Background: The reactions with the neutron-rich 48 Ca beam and actinide targets resulted in detection of new super-heavy (SH) nuclides with Z = 104 − 118. The unambiguous identification of the new isotopes, however, still poses a problem because their α-decay chains terminate by spontaneous fission (SF) before reaching the known region of the nuclear chart. The understanding of the competition between α-decay and SF channels in SH nuclei is, therefore, of crucial importance for our ability to map the SH region and assess its extent.Purpose: We perform self-consistent calculations of the competing decay modes of even-even SH isotopes with 108 ≤ Z ≤ 126 and 148 ≤ N ≤ 188. Methods:We use the state-of-the-art computational framework based on self-consistent, symmetry-unrestricted nuclear density functional theory capable of describing the competition between nuclear attraction and electrostatic repulsion. We apply the SkM* Skyrme energy density functional. The collective mass tensor of the fissioning superfluid nucleus is computed by means of the cranking approximation to the adiabatic time-dependent Hartree-Fock-Bogoliubov (HFB) approach. This work constitutes the very first systematic self-consistent study of spontaneous fission in the SH region, carried out at a full HFB level, that simultaneously takes into account both triaxiality and reflection asymmetry.Results: Breaking axial symmetry and parity turns out to be crucial for a realistic estimate of collective action; it results in lowering SF lifetimes by more than seven orders of magnitude in some cases. We predict two competing SF modes: reflection-symmetric and reflection-asymmetric. Conclusions:The shortest-lived SH isotopes decay by SF; they are expected to lie in a narrow corridor formed by 280 Hs, 284 Fl, and 284 118 Uuo that separates the regions of SH nuclei synthesized in "cold fusion" and "hot fusion" reactions. The region of long-lived SH nuclei is expected to be centered on 294 Ds with a total half-life of ∼1.5 days. Our survey provides a solid benchmark for the future improvements of self-consistent SF calculations in the region of SH nuclei.

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.

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 256 Fm. 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 perturbative cranking approximations to ATDHFB, and within GOA, are discussed.

Background: 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.Purpose: 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 framework. Methods:We employ the superfluid nuclear density functional theory with the Skyrme energy density functional 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 Fm, where the effect of triaxiality on the fission barrier is large, the geometry of fission pathway in the space of shape degrees of freedom is weakly impacted by pairing. This is not the case for 240 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.PACS numbers: 24.75.+i, 25.85.Ca, 21.60.Jz, 21.30.Fe, 27.90.+b Introduction -Nuclear fission is a fundamental phenomenon that is a splendid example of a large-amplitude collective motion of a system in presence of many-body tunneling. The corresponding equations involve potential, dissipative, and inertial terms [1]. The individualparticle motion gives rise to shell effects that influence the fission barriers and shapes on the way to fission, and also strongly impact the inertia tensor through the crossings of single-particle levels and resulting configuration changes [2][3][4]. The residual interaction between crossing configurations is strongly affected by nucleonic pairing: the larger pairing gap ∆ the more adiabatic is the collective motion [5][6][7][8][9].

The spontaneous fission lifetime of 264 Fm 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 underlying collective inertia. With the non-perturbative mass parameters, the dynamic fission pathway becomes strongly triaxial and it approaches the static fission valley. On the other hand, when the standard perturbative cranking inertia tensor is used, axial symmetry is restored along the path to fission; an effect that is an artifact of the approximation used. Introduction.-The spontaneous fission (SF) of a nucleus plays important role in many areas of science and applications [1][2][3]. In particular, it determines the stability of the heaviest and superheavy elements [4,5] and it impacts the formation of heavy elements at the final stages of the r-process through the recycling mechanism [6][7][8]. Therefore, a capability of theory to predict SF lifetimes in a reliable way is essential.The main ingredients for a theoretical determination of SF lifetimes are the collective potential and inertia tensor. For heavy systems, these quantities can be calculated by using the self-consistent mean field theory based on the energy density functional [9]. The potential energy surface (PES) is obtained by solving constrained HartreeFock-Bogoliubov equations (HFB) in a multidimensional space of collective coordinates. The collective inertia (or mass) tensor is obtained from the self-consistent densities by employing the adiabatic time-dependent HFB approximation (ATDHFB) [10][11][12]. Since SF is a quantummechanical tunneling process and the fission barriers are usually both high and wide, the SF lifetime is obtained semi-classically by minimizing the fission action integral in the collective space.The main objective of this work is to study SF by combining the microscopic density functional input with the sophisticated action minimization techniques. We demonstrate that the predicted SF pathway strongly depends on the choice of the collective inertia. In particular, in the commonly used perturbative cranking approximation, the variations of mass parameters due to level crossings (configuration changes) are underestimated; this re-

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This is a short review of methods and results of calculations of fission barriers and fission halflives 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 to the nuclear fission dynamics. We also discuss the predictive power of Skyrme functionals applied to key properties of the fission path of 266 Hs. It applies the standard techniques of error estimates in the framework of a χ 2 analysis.

Broyden's method, widely used in quantum chemistry electronic-structure calculations for the numerical solution of nonlinear equations in many variables, is applied in the context of the nuclear many-body problem. Examples include the unitary gas problem, the nuclear density functional theory with Skyrme functionals, and the nuclear coupled-cluster theory. The stability of the method, its ease of use, and its rapid convergence rates make Broyden's method a tool of choice for large-scale nuclear structure calculations.