Fission Path Research Articles

Time-dependent Density Functional Studies of Nuclear Quantum Dynamics in Large Amplitudes

The time-dependent density functional theory (TDDFT) provides a unified description of the structure and reaction. The linear approximation leads to the random-phase approximation (RPA) which is capable of describing a variety of collective motion in a harmonic regime. Beyond the linear regime, we present applications of the TDDFT to nuclear fusion and fission reaction. In particular, the extraction of the internuclear potential and the inertial mass parameter is performed using two different methods. A fusion hindrance mechanism for heavy systems is investigated from the microscopic point of view. The canonical collective variables are determined by the adiabatic self-consistent collective coordinate method. Preliminary results of the spontaneous fission path, the potential, and the collective mass parameter are shown for 8Be --> alpha+alpha.

Pairing-induced speedup of nuclear spontaneous fission

Background: Collective inertia is strongly influenced at the level crossing at which the quantum system changes its microscopic configuration diabatically. Pairing correlations tend to make the large-amplitude nuclear collective motion more adiabatic by reducing the effect of these configuration changes. Competition between pairing and level crossing is thus expected to have a profound impact on spontaneous fission lifetimes.Purpose: To elucidate the role of nucleonic pairing on spontaneous fission, we study the dynamic fission trajectories of $^{264}\mathrm{Fm}$ and $^{240}\mathrm{Pu}$ using the state-of-the-art self-consistent framework.Methods: We employ the superfluid nuclear density functional theory with the Skyrme energy density functional ${\mathrm{SkM}}^{*}$ and a density-dependent pairing interaction. Along with shape variables, proton and neutron pairing correlations are taken as collective coordinates. The collective inertia tensor is calculated within the nonperturbative cranking approximation. The fission paths are obtained by using the least action principle in a four-dimensional collective space of shape and pairing coordinates.Results: Pairing correlations are enhanced along the minimum-action fission path. For the symmetric fission of $^{264}\mathrm{Fm}$, where the effect of triaxiality on the fission barrier is large, the geometry of the fission pathway in the space of the shape degrees of freedom is weakly impacted by pairing. This is not the case for $^{240}\mathrm{Pu}$, where pairing fluctuations restore the axial symmetry of the dynamic fission trajectory.Conclusions: The minimum-action fission path is strongly impacted by nucleonic pairing. In some cases, the dynamical coupling between shape and pairing degrees of freedom can lead to a dramatic departure from the static picture. Consequently, in the dynamical description of nuclear fission, particle-particle correlations should be considered on the same footing as those associated with shape degrees of freedom.

Spontaneous fission withβ-parameterized quasimolecular shape

In the framework of a generalized liquid drop model (GLDM), a quasimolecular mechanism is introduced to describe the deformation of a nucleus in the procedure of nuclear fission or fusion. In order to more appropriately evaluate the shell correction on the fission or fusion path, the quasimolecular shape is described in terms of deformation parameters beta (i. e., so-called beta parameterized) by a transformation. For symmetric fission it is done analytically, whereas for asymmetric fission it is performed in a pure numerical way. Thereafter, for each quasimolecular shape, the shell correction can be calculated by the Strutinsky method where the single-particle energies are derived from a shell model in an axially deformed Woods-Saxon potential with the beta-parameterized quasimolecular shape. We then use this recipe to predict the half-lives of several spontaneous fission channels for some heavy nuclei, and the results are in agreement with the experimental data.

Dynamic versus static fission paths with realistic interactions

The properties of dynamic (least action) fission paths are analyzed and compared to the ones of the more traditional static (least energy) paths. Both the BCPM and Gogny D1M energy density functionals are used in the calculation of the HFB constrained configurations providing the potential energy and collective inertias. The action is computed as in the WKB method. A full variational search of the least-action path over the complete variational space of HFB wave functions is cumbersome and probably unnecessary if the relevant degrees of freedom are identified. In this paper, we consider the particle number fluctuation degree of freedom that explores the amount of pairing correlations in the wave function. For a given shape, the minimum action can be up a factor of three smaller than the action computed for the minimum energy state with the same shape. The impact of this reduction on the lifetimes is enormous and dramatically improves the agreement with experimental data in the few examples considered.

Microscopic description of fission in neutron-rich plutonium isotopes with the Gogny-D1M energy density functional

The most recent parametrization D1M of the Gogny energy density functional is used to describe fission in the isotopes $^{232-280}$ Pu. We resort to the methodology introduced in our previous studies [Phys. Rev. C \textbf{88}, 054325 (2013) and Phys. Rev. C \textbf {89}, 054310 (2014)] to compute the fission paths, collective masses and zero point quantum corrections within the Hartree-Fock-Bogoliubov framework. The systematics of the spontaneous fission half-lives t$_{SF}$, masses and charges of the fragments in Plutonium isotopes is analyzed and compared with available experimental data. We also pay attention to isomeric states, the deformation properties of the fragments as well as to the competition between the spontaneous fission and $\alpha$-decay modes. The impact of pairing correlations on the predicted t$_{SF}$ values is demonstrated with the help of calculations for $^{232-280}$Pu in which the pairing strengths of the Gogny-D1M energy density functional are modified by 5 $\%$ and 10 $\%$, respectively. We further validate the use of the D1M parametrization through the discussion of the half-lives in $^{242-262}$Fm. Our calculations corroborate that, though the uncertainties in the absolute values of physical observables are large, the Gogny-D1M Hartree-Fock-Bogoliubov framework still reproduces the trends with mass and/or neutron numbers and therefore represents a reasonable starting point to describe fission in heavy nuclear systems from a microscopic point of view.

Study of Fission Barrier Heights of Uranium Isotopes by the Macroscopic-Microscopic Method

Potential energy surfaces of uranium nuclei in the range of mass numbers 229 through 244 are investigated in the framework of the macroscopic-microscopic model and the heights of static fission barriers are obtained in terms of a double-humped structure. The macroscopic part of the nuclear energy is calculated according to Lublin—Strasbourg-drop (LSD) model. Shell and pairing corrections as the microscopic part are calculated with a folded-Yukawa single-particle potential. The calculation is carried out in a five-dimensional parameter space of the generalized Lawrence shapes. In order to extract saddle points on the potential energy surface, a new algorithm which can effectively find an optimal fission path leading from the ground state to the scission point is developed. The comparison of our results with available experimental data and others' theoretical results confirms the reliability of our calculations.

Microscopic description of fission in uranium isotopes with the Gogny energy density functional

The most recent parametrizations D1S, D1N, and D1M of the Gogny energy density functional are used to describe fission in the isotopes ${}^{232--280}U$. Fission paths, collective masses, and zero-point quantum corrections, obtained within the constrained Hartree-Fock-Bogoliubov approximation, are used to compute the systematics of the spontaneous fission half-lives ${t}_{\mathrm{SF}}$, the masses and charges of the fission fragments, and their intrinsic shapes. The Gogny-D1M parametrization has been benchmarked against available experimental data on inner and second barrier heights, excitation energies of the fission isomers, and half-lives in a selected set of Pu, Cm, Cf, Fm, No, Rf, Sg, Hs, and Fl nuclei. It is concluded that D1M represents a reasonable starting point to describe fission in heavy and superheavy nuclei. Special attention is also paid to understand the uncertainties in the predicted ${t}_{\mathrm{SF}}$ values arising from the different building blocks entering the standard semiclassical Wentzel-Kramers-Brillouin formula. Although the uncertainties are large, the trend with mass or neutron numbers are well reproduced and therefore the theory still has predictive power. In this respect, it is also shown that modifications of a few percent in the pairing strength can have a significant impact on the collective masses leading to uncertainties in the ${t}_{\mathrm{SF}}$ values of several orders of magnitude.

Formation and dynamics of fission fragments

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 nonadiabatic evolution is computed using the time-dependent Hartree-Fock method, starting from an adiabatic configuration where the fragments have acquired their identity. It is shown that dynamics has an important effect on the kinetic and excitation energies of the fragments. The vibrational modes of the fragments in the post-scission evolution are also analyzed.

Fission in Rapidly Rotating Nuclei

We study the effect of rotation in fission of the atomic nucleus256Fm using an independent-particle shell modelwith the mean field represented by a deformed Woods-Saxon potential and the shapes defined through the Cassinian oval parametrization. The variations of barrier height with increasing angular momentum, appearance of double hump in fission path are analysed. Our calculations explain the appearance of double hump in fission path of 256Fm nucleus. The second minimum vanishes with increase in angular momentum which hints that the fission barrier disappears at large spin.

Multidimensionally-constrained relativistic mean-field models and potential-energy surfaces of actinide nuclei

Background: Many different shape degrees of freedom play crucial roles in determining the nuclear ground state and saddle point properties and the fission path. For the study of nuclear potential energy surfaces, it is desirable to have microscopic and self-consistent models in which all known important shape degrees of freedom are included.Purpose: By breaking both the axial and the spatial reflection symmetries simultaneously, we develop multidimensionally-constrained relativistic mean field (MDC-RMF) models.Methods: The nuclear shape is assumed to be invariant under the reversion of $x$ and $y$ axes, i.e., the intrinsic symmetry group is ${V}_{4}$ and all shape degrees of freedom ${\ensuremath{\beta}}_{\ensuremath{\lambda}\ensuremath{\mu}}$ with even $\ensuremath{\mu}$, such as ${\ensuremath{\beta}}_{20}$, ${\ensuremath{\beta}}_{22}$, ${\ensuremath{\beta}}_{30}$, ${\ensuremath{\beta}}_{32}$, ${\ensuremath{\beta}}_{40}$, $...,$ are included self-consistently. The single-particle wave functions are expanded in an axially deformed harmonic oscillator (ADHO) basis. The RMF functional can be one of the following four forms: the meson exchange or point-coupling nucleon interactions combined with the nonlinear or density-dependent couplings. The pairing effects are taken into account with the BCS approach.Results: The one-, two, and three-dimensional potential energy surfaces of ${}^{240}$Pu are illustrated for numerical checks and for the study of the effect of the triaxiality on the fission barriers. Potential energy curves of even-even actinide nuclei around the first and second fission barriers are studied systematically. Besides the first ones, the second fission barriers in these nuclei are also lowered considerably by the triaxial deformation. This lowering effect is independent of the effective interactions used in the RMF functionals. Further discussions are made about different predictions on the effect of the triaxiality between the macroscopic-microscopic and MDC-RMF models, possible discontinuities on PES's from self-consistent approaches, and the restoration of broken symmetries.Conclusions: MDC-RMF models give a reasonably good description of fission barriers of even-even actinide nuclei. It is important to include both the nonaxial and the reflection asymmetric shapes simultaneously for the study of potential energy surfaces and fission barriers of actinide nuclei and of those in unknown mass regions such as, e.g., superheavy nuclei.

(A = 10)-Accompanied spontaneous ternary fission of californium isotopes

The ternary fission of 238-256Cf even-A isotopes is studied in the framework of the three-cluster model within the spherical approximation. The study is confined to those decays in which the accompanied light charged particle is of mass A = 10. The most probable ternary fission path is obtained as the one that has a peak in the Q-value and a minimum in the driving potential, with respect to the mass and charge asymmetries. Assuming both equatorial and collinear configurations of the different emissions of the light charged particles with A = 10, the ternary fission of 252Cf is found to be most favored with 10Be. The predicted favorable fragmentation channels of the 10Be-accompanied ternary fission of 238-256Cf, even-A, isotopes are discussed in detail. The closed-shell structure of the produced heaviest fragment is found to play a key role for the detected most favorable channels. A nucleus with a closed neutron shell or even doubly closed shell always appears as the heaviest nucleus in the favored channel of the ternary fission of all mentioned isotopes. As the isospin asymmetry of the isotope increases, the width of its cold fission valley around the symmetric mass region decreases, with more shifting towards the centre of the symmetric region. Neither the most favorable ternary emissions nor the behavior of the ternary fragmentation potential changed with the type of nuclear interaction potential (Yukawa, folding potential with Migdal force, and proximity 77). Also, almost no change is observed according to the method of emission of the fragments (equatorially or collinearly) or when the calculations are performed without inserting the nuclear interaction potential.

Towards Predictions of Fission Probabilities for Astrophysics Applications

Abstract In almost all the evaluations of the neutron-induced fission cross section for nuclei involved in nuclear applications, a double- humped fission barrier described by two independent inverted parabolas is considered. In addition, the continuum transition states are described by a simple Fermi-gas type nuclear level density expression for which the formulation as well as the parametrization remain highly uncertain. For these reasons, these approaches can be considered as having a rather poor predictive power and are not recommended for applications requiring a proper description of fission for nuclei far from stability, such as nuclear astrophysics. Hartree-Fock-Bolgolyubov (HFB) calculation are now available and can provide all the nuclear ingredients required to describe the fission path from the equilibrium deformation up to the nuclear scission point. The aim of this contribution is to apply the basic features of the optical model for fission, using the full microscopic information obtained from HFB models to calculate neutron- induced fission cross sections on selected actinide nuclei. This approach includes not only the details of the energy surface along the fission path, but also the estimate of the nuclear level density derived within the combinatorial approach on the basis of the same HFB single-particle properties, in particular at the fission saddle points. The sensitivity of the calculated fission cross sections to different model approximations is illustrated and the predictive power of such a microscopic approach tested. It is also shown that the various inputs can be tuned to reproduce at best experimental data in one unique coherent framework, so that in a close future it should become possible to make, on the basis of such models, accurate fission cross-section calculations and the corresponding estimates for nuclei, energy ranges or reaction channels for which no data exist. Finally, the same model is applied to the calculation of the fission probabilities, including β-delayed fission, spontaneous fission and neutron-induced fission, for all nuclei potentially produced by the r-process nucleosynthesis. Nucleosynthesis calculations resulting from the decompression of neutron star matter are performed and the sensitivity of the abundance calculation with respect to nuclear fission uncertainties (including fission probabilities and fission fragment distributions) is studied.

Fission of actinides through quasimolecular shapes

The potential energy of heavy nuclei has been calculated in the quasimolecular shape path from a generalized liquid drop model including the proximity energy, the charge and mass asymmetries and the microscopic corrections. The potential barriers are multiple-humped. The second maximum is the saddle-point. It corresponds to the transition from compact one-body shapes with a deep neck to two touching ellipsoids. The scission point lies at the end of an energy plateau well below the saddle-point and where the effects of the nuclear attractive forces between two separated fragments vanish. The energy on this plateau is the sum of the kinetic and excitation energies of the fragments. The shell and pairing corrections play an essential role to select the most probable fission path. The potential barrier heights agree with the experimental data and the theoretical half-lives follow the trend of the experimental values. A third peak and a shallow third minimum appear in asymmetric decay paths when one fragment is close to a double magic quasi-spherical nucleus, while the smaller one changes from oblate to prolate shapes.

Two-mode fission – experimental verification and characterization of two fission-modes

AbstractExperimental verification and characterization of the two-mode fission are reviewed. The presence of two independent deformation-paths in low energy fission of actinides is demonstrated by studying correlation among saddle-point configurations, scission-point configurations, and mass-yield distributions; the elongated scission configuration is related with the fission process that goes over a higher threshold energy and results in a symmetric mass-division mode, while the compact scission configuration with the process that experiences a lower threshold ends up with an asymmetric mass-division mode. Based on an extensive systematic analysis of scission properties in a wide range of actinide fission, the bimodal fission observed in the spontaneous fission of the heavy actinides is interpreted as the result of the presence of two fission paths, namely, the ordinary asymmetric fission path and a strongly shell-influenced symmetric mode.

Fission half-lives of superheavy nuclei in a microscopic approach

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$_\alpha$-factors and lifetimes with respect to fission and $\alpha$-decay as well as other observables are discussed. Larger configuration spaces and more general HFB wave functions as compared to previous studies provide a very good agreement with the experimental data.

Fission properties forr-process nuclei

We present a systematics of fission barriers and fission lifetimes for the whole landscape of superheavy elements (SHE), i.e., nuclei with $Z\ensuremath{\ge}100$. The fission lifetimes are also compared with the $\ensuremath{\alpha}$-decay half-lives. The survey is based on a self-consistent description in terms of the Skyrme-Hartree-Fock (SHF) approach. Results for various different SHF parametrizations are compared to explore the robustness of the predictions. The fission path is computed by quadrupole constrained SHF. The computation of fission lifetimes takes care of the crucial ingredients of the large-amplitude collective dynamics along the fission path, as self-consistent collective mass and proper quantum corrections. We discuss the different topologies of fission landscapes which occur in the realm of SHE (symmetric versus asymmetric fission, regions of triaxial fission, bimodal fission, and the impact of asymmetric ground states). The explored region is extended deep into the regime of very neutron-rich isotopes as they are expected to be produced in the astrophysical $r$ process.

Potential energy surfaces of actinide nuclei from a multidimensional constrained covariant density functional theory: Barrier heights and saddle point shapes

For the first time the potential energy surfaces of actinide nuclei in the $(\beta_{20}, \beta_{22}, \beta_{30})$ deformation space are obtained from a multi-dimensional constrained covariant density functional theory. With this newly developed theory we are able to explore the importance of the triaxial and octupole shapes simultaneously along the whole fission path. It is found that besides the octupole deformation, the triaxiality also plays an important role upon the second fission barriers. The outer barrier as well as the inner barrier are lowered by the triaxial deformation compared with axially symmetric results. This lowering effect for the reflection asymmetric outer barrier is 0.5 $\sim$ 1 MeV, accounting for $10 \sim 20%$ of the barrier height. With the inclusion of the triaxial deformation, a good agreement with the data for the outer barriers of actinide nuclei is achieved.

Analysis of the (n,f) Reaction in the Plutonium Isotopes

This paper describes the modified Hauser-Feshbach formalism used to com- pute accurately fission cross sections for low-energy neutrons (from a few keV up to 5.5 MeV) in presence of intermediate structure in the second well. Application to the large plutonium isotope family (236 to 244) has been made with in particular reliable predic- tions of the cross sections of the short-lived nuclides. Special attention is paid to the choice of the model parameters entering in the calculations. The decay of a compound nucleus by fission is a complex multi-dimensional problem, whose de- scription still requires many model parameters. The elucidation and physical understanding of these parameters is an ongoing research process, and this paper reports our progress in the study of the fis- sion cross-sections for a long sequence of plutonium isotopes. Our ultimate goal is to replace many fitting parameters ubiquitous in current data evaluations with physically sound theory with fewer pa- rameters. If successful, such an approach would lead to reliable predictions of the fission cross-sections of nuclides for which measurements are unavailable or very poor, while at the same time deepen our understanding of the fission process. Here we present results obtained with a modern version of the AVXSF code (AVerage Cross Sec- tion of Fission), which treats in great detail the intermediate structure due to the presence of a sec- ond well on the fission path. The correct modeling of the class-II states properties and in particular, their coupling with the levels present in the first well as well as to the continuum, is crucial to infer meaningful barrier heights. Experimentally, those barrier heights can be inferred from an analysis of low-energy neutron-induced fission cross sections in the case of fertile nuclei, and transfer reactions leading to fission, e.g., ( t, p f ), in the case of fissile nuclei. The AVXSF code can treat both situations adequately in order to extract fission barrier heights. In the following, the theory and main equations implemented in the AVXSF code are introduced. Approximate formulae valid in specific situations are also discussed. A Monte Carlo option is used when analytic approximations fail and a full sampling of the level characteristics distributions is needed. We then present preliminary results on the neutron-induced fission cross sections obtained for a suite of plutonium isotopes, for which a consistent set of model input parameters was devel- oped. Our analysis encompasses neutron incident energies from a few keV up to 5.5 MeV, just below the onset of second-chance fission. The importance of fission barrier heights, nucleon pairing gaps and level densities is emphasized. This paper concludes on some examples of predicted cross sections with some reliability for the short-lived nuclides of the Pu series.

Microscopic description of cluster radioactivity in actinide nuclei

Cluster radioactivity is the emission of a fragment heavier than $\alpha$ particle and lighter than mass 50. The range of clusters observed in experiments goes from $^{14}$C to $^{32}$Si while the heavy mass residue is always a nucleus in the neighborhood of the doubly-magic $^{208}$Pb nucleus. Cluster radioactivity is described in this paper as a very asymmetric nuclear fission. A new fission valley leading to a decay with large fragment mass asymmetry matching the cluster radioactivity products is found. The mass octupole moment is found to be more convenient than the standard quadrupole moment as the parameter driving the system to fission. The mean-field HFB theory with the phenomenological Gogny interaction has been used to compute the cluster emission properties of a wide range of even-even actinide nuclei from $^{222}$Ra to $^{242}$Cm, where emission of the clusters has been experimentally observed. Computed half-lives for cluster emission are compared with experimental results. The noticeable agreement obtained between the predicted properties of cluster emission (namely, clusters masses and emission half-lives) and the measured data confirms the validity of the proposed methodology in the analysis of the phenomenon of cluster radioactivity. A continuous fission path through the scission point has been described using the neck parameter constraint.

Fission properties of the Barcelona–Catania–Paris energy density functional

Fission properties of the Barcelona–Catania–Paris (BCP) energy density functional are explored by performing constrained mean field Hartree-Fock-Bogoliubov (HFB) calculations along the fission path. These calculations provide us with the quantities required to estimate the spontaneous fission half lives and fragment mass distribution. The results obtained are compared to experimental data and other calculations.