⚛️ nuclear theory

R-process nuclei in the rare-earth region above N=126 in Relativistic Hartree-Bogoliubov theory: A preponderance of shape-coexistence in nuclei

This study utilizes relativistic Hartree-Bogoliubov theory to investigate nuclei beyond N=126, revealing that shape coexistence and enhanced stability in the neutron-rich region significantly influence the r-process nucleosynthesis and shift the neutron drip line.

Original authors: M. M. Sharma

Published 2026-10-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: M. M. Sharma

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

Deep in the heart of the universe, where stars collide and space-time ripples, a cosmic alchemy is taking place. This is the rapid neutron-capture process, or r-process, a violent event that forges more than half of all the heavy elements in existence, from the gold in a wedding ring to the uranium in a power plant. In these extreme environments, atomic nuclei are bombarded by a flood of neutrons, forcing them to grow heavier and heavier until they become unstable and decay into the elements we know. To understand how the universe creates these heavy elements, scientists must map the landscape of atomic nuclei, particularly those that are incredibly neutron-rich and far from the stability found in ordinary matter. A critical region for this study lies just beyond a specific "magic number" of neutrons, a point where the internal structure of the nucleus is particularly robust. The question that has long puzzled physicists is how these nuclei behave when pushed to their absolute limits, right at the edge of existence where they can no longer hold onto extra neutrons.

A recent study by physicist M.M. Sharma at Kuwait University takes a fresh look at this mysterious region, focusing on nuclei with more than 126 neutrons. These are the building blocks that eventually form the heavy elements found in the third peak of cosmic abundance, around a mass number of 195. Using advanced computer simulations based on the theory of how particles interact at the subatomic level, the researcher explored the ground-state properties of these exotic nuclei. The goal was to see how the shape of these atomic cores changes as they absorb more and more neutrons, and to determine where the line of stability actually ends. By comparing three different mathematical models of nuclear forces, the study aimed to find the most accurate way to predict the behavior of matter in these extreme conditions, a task that is currently impossible to perform in a laboratory because such nuclei are too fleeting to be caught and measured.

The investigation revealed that as these nuclei accumulate neutrons, they do not simply sit still; they begin to stretch and change shape. Initially spherical, the nuclei start to elongate into a football-like shape as they gain neutrons. However, the story does not end there. As more neutrons are added, the nuclei undergo a dramatic transformation, flipping from a stretched, elongated shape to a flattened, pancake-like shape. What makes this discovery particularly striking is the prevalence of "shape coexistence." In many of these neutron-rich nuclei, the elongated shape and the flattened shape exist in a state of near-equal energy, meaning the nucleus can essentially be both shapes at once, or switch between them with very little effort. This phenomenon is most common in the heavier elements of the rare-earth group, where the competition between these two shapes creates a complex and dynamic landscape just before the nucleus reaches its breaking point.

This shape-shifting behavior has a profound effect on how long these nuclei can survive. The study found that the onset of deformation actually grants the nuclei extra stability, allowing them to hold onto neutrons that they would otherwise lose. This effect pushes the "neutron drip line"—the theoretical boundary where a nucleus can no longer bind any more neutrons—much further out than previously expected. For the heaviest elements studied, this boundary extends all the way to a new magic number of 184 neutrons, whereas for lighter elements in the same group, the boundary appears much sooner. The research suggests that the ability of these nuclei to deform and exist in multiple shapes simultaneously acts as a stabilizing force, effectively stretching the limits of matter in the neutron-rich regions of the periodic table.

To reach these conclusions, the researcher compared the results of three different theoretical models, each using a slightly different set of rules to describe how the forces inside the nucleus work. One model, which avoids certain complex interactions between the particles, proved to be the most consistent with established mass formulas used by the broader scientific community. This specific model predicted that the neutron drip line for elements like hafnium reaches the magic number of 184, while for elements like ytterbium, it stops short at around 172. In contrast, other models predicted that the drip line would be reached much earlier, or that the nuclei would be significantly more tightly bound than they actually are. The study explicitly rules out the idea that the strong shell effects, which make certain numbers of neutrons particularly stable, disappear as the nuclei become more extreme. Instead, the simulations show that these shell effects remain robust, continuing to provide a strong foundation for the nucleus even as it approaches the edge of stability.

The implications of these findings reach back to the cosmic origins of the elements. The shape of a nucleus influences how quickly it decays, and the presence of shape coexistence means that the decay rates could be different than previously thought. If a nucleus can exist in two shapes, its lifetime might be a mix of the lifetimes of both shapes, potentially altering the speed at which heavy elements are synthesized in the cosmos. The study suggests that the interplay between deformation and stability creates a "ridge" of extra stability in the neutron-rich region, creating a path for the r-process to travel further into the unknown than standard models predict. This could significantly change our understanding of the abundance of heavy elements in the universe, particularly around the third peak of cosmic distribution.

Ultimately, this work provides a clearer picture of the behavior of matter under the most extreme conditions imaginable. It confirms that the atomic nucleus is not a rigid, unchanging sphere, but a fluid entity capable of dramatic shape changes that can extend its life and alter its path through the cosmos. By identifying the specific conditions under which these shapes coexist and how they influence stability, the study offers a refined map for future explorations of the r-process. While the results are derived from sophisticated computer simulations rather than direct observation, the consistency of the findings across different models and their alignment with known mass formulas gives the scientific community a strong basis for revising how they calculate the birth of heavy elements in the universe. The discovery of this widespread shape coexistence near the neutron drip line adds a new layer of complexity to our understanding of nuclear physics, suggesting that the universe's ability to create heavy elements is even more intricate and resilient than previously imagined.

Technical Summary: R-process nuclei in the rare-earth region above N = 126 in Relativistic Hartree-Bogoliubov theory

Problem Statement
The paper investigates the shell structure and ground-state properties of extremely neutron-rich nuclei in the rare-earth region ( $Z = 58-72$ ) beyond the magic neutron number $N = 126$ . These nuclei are relevant to the third peak ( $A \approx 195$ ) of the $r$ -process nuclear abundances. Due to their extreme isospin, these nuclei are experimentally inaccessible, necessitating reliance on theoretical predictions. A critical issue in this domain is the potential quenching of shell effects near the $r$ -process path and the behavior of nuclear shapes (deformation) as one approaches the neutron drip line. Previous studies have suggested shell quenching at $N=82$ and $N=126$ , or conversely, strong shell effects, leading to uncertainties in $r$ -process nucleosynthesis simulations. Furthermore, the interplay between nuclear deformation and $\beta$ -decay rates significantly influences nuclear abundances, yet the specific evolution of shapes (prolate vs. oblate) and shape coexistence in this extreme region remains to be fully characterized within relativistic frameworks.

Methodology
The study employs the Deformed Relativistic Hartree-Bogoliubov (DRHB) theory, which utilizes a pairing mechanism based on the Bogoliubov quasi-particle scheme within a deformed basis. The calculations assume axial symmetry, leading to spheroidal shapes. The fermionic and mesonic wavefunctions are expanded in a basis of 20 harmonic oscillator shells.

To assess the robustness of predictions, the author performs a comparative study using three distinct Lagrangian models within the Relativistic Mean-Field (RMF) framework:

NL-SV1: A model featuring nonlinear couplings of both $\sigma$ and $\omega$ mesons (specifically a quartic $\omega$ coupling).
NL3: The standard nonlinear $\sigma$ -model with self-interactions of the $\sigma$ meson.
SVI-2: A Scalar-Vector Interaction model that removes self-interactions of meson fields, relying instead on $\sigma$ - $\omega$ meson-meson interactions.

For the pairing channel, the Gogny force D1S is used, with pairing strength calibrated against experimental binding energies of Sn isotopes. The calculated ground-state properties (binding energies, quadrupole deformations $\beta_2$ , and two-neutron separation energies $S_{2n}$ ) are compared against macroscopic-microscopic mass formulas (FRDM-2012 and HFB-24) and the recent DRHBc mass formula.

Key Contributions and Results

Binding Energies and Model Comparison:
- In the region near $N=126$ , all three RMF models show reasonable agreement with mass formulas. However, significant divergences appear as nuclei approach the neutron drip line ( $N > 150$ ).
- The NL3 model consistently overbinds nuclei, with deviations increasing significantly as the drip line is approached. Specifically, for Er isotopes, NL3 overestimates HFB-24 values by 4.0 MeV at $N=140$ , 6.9 MeV at $N=150$ , and 11.1 MeV at $N=160$ .
- The SVI-2 model demonstrates excellent agreement with the HFB-24 mass formula across the isotopic chains, particularly for heavier elements ( $Z > 60$ ), with root-mean-square deviations as low as 0.7 MeV for Er isotopes.
- The NL-SV1 model also agrees well with HFB-24 up to $N \approx 150$ , though it begins to overbind slightly compared to SVI-2 in the extreme region.
- The DRHBc mass formula significantly overestimates binding energies (by up to 27 MeV for Hf at $N=180$ ) compared to the RMF models and HFB-24.
Neutron Drip Line Extension:
- The onset of deformation for $N \ge 130$ induces enhanced stability, shifting the neutron drip line significantly farther into the neutron-rich region compared to predictions from mass formulas.
- For Hf ( $Z=72$ ), the RMF models predict the drip line extends to the next magic number, $N=184$ . For lighter chains (e.g., Sm, $Z=62$ ), the drip line is extended to $N \approx 152-154$ .
- The SVI-2 model predicts the drip line closest to mass formula predictions (e.g., $N \approx 170$ for Yb), whereas NL3 and NL-SV1 extend it further (e.g., $N \approx 174-176$ for Yb) due to stronger binding.
Shape Coexistence and Deformation Evolution:
- A dominant feature in the region between $N=126$ and $N=184$ is the preponderance of shape coexistence between well-deformed prolate and oblate shapes in the ground state.
- Nuclei generally start spherical at $N=126$ , acquire a prolate deformation ( $\beta_2 \approx 0.12$ ) around $N=134$ , and reach maximum deformation ( $\beta_2 \approx 0.30-0.40$ ) near the middle of the shell ( $N \approx 150-158$ ).
- As neutrons are added further, a smooth transition to an oblate ground state occurs for most chains (except Hf), often passing through a region of near-degeneracy where prolate and oblate states are separated by less than 0.2 MeV.
- For Hf ( $Z=72$ ), the shell closure at $N=184$ prevents the transition to an oblate shape; the nuclei remain prolate until the drip line, where they become spherical.
- The SVI-2 model predicts fewer cases of shape coexistence compared to NL-SV1 and NL3, suggesting a "stiffer" shell structure in the middle of the shell.
Shell Effects at N = 126:
- The study confirms that shell effects at $N=126$ remain robust and strong even as the neutron drip line is approached. There is no evidence of shell quenching in the RMF models.
- The shell gap $\Delta$ at $N=126$ shows only a marginal decrease for lighter chains (approaching the drip line) but remains large ( $\approx 5$ MeV for SVI-2).
- RMF models predict shell gaps significantly larger (by 50-70%) than those predicted by mass formulas (FRDM-2012, HFB-24) in the mid-shell region, though the values converge near the drip line.
Single-Particle Levels:
- The enhanced stability and the "ridge" in two-neutron separation energies ( $S_{2n}$ ) are attributed to the deformation-induced lowering of high- $l$ , low- $\Omega$ neutron single-particle levels (e.g., components of $1i_{13/2}$ , $1h_{11/2}$ ). This lowers the center-of-gravity of the neutron levels, providing additional binding energy that delays the onset of the drip line.

Significance and Claims
The paper claims that the investigation provides a comprehensive theoretical picture of the rare-earth region relevant to the $r$ -process third peak. The primary significance lies in the demonstration that:

Shape Coexistence is Prevalent: The ground states of many nuclei in this region exhibit coexistence of prolate and oblate shapes, often with near-degeneracy. This phenomenon, which is difficult to capture with semi-empirical mass formulas, is a robust prediction of the DRHB theory.
Enhanced Stability: The onset of deformation creates a region of enhanced stability, significantly extending the neutron drip line (e.g., to $N=184$ for Hf) compared to mass formula predictions.
Model Dependence: The choice of Lagrangian model significantly impacts predictions in the extreme region. The SVI-2 model, which lacks meson self-interactions, yields results most consistent with the refined HFB-24 mass formula, suggesting it may be a viable candidate for constructing mass formulas for extreme nuclei.
Implications for Nucleosynthesis: The predicted oblate ground states for drip-line nuclei (in most chains) imply longer $\beta$ -decay half-lives, which could resist nucleosynthesis flow. However, the presence of shape coexistence (mixing prolate and oblate configurations) could accelerate the process. The author concludes that reaction network calculations incorporating these deformation-dependent $\beta$ -decay rates are necessary to realistically model $r$ -process abundances near the third peak.

The work reinforces the absence of shell quenching at $N=126$ in the RMF framework and highlights the critical role of deformation in determining the limits of nuclear existence in the $r$ -process path.

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