hamiltonian kernels to do the selection and helps to reduce the numerical instabilities. The performance of the method is analyzed in detail in the ground state and $0^{+}$ excited states of some selected nuclei computed with the Gogny energy density functional…
