State-of-the-art macroscopic-microscopic calculations reproduce spontaneous-fission half-lives of actinide and superheavy nuclei to within about three orders of magnitude, but only after multiplying the hydrodynamical irrotational-flow (Werner-Wheeler) collective-inertia tensor by an empirical global factor β=5, fitted by χ² minimization to 39 measured actinide half-lives. The open problem is to give a microscopic account of this effective inertia: derive the global scaling from microscopic nuclear dynamics such as adiabatic time-dependent Hartree-Fock-Bogoliubov theory, the generator coordinate method, or cranking, or provide a tractable microscopic replacement for the tensor. The fission-barrier region is of particular concern, since shell-induced fluctuations there make the hydrodynamical tensor suspect and the available microscopic approximations disagree with one another by up to an order of magnitude.

Background

Spontaneous fission (SF) is the decay of a heavy nucleus by quantum tunneling through the fission barrier — the barrier in the potential-energy surface (PES) separating the ground state from the scission configuration at which the nucleus splits. Because the half-life depends exponentially on the tunneling action, it is extremely sensitive both to the PES and to the collective inertia tensor that measures how the nucleus resists slow shape deformation. In the widely used macroscopic-microscopic approach, the PES combines a macroscopic liquid-drop-type energy (here the Lublin-Strasbourg Drop), a shell correction from a folded mean-field single-particle spectrum (Yukawa folding), and BCS (Bardeen–Cooper–Schrieffer) pairing correlations; the nuclear shape is described by a Fourier shape parametrization with four collective coordinates (elongation, mass asymmetry, neck, non-axiality). The half-life is then obtained in the semiclassical WKB (Wentzel–Kramers–Brillouin) tunneling approximation, T1/2∝exp⁡(2S)T_{1/2}\propto\exp(2S), where the action SS is minimized over paths through the multidimensional deformation space (the least-action path). The integrand of SS contains the collective inertia tensor BijB_{ij}. For BijB_{ij} the calculation of Ref. [1] uses the irrotational-flow (Werner-Wheeler) hydrodynamical mass — the inertia of an incompressible, frictionless fluid with a curl-free velocity field, introduced into dynamical fission calculations by Davies, Sierk, and Nix — which is cheap and smooth over the deformation grid and works well near scission (for example for fragment mass distributions), but is purely macroscopic and blind to single-particle structure. Microscopic alternatives compute BijB_{ij} from the response of the quasiparticles of a Hartree-Fock-Bogoliubov (HFB) mean field: the cranking (adiabatic perturbative) mass, the adiabatic time-dependent HFB (ATDHFB) mass, or the generator coordinate method with the Gaussian overlap approximation (GCM+GOA). These are known to differ from the hydrodynamical tensor by a factor of about 3-5 on average near compact shapes, precisely where the fission barrier lies (A. Góźdź, K. Pomorski, M. Brack, E. Werner, 1985), and pairing correlations lower the inertia roughly as 1/Δ21/\Delta^2, where Δ\Delta is the pairing gap, so that pairing dynamics feeds directly into the inertia. Fully microscopic frameworks — Gogny energy-density-functional calculations, based on a finite-range nuclear interaction (M. Warda and J. L. Egido, 2012) and nuclear density functional theory with ATDHFB inertias (A. Staszczak, A. Baran, W. Nazarewicz, 2013) — reproduce SF half-lives without any globally fitted inertia factor, showing that the microscopic theory contains the needed information, but at substantial computational cost and with their own inertia approximations; a recent review (N. Schunck and L. M. Robledo, 2016) identifies the collective inertia, and the computing it requires, as a central difficulty of microscopic fission theory. Marin Blanco et al. [1] found that within their macroscopic-microscopic framework the hydrodynamical tensor had to be multiplied by a single global factor β, fixed by χ² minimization to β=5 against the 39 measured actinide (thorium to fermium) half-lives, to reproduce empirical SF half-lives of actinide and superheavy nuclei to within about three orders of magnitude; they explicitly flag that the hydrodynamical tensor may not be reliable in the barrier region, where microscopic shell-induced fluctuations matter, and note that improved microscopic approaches to the collective inertia are numerically very costly.

Problem Statement

Give a microscopic, computationally tractable account of the effective collective inertia that governs spontaneous-fission barrier tunneling. Concretely: derive, from microscopic nuclear dynamics — cranking, adiabatic time-dependent Hartree-Fock-Bogoliubov (ATDHFB), or the generator coordinate method (GCM, with or without the Gaussian overlap approximation) — the effective inertia entering the WKB least-action expression for spontaneous-fission half-lives, and in particular the physical origin and value of the global scaling of the irrotational-flow (Werner-Wheeler) tensor by β≈5 that macroscopic-microscopic calculations currently fit to actinide half-lives, or else provide a tractable microscopic replacement for that tensor at a computational cost comparable to that of the irrotational-flow tensor itself on the same deformation grid, so that the systematic calculations over dense multidimensional deformation grids and many nuclei it currently enables remain feasible. The first alternative requires a microscopic derivation of the origin of the β≈5 scaling, not merely its empirical recovery; the second requires a replacement for the tensor usable in the WKB least-action expression within that cost. The fission-barrier region is of particular concern: there, shell-induced fluctuations make the smooth hydrodynamical tensor suspect, and the available microscopic approximations (cranking, ATDHFB, GCM+GOA, and quasiparticle random-phase approximation (QRPA)-type schemes including dynamical residual interactions and dynamical pairing) disagree with one another by up to an order of magnitude. Existing fully microscopic energy-density-functional least-action calculations (Gogny-based, DFT+ATDHFB, BSkG3) that bypass the hydrodynamical tensor entirely reproduce measured spontaneous-fission half-lives without any globally fitted inertia scaling factor, but they do not explain the origin of the empirical β≈5 scaling of the hydrodynamical tensor and do not supply a microscopic inertia for the macroscopic-microscopic framework.

References

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