Fission of

Nuclear fission plays an important role in fundamental and applied science, from astrophysics to nuclear engineering, yet it remains a major challenge to nuclear theory. Theoretical methods used so far to compute fission observables rely on symmetry-breaking schemes where basic information on the number of particles, angular momentum, and parity of the fissioning nucleus is lost. In this Letter, we analyze the impact of restoring broken symmetries in the benchmark case of ^{240}Pu.

Microscopic theory of nuclear fission: a review.

This article reviews how nuclear fission is described within nuclear density functional theory. A distinction should be made between spontaneous fission, where half-lives are the main observables and quantum tunnelling the essential concept, and induced fission, where the focus is on fragment properties and explicitly time-dependent approaches are often invoked. Overall, the cornerstone of the density functional theory approach to fission is the energy density functional formalism. The basic tenets of this method, including some well-known tools such as the Hartree-Fock-Bogoliubov (HFB) theory, effective two-body nuclear potentials such as the Skyrme and Gogny force, finite-temperature extensions and beyond mean-field corrections, are presented succinctly. The energy density functional approach is often combined with the hypothesis that the time-scale of the large amplitude collective motion driving the system to fission is slow compared to typical time-scales of nucleons inside the nucleus. In practice, this hypothesis of adiabaticity is implemented by introducing (a few) collective variables and mapping out the many-body Schrödinger equation into a collective Schrödinger-like equation for the nuclear wave-packet. The region of the collective space where the system transitions from one nucleus to two (or more) fragments defines what are called the scission configurations. The inertia tensor that enters the kinetic energy term of the collective Schrödinger-like equation is one of the most essential ingredients of the theory, since it includes the response of the system to small changes in the collective variables. For this reason, the two main approximations used to compute this inertia tensor, the adiabatic time-dependent HFB and the generator coordinate method, are presented in detail, both in their general formulation and in their most common approximations. The collective inertia tensor enters also the Wentzel-Kramers-Brillouin (WKB) formula used to extract spontaneous fission half-lives from multi-dimensional quantum tunnelling probabilities (For the sake of completeness, other approaches to tunnelling based on functional integrals are also briefly discussed, although there are very few applications.) It is also an important component of some of the time-dependent methods that have been used in fission studies. Concerning the latter, both the semi-classical approaches to time-dependent nuclear dynamics and more microscopic theories involving explicit quantum-many-body methods are presented. One of the hallmarks of the microscopic theory of fission is the tremendous amount of computing needed for practical applications. In particular, the successful implementation of the theories presented in this article requires a very precise numerical resolution of the HFB equations for large values of the collective variables. This aspect is often overlooked, and several sections are devoted to discussing the resolution of the HFB equations, especially in the context of very deformed nuclear shapes. In particular, the numerical precision and iterative methods employed to obtain the HFB solution are documented in detail. Finally, a selection of the most recent and representative results obtained for both spontaneous and induced fission is presented, with the goal of emphasizing the coherence of the microscopic approaches employed. Although impressive progress has been achieved over the last two decades to understand fission microscopically, much work remains to be done. Several possible lines of research are outlined in the conclusion.

Uncertainty quantification for nuclear density functional theory and information content of new measurements.

Statistical tools of uncertainty quantification can be used to assess the information content of measured observables with respect to present-day theoretical models, to estimate model errors and thereby improve predictive capability, to extrapolate beyond the regions reached by experiment, and to provide meaningful input to applications and planned measurements. To showcase new opportunities offered by such tools, we make a rigorous analysis of theoretical statistical uncertainties in nuclear density functional theory using Bayesian inference methods. By considering the recent mass measurements from the Canadian Penning Trap at Argonne National Laboratory, we demonstrate how the Bayesian analysis and a direct least-squares optimization, combined with high-performance computing, can be used to assess the information content of the new data with respect to a model based on the Skyrme energy density functional approach. Employing the posterior probability distribution computed with a Gaussian process emulator, we apply the Bayesian framework to propagate theoretical statistical uncertainties in predictions of nuclear masses, two-neutron dripline, and fission barriers. Overall, we find that the new mass measurements do not impose a constraint that is strong enough to lead to significant changes in the model parameters. The example discussed in this study sets the stage for quantifying and maximizing the impact of new measurements with respect to current modeling and guiding future experimental efforts, thus enhancing the experiment-theory cycle in the scientific method.

Visualizing Nuclear Scission through a Multifield Extension of Topological Analysis.

In nuclear science, density functional theory (DFT) is a powerful tool to model the complex interactions within the atomic nucleus, and is the primary theoretical approach used by physicists seeking a better understanding of fission. However DFT simulations result in complex multivariate datasets in which it is difficult to locate the crucial `scission' point at which one nucleus fragments into two, and to identify the precursors to scission. The Joint Contour Net (JCN) has recently been proposed as a new data structure for the topological analysis of multivariate scalar fields, analogous to the contour tree for univariate fields. This paper reports the analysis of DFT simulations using the JCN, the first application of the JCN technique to real data. It makes three contributions to visualization: (i) a set of practical methods for visualizing the JCN, (ii) new insight into the detection of nuclear scission, and (iii) an analysis of aesthetic criteria to drive further work on representing the JCN.

Island of rare Earth nuclei with tetrahedral and octahedral symmetries: possible experimental evidence.

Calculations using realistic mean-field methods suggest the existence of nuclear shapes with tetrahedral Td and/or octahedral Oh symmetries sometimes at only a few hundreds of keV above the ground states in some rare earth nuclei around 156Gd and 160Yb. The underlying single-particle spectra manifest exotic fourfold rather than Kramers's twofold degeneracies. The associated shell gaps are very strong, leading to a new form of shape coexistence in many rare earth nuclei. We present possible experimental evidence of the new symmetries based on the published experimental results--although an unambiguous confirmation will require dedicated experiments.

Nuclear tetrahedral symmetry: possibly present throughout the periodic table.

More than half a century after the fundamental, spherical shell structure in nuclei had been established, theoretical predictions indicated that the shell gaps comparable or even stronger than those at spherical shapes may exist. Group-theoretical analysis supported by realistic mean-field calculations indicate that the corresponding nuclei are characterized by the TD(d) ("double-tetrahedral") symmetry group. Strong shell-gap structure is enhanced by the existence of the four-dimensional irreducible representations of TD(d); it can be seen as a geometrical effect that does not depend on a particular realization of the mean field. Possibilities of discovering the TD(d) symmetry in experiment are discussed.