Principled quantified extrapolation of the nuclear landscape to the particle drip lines

The chart of atomic nuclides is bounded by proton and neutron drip lines, where nuclei lose stability to nucleon emission; only about 3000 of the several thousand nuclei predicted to exist have been observed, and for heavy elements the neutron drip line is purely a theoretical construct. Neufcourt, Cao, Giuliani, Nazarewicz, Olsen, and Tarasov (Phys. Rev. C 101, 044307, 2020) combined several global nuclear mass models with Gaussian-process corrections to model residuals and Bayesian model averaging (including a heuristic local variant, BMA(n+p)) to produce the first landscape-wide quantified predictions of separation energies, existence probabilities, drip-line locations, and the total count of bound nuclei (7708±534 for Z,N≥8, Z≤119). They explicitly left open the statistical and physical foundations of this program: a general theory of local model averaging, control of predictive uncertainty when extrapolating two-to-three times beyond the fitted domain where Gaussian-process corrections lose power, uncompensated systematic deficiencies of the underlying mass models, and the calibration and tightening of drip-line credibility intervals as new data arrive.

Background

The chart of nuclides maps all atomic nuclei by proton number $Z$ and neutron number $N$; its boundaries are the proton and neutron drip lines, beyond which a nucleus is unstable to prompt nucleon emission. Experimentally, the proton drip line has been reached up to neptunium ($Z=93$), but the neutron drip line has been delineated only for light nuclei up to neon ($Z=10$); for heavier elements its location is a theoretical extrapolation. The natural observables are the one- and two-nucleon separation energies, $S_{1n}(Z,N)$ and $S_{2n}(Z,N)$ (and their proton analogues), defined as finite differences of binding energies between neighboring isotopes or isotones; a nucleus is particle-bound when the relevant separation energies are positive, and the drip line is where they cross zero. Quantified drip-line predictions matter for the astrophysical rapid neutron-capture (r-) process, which operates near the neutron drip line. The only microscopic theories that scale to the whole chart are global mass models: nuclear density functional theory (DFT) built on energy density functionals (EDFs) — phenomenological functionals of nuclear density fitted to masses and radii, in Skyrme, Gogny, and covariant families — together with micro-macro models — macroscopic liquid-drop energies corrected by microscopic single-particle shell effects — such as the finite-range droplet model. Different calibrated EDFs disagree substantially for unmeasured nuclei, and their errors far from stability are systematic (structural), not random. To combine models and quantify uncertainty, Neufcourt and collaborators introduced a Bayesian pipeline: Gaussian-process (GP) regression — a Bayesian nonparametric model that treats each model's residuals to measured masses as a correlated random field — corrects each mass model, and Bayesian model averaging (BMA) combines the corrected models with data-driven weights. For the whole landscape, the 2020 study used local weight functions: BMA($n$) informed by existence data on neutron-rich nuclei, BMA($p$) informed by two-proton emitters, and BMA($n+p$), which retains each variant on the side of the beta-stability line (the valley of nuclei stable against beta decay) where it is designed to perform best (weights stitched together with a Heaviside step). This yielded, e.g., $7708\pm534$ particle-bound nuclei with $Z,N\ge8$ and $Z\le119$ (95% credibility interval $[6688,8440]$) and per-nucleus existence probabilities (posterior probabilities that a given nucleus is particle-bound); an earlier estimate with uniform model mixing was Erler et al. (Nature 486, 509, 2012). The authors stressed that the landscape prediction is 'timestamped' — a dynamical prediction to be updated with new data. The same authors acknowledged what the framework does not deliver. First, BMA($n+p$) is 'an innovative attempt for a principled local model averaging', called for in earlier work (Kejzlar et al., arXiv:1904.04793; Neufcourt et al., 2018–2020), not a theory: general statistical foundations for local model averaging of models with different domains of validity remain to be established. Second, machine learning cannot compensate for unknown systematic model deficiencies when extrapolating far beyond the fitted domain — there, predictions must rely on the quality of the underlying nuclear modeling (the related program of reducing EDF systematics, including ab initio input, proceeds separately; ab initio many-body methods currently cover only limited regions of the chart). Third, because the extrapolation range is two-to-three times larger than the fitted range of the GP correlation effects, the GP correction is expected to be relatively limited apart from a mean shift; the authors argue that an honest posterior predictive distribution matters more than the correction to the mean, and that a more elaborate GP tail model would offer no obvious advantage unless substantiated by physics — which, at present, it is not. Fourth and fifth, predictions need updating and testing as new mass and existence data arrive, with special discriminating power in the neutron-rich regions $Z=28$–$30$, $Z=42$–$48$, and $Z=64$–$66$, where the one-neutron drip line is within experimental reach (the Facility for Rare Isotope Beams is expected to reach it up to $Z=42$) and model disagreement on the drip-line position is appreciable. The open question is whether these gaps can be closed — whether statistical learning, properly founded and physically substantiated, can deliver trustworthy quantified limits of nuclear binding.

Problem Statement

Can quantified extrapolation of nuclear binding from measured nuclei to the proton and neutron drip lines be placed on principled, physically substantiated statistical foundations? Given a set of imperfect global mass models (density-functional-theory mass tables and micro-macro models) together with all experimental mass and existence data, construct and validate a model-averaging/mixing framework that jointly achieves: (i) a general, theoretically grounded treatment of local (domain-dependent) model averaging — replacing heuristic weight constructions such as BMA($n+p$) — for models whose domains of validity differ; (ii) combined predictions of one- and two-nucleon separation energies and of nuclear existence probabilities whose uncertainties grow in a controlled, physics-substantiated way in far extrapolation, beyond roughly two-to-three times the fitted correlation range of the statistical corrections, where the correction to the mean is of lesser importance than the honesty of the posterior predictive distributions; (iii) robustness to the systematic deficiencies of the underlying models, which statistical learning alone cannot compensate far beyond the training domain; and (iv) posterior credibility intervals for drip-line locations and for the total number of bound nuclei (reference point: $7708\pm534$ particle-bound nuclei with $Z,N\ge8$, $Z\le119$, 95% interval $[6688,8440]$) that are demonstrably calibrated and tighten as new experimental data arrive, notably the existence and mass data in the neutron-rich regions $Z=28$–$30$, $Z=42$–$48$, and $Z=64$–$66$ where the one-neutron drip line is within experimental reach. These elements are one scientific objective — trustworthy, validated quantification of the limits of nuclear binding — and they are conjunctive rather than independently sufficient: an answer that satisfies only some of (i)–(iv), that delivers point accuracy without calibrated uncertainties, or that provides calibrated uncertainties only on a subregion of the landscape without a principled general framework, does not meet it.

References

1. L. Neufcourt, Y. Cao, S. A. Giuliani, W. Nazarewicz, E. Olsen, and O. B. Tarasov, "Quantified limits of the nuclear landscape," Phys. Rev. C 101, 044307 (2020). DOI: 10.1103/PhysRevC.101.044307, https://doi.org/10.1103/PhysRevC.101.044307 (preprint arXiv:2001.05924, https://arxiv.org/abs/2001.05924)
2. J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. M. Perhac, and M. Stoitsov, "The limits of the nuclear landscape," Nature 486, 509 (2012). DOI: 10.1038/nature11188, https://doi.org/10.1038/nature11188
3. V. Kejzlar, L. Neufcourt, T. Maiti, and F. Viens, "Bayesian averaging of computer models with domain discrepancies: a nuclear physics perspective," arXiv:1904.04793 (2019), https://arxiv.org/abs/1904.04793
4. L. Neufcourt, Y. Cao, W. Nazarewicz, E. Olsen, and F. Viens, "Neutron drip line in the Ca region from Bayesian model averaging," Phys. Rev. Lett. 122, 062502 (2019). DOI: 10.1103/PhysRevLett.122.062502, https://doi.org/10.1103/PhysRevLett.122.062502
5. L. Neufcourt, Y. Cao, S. A. Giuliani, W. Nazarewicz, E. Olsen, and O. B. Tarasov, "Beyond the proton drip line: Bayesian analysis of proton-emitting nuclei," Phys. Rev. C 101, 014319 (2020). DOI: 10.1103/PhysRevC.101.014319, https://doi.org/10.1103/PhysRevC.101.014319
6. V. Kejzlar, L. Neufcourt, and W. Nazarewicz, "Local Bayesian Dirichlet mixing of imperfect models," Scientific Reports 13, 19600 (2023). DOI: 10.1038/s41598-023-46568-0, https://doi.org/10.1038/s41598-023-46568-0
7. P. Giuliani, K. Godbey, V. Kejzlar, and W. Nazarewicz, "Model orthogonalization and Bayesian forecast mixing via principal component analysis," Phys. Rev. Research 6, 033266 (2024). DOI: 10.1103/PhysRevResearch.6.033266, https://doi.org/10.1103/PhysRevResearch.6.033266
8. A. Sharma, N. Schunck, and K. Wendt, "Bayesian model mixing with multireference energy density functional," Phys. Rev. Research 7, 023179 (2025). DOI: 10.1103/PhysRevResearch.7.023179, https://doi.org/10.1103/PhysRevResearch.7.023179
9. C. M. Ireland, G. Bollen, S. E. Campbell, X. Chen, H. Erington, N. D. Gamage, K. Godbey, A. M. Houff, C. Izzo, B. Knight, S. Lalit, E. Leistenschneider, E. M. Lykiardopoulou, F. M. Maier, W. Nazarewicz, R. Orford, W. S. Porter, C. Quick, A. Ravlić, M. Redshaw, P.-G. Reinhard, R. Ringle, S. Schwarz, C. S. Sumithrarachchi, A. A. Valverde, and A. C. C. Villari, "The mass of $^{101}$Sn and Bayesian extrapolations to the proton drip line," arXiv:2510.11815 (2025), https://arxiv.org/abs/2510.11815
10. B. Knight, S. Lalit, P. Giuliani, K. Godbey, W. Nazarewicz, A. Ravlić, and P.-G. Reinhard, "Beyond Constant Error: Heteroscedastic Bayesian Model Combination for Modeling Unmeasured Nuclei," arXiv:2607.14039 (2026), https://arxiv.org/abs/2607.14039
11. R. Navarro Pérez and N. Schunck, "Controlling extrapolations of nuclear properties with feature selection," Phys. Lett. B 829, 137336 (2022). DOI: 10.1016/j.physletb.2022.137336, https://doi.org/10.1016/j.physletb.2022.137336
12. W.-J. Huang, K. Fujii, and H.-Z. Liang, "Taming nuclear mass models with Gaussian processes," Nuclear Science and Techniques 37, 150 (2026). DOI: 10.1007/s41365-026-01939-w, https://doi.org/10.1007/s41365-026-01939-w
13. Q. Z. Chai, J. C. Pei, Na Fei, and D. W. Guan, "Constraints on the neutron drip line with the newly observed $^{39}$Na," Phys. Rev. C 102, 014312 (2020). DOI: 10.1103/PhysRevC.102.014312, https://doi.org/10.1103/PhysRevC.102.014312
14. A. Hamaker, E. Leistenschneider, R. Jain, G. Bollen, S. A. Giuliani, K. Lund, W. Nazarewicz, L. Neufcourt, C. R. Nicoloff, D. Puentes, R. Ringle, C. S. Sumithrarachchi, and I. T. Yandow, "Precision mass measurement of lightweight self-conjugate nucleus $^{80}$Zr," Nature Physics 17, 1408–1412 (2021). DOI: 10.1038/s41567-021-01395-w, https://doi.org/10.1038/s41567-021-01395-w
15. S. R. Stroberg, J. D. Holt, A. Schwenk, and J. Simonis, "Ab initio limits of atomic nuclei," Phys. Rev. Lett. 126, 022501 (2021). DOI: 10.1103/PhysRevLett.126.022501, https://doi.org/10.1103/PhysRevLett.126.022501
16. V. Kejzlar, S. Bhattacharya, M. Son, and T. Maiti, "Black Box Variational Bayesian Model Averaging," The American Statistician 77, 85–96 (2023). DOI: 10.1080/00031305.2022.2058611, https://doi.org/10.1080/00031305.2022.2058611