Cluster emission — the spontaneous decay of a heavy nucleus by emission of a fragment heavier than an alpha particle but lighter than typical fission fragments — is experimentally established only in the trans-lead region (221^{221}Fr–242^{242}Cm), whereas the nuclei traversed by the r-process of rapid neutron capture are far more neutron-rich. The first study to insert cluster-emission rates into neutron-star-merger r-process network calculations (Thomas et al., arXiv:2608.24499, 2026) used four actinide-fitted analytical half-life formulas, motivated by five tentative high-energy ISOLDE events attributed to 230^{230}Ra emitting 22^{22}O with branching ratio (4.3±1.9)×10−9(4.3\pm1.9)\times10^{-9}, and found no change in the final abundance distributions. That verdict rests on phenomenological extrapolations whose neutron-rich predictions spread over roughly six orders of magnitude and which the tentative 230^{230}Ra events contradict by at least fifteen orders of magnitude in half-life. The open problem is to determine cluster-emission rates for neutron-rich heavy nuclei with reliability beyond these formulas — through microscopic mean-field calculations and experimental anchors — and thereby to settle whether cluster emission materially affects r-process nucleosynthesis.

Background

The rapid neutron-capture (r-)process produces about half of all nuclei heavier than iron. It proceeds in hot, neutron-dense ejecta, and neutron-star mergers are an established site, confirmed by the gravitational-wave event GW170817 and its kilonova afterglow powered by the radioactive decay of freshly synthesized r-process material. Nucleosynthesis calculations follow a reaction network of thousands of (mostly neutron-rich) nuclei coupled by neutron capture, photodisintegration, and decay. Besides beta and alpha decay, fission modes — spontaneous, beta-delayed, and neutron-induced — must be included, because fission of heavy actinides terminates the neutron-capture path and recycles material back to lighter seeds ('fission recycling'), leaving a direct imprint on the final abundance pattern. Any decay mode whose rates compete along the r-process path can likewise reshape the abundances, so network builders must justify which channels are excluded.

Cluster emission (also called cluster radioactivity) is an exotic decay mode intermediate between alpha decay and fission, predicted theoretically in 1980 and first observed in 1984 in the decay 223^{223}Ra → 14^{14}C (Rose and Jones). In it, a parent nucleus emits a light nucleus heavier than an alpha particle (observed clusters range from 14^{14}C to 34^{34}Si) and correspondingly lighter than a fission fragment. All confirmed emitters lie in the trans-lead region from 221^{221}Fr to 242^{242}Cm; favored decays produce daughters near the doubly magic (closed-shell, Z=82, N=126) nucleus 208^{208}Pb, and measured branching ratios relative to alpha decay span 10−410^{-4} to 10−1610^{-16}. Because of these tiny branchings, cluster emission has conventionally been ignored in astrophysical nucleosynthesis networks. The question was reopened for very heavy nuclei by superheavy-region predictions (Saxena et al. 2024) of sub-second cluster-emission half-lives for oganesson isotopes, rates that would outcompete alpha, beta, and fission there — although the r-process path itself runs through neutron-rich, not neutron-deficient, nuclei.

Practical rate estimates rest on semi-empirical formulas in the Geiger–Nuttall tradition, relating the decay Q-value (the energy released, QQ, fixed by the parent and fragment masses) to the partial half-life, under a picture of a preformed cluster tunneling through the combined nuclear-plus-Coulomb barrier. Four such formulas are in current use: the Universal Decay Law (UDL, Qi et al. 2009), the modified BKAG formula (MBKAG, Jain et al. 2023), the New Geiger–Nuttall law (NGN, Qi L.-J. et al. 2023), and the refitted Royer formula (RRF, Saxena et al. 2024, which includes generalized-liquid-drop predictions for superheavy nuclei in its fit). All are fitted essentially to trans-lead actinide data; where experimental masses are lacking, Q-values come from mass models such as the BSkG3 Skyrme energy-density-functional model (Grams et al. 2023). Far from the fit region the formulas diverge strongly from one another.

The source study (Thomas et al. 2026) is the first to place cluster-emission rates inside an r-process network. It anchors on five high-energy events (20–40 MeV) recorded in a mass-A=230 silicon spectrum during an ISOLDE campaign (ISOLDE is CERN's on-line radioactive-ion-beam facility) on beta-delayed fission of 230,232,234^{230,232,234}Ac (parent experiment: Bara et al. 2025). Beam contamination, electronic noise, alpha-pulse summing, and degraded fission fragments were excluded quantitatively. Under the assumption that the events are cluster emission, the most likely channel is 230^{230}Ra → 22^{22}O + 208^{208}Hg, with branching ratio (4.3±1.9)×10−9(4.3\pm1.9)\times10^{-9} and partial half-life (1.31±0.60)×1012(1.31\pm0.60)\times10^{12} s. The four formulas, however, predict 102710^{27}–103310^{33} s for this decay — at least fifteen orders of magnitude longer than the events imply, with the formulas themselves spread over six orders. Cluster-emission rates for all neutron-rich nuclei with 82<Z≤11882<Z\leq118 on the r-process network (all open cluster channels with half-lives below the age of the universe) were then folded into post-processing nucleosynthesis calculations — reaction networks evolved on precomputed neutron-star-merger ejecta trajectories of density and temperature (Just et al. 2023). Even with the most aggressive formula — RRF, which the 294^{294}Og comparison suggests overestimates cluster emission and therefore serves as an upper limit — the isotopic abundance distributions were indistinguishable from calculations without cluster emission.

Two neighboring lines of work mark what is missing. First, a microscopic route exists: describing cluster emission as extremely asymmetric fission within Hartree–Fock–Bogolyubov calculations based on the Gogny energy density functional (a self-consistent nuclear mean-field method with pairing, using the Gogny effective nucleon–nucleon interaction; Warda and Robledo 2011; Warda, Zdeb, and Rodríguez-Guzmán 2026, who mapped cluster-radioactivity fission valleys along isotopic and isotonic chains), but no microscopic cluster-emission half-life set for neutron-rich nuclei exists — by contrast, microscopic spontaneous-fission half-lives across the nuclear chart became available only in 2026 (Sánchez-Fernández et al.). Second, the r-process environment is hot (gigakelvin temperatures): thermal population of excited states can shorten cluster-decay half-lives by orders of magnitude (Kelkar et al. 2022; Rojas-Gamboa et al. 2022, who built an excitation-energy-dependent decay law, and Rojas-Gamboa et al. 2024 for alpha decay), an effect absent from ground-state network inputs. Whether cluster emission matters for the r-process is therefore answered today only conditionally, on rate extrapolations that neither the tentative 230^{230}Ra observation nor any microscopic calculation yet supports.

Problem Statement

Determine whether cluster emission — spontaneous emission of a nuclear fragment heavier than an alpha particle but lighter than fission fragments — from the neutron-rich heavy nuclei traversed by the r-process materially affects the isotopic abundances produced in neutron-star-merger nucleosynthesis, on the basis of decay rates whose reliability extends to the neutron-rich region. This single objective comprises three jointly supporting parts: (i) establish cluster-emission partial half-lives and branching ratios for neutron-rich nuclei with 82<Z≤11882<Z\leq118 on the r-process network with a validated microscopic or benchmarked model, and quantify how far the actinide-fitted analytical formulas (the Universal Decay Law, the modified BKAG formula, the New Geiger–Nuttall law, and the refitted Royer formula) can be trusted in this region; (ii) include these rates in r-process network calculations over representative neutron-star-merger trajectories and determine, with uncertainty quantified across rate and mass models, whether the resulting abundance distributions differ materially from calculations without cluster emission; and (iii) resolve the status of the candidate observation motivating the question — the five high-energy events in an ISOLDE (CERN radioactive-beam facility) mass-A=230 spectrum tentatively attributed to 230^{230}Ra emitting 22^{22}O with branching ratio (4.3±1.9)×10−9(4.3\pm1.9)\times10^{-9}, in tension with all four formulas by at least fifteen orders of magnitude in half-life — either by dedicated experiment or by a microscopic calculation of the 230^{230}Ra cluster-emission channel. Parts (i) and (ii) jointly define the objective; part (iii) is optional.

References

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