Nuclear Fission: from more phenomenology and adjusted parameters to more fundamental theory and increased predictive power

TL;DR

Recent advances in theory and computational power have enabled a near-microscopic understanding of nuclear fission, moving beyond phenomenological models towards more fundamental, predictive descriptions of the process.

Contribution

The paper discusses how the extension of TDDFT to superfluid systems and powerful supercomputers are advancing microscopic nuclear fission modeling.

Findings

Improved accuracy of nuclear energy density functionals.

Potential to predict fission fragment properties without simplified models.

Progress towards fully microscopic fission simulations.

Abstract

Two major recent developments in theory and computational resources created the favorable conditions for achieving a microscopic description of nuclear fission almost eighty years after its discovery in 1939 by Hahn and Strassmann (1930). The first major development was in theory, the extension of the Time-Dependent Density Functional Theory (TDDFT) to superfluid fermion systems. The second development was in computing, the emergence of powerful enough supercomputers capable of solving the complex systems of equations describing the time evolution in three dimensions without any restrictions of hundreds of strongly interacting nucleons. Even though the available nuclear energy density functionals (NEDFs) are phenomenological still, their accuracy is improving steadily and the prospects of being able to perform calculations of the nuclear fission dynamics and to predict many properties of the fission fragments, otherwise not possible to extract from experiments, are within reach, all without making recourse anymore to uncontrollable assumptions and simplified phenomenological models.

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Full text

Meitner and Frisch Nature_1939r and Bohr and Wheeler Nature_1939r1 ; PhysRev_1939r interpreted the neutron induced fission of uranium observed by Hahn and Strassmann Natur_1939r as the Coulomb-driven division of a classically charged liquid drop in competition with the surface tension of the liquid drop and they obtained a good back-of-the-envelope estimated of the total kinetic energy (TKE) of the fission fragments (FFs). Unlike the $\alpha$-decay and spontaneous fission observed later Flerov , induced fission occurs above the barrier, therefore is to a large extent a classical process. The impinging neutron on uranium leads to the formation of an excited state, the compound nucleus, in a region of the nuclear spectrum where the level density is extremely high and the nucleus is thus relatively hot. The nucleus evolves for a very long time from its ground state shape towards the top of the fission, where the Coulomb repulsion overcomes surface tension and the nucleus starts its descent to scission, where the two FFs are formed. The top of the fission barrier is energetically barely below the neutron threshold. Although the nucleus is already quite elongated at this point, it has relatively little intrinsic excitation energy and is relatively cold.

It was soon realized that in nuclei nucleons form shells and behave in many instances as independent particles, like electrons in atoms. In other words nucleons live on quantized orbits Mayer ; Jensen , and the spin-orbit interaction plays a critical role in the formation of the nuclear shells and in the formation of the FFs Meitner . Hill and Wheeler PhysRev_1953r discussed how the liquid drop deformation energy emerges from a quantum mechanical approach based on considering the quantized single-particle motion of nucleons in a slowly deforming potential well. In their approach, the liquid drop potential deformation energy was, in the first approximation, an envelope of many intersecting parabolas caused by single-particle level crossings, see Figure 1. At single-particle level crossings of the last occupied level, nucleons jump from one level to another, in order to maintain the sphericity of the Fermi sphere. Without such a redistribution of nucleons at the Fermi level, the Fermi sphere would become oblate when a nucleus deforms on the way to scission into two fragments, while the spatial shape of the nucleus would become ever more prolate. This would lead to a volume excitation energy of the nucleus. This is inconsistent with the fact that nuclei are saturating systems with a surface tension that can only deform by changing the shape of their surface while maintaining their volume constant. Hence, only the Coulomb and the surface contributions to the total energy changes. Each single-particle level is typically double degenerate, due to Kramers degeneracy, and nucleons have to jump in pairs, otherwise the nuclear shape evolution towards scission would be hindered Bertsch:1980 ; PhysRevLett_1997r . In this case, nucleons would follow diabatic levels instead of adiabatic ones, as was indeed established in recent TDHF simulations of fission dynamics arxiv_2015r ; arxiv_2015r1 . Pairing correlations, while relatively weak in nuclei, is very effective at promoting simultaneously two nucleons from time-reverse orbits into other time-reverse orbits and thus it greatly facilitates the evolution of the nuclear shape towards scission.

It was established later that single-particle level bunching exists in nuclear systems not only in the case of spherical nuclei (as in the case of atoms), but also in deformed and highly deformed nuclei. At first this phenomenon was experimentally observed in the case of fission isomers at very large elongations Polikanov ; RMP_1972r ; RMP_1980r and subsequently in the case of superdeformed nuclei Twin . The existence of nucleonic shells at large deformations results in a potential energy deformation surface with significant maxima and minima, which are otherwise absent in the case of a classical charged liquid drop. The aforementioned level crossings cause discontinuities in the potential energy surface. Some of these discontinuities can be removed when pairing correlations are present, since their effect is to avoid level crossings.

Overall fission dynamics is a very complex process, which still did not reach a fully microscopic description arxiv_1511r in spite of almost eight decades of effort. The formation of FFs was and still is one of the most debated topics in phenomenology of nuclear fission. In the absence of a microscopic description and because of the enormous challenges arising from solving the necessarily time-dependent Schrödinger equation for hundreds of strongly interacting nucleons, many phenomenological models, both quantum and classical in nature, have been suggested over the years. Unfortunately most of them lack a solid microscopic underpinning and for that reason their predictive power is typically limited to the processes used to construct these models.

The bulk behavior of the deformation potential energy surface is determined by the surface tension and Coulomb energy, which are rather well-known and understood, and they are the main driving force leading to fission as in liquid drop models. The underlying shell structure imprints hills and valleys on the overall charged liquid drop energy landscape RMP_1972r ; RMP_1980r . On the way to scission configuration, nucleons have to go through a large number of redistributions at the single-particle levels crossing around the Fermi level, in order to maintain the spherical symmetry of their local momentum distribution.

Overall, the deformation potential energy surface acquires a profile somewhat similar to that of an uneven mountain, with little hills and valleys and covered by trees, and the evolution of the nuclear shape is in the end similar to the erratic motion of a pinball, not straight down the hill, but rather left and right, bouncing (mostly elastically) from the many obstacles on the way to the bottom of the valley, where the pinball breaks up. In the last stages of this complex evolution, the independent character of the nucleons inside the nucleus plays again a critical role, since magic closed shells control the nuclear shape evolution and the splitting leading to the formation of FFs.

Since fission dynamics is a truly non-stationary phenomenon an extension of the TDDFT hk ; ks ; monograph1 ; monograph2 was needed ARNPS__2013 and this approach was dubbed the Time-Dependent Superfluid Local Density Approximation (TDSLDA). In the case of many strongly interacting particles the Schrödinger equation has to be replaced with the TDDFT, which is mathematically equivalent to the Schrödinger equation for one body observables and the only theoretical framework suitable to describe the structure and dynamics of quantum many-body systems, if the corresponding energy density functional (the equivalent of the potential in the Schrödinger equation) is known in sufficient detail. TDSLDA allows the evaluation of FFs properties, which are otherwise impossible to either measure or compute, such as FFs excitation energies. As our simulations within TDSLDA demonstrate Bulgac:2016 , see Figure 2, the nucleus separates typically into two FFs, one bigger and the other somewhat smaller. In our calculations, we used in the particle-hole channel a phenomenological NEDF NuclPhys_1998 , complemented with pairing correlations described as in Refs. PRL__2002 ; PRL__2003a . The larger fragment fragment, which has properties very similar to the energetically very stable double-magic 132Sn, emerges with an almost spherical shape, while the lighter fragment emerges in an elongated configuration, with a ratio of the major to minor axes close to 3/2. To much of our surprise, the properties of the FFs in these simulations Bulgac:2016 are very close experimental measurements, even though no effort was put into reproducing observations. We attribute this to the fact that the NEDF reproduces bulk nuclear properties relatively well. The biggest surprise, however, was the observation that the time from saddle-to-scission was about an order of magnitude larger than ever estimated previously in the literature. This was ulteriorly confirmed by independent simulations Tanimura .

The right panel of Figure 2 serves as an illustration of the crucial role played by pairing correlations in fission dynamics. In the left panel one can see that the pairing field on the way from saddle-to-scission does fluctuate noticeably in magnitude and phase. Strictly speaking, therefore, the pairing field during its time evolution stops being a superfluid condensate of Cooper pairs, which otherwise would show a uniform phase. By increasing artificially the strength of the pairing field, we observe that fission dynamics is accelerated significantly, see right panel in Figure 2. In that case, the pairing field shows the expected characteristics of a slowly-evolving superfluid condensate, the nuclear fluid behaving almost like a perfect or ideal fluid, where dissipation is practically absent, and the two FFs emerge macroscopically entangled now Bulgac:2017 .

At FUSION17 workshop W. Mittig mittig discussed the very intriguing possibility of the existence of long-lived nuclear systems with very large charges, $Z\approx 200$ or even higher. One can contemplate the collision of two heavy nuclei with an energy just at the Coulomb barrier, when fusion is very likely to occur. As in the case of fission dynamics, immediately after fusion many collective degrees of freedom of the combined system are likely to get excited, leading to a sharing of the energy from the incoming collision channel. This combined nuclear system might thus survive for a relatively long time, but eventually the system will decay into two almost equal fragments, as the combined system would likely equilibrate, even in the case of asymmetric collisions.

A major limitation of the DFT is a lack of a method to evaluate two-body observables, such as the width of the FFs mass and charge yields and of the total TKE of the FFs. Within the TDHF approach one can use the Balian and Vénéroni Balian prescription or the more involved stochastic mean-field model introduced by Ayik Ayik , which reproduces the widths evaluated within the Balian and Vénéroni procedure. Ayik’s model is computationally more costly, but is the first attempt to describe quantum mechanically within TDDFT the full distribution of various observables, which in case of fission would amount to the FFs mass, charge yields, and TKE distributions. While appealing, Ayik’s model is nevertheless phenomenological in nature (as is random matrix theory). It has been applied recently to the case of the spontaneous fission of 258Fm Tanimura in an approach characterized by the authors somewhat optimistically as “the first time a fully microscopic description of the fragment TKE distribution after fission and gives unique microscopic information on the fission process.” The main difference of the approach of Ref. Tanimura and TDHF calculations performed in the past is in choosing an ad hoc ensemble of stochastic initial conditions, instead of a single initial nuclear shape past the outer fission barrier. In the case of spontaneous fission the nucleus emerges from under the barrier likely in its ground state corresponding to that instantaneous shape with $Q_{20}\approx 100$ barn for 258Fm Tanimura . These authors Tanimura however chose to start their fission dynamics much further, when $Q_{20}$ is either 125 or 160 barn, at which point they excite their nucleus to an energy corresponding to the ground state. This stochastic ensemble has fluctuations of all possible observables $(N,Z,E)$, center-of-mass velocity, intrinsic currents, total angular momentum, parity, etc. of uncharacterized distributions.

When the nucleus exits from under the barrier it is expect to start rolling down towards scission configurations rather slowly and if intrinsic degrees of freedom get excited, one naturally would expect a thermal ensemble for the nuclear shapes with $Q_{20}$ = 125 or 160 barn, rather that the stochastic ensemble suggested by Ayik Ayik . There is no theoretical argument presented in Ref. Tanimura to choose as a starting point of the dynamical simulations the arbitrary deformations $Q_{20}$ = 125 or 160 barn, by which time the nucleus acquired a non-negligible collective kinetic energy not explicitly accounted for in this approach. It would be more natural to start the simulation at the exact configuration where the nucleus emerged from under the barrier at $Q_{20}\approx$ 100 barn. In that case the size of the “quantum fluctuations” would be zero and the nucleus would have no intrinsic excitation energy and zero collective momentum. But that would deprive the authors from the ability to generate a FFs distribution. The authors even establish that if they were to start their simulations closer to the scission configurations their results would be quite different, thus precluding this approach of its predictive power. These authors conclusion that simulations with and without “dynamical pairing" lead to similar results also appear quite hasty, as in both cases one starts with an ensemble of single Slater determinants, which probably share some similar fluctuations. One can safely conclude that the FFs distribution definitely depend strongly on the initial conditions within such a stochastic approach. A more natural approach would be a TDGCM approach advocated by Regnier et al. Regnier to specify the distribution of the initial conditions in a TDDFT simulation of fission dynamics.

The predictive power of a TDDFT approach depends critically on the quality of the NEDFs, which are phenomenological so far and depend on a rather large number of fitting parameters (typically 14 and sometimes even more), some of them rather strongly correlated with each other. This points to the fact that there are elements of NEDF that are still poorly understood and there is a need for better constrained NEDFs, either phenomenologically or microscopically. We have developed a qualitatively new NEDF NEDF , which contains terms previously not considered in literature, which depends on only 7 uncorrelated parameters, with a rms for masses of even-even nuclei from the AME2012 ame2012 of 1.74 MeV and for charge radii of 0.034 fm. The potential energy surface for 240Pu predicted by SeaLL1, see Figure 3, has potential energy barriers comparable to those obtained in FRLDM LDM , though a relatively low energy of the fission isomer, and a nice single-particle spectrum of 208Pb, see Figure 4.

This work was supported in part by U.S. Department of Energy (DOE) Grant No. DE-FG02-97ER41014 and by the U.S. Department of Energy under Contract Nos. DE-AC52-07NA27344 (Lawrence Livermore National Laboratory). This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725. It also used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Some of the calculations were performed on Moonlight at LANL. Computing support for this work also came from the Lawrence Livermore National Laboratory (LLNL) Institutional Computing Grand Challenge program.