The highest tightly bound nuclei and the stability prediction of their isobaric nuclei versus beta decay in the liquid drop model

Technical Summary: The Highest Tightly Bound Nuclei and Stability Prediction in the Liquid Drop Model

Problem Statement
The study addresses the ongoing debate regarding the identification of the most tightly bound nucleus and the accuracy of theoretical models in predicting nuclear masses and beta-decay stability. While literature often cites $^{56}\text{Fe}$ as the most tightly bound nucleus, experimental data from the AME2020 compilation indicates that $^{62}\text{Ni}$, $^{58}\text{Fe}$, and other mid-mass nuclei exhibit higher average binding energies per nucleon. Furthermore, the authors investigate the limitations of the standard five-term Bethe–Weizsäcker mass formula (BWMF-5T) and whether extending the model with additional terms and optimizing coefficients through regional versus global fitting procedures yields superior predictive accuracy for these specific nuclei.

Methodology
The authors employ a semi-empirical approach based on the Liquid Drop Model (LDM). The methodology involves three primary components:

1. Model Formulation: The study utilizes the standard five-term BWMF (volume, surface, Coulomb, asymmetry, and pairing) and extends it to a ten-term formulation (BWMF-10T) by incorporating five additional terms: the Wigner term, Coulomb-exchange correction, surface-symmetry correction, curvature energy term, and a shell correction term.
2. Data Fitting and Optimization: Coefficients for both the 5-term and 10-term formulas are determined using a least-squares fitting procedure against experimental mass data from the AME2020 compilation. The fitting is performed across multiple datasets to assess sensitivity:
   • A global set of 2,457 measured nuclei.
   • Successive subsets of 500 nuclei.
   • A regional set of 366 even–even nuclei spanning the oxygen to gadolinium mass region.
3. Stability Analysis: Using the optimized coefficients, the authors calculate the binding energy per nucleon for the six most tightly bound candidates ($^{62}\text{Ni}$, $^{58}\text{Fe}$, $^{56}\text{Fe}$, $^{60}\text{Ni}$, $^{54}\text{Cr}$, and $^{64}\text{Ni}$). Additionally, they analyze beta-decay stability within isobaric families for mass numbers $A = 54, 56, 58, 60, 62,$ and $64$ by deriving the mass parabola and determining the most stable isobar ($Z_0$) via differentiation of the mass expression.

Key Contributions

• Comparative Coefficient Analysis: The paper systematically compares coefficient values derived from regional fits (mid-mass nuclei) against global fits (entire nuclear landscape). It demonstrates that coefficients are highly sensitive to the size and diversity of the dataset used for fitting.
• Extended Model Evaluation: The study evaluates the impact of adding five specific terms to the standard mass formula, analyzing how these terms influence the symmetry energy coefficient and the overall fit quality.
• Beta-Decay Stability Prediction: The authors apply the refined formulas to predict the stability trends of isobaric chains, explicitly calculating the $Q$-values for $\beta^-$ and $\beta^+$ transitions and identifying the most stable nuclei based on mass minimization.

Results

• Coefficient Sensitivity: The fitted coefficients for the basic five terms show a systematic increase in magnitude as the dataset expands from 500 to 2,457 nuclei. For instance, the symmetry energy coefficient increases significantly when moving from a regional fit to a global fit.
• Regional vs. Global Accuracy: Regional fits (specifically for the oxygen–gadolinium region) significantly outperform global fits when applied to mid-mass nuclei. Global fits, which include heavy and superheavy nuclei, introduce systematic deviations when predicting the binding energies of lighter, tightly bound nuclei.
• Binding Energy Predictions:
  • Theoretical calculations using regional coefficients show excellent agreement with experimental binding energies for $^{62}\text{Ni}$, $^{58}\text{Fe}$, and $^{64}\text{Ni}$.
  • The relative error ($\Delta B/B$) for the most tightly bound nuclei remains below 0.26% for regional fits, whereas global fits yield larger deviations.
  • The study confirms $^{62}\text{Ni}$ as having the highest experimentally observed binding energy per nucleon among known nuclides, consistent with AME2020 data.
• Shell Effects: The analysis suggests that the high binding energy of $^{62}\text{Ni}$ is linked to its proton number ($Z=28$, a magic number) and neutron configuration, while $^{58}\text{Fe}$ benefits from a closed neutron sub-shell ($N=32$).
• Beta Stability: The derived mass parabolas correctly identify the stable isobars for the selected mass chains, with the pairing term successfully accounting for the energy differences between odd-$Z$ and even-$Z$ transitions.

Significance and Claims
The paper claims that the primary significance of this work lies in demonstrating that regional mass fitting provides a superior description of mid-mass nuclei compared to global fitting approaches. The authors assert that the standard Bethe–Weizsäcker formula, when refined with regional coefficients, offers a high-accuracy approximation of atomic masses for the most tightly bound nuclei.

The study concludes that while the Liquid Drop Model is a powerful macroscopic tool, its predictive accuracy for specific regions of the nuclear chart is heavily dependent on the dataset used for coefficient optimization. The authors modestly note that while the model predicts $^{62}\text{Ni}$ as the most tightly bound, abundance in nature (such as the prevalence of $^{56}\text{Fe}$) is governed by stellar nucleosynthesis processes rather than binding energy alone. The work highlights the need for further experimental measurements of light nickel isotopes to clarify shell gaps at $Z=28$, but does not propose new experimental setups itself.