Ground-State 2p Radioactivity: Mechanisms & Models

• Ground-state two-proton radioactivity is a decay mode in extremely proton-rich nuclei where two protons are emitted simultaneously under the strict energy condition (S2p < 0 and Sp > 0).
• The decay proceeds via a genuine three-body tunneling process where nuclear, Coulomb, and centrifugal barriers shape the lifetime and proton-proton correlations, illuminating shell structure and deformation effects.
• Experimental studies using silicon detectors and gaseous time projection chambers have accurately measured decay parameters in benchmark emitters like 48Ni and 54Zn, thereby validating theoretical models and global systematics.

Searching arXiv for recent and foundational papers on ground-state two-proton radioactivity to support the article. Ground-state two-proton radioactivity is a decay mode of extremely proton-rich nuclei in which the parent ground state is unbound with respect to emission of two protons but not to emission of a single proton, so that decay proceeds by direct emission of a correlated $p+p$ pair from the ground state. In the standard Goldansky formulation this corresponds to $S_{2p}<0$ and $S_p>0$, or equivalently $Q_{2p}>0$ and $Q_p<0$; a less restrictive operational criterion used in global surveys is $Q_{2p}>0$ together with $Q_p<0.2\,Q_{2p}$ to suppress sequential one-proton decay (Saxena et al., 2017, Wang et al., 2020, Olsen et al., 2013). The phenomenon occupies a distinct position among proton-rich decay modes: it is neither ordinary one-proton radioactivity nor $\beta$-delayed $2p$ emission, and in its genuine form it is a three-body tunneling process whose lifetime, energy release, and proton-proton correlations encode shell structure, pairing, deformation, and continuum coupling (Zhou et al., 2022).

1. Definition and classification

Ground-state $2p$ radioactivity is defined by the coexistence of energetic openness for two-proton emission and closure, or strong suppression, of the one-proton channel. In the notation used across the literature, the one- and two-proton separation energies are

$S_{2p}<0$0

$S_{2p}<0$1

so that true ground-state $S_{2p}<0$2 emission corresponds to

$S_{2p}<0$3

or, equivalently,

$S_{2p}<0$4

(Saxena et al., 2017, Zhu et al., 2022). A more permissive criterion used in landscape studies,

$S_{2p}<0$5

separates true or simultaneous $S_{2p}<0$6 emission from sequential $S_{2p}<0$7 decay through an intermediate state with a sufficiently open one-proton channel (Wang et al., 2020, Olsen et al., 2013).

This decay mode must be distinguished from two related processes. Sequential two-proton emission requires that one-proton emission already be open, so that the decay chain proceeds through a real intermediate nucleus. By contrast, $S_{2p}<0$8-delayed $S_{2p}<0$9 emission originates in $S_p>0$0 decay or electron capture to an excited state of the daughter, followed by proton emission from that excited state rather than from the parent ground state (Saxena et al., 2017, Pomorski et al., 2014). The distinction is experimentally consequential because topology, timing, and energy sharing differ between prompt ground-state $S_p>0$1 emission, sequential decay, and $S_p>0$2-delayed channels.

The review literature places ground-state $S_p>0$3 decay within a broader taxonomy of prompt and radioactive two-proton emitters. It treats $S_p>0$4Be as the lightest two-proton ground-state emitter in Goldansky’s original sense, while medium-mass nuclei such as $S_p>0$5Fe, $S_p>0$6Ni, $S_p>0$7Zn, and $S_p>0$8Kr are the canonical long-lived ground-state $S_p>0$9 radioemitters with measurable half-lives (Casal, 2018, Zhou et al., 2022). This suggests two regimes: very light systems with prompt three-body breakup characteristics, and heavier systems where barrier penetration produces measurable radioactivity.

2. Energetics, barriers, and half-lives

The existence of ground-state $Q_{2p}>0$0 radioactivity depends first on mass differences and then on barrier penetration. Even when $Q_{2p}>0$1, the two protons must tunnel through an effective barrier generated by the nuclear mean field, Coulomb repulsion, and, where relevant, centrifugal terms. In one common schematic form,

$Q_{2p}>0$2

with $Q_{2p}>0$3 the attractive nuclear potential, $Q_{2p}>0$4 the Coulomb potential, and $Q_{2p}>0$5 a reduced mass (Saxena et al., 2017). In cluster-based formulations the same structure appears as

$Q_{2p}>0$6

or as a macroscopic barrier built from Coulomb, proximity, and centrifugal terms (Zhu et al., 2022, Cui et al., 2020).

The lifetime is then governed by an exponentially sensitive penetrability. In WKB-based descriptions the penetrability is written as

$Q_{2p}>0$7

with turning points determined by $Q_{2p}>0$8 (Zhu et al., 2022). Cluster models typically factorize the decay constant as

$Q_{2p}>0$9

where $Q_p<0$0 is a preformation or spectroscopic factor and $Q_p<0$1 an assault frequency (Zhu et al., 2022, Cui et al., 2020). A related two-potential approach gives

$Q_p<0$2

with $Q_p<0$3 the normalization factor of the internal quasi-bound wave function (Pan et al., 2021).

Global semi-empirical systematics recover the same barrier-penetration logic. A compact Geiger–Nuttall-type relation proposed for $Q_p<0$4 radioactivity is

$Q_p<0$5

which reproduces known $Q_p<0$6 half-lives with order-of-magnitude accuracy and makes explicit the combined roles of daughter charge, decay energy, and angular momentum (Liu et al., 2020). A Skyrme-Hartree-Fock two-potential study further reports a standard deviation of $Q_p<0$7 between measured and calculated half-lives for true $Q_p<0$8 emitters, while a screened-electrostatic-barrier model gives $Q_p<0$9 for the same class of nuclei (Pan et al., 2021, Zou et al., 2021). These results suggest that, at the level of systematics, barrier penetration with a modest structure input captures much of the gross half-life behavior.

3. Three-body dynamics and decay mechanisms

Ground-state $Q_{2p}>0$0 radioactivity is not exhausted by a diproton-cluster picture. Modern theory treats it as a genuine three-body problem in which structure and continuum dynamics are inseparable. In hyperspherical or Jacobi-coordinate formulations, the decay is described as daughter $Q_{2p}>0$1, with observables expressed in terms of energy-sharing variables and angles between Jacobi momenta (Zhou et al., 2022). For $Q_{2p}>0$2Ni, for example, the reconstructed decay kinematics were transformed to the Jacobi “T” system with

$Q_{2p}>0$3

and parameterized by

$Q_{2p}>0$4

which separate the proton-proton relative motion from the recoil against the daughter (Pomorski et al., 2014).

The central mechanistic distinction is between true simultaneous $Q_{2p}>0$5 emission, sequential $Q_{2p}>0$6 emission through an intermediate resonance, and the transitional or democratic regime where both descriptions mix. A dedicated study of $Q_{2p}>0$7-$Q_{2p}>0$8-shell nuclei shows that the relevant control parameters are the total $Q_{2p}>0$9 decay energy $Q_p<0.2\,Q_{2p}$0, the energy $Q_p<0.2\,Q_{2p}$1 of the core$Q_p<0.2\,Q_{2p}$2 ground-state resonance of the intermediate subsystem, and its width $Q_p<0.2\,Q_{2p}$3 (Golubkova et al., 2016). In that formulation,

$Q_p<0.2\,Q_{2p}$4

and the character of the decay changes as $Q_p<0.2\,Q_{2p}$5 crosses the sequential threshold and as $Q_p<0.2\,Q_{2p}$6 broadens. The improved direct-decay model introduced there reproduces three-body correlations and shows that transition dynamics can be used to extract properties of the intermediate core$Q_p<0.2\,Q_{2p}$7 resonance from measured $Q_p<0.2\,Q_{2p}$8 correlations (Golubkova et al., 2016).

Time-dependent and Gamow-basis approaches further refine this picture. The Gamow coupled-channel treatment of $Q_p<0.2\,Q_{2p}$9Kr demonstrates that deformation and core excitations can change the valence-proton orbital content from high-$\beta$0 configurations to low-$\beta$1 Nilsson components, dramatically increasing the $\beta$2 width and resolving the unexpectedly short measured lifetime (Wang et al., 2018). In that case, the calculated angular proton-proton correlations show a competition between $\beta$3 and $\beta$4 modes rather than a purely diproton or purely sequential limit (Wang et al., 2018). This suggests that the asymptotic decay pattern depends sensitively on configuration mixing, pairing, and deformation, not only on Q values.

4. Experimental methods and benchmark emitters

Experimentally, ground-state $\beta$5 radioactivity has been studied with two complementary classes of detectors: implantation detectors based on silicon and gaseous time projection chambers with optical or electronic readout. Silicon detectors established the existence of several $\beta$6 emitters through implantation–decay spectroscopy, but gaseous TPC systems made it possible to reconstruct individual proton tracks and extract angular and energy correlations event by event (Pomorski et al., 2014, Kubiela et al., 29 Sep 2025).

The Optical Time Projection Chamber used for $\beta$7Ni had an active volume of $\beta$8, filled with approximately $\beta$9 Ar, $2p$0 He, and $2p$1 N$2p$2 at 1 atm, with a drift velocity $2p$3 and GEM-based light amplification recorded by a CCD and a photomultiplier (Pomorski et al., 2014). That experiment recorded six decays of $2p$4Ni, including four events of two-proton ground-state radioactivity. Full 3D reconstruction yielded a weighted average

$2p$5

a total half-life

$2p$6

and branching ratios

$2p$7

with the $2p$8 branch dominating (Pomorski et al., 2014). The four reconstructed opening angles, between about $2p$9 and $2p$0, and the low-$2p$1 Jacobi points support a true three-body decay with substantial proton-proton correlation rather than a purely sequential mechanism (Pomorski et al., 2014).

For $2p$2Zn, the Warsaw OTPC detected and reconstructed five $2p$3 radioactivity events. The deduced half-life was

$2p$4

and the weighted-average decay energy was

$2p$5

(Kubiela et al., 29 Sep 2025). The angular information is especially notable: the combination of the new data with earlier measurements suggests a flat distribution of the opening angle $2p$6, in contrast to the asymmetric small-angle-enhanced distribution measured for $2p$7Fe (Kubiela et al., 29 Sep 2025). This suggests structurally distinct $2p$8 dynamics on opposite sides of the $2p$9 shell closure.

The same OTPC methodology has been used to validate reconstruction procedures on neighboring $S_{2p}<0$00 and $S_{2p}<0$01 emitters. In the $S_{2p}<0$02Ni work, decays of $S_{2p}<0$03Cr and $S_{2p}<0$04Fe provided energy-calibration benchmarks, while in the later Zn-region study the chamber resolved $S_{2p}<0$05 and $S_{2p}<0$06 channels in $S_{2p}<0$07Zn, $S_{2p}<0$08Zn, and $S_{2p}<0$09Cu, including the first observation of $S_{2p}<0$10-delayed $S_{2p}<0$11 emission in $S_{2p}<0$12Zn (Pomorski et al., 2014, Kubiela et al., 29 Sep 2025). This strengthens confidence in TPC-based $S_{2p}<0$13 kinematics for genuine ground-state emitters.

5. Structure effects: pairing, resonance, deformation, and halo

The occurrence of ground-state $S_{2p}<0$14 radioactivity is strongly shaped by pairing and by the character of near-threshold single-particle orbitals. In an RMF+BCS study of proton-rich nuclei with $S_{2p}<0$15, the Goldansky condition $S_{2p}<0$16 identified a set of candidate $S_{2p}<0$17 emitters including $S_{2p}<0$18Ti, $S_{2p}<0$19Cr, $S_{2p}<0$20Fe, $S_{2p}<0$21Ni, $S_{2p}<0$22Zn, $S_{2p}<0$23Ge, $S_{2p}<0$24Se, $S_{2p}<0$25Kr, $S_{2p}<0$26Sr, and $S_{2p}<0$27Zr (Singh et al., 2013). That work emphasized that low-lying proton resonances, such as $S_{2p}<0$28 in $S_{2p}<0$29Cr and $S_{2p}<0$30, $S_{2p}<0$31, and $S_{2p}<0$32 in $S_{2p}<0$33Ge, acquire pairing gaps of order $S_{2p}<0$34 MeV and behave structurally like bound orbitals. This extends the effective drip line and enables metastable even-$S_{2p}<0$35 nuclei with $S_{2p}<0$36 but $S_{2p}<0$37 (Singh et al., 2013). A plausible implication is that pairing through resonant states is not merely a correction to the mass surface but part of the mechanism that produces measurable $S_{2p}<0$38 radioactivity.

In lighter systems, the literature links $S_{2p}<0$39 radioactivity to extended proton densities and halo-like structure. A theoretical study of nuclei with $S_{2p}<0$40–34 identifies $S_{2p}<0$41Mg, $S_{2p}<0$42Si, $S_{2p}<0$43S, $S_{2p}<0$44Ar, and $S_{2p}<0$45Ca as promising ground-state $S_{2p}<0$46 emitters with $S_{2p}<0$47 and $S_{2p}<0$48, and relates this to extended charge-density tails, increased charge radii, weakly bound valence protons near the Fermi surface, and occupancy of low-$S_{2p}<0$49 orbitals such as $S_{2p}<0$50 in $S_{2p}<0$51Si (Saxena et al., 2017). In $S_{2p}<0$52Si the calculated Coulomb and centrifugal barriers produce a quasi-bound valence-proton configuration, explaining how a $S_{2p}<0$53-unbound nucleus can still persist long enough to be studied (Saxena et al., 2017).

The deformation dependence of $S_{2p}<0$54 half-lives has also been treated semi-empirically. One formula introduces an explicit $S_{2p}<0$55 dependence,

$S_{2p}<0$56

with fitted parameters and $S_{2p}<0$57, and achieves an RMSE of about $S_{2p}<0$58 on a set of measured $S_{2p}<0$59 decays (Saxena et al., 2022). That work finds shape coexistence in several $S_{2p}<0$60 emitters and candidates, with prolate ground states predominating. This suggests that deformation affects $S_{2p}<0$61 decay not only through barrier geometry but also through changes in shell structure and pairing near the Fermi surface.

6. Global systematics and the nuclear landscape

Global surveys show that ground-state $S_{2p}<0$62 radioactivity is a generic feature of even-$S_{2p}<0$63 nuclei beyond the two-proton drip line, but its measurable domain is limited by competition from other channels and by rapidly varying half-lives. A DFT-based landscape study using several Skyrme functionals concluded that candidates for true $S_{2p}<0$64 radioactivity satisfying both energy and half-life constraints occur only up to tellurium, while sequential $S_{2p}<0$65 emission is expected in every even-$S_{2p}<0$66 isotope above Te except Xe, where $S_{2p}<0$67 decay dominates (Olsen et al., 2013). The same survey singled out $S_{2p}<0$68Ge, $S_{2p}<0$69Se, $S_{2p}<0$70Kr, and $S_{2p}<0$71Te as especially interesting candidates close to then-current experimental reach, and identified $S_{2p}<0$72Te and $S_{2p}<0$73Hf as cases where competition between $S_{2p}<0$74 and $S_{2p}<0$75 decay may be observable (Olsen et al., 2013).

Mass-model dependence remains substantial. A later study combining $S_{2p}<0$76 values from WS4, FRDM, KTUY, and HFB29 with GLDM half-life estimates found that probable $S_{2p}<0$77 candidates are concentrated in nuclei beyond the proton drip line with $S_{2p}<0$78 or $S_{2p}<0$79 for all models, whereas only HFB29 predicts a group of heavier candidates beyond $S_{2p}<0$80, including $S_{2p}<0$81Te, $S_{2p}<0$82Xe, $S_{2p}<0$83Ba, $S_{2p}<0$84Ce, and $S_{2p}<0$85Ce (Wang et al., 2020). For those nuclei, competition with $S_{2p}<0$86 decay depends sensitively on the chosen mass model; within HFB29, $S_{2p}<0$87Te, $S_{2p}<0$88Ba, and $S_{2p}<0$89Ce prefer $S_{2p}<0$90 radioactivity, whereas $S_{2p}<0$91Xe and $S_{2p}<0$92Ce prefer $S_{2p}<0$93 decay (Wang et al., 2020). This suggests that progress in mass determination remains central to any extension of $S_{2p}<0$94 radioactivity into heavier regions.

Global half-life calculations with phenomenological models broadly agree for moderate $S_{2p}<0$95, but diverge strongly below $S_{2p}<0$96 MeV. In the generalized liquid-drop model, the most promising true $S_{2p}<0$97 candidates based on AME2016 were $S_{2p}<0$98Si, $S_{2p}<0$99Ca, $S_p>0$00Ti, and $S_p>0$01Cr, with predicted half-lives from about $S_p>0$02 s to $S_p>0$03 s, while heavier candidates such as $S_p>0$04Ni, $S_p>0$05Zn, $S_p>0$06Ge, and $S_p>0$07Se were predicted to be effectively unobservable as $S_p>0$08 radioactivity because of extremely long half-lives (Cui et al., 2020). A Coulomb-and-proximity-potential study reached a similar conclusion and emphasized that low-$S_p>0$09 cases such as $S_p>0$10Ni, $S_p>0$11Zn, and $S_p>0$12Se are especially model dependent (Zhu et al., 2022). This suggests that the experimentally accessible landscape is narrower than the energetically allowed one.

Ground-state $S_p>0$13 radioactivity thus occupies a well-defined but structurally rich region of the proton-rich chart. Its existence requires the Goldansky energetic condition; its lifetime reflects barrier penetration, pairing, and configuration mixing; and its correlations expose whether the decay is diproton-like, democratic, or transitional. The combined experimental and theoretical record indicates that it is best treated as an open-quantum-system problem in which masses, shell evolution, deformation, and continuum coupling are equally fundamental (Zhou et al., 2022).