⚛️ nuclear theory

Impact of perturbative tensor interactions on the spontaneous fission half-lives of superheavy nuclei

This study demonstrates that incorporating a perturbative tensor term into the Gogny-D1S force significantly reduces the first fission barrier height in superheavy nuclei, thereby decreasing theoretical spontaneous fission half-lives and improving their agreement with experimental data.

Original authors: R. Rodriguez-Guzman, A. Rakhmankulov, L. M. Robledo, R. N. Bernard

Published 2026-08-13

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: R. Rodriguez-Guzman, A. Rakhmankulov, L. M. Robledo, R. N. Bernard

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the atomic nucleus as a tiny, wobbly drop of liquid, held together by a sticky force that fights against the electric repulsion trying to blow it apart. For heavy atoms, this balance is so delicate that they can spontaneously split in two, a process called nuclear fission. It's like a stretched rubber band that suddenly snaps; the energy released is immense. But for the heaviest, most unstable elements we can create in a lab—superheavy nuclei—this snapping happens so fast that they vanish almost instantly. Scientists have been trying to predict exactly how long these atomic giants last before they split. To do this, they use complex computer models that map out the "energy landscape" of the nucleus. Think of this landscape as a mountain range where the nucleus sits in a valley (its stable state) and must climb over a mountain peak (the fission barrier) to fall apart. The higher the mountain, the longer the nucleus survives.

Recently, physicists have been refining the rules of this game. They know that inside the nucleus, protons and neutrons interact in complicated ways, and one specific type of interaction, called the "tensor force," has been a bit of a mystery. It's like a subtle magnetic tug between the particles that depends on how they are oriented. While standard models have mostly ignored this tug, new theories suggest it might be the missing piece of the puzzle. The big question is: Does adding this tiny, subtle force change the height of the mountain peaks enough to explain why some superheavy atoms live longer or shorter than our current models predict?

This paper takes a deep dive into that question by running detailed simulations on a series of superheavy nuclei, ranging from Nobelium to Darmstadtium. The researchers compared two versions of their nuclear model: the standard version (called D1S) and a new version that includes the perturbative tensor term (called D1ST). They found that adding this tensor force doesn't change the "glue" holding the nucleus together (pairing properties) or the weight of the nucleus as it moves (collective inertia). However, it does something dramatic to the mountain itself: it lowers the height of the first mountain peak (the inner fission barrier) significantly.

Because the mountain is lower, the nucleus can "tunnel" through it much more easily, just as a ball would roll down a shorter hill faster than a taller one. The results show that when the tensor force is included, the predicted lifetimes of these superheavy nuclei become much shorter. In fact, for many of the elements studied, the new predictions with the tensor term match experimental data much better than the old models did. For example, for the nucleus Rutherfordium-260, the standard model predicted a lifetime that was too long, but the new model with the tensor term predicted a lifetime of about $10^{-1.08}$ seconds, which is very close to the experimental value of $10^{-1.69}$ seconds. The paper concludes that this tensor interaction is a crucial ingredient for accurately describing how these heavy atoms fall apart, and the authors plan to use this improved understanding to study even heavier elements in the future.

Technical Summary: Impact of Perturbative Tensor Interactions on Spontaneous Fission Half-Lives of Superheavy Nuclei

Problem Statement
The accurate microscopic description of spontaneous fission in heavy and superheavy nuclei remains a significant challenge in nuclear structure physics. While the standard approach utilizes the mean-field Hartree-Fock-Bogoliubov (HFB) approximation combined with a semi-classical tunneling description (WKB formalism), existing models often neglect the tensor component of the nuclear interaction. This tensor force, associated with one-pion exchange, influences single-particle distributions near the Fermi level and shell effects, which are critical for the stability of superheavy elements. Previous studies indicated that tensor correlations could significantly alter fission properties in actinides, but a systematic analysis of their impact on the spontaneous fission half-lives ( $t_{SF}$ ) of superheavy nuclei (atomic number $Z > 100$ ) using Gogny-type interactions was lacking.

Methodology
The authors employed a constrained HFB framework to analyze a series of even-even isotopes ranging from Nobelium (No, $Z=102$ ) to Darmstadtium (Ds, $Z=110$ ). The study compared two parametrizations of the Gogny energy density functional (EDF):

Gogny-D1S: The standard parametrization widely used in fission studies.
Gogny-D1ST: A modified version where a complete, long-range tensor term is added perturbatively to the D1S force. The tensor parameters were adjusted to reproduce specific neutron single-particle energies in $^{48}$ Ca, while other central, spin-orbit, and density-dependent terms remained identical to D1S.

The calculations involved:

Constrained HFB: Minimizing the energy density functional under constraints on multipole operators, specifically the axial quadrupole moment ( $\hat{Q}_{20}$ ), to map the potential energy surfaces (PES).
Collective Quantities: Calculation of collective potentials $V(Q_{20})$ (including zero-point rotational and vibrational corrections) and collective inertias $M(Q_{20})$ using both the Generator Coordinate Method (GCM) in the Gaussian Overlap Approximation (GOA) and the Adiabatic Time-Dependent HFB (ATDHFB) scheme.
Half-Life Calculation: Determination of spontaneous fission half-lives using the Wentzel-Kramers-Brillouin (WKB) formalism, integrating the action over the fission path defined by the collective potential and inertia.

Key Contributions and Results
The study systematically investigated the isotopic chains of No, Rf, Sg, Hs, and Ds. The primary findings include:

Reduction of Fission Barriers: The inclusion of the perturbative tensor term (D1ST) consistently reduces the height of the first (inner) fission barrier compared to the standard D1S results. For the representative nucleus $^{260}$ Rf, the inner barrier height decreased from 12.02 MeV (D1S) to 8.93 MeV (D1ST). This reduction was observed across all studied isotopic chains, with differences ( $\delta B_{I,GCM}$ ) ranging from approximately 0.7 MeV to 4.4 MeV depending on the element and neutron number.
Minimal Impact on Pairing and Inertia: The tensor term had a negligible effect on the pairing properties (proton and neutron pairing interaction energies) and the collective inertias. The collective masses calculated with D1ST remained largely similar to those of D1S, indicating that the changes in $t_{SF}$ are driven primarily by the modification of the potential energy landscape (barrier height) rather than dynamic mass effects.
Single-Particle Structure: The tensor term altered the relative positions of single-particle orbitals, particularly affecting the shell gaps at the ground state and fission isomer configurations. However, the overall topology of the fission paths (locations of minima, barriers, and isomers in terms of quadrupole deformation) remained qualitatively similar between the two parametrizations.
Agreement with Experimental Data: The most significant result is the impact on predicted half-lives. The standard D1S calculations consistently overestimated the spontaneous fission half-lives for superheavy nuclei, often by several orders of magnitude. The inclusion of the tensor term (D1ST) significantly reduced the predicted $t_{SF}$ values, bringing them into much closer agreement with available experimental data. For example, in the Rf isotopic chain, the D1ST predictions achieved almost quantitative agreement with experimental values for $^{254-262}$ Rf, whereas D1S results were substantially higher.
Robustness of Trends: The observed trends were consistent across different theoretical schemes (GCM vs. ATDHFB). While ATDHFB calculations yielded systematically larger half-lives due to larger collective masses, the relative reduction in $t_{SF}$ caused by the tensor term persisted in both frameworks.

Significance and Claims
The paper concludes that the tensor interaction is a crucial ingredient in the microscopic description of fission for superheavy nuclei. The authors claim that the perturbative addition of the tensor term to the Gogny-D1S force provides a mechanism to correct the systematic overestimation of fission lifetimes found in standard mean-field approaches. By lowering the inner fission barrier heights without significantly altering collective inertias, the tensor term offers a pathway to more accurate theoretical predictions that align better with experimental observations in the region $Z > 100$ .

The authors note that their study is restricted to nuclei with $Z < 110$ and utilizes a perturbative tensor term. They acknowledge that a fully integrated tensor term within the EDF fitting protocol (as proposed in recent literature) might yield further improvements, and they plan to extend their study to nuclei with $Z=112-120$ and very neutron-rich isotopes in future work. The paper does not claim to resolve all discrepancies with experimental data (e.g., the sudden drop in half-life for $^{258}$ No remains unexplained by their model), but it establishes the tensor force as a necessary component for refining fission models in the superheavy region.

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