Measurement in experiments

Generally, the half-lives of proton-rich nuclei decrease under the increase in proton-neutron asymmetry, and the span ranges from 10¹ to 10^-22 s. Therefore, different production mechanisms, detection methods, and technologies [108-112] are used for such nuclei in experiments, depending on the half-life. In early years, light particle-emission nuclei (e.g., ⁶Be and ¹²O) were produced by transfer reactions with protons or ³He beams [23, 26]. Nowadays, the production of exotic nuclei is performed using two main technologies: (i) in-flight projectile fragmentation (PF) and (ii) isotope separation on-line (ISOL). A PF radioactive beam device using heavy ions as the incident beam was installed in Berkeley Laboratory in the 1970s and has been widely used since then; this allows heavier nuclei to be accelerated [113]. In this method, the short-lived radionuclides are produced via fragmentation of the incident nucleus on a thin target. The momentum direction of the produced radionuclide is strongly consistent with the direction of the incident beam. Then, the generated radionuclides is selected and transferred via a fragment separator to a secondary reaction zone, where different experiments can be performed using different secondary target and detector setups. PF can produce nuclei with very short half-lives or close to the proton dropline; however, the secondary-beam quality is relatively low. When studying the decay of nuclei with very short half-lives, the primary detection method is the in-flight decay one, because most nuclei decay before they can pass through the separation and purification spectrometer. ISOL is a radioactive nuclear beam device developed in the 1950s; it was first proposed and verified by O. Kofoed-Hansen and K.O. Nielsen of Copenhagen University, Denmark, in 1951 [114]. They bombarded a uranium target with a neutron beam, causing it to fragment and produce the neutron-rich nuclides ^89,90,91Kr and ^89,90,91Rb. Then, the generated radionuclides were diffused and extracted to the ion source for ionization, by heating the thick target. Subsequently, the ionized radionuclides were extracted for mass spectrometry and electromagnetic separation processes; finally, the separated and purified radioactive beams were transferred into the secondary reaction zone to perform the experiment. In ISOL, the generation, ionization, separation, and experiment procedures of the radioisotopes are carried out continuously. One attractive feature of ISOL facilities is their extraordinarily good secondary-beam quality, which offers a very low cross-contamination (below 10^-4); however, it is difficult to produce short-lived nuclides. The main detection method for the decay of nuclei produced by ISOL is the implantation method, because the half-life is considerably longer than the flight time in the spectrometer after generation. The measurement methodology also depends on different circumstances, including the half-life, decay mechanism, and experimental objective. Here, we use 2p emission as an example to demonstrate the advantages and disadvantages of different measurement techniques, as well as the corresponding scopes of applications.

In-flight decay technique. — As discussed, in-flight decay constitutes an effective method that can be used to measure the decay of 2p emitters with half-lives of less than a pico-second. In this case, the unstable nuclei decay shortly after being produced by the radioactive beam facilities. Among the ground-state 2p emitters discovered thus far, ¹⁹Mg was investigated using this technique [87]. Meanwhile, it also consistutes the main experimental technique for studying excited 2p emissions.

In the following, the 2p emission of excited state ²²Mg [79] is introduced as an application of in-flight decay for measuring the decay of very short-lived proton-rich nuclei. The experiment was performed using the projectile fragment separator (RIPS) beam line at the RI Beam Factory (RIBF), operated by the RIKEN Nishina Center and the Center for Nuclear Study, University of Tokyo. A primary beam of 135 A MeV ²⁸Si was used to produce secondary ²³Al and ²²Mg beams with incident energies of 57.4 A MeV and 53.5 A MeV, respectively, in the center of the carbon reaction target. Beyond the reaction target, five layers of silicon detectors and three layers of plastic hodoscopes were positioned as shown in Fig. 4. The first two layers of Si-strip detectors were located 50 cm downstream of the target and used to measure the emission angles of the fragment and protons. Three layers of 3×3 single-electrode Si were used as ΔE-E detectors for the fragment. The three layers of plastic hodoscopes were located 3 m downstream of the target and were used as ΔE and E detectors for protons. The time of flight of the protons was measured using the first layer. Clear particle identifications were obtained using this setup for the kinematically complete three-body decays. The momenta and emission angles for protons and the residue were determined by analyzing the detector signals. The excitation energy (E^*) of the incident nucleus was reconstructed from the difference between the invariant mass of the three-body system and the mass of the mother nucleus in the ground state; this provides useful information with which to determine the decay mechanism of 2p emission. These experimental studies indicate that the highly excited state of ²²Mg may produce a diproton emission [79].

Fig. 4Full-screen View

(Color online) Sketch of the detector setup for in-flight decay. See Refs. [79, 80] for details.

Implantation decay method. — Implantation decay is one of the most popular methods used to experimentally measure long-lived nuclei. In terms of 2p decay, it is particularly useful for the β-delayed decays of proton-rich nuclei, because of the long half-life of β decay (as governed by the weak interaction). As an example, we introduce an experiment involving the β-delayed 2p emission of ²²Al, performed at the Radioactive Ion Beam line in Lanzhou (RIBLL) [76, 115, 116]. A sketch of the detector setup for implantation decay is shown in Fig. 5. The emitted nucleus ²²Al produced by RIBLL first passes through two scintillation detectors (located at the first and second focal planes T1 and T2, respectively). The energy of the secondary beam is adjusted by selecting a combination of aluminum degraders with different thicknesses through the stepper motors; thus, most of the ²²Al nuclei can be injected and halted at the decay position. Three silicon detectors with thicknesses of 300 μm are installed after the aluminum degraders, to measure the energy loss (ΔE) of the beam ions. The identification of incident ions is performed using a ΔE-TOF. Then, the incident ions are injected and stopped using a double-sided silicon strip detector (DSSD) set with a 42.5∘ tilt angle, to increase the effective thickness. The subsequent decays are measured and correlated to the preceding implantations using the position and time information, and the emitted charged particles with sufficient energies to escape the centered DSSD are measured by the detection array, which is composed of four DSSDs and four quadrant silicon detectors (QSDs). In addition, five high-purity Germanium (HPGe) γ-ray detectors outside the silicon detectors are used to detect the γ-rays emitted by the decay, and several QSDs are also installed to measure anti-coincidences with β particle detections. In the implantation decay experiment, the radioactive nuclei are injected into silicon detectors and then decay via β-delayed 1p or 2p emission after a certain time. Both processes produce corresponding signals in the silicon detector. Moreover, it should be noted that the energy loss of the radionuclide in silicon detectors is relatively high (several hundreds of MeV) compared to the energy of protons emitted during the decay (generally only a few MeV). The different energy scales makes it difficult to measure the signals with the same magnification. To solve this problem, each output channel of the preamplifiers for the three DSSDs is split into two parallel electronic chains with low and high gains, to measure both the implantation events with energies up to hundreds of MeV and the decay events with energies on the order of hundreds of keV or less. One uses a relatively small amplification factor to measure the signal of injected ions; the other one measures the emitted protons. Owing to the granularity and position resolution of DSSDs, the opening angle and relative momentum correlations of two emitted protons can be easily reconstructed. Moreover, the decay channel and half-life can also be obtained. The decay half-life corresponds to the entire process, including the β decay [76].

Fig. 5Full-screen View

(Color online) Sketch of the detector setup for implantation decay. See Ref. [76] for details.

Time projection chamber. — The opening angle and relative momentum correlations can be reconstructed from the signals measured in the experiment via the two abovementioned methods; however, the physical process of decay cannot be directly observed. Time projection chambers (TPCs) are an effective detection technology that can intuitively “see’’ the decay process and particle tracks. TPCs are a type of particle track detector widely used in nuclear and particle physics to measure the three-dimensional trajectory and energy information of particles [117]. For the decay from proton-rich nuclei (e.g., ⁴⁵Fe), the traces of two valence protons are measured using an optical time projection chamber-charge coupled device (OTPC-CCD) method [93]. A sketch of the OTPC-CCD detector setup is shown in Fig. 6. The entire geometric volume of the OTPC is 20×20×15 cm³; this is filled with a gas mixture of 49% He, 49% Ar, 1% N₂, and 1% CH₄ at atmospheric pressure. The protons emitted from proton-rich nuclei typically have an energy ranging from a few hundred keV to a few MeV. They can be absorbed in gas-detector-based TPCs, and they exhibit clear tracks. For ⁴⁵Fe, the two protons emitted in the 2p decay process share an energy of 1.15 MeV. In the most probable case (i.e., of equal energy division), the proton with an energy of 0.5 MeV has a sufficiently long track (2.3 cm) in the counting gas of the OTPC. However, for a 4 MeV proton (as emitted in the β-delayed 2p decay of certain proton-rich nuclei), the length of the proton track is 50 cm, which means the protons will escape the chamber. Therefore, the lower limit of the proton energy can only be determined via TPC if the proton energy is too high [93]. After the incident nucleus decays in the detector, the ultraviolet signals generated by ionization are converted to visible light signals by a wave-length-shifter. Then, these signals are recorded by a CCD camera and a photomultiplier tube (PMT) through a glass window. The sampling frequency of the PMT is 50 MHz, which can record the time of signal drift and yields the spatial Z-direction information of particles. As an example, the results of the β-delayed 1p, 2p, and 3p emissions of ⁴³Cr, as measured by OTPC-CCD in the experiment at NSCL, are shown in Fig. 7; here, the track of the emitted proton can be intuitively seen [118].

Fig. 6Full-screen View

(Color online) Sketch of the detector setup for the optical time projection chamber-charge coupled device (OTPC-CCD) [93].

Fig. 7Full-screen View

(Color online) Snapshots of β-delayed particle emission from ⁴³Cr [118]: (a) proton emission, (b) 2p emission, and (c) three-proton emission. See Ref. [118] for details.

The abovementioned three experimental methods for measuring the 2p emission of proton-rich nuclei can be adapted to different conditions. The method of in-flight decay is useful for measuring decay processes with a short half-life (e.g., the 2p decay from light nuclei). The implantation decay method is suitable for the measurement of long-lived nuclei, especially for β-delayed decay. TPCs place no clear requirements upon the half-life of decay: both short half-life and long half-life (e.g., β-delayed) decays can be measured. In particular, the OTPC-CCD can be used to observe decay processes and particle traces. The energy of emitted protons can also be obtained by the OTPC-CCD; however, the precision is generally lower than that measured by silicon detectors. In recent years, detection technology has developed rapidly [119, 120]. Similar to its development in the field of nuclear electronics, waveform sampling technology is becoming increasingly important in data acquisition for nuclear experiments [121]. If these new techniques are used in experiments, the original signal of the detector can be completely saved, the data can be analyzed and processed offline, and the applicability of the above three methods can be extended.

Identification of 2 p emission mechanism

As shown in Fig. 1, three mechanisms are available for 2p emission. Typically, the emission time or pp correlation of two emitted protons can differ between these decay processes; this can be used to distinguish the decay mechanisms of 2p emissions. For a 2p decay from the ground or low-lying states of the light nuclei, the energy for the relative motions of two valence protons is compatible with the decay energy Q_2p. Moreover, the Coulomb barrier is relatively small, and the diproton structure can be distorted by Coulomb repulsion after tunneling. As a result, the energy/momentum and angular correlation can be widely distributed, even in the cases exhibiting diproton emission. ⁶Be is one such case; it is thought to exhibit both diproton and large-angle (three-body) emissions compared with the theoretical calculations [122-125]; meanwhile, the observed pp correlation has a wide energy distribution [18, 24, 98]. Similar situations occur in ^11,12O [50, 51, 126-128]. However, one can still extract useful information from these observables. We take the ground-state 2p emitter ¹⁶Ne as an example: Although from the perspective of energy, its first excited (J^π=2⁺) state exhibits sequential decay, the measured correlation pattern shows aspects of both sequential and diproton-like decay (see Fig. 8 and Refs. [102, 103]). This suggests that interference between these processes is responsible for the observed features, which can be described in terms of a “tethered decay” mechanism.

Fig. 8Full-screen View

(Color online) Comparison of the experimental Jacobi-Y (a) and -T (d) correlation distributions against those of the three-body model [(b) and (e)] and those from a sequential decay simulation [(c) and (f)]. The effects of the detector efficiency and resolution upon the theoretical distributions have been included via Monte Carlo simulations. (g) The relative orientations and magnitudes of the two-proton velocity vectors for the peak regions indicated by blue circles in Panel (a). See Ref. [103] for details.

For the 2p decay produced by the highly-excited or mid-heavy nuclei, the diproton peak in the energy/momentum correlation can be more pronounced. For example, experimental measurements of ¹⁸Ne show a clear diproton emission mechanism in the 6.15 MeV excited state [75]. In the study of 2p emissions for excited states of ^27,28P and ^28,29S, conducted at RIBLL, a diproton emission mechanism was also identified using the same method [81-83]. In the following, we present in detail a method for identifying the 2p decay mechanism.

As described above, in 1983, a β-delayed decay experiment performed at the LBNL identified a 2p emission in the 14.044 MeV excited state of ²²Mg; however, the emission mechanism could not be clearly identified because the statistics for the relative angle distribution were too low [129]. To elucidate this decay mechanism, the 2p emissions of two excited proton-rich nuclei, ²³Al and ²²Mg, were recently measured using the RIPS at the RIKEN RI Beam Factory [79]. The in-flight decay introduced above was adopted, and a sketch of the detector setup is shown in Fig. 4. The relative momenta and opening angle between the two emitted protons decayed from ²³Al and ²²Mg and the excitation energies of the decay nuclei were measured. Figure 9(a),(b) shows the results obtained by comparing the experimental data and theoretical simulation results for different mechanisms; the highly excited states of ²³Al were dominated by the sequential emissions or three-body (large-angle) emissions in different energy ranges. However, as shown in Fig. 9(c),(d), around the 14.044 MeV excited state of ²²Mg, a 30% probability that the valence protons would decay as diprotons was observed; the other 70% were likely to undergo sequential or three-body (large-angle) emission. Meanwhile, the other low-excitation states ²²Mg were basically governed by three-body (large-angle) or sequential emission. Consequently, the experiment quantitatively determined the 2p decay mechanism for the 14.044 MeV excited state of ²²Mg [79]; this is useful for further studying the properties of excited 2p emissions. As a different experimental method, the implantation decay was used to measure the β-delayed 2p emission of ²²Al in RIBLL [76]; this confirms the diproton emission from the 14.044 MeV excited state of ²²Mg.

Fig. 9Full-screen View

The (a) relative momenta and (b) opening angles between the two protons emitted from the exited states of ²³Al; the (c) relative momentum and (d) opening angle between the two protons emitted from the exited states of ²²Mg. See Ref. [79] for details.

To further analyze the 2p decay mechanism, simulation tools are required. The two-particle intensity interferometry (HBT) [130] method is widely used to obtain the emission source size and emission time for middle- and high-energy nuclear reactions [131]. For the simultaneous three-body and sequential emission, the time scales can differ slightly, because the former proceeds via simultaneous emission and the latter exhibits a certain time difference. To clearly distinguish the mechanism of three-body (large-angle) and sequential emission, the HBT method was used in the above study investigating the 2p emission of ²³Al and ²²Mg [80]. Assuming that the first proton is emitted at time t₁=0 and the second proton is emitted at time t₂=t, the space and time profile of the Gaussian source can be simulated according to the function S(r,t) ∼ $\text{exp}(-r^2/2r_0^2-t/\tau )$ in the CRAB code [80]. r₀ refers to the radius of the Gaussian source, and τ refers to the lifetime for the emission of the second proton, which starts from the emission time of the first proton. r and t denote the position and time of proton emission, respectively; they were randomly selected. The agreement between the experimental data and CRAB calculations for different r₀ and τ values was evaluated by determining the value of the reduced χ², as shown in Fig. 10. The results show that the ranges of the source size parameters were obtained as r₀ = 1.2 ~ 2.8 fm and r₀ = 2.2 ~ 2.4 fm for ²³Al and ²²Mg, respectively. Although the source sizes differ between these two experiments, they have essentially identical effects when compared to the size of the nucleus itself. The ranges of the emission time parameters were obtained as τ = 600 ~ 2450 fm/c and τ = 0 ~ 50 fm/c for ²³Al and ²²Mg, respectively. The large difference in emission time between the two nuclei means that the 2p emission mechanism of ²³Al is primarily sequential emission, owing to the long emission-time difference. Meanwhile, the emission mechanism for ²²Mg is almost that of three-body emission. Combining this with the relative momenta and opening angles of the two emitted protons (as discussed above), the decay mechanism of 2p emission can be clearly identified [80, 132].

Fig. 10Full-screen View

(Color online) Contour plot of the reduced χ² obtained by fitting the proton-proton momentum correlation function using the CRAB calculation [80]: (a) ²³Al → p + p + ²¹Na and (b) ²²Mg → p + p + ²⁰Ne.

Meanwhile, the HBT method has been used to analyze experimental data from the two-neutron (2n) decay, which has also come to represent a frontier of radioactive nuclear beam physics in recent years [133]. The 2n emission of ¹⁸C and ²⁰O have been measured at GSI, and the measured momentum correlation for the emitted neutrons was analyzed using the HBT method [134]. The results also indicate that the time scales substantially differ between simultaneous and sequential decay.

So far, only a few ground-state 2p emitters have been observed with relatively low statistics; hence, the primary goal is to explore more 2p emitter candidates. Meanwhile, the study of the 2p decay mechanism and other characteristics has also increased the demand for high-quality experimental data. According to the theory prediction (see the discussions below and Refs. [22, 21, 46] for details), ground-state 2p decay is widespread in the light and mid-heavy nuclei beyond the proton dripline. However, most of these extremely unstable nuclei are difficult to produce via experiments, and the 2p emission is sensitive to impacts from other decay channels. The construction of a new generation of high-performance accelerators and the development of novel detection technology [e.g., the Facility for Rare Isotope Beams (FRIB) in the United States, the High Intensity Accelerator Facility (HIAF) in China, and the Rare Isotope Accelerator Complex for On-line Experiment (RAON) in South Korea) will provide better experimental conditions for the further study of radioactivity in exotic nuclei, including 2p and 2n decay.

Towards quantified prediction of the 2 p landscape

Exploring the candidate 2p emitters using density functional theory. — The ground-state 2p decay usually occurs beyond the 2p dripline; here, single-proton emission is energetically forbidden or strongly suppressed owing to the odd-even binding energy staggering attributable to the pairing effect [17-19, 135-137]. Meanwhile, the presence of the Coulomb barrier has a confining effect upon the nucleonic density; hence, relatively long-lived 2p emitters should be expected. So far, the 2p decay has been experimentally observed in a few light- and medium-mass nuclides with Z ≤ 36. This raises an urgent demand for theoretical studies to identify further candidates, to better understand the properties of this exotic process [39, 40, 43]. To this end, density functional theory (DFT) represents an effective tool offering a microscopic framework and moderate computational costs; it has been widely and successfully applied to describe the bulk properties of nuclei across the entire nuclear landscape [6, 138, 139].

The candidate ground-state 2p emitters have been explored using the self-consistent Hartree-Fock-Bogoliubov (HFB) equations [21, 22], in which the binding energies for odd-N isotopes are determined by adding the computed average pairing gaps to the binding energy of the corresponding zero-quasiparticle vacuum obtained by averaging the binding energies of even-even neighbors. Because of the Coulomb effect, all the Skyrme energy density functionals in Ref. [21, 22] produced a very consistent prediction for the 2p dripline. The corresponding mean value and uncertainty are shown in Fig. 11.

Fig. 11Full-screen View

(Color online) The landscape of ground-state 2p emitters. The mean 2p dripline (thick black line) and its uncertainty (grey) were obtained in Ref. [6] by averaging the results of six interaction models. The known proton-rich even-even nuclei are denoted by yellow squares, stable even-even nuclei are denoted by black squares, and parts of the known 2p emitters are denoted by stars. The current experimental reach for even-Z nuclei (including odd-A systems) [140] is indicated by a dotted line. The average lines N_av(Z) of 2p emission for the diproton model (dashed line) and direct-decay model (dash-dotted line) are shown. The energetic condition for the true 2p decay is illustrated in the inset. See Ref. [21, 22] for details.

In Ref. [21, 22], the 2p decay candidates were selected according to the decay energy criteria Q_2p > 0 and Qp < 0. In this case, single-proton or sequential decay are supposed to be energetically forbidden for medium-mass nuclides (see the inset of Fig. 11). Meanwhile, the decay half-lives are estimated via both direct [19] and diproton decay [56, 57] models and compared with those of alpha decay.

As a result, this 2p decay mode was shown to not represent an isolated phenomenon but rather a typical feature for proton-unbound isotopes with even atomic numbers. Almost all elements between argon and lead can have 2p-decaying isotopes (see Fig. 12); in this range, two regions are most promising for experimental observation: one extends from germanium to krypton and the other is located just above tin. For those nuclei with Z ≤ 82, α decay begins to dominate.

Fig. 12Full-screen View

(Color online) The predictions for the (a) direct-decay model and (b) diproton model for the ground-state 2p radioactivity. For each value of Z ≥ 18, neutron numbers N of predicted proton emitters are shown in comparison with the average 2p dripline of Ref. [6], as shown in Fig. 11. The model multiplicity m(Z,N) is indicated by the legend. The candidates for competing 2p and α decay are denoted by stars. The current experimental reach of Fig. 11 is denoted by the dotted line. See Ref. [21, 22] for details.

Prediction with uncertainty quantification. — To identify the uncertainty in the different theoretical models, and to provide quantitative predictions for further experimental studies, Bayesian methodology has been adopted to combine several Skyrme and Gogny energy density functionals [46]. In the framework of Ref. [46], predictions were obtained for each model 𝒨k using Bayesian Gaussian processes trained upon separation-energy residuals with respect to the experimental data [141, 142] and combined via Bayesian model averaging. The corresponding posterior weights conditional to the data y are given by [143-145] $p({\mathcal{M}}_k|y)=\frac{p(y|{\mathcal{M}}_k)\pi ({\mathcal{M}}_k)}{{\displaystyle {\sum }_{\mathcal{l}=1}^{K}p(y|{\mathcal{M}}_{\mathcal{l}})\pi ({\mathcal{M}}_{\mathcal{l}})}},$ (1) where π($\mathcal{M}$k) are the prior weights and p(y|$\mathcal{M}$k) denotes the evidence (integrals) obtained by integrating the likelihood over the parameter space.

Within this methodology, a good agreement was obtained between the averaged predictions of statistically corrected models and experiments [7]. In particular, for this 2p decay topic, Ref. [46] provides the quantified model results for 1p and 2p separation energies and the derived probabilities for proton emission (see Fig. 13); this indicates promising candidates for 2p decay and will be tested using experimental data from rare-isotope facilities.

Fig. 13Full-screen View

(Color online) Probability of true 2p emission for even-Z proton-rich isotopes. BMA-I and BMA-II are the two strategies used to determine model weights [46]. The color indicates the posterior probability of 2p emission (i.e., given that Q_2p > 0 and Q_1p < 0) according to the posterior average models. For each proton number, the relative neutron number N₀(Z)-N is shown, where N₀(Z) is the neutron number of the lightest proton-bound isotope for which an experimental 2p separation energy value is available. The dotted line denotes the predicted dripline (corresponding to pex = 0.5). The observed nuclei are marked by stars (⁴⁵Fe, ⁴⁸Ni, ⁵⁴Zn, and ⁶⁷Kr are denoted by closed stars); those within FRIB’s experimental reach are denoted by dots. See Ref. [46] for details.

Towards a comprehensive description of three-body decay

One of the most fundamental problems of 2p emission is the decay mechanism. In contrast to sequential decay, in which the radioactive process occurs via the intermediate state of the neighboring nucleus, a true (direct) 2p decay occurs when two protons are emitted simultaneously from the mother nucleus. As shown above, to determine whether a nucleus with charge number Z and mass number A produces a true 2p decay, one commonly used selection method (also introduced by Goldansky [15]) is to use the decay energy criteria Q_2p > 0 and Qp < 0. This method is effective because the only information needed is the nuclear binding energies. However, the situation is considerably more complicated for realistic cases, as discussed in Ref. [17, 18, 123, 146] (see Fig. 14).

Fig. 14Full-screen View

Energy conditions for different modes of 2p emission: (a) typical situation for decays of excited states (both 1p and 2p decays are possible), (b) sequential decay via narrow intermediate resonance, and (c) true 2p decays. Cases (d) and (e) indicate “democratic” decays, in which the neighboring nucleus has a relatively large decay width. The gray dotted arrows in (c) and (d) indicate the “decay path” through the states available only as virtual excitations. See Ref. [18] for details.

The energy criterion might not be very strict for 2p decays originating from the excited states or in the light-mass region. For the former case [as shown in Fig. 14(a)], the decay energy is typically quite large; in such instances, numerous decay channels are permitted and the decay mechanism can only be roughly estimated using the theoretical structure information or angular momentum transition. For the latter case shown in Fig. 14(d,e), the mother and neighboring nuclei may have large decay widths owing to the small Coulomb barrier [see Fig. 14(d,e)]. Consequently, both direct and sequential decay processes are allowed; this is often referred to as “democratic decay” [18, 24]. ⁶Be and the recently discovered ^11,12O are two such cases [50, 100, 147]. Moreover, it has been argued that, even though the single-proton decay is energetically forbidden, the mother nuclei can be still be influenced by the intermediate state of the neighboring nuclei via a virtual excitation [18].

All of these results indicate that the 2p emitter strongly depends on the properties of its neighboring and daughter nuclei, and each one must be studied carefully to provide a comprehensive and accurate description. Those direct or diproton models (which treat two valence protons as a pair for simplicity) are useful for systematical calculations but might not be very helpful for understanding the 2p mechanism. This will also introduce large uncertainties in the decay property studies. This complicated 2p process is a consequence of the interplay between the internal nuclear structure, asymptotic behavior, and continuum; hence, these effects must be precisely treated.

Moreover, interest in this exotic decay process has been invigorated by proton-proton correlation measurements following the decay of ⁴⁵Fe [148], ¹⁹Mg [101], and ⁴⁸Ni [71]; these have demonstrated the unique three-body features of the process and (in terms of theory) the prediction sensitivity for the angular momentum decomposition of the 2p wave function. The high-quality 2p decay data have necessitated the development of comprehensive theoretical approaches that can handle the simultaneous description of structural and reaction aspects of the problem [17, 18].

Configuration interaction. — As one of the most successful models in nuclear theory, configuration interaction (CI) (also known as the interacting shell model) is widely applied in spectroscopic studies across the nuclear landscape. Applying CI to study 2p decay can help us to better understand the internal structure information and its impact upon the 2p decay width.

The regular CI model, because of its harmonic-oscillator single-particle basis, cannot capture the continuum effect and three-body asymptotic behaviors in the presence of long-range Coulomb interactions. The former phenomena can be well incorporated into the framework of the shell model embedded in the continuum (SMEC) [149, 150], which has been successfully applied to describe the 2p decay from the $1_2^{-}$ state in ¹⁸Ne and other ground-state 2p emitters (see Fig. 15 and Refs. [151, 152]). Regarding the latter phenomena, the contribution of the decay width from the three-body dynamics has been estimated using a hybrid model [153].

Fig. 15Full-screen View

(a) The calculated half-life for sequential 2p decay from the ground-state J^π = 3/2⁺ in ⁴⁵Fe as a function of Qp (circle), as well as the half-life for 1p decay (squares). The dashed-dotted line shows the half-life for the diproton decay. The results were obtained via SMEC calculations. (b) The measured half-life for the ground-state of ⁴⁵Fe compared with different theoretical calculations. See Refs. [18, 152] for details.

In this hybrid framework (as shown in Ref. [153]), we can investigate the spectroscopic amplitude via the two-nucleon decay amplitudes (TNAs) for the removal of two protons from the initial state $|A{\omega }^{\text{'}}{J}^{\text{'}}\rangle$, leaving it in the final state $\langle (A-2)\omega J|$ given by the reduced matrix element, as $\text{TNA}\left(q_a\mathrm{,}{q}_b\right)=\frac{\langle (A-2)\omega J\left|\right|{\left[{\tilde{a}}_{q_a}\otimes {\tilde{a}}_{q_b}\right]}^{J_o}\left|\right|A{\omega }^{\text{'}}{J}^{\text{'}}\rangle }{\sqrt{\left(1+{\delta }_{q_a{q}_b}\right)(2J+1)}}\mathrm{,}$ (2) where ${\tilde{a}}_q$ is an operator that destroys a proton in the orbital q, q denotes the (n,𝒷, j) quantum numbers of the transferred proton, and ω contains other quantum numbers. For simplicity, only the case with J′ = J = Jo = 0 and qa = qb = q has been considered. Notably, this spin-singlet nucleon pair might not be spatially close. The corresponding 2p decay width is estimated in two extreme conditions. One assumes that the decay for each orbital q is not correlated with the others. Consequently, the 2p decay width can be calculated from an incoherent sum: ${\text{Γ}}_i\left(Q_{2p}\right)={\displaystyle \sum _q}{\text{Γ}}_s\left(Q_{2p}\mathrm{,}{\mathcal{l}}^2\right){\left[\text{TNA}\left(q^2\right)\right]}^2\mathrm{,}$ (3) where Γs is the partial decay width for a single channel. The other extreme situation is where all amplitudes combine coherently as in two-nucleon transfer reactions. This gives ${\text{Γ}}_c\left(Q_{2p}\right)={\left[{\displaystyle \sum _q}\sqrt{{\text{Γ}}_s\left(Q_{2p}\mathrm{,}{\mathcal{l}}^2\right)}\text{TNA}\left(q^2\right)\right]}^2\mathrm{,}$ (4) in which the phase is taken as positive for all terms.

As shown in Table 2, the calculated decay widths incorporating three-body dynamics are substantially improved and agree well with the experimental data. Meanwhile, as discussed in Ref. [153], it is important to include small s² components in the decay. These components with low-𝒷 orbitals are crucial for the decay process (owing to their small centrifugal barriers) and are often introduced via the continuum coupling or core/cross-shell excitations. Moreover, the half-lives of ⁶⁷Kr retain a large discrepancy between the experimental data and theoretical predictions [153]. As a recently observed 2p emitter, the half-life of ⁶⁷Kr is systematically over-estimated by various theoretical models [43, 154]. It seems that such a short lifetime can only be reproduced by using a large amount of s- or p-wave components [153, 155], which indicates that deformation effects or certain other factors might be missing from the current theoretical or experimental frameworks [49, 153].

Three-body model. — To study the configuration and correlations of valence protons, a theoretical model was developed from a different basis: the three-body nature of 2p decay. In these few-body frameworks, a two-particle emitter can be viewed as a three-body system, comprising a core (c) representing the daughter nucleus and two emitted nucleons (n₁ and n₂). The i-th cluster (i=c,n₁,n₂) contains the position vector ri and linear momentum ki [31, 123, 126, 156-159].

Two types of coordinates are commonly used to construct the three-body framework. As shown in Fig. 16, one is the cluster-orbital shell model (COSM) [160] and the other is Jacobi coordinates [126]. The former uses single-particle coordinates (rn - rc) with a recoil term to handle the extra energy introduced by center-of-mass (c.m.) motion [160, 161]. In this framework, it is easy to calculate matrix elements and extend them to many-body systems. However, because all the coordinates of the valence nucleons are measured with respect to the core, we must be very careful when treating the asymptotic observables. The latter coordinates are also known as the relative coordinates; they are expressed as $\begin{array}{ll}\text{x}\hfill & =\sqrt{{\mu }_{\text{x}}}(\text{r}{}_{i_1}-\text{r}{}_{i_2})\mathrm{,}\hfill \\ \text{y}\hfill & =\sqrt{{\mu }_y}\left(\frac{A_{i_1}\text{r}{}_{i_1}+A_{i_2}\text{r}{}_{i_2}}{A_{i_1}+A_{i_2}}-\text{r}{}_{i_3}\right)\mathrm{,}\hfill \\ \hfill & \hfill \end{array}$ (5) where i₁=n₁, i₂=n₂, i₃=c for T-coordinates and i₁=n₂, i₂=c, i₃=n₁ for Y-coordinates, see Fig. 16. In Eq. (5) Ai is the i-th cluster mass number, and ${\mu }_{\text{x}}=\frac{A_{i_1}{A}_{i_2}}{A_{i_1}+A_{i_2}}$ and ${\mu }_y=\frac{(A_{i_1}+A_{i_2})A_{i_3}}{A_{i_1}+A_{i_2}+A_{i_3}}$ are the reduced masses associated with x and y, respectively. The Jacobi coordinates automatically eliminate c.m. motion and allow for the exact treatment of the asymptotic wave functions. Hence, it is widely used in nuclear reactions and in the description of other asymptotic properties. However, the complicated coupling/transformation coefficient and anti-symmetrization prevent it from being extended to larger fermionic systems.

Fig. 16Full-screen View

(Color online) Schematic diagram for different coordinates of a core + nucleon + nucleon system: (a) Jacobi T-type, Y-type; (b) cluster-orbital shell model coordinates; and (c) the corresponding momentum scheme. A is the mass number, and k₁, k₂, and kc are the momenta of the two nucleons and core, respectively, in the center-of-mass coordinate frame.

The benchmarking between COSM and Jacobi coordinates has been performed in Ref. [162], in which both weakly bound and unbound systems (⁶He, ⁶Li, ⁶Be, and ²⁶O) were investigated using the Berggren ensemble technique [163]. Consequently, the COSM-based Gamow shell model (GSM) and Jacobi-based Gamow coupled-channel (GCC) method gave practically identical results for the spectra, decay widths, and coordinate-space angular distributions of weakly bound and resonant nuclear states [162]. Furthermore, it has been shown that the Jacobi coordinates capture cluster correlations (e.g., dineutron and deuteron-type correlations) more efficiently.

The advantage of the three-body model is that it can capture the configuration of valence protons and the asymptotic correlations of the two-nucleon decay. For example, as a typical light-mass 2p emitter, ⁶Be has been considered to feature a “democratic” decay mode, attributable to the large width of the ground state of its neighboring nucleus ⁵Li (Qp = 1.97 MeV, Γ = 1.23 MeV). The density distribution of ⁶Be has been studied in various few-body models; it shows two maxima associated with diproton and cigarlike configurations (see Fig. 17). This accords with the angular densities obtained by the three-body model [31, 162] and GSM [164]. This angular density can be calculated using $\rho \left(\theta \right)={\displaystyle \int }\langle \Psi \left|\delta \left(r_1-r\right)\delta \left(r_2-r^{\text{'}}\right)\delta \left({\theta }_{12}-\theta \right)\right|\Psi \rangle \text{d}r\text{d}{r}^{\text{'}}\mathrm{,}$ (6) which primarily represents the average opening angle of the two valence protons inside the nucleus. Notably, the angular density is defined in coordinate space; it cannot be directly observed in experiments.

Fig. 17Full-screen View

(Color online) (a) Calculated 2p density distribution (marked by contours) and 2p flux (denoted by arrows) in the ground state of ⁶Be in Jacobi coordinates pp and core-pp. The thick dashed line marks the inner turning point of the Coulomb-plus-centrifugal barrier. The maxima marked by filled and open stars correspond to diproton and cigarlike structures, respectively. (b) Two-nucleon angular densities (total and spin-triplet S=1 channels) in the ground-state configurations of ⁶Be, as obtained in GCC and GSM. See Refs. [51, 162] for details.

Instead, we can look at the configuration evolution of the internal structure during decay by using the flux current $j=\text{I}m({\Psi }^{†}\nabla \Psi )\hslash /m$, which shows how the two valence protons evolve within a given state wave function Ψ. In the case of ⁶Be (as shown in Fig. 17), a competition between diproton and cigarlike configurations occurs inside the inner turning point of the Coulomb-plus-centrifugal barrier associated with the core-proton potential [51]. Near the origin, the dominant diproton configuration tends to evolve toward the cigarlike configuration, owing to the repulsive Coulomb interaction and the Pauli principle. On the other hand, near the surface, the direction of the flux extends from the cigarlike maximum toward the diproton maximum, tunnelling through the barrier. Moreover, at the peak of the diproton configuration (located near the barrier), the direction of the flux is almost aligned with the core-2p axis, indicating a clear diproton-like decay. Beyond the potential barrier, the two emitted protons tend to gradually separate under the repulsive Coulomb interaction. The behavior of the two protons below the barrier can be understood in terms of the influence of pairing, which favors low angular momentum amplitudes; hence, it effectively lowers the centrifugal barrier and increases the probability that the two protons decay by tunnelling [123, 124, 162, 165].

The three-body model also successfully reproduced the asymptotic pp correlations observed in experiments. This nucleon-nucleon correlation indicates the relative energy and opening angle of the emitted protons in the asymptotic region. To elucidate the characteristics of nucleon-nucleon correlation, it is convenient to introduce the relative momenta: $\begin{array}{ll}\text{k}{}_{\text{x}}\hfill & ={\mu }_{\text{x}}\left(\frac{\text{k}{}_{i_1}}{A_{i_1}}-\frac{\text{k}{}_{i_2}}{A_{i_2}}\right)\mathrm{,}\hfill \\ \text{k}{}_y\hfill & ={\mu }_y\left(\frac{\text{k}{}_{i_1}+\text{k}{}_{i_2}}{A_{i_1}+A_{i_2}}-\frac{\text{k}{}_{i_3}}{A_{i_3}}\right)\mathrm{.}\hfill \\ \hfill & \hfill \end{array}$ (7) No c.m. motion occurs; hence, it is easy to see that ${\displaystyle {\sum }_i}\text{k}{}_i=0$, and ky is aligned in the opposite direction to $k{}_{i_3}$. θk and ${{\theta }^{\prime }}_k$ denote the opening angles of (kx, ky) in Jacobi-T and -Y coordinates, respectively (see Fig. 16). The kinetic energy of the relative motions of the emitted nucleons is given by $E_{pp/nn}=\frac{{\hslash }^2{k}_{\text{x}}^2}{2{\mu }_{\text{x}}}$, and E_core-p/n denotes that of the core-nucleon pair. Therefore, the T-type (θk, Epp/nn) and Y-type (${{\theta }^{\prime }}_k$, E_core-p/n) distributions reveal the nucleon-nucleon correlation and structural information regarding the mother nucleus. Finally, the total momentum k is defined as $\sqrt{\frac{k_{\text{x}}^2}{{\mu }_{\text{x}}}+\frac{k_y^2}{{\mu }_y}}$, which at later times approaches the limit $\frac{\sqrt{2m{Q}_{2p/2n}}}{\hslash }$, where Q_2p/2n is the two-nucleon decay energy given by the binding energy difference between parent and daughter nuclei.

The nucleon-nucleon correlation is important not only for the experimental accessibility but also for the valuable information it carries. For instance, the study of short-range correlations can be used to analyze the fundamental structures of nucleonic pairs at very high relative momenta [166]. Meanwhile, the long-range correlations of 2p decay reveal ground-state properties and the pairing effects of atomic nuclei. Moreover, the decay dynamics and long-range correlations are strongly influenced by both initial-state and final-state interactions [167-169]; this allows us to gain more insight into the connection between the internal structure and decay properties, including asymptotic correlations.

As shown in Fig. 18, the asymptotic nucleon-nucleon correlations of ⁶Be, ⁴⁵Fe, and several other 2p emitters have been reproduced or predicted using the three-body model [18, 122]. Moreover, from these correlations, one can roughly determine how the valence protons are emitted, as well as the corresponding configurations. For example, in the case of ⁴⁵Fe, the small-angle emission dominates in the asymptotic region, which may correspond to a diproton decay. Meanwhile, the situation in ⁶Be is considerably more complicated, with numerous possible configurations. However, because the emitted protons are influenced by the centrifugal barrier and long-range Coulomb interactions, one must be careful that these configurations properly reflect the circumstances in the asymptotic region, which differ from those inside the nucleus. To better understand how the inner structure evolves into the asymptotic configurations and nucleon-nucleon correlations, several time-dependent frameworks have been developed (see the discussions below and Refs. [124, 125] for details).

Fig. 18Full-screen View

(Color online) Asymptotic nucleon-nucleon correlation for the ground-state 2p decay of (a) ⁶Be and (b) ⁴⁵Fe, represented in Jacobi-T and -Y coordinates (left and right columns, respectively). ϵ denotes the energy ratio between the subsystem and total Q_2p, θk is the momentum-space opening angle in Jacobi coordinates. The experimental data are also shown. See Ref. [122] for details.

One main drawback of the regular three-body model is the lack of structural information regarding the daughter nucleus, which is usually treated as a frozen core. To this end, several efforts have been made towards a more microscopic description of the core wave function [171-174]. For 2p emission, the GCC method has been extended to a deformed case [162, 170], which allows a pair of nucleons to couple to the collective states of the core via nonadiabatic coupling. Consequently, the total wave function of the parent nucleus can be written as ${\Psi }^{J\pi }={\displaystyle {\sum }_{J_p{\pi }_p{j}_c{\pi }_c}}{\left[{\Phi }^{J_p{\pi }_p}\otimes {\varphi }^{j_c{\pi }_c}\right]}^{J\pi }$, where ${\Phi }^{J_p{\pi }_p}$ and ${\varphi }^{j_c{\pi }_c}$ are the wave functions of the two valence protons and core, respectively. The wave function of the valence protons ${\Phi }^{J_p{\pi }_p}$ is expressed in Jacobi coordinates and expanded using the Berggren basis [163, 175], which is defined in the complex-momentum k space. Because the Berggren basis is a complete ensemble that includes bound, Gamow, and scattering states, it provides the correct outgoing asymptotic behavior to describe the 2p decay, and it effectively allows nuclear structures and reactions to be treated on the same footing.

As discussed above, the lifetime of ⁶⁷Kr can be influenced by deformation effects [49]. Indeed, studies of single-proton (1p) emitters [172-174, 176-181] have demonstrated the impact of rotational and vibrational couplings on 1p half-lives. The corresponding influence in the 2p decay of ⁶⁷Kr has been studied using the deformed GCC method [170]. Figure 19(a) shows the proton Nilsson levels (labeled with asymptotic quantum numbers Ω[Nn_zΛ]) of the Woods-Saxon core-p potential. When the deformation of the core increases, a noticeable oblate gap at Z=36 opens up, attributable to the down-sloping 9/2[404] Nilsson level originating from the 0g_9/2 shell. This gap is responsible for the oblate ground-state shapes of proton-deficient Kr isotopes [182-184]. The structure of the valence proton orbital changes from the 9/2[404] (𝒷=4) state at smaller oblate deformations to the 1/2[321] orbital, which has a large 𝒷=1 component. This transition can dramatically change the centrifugal barrier and decay properties of ⁶⁷Kr. Figure 19(b) shows the 2p decay width predicted in the two limits of the rotational model: (i) the 1/2[321] level belongs to the core, and the valence protons primarily occupy the 9/2[404] level; (ii) the valence protons primarily occupy the 1/2[321] level. In reality, the core is not rigid; hence, proton pairing is expected to produce a diffused Fermi surface, and the transition from (i) to (ii) becomes gradual, as indicated by the shaded area in Fig. 19(b). The decreasing 𝒷 content of the 2p wave function results in a dramatic increase in the decay width. At the deformation ${\beta }_2\approx -0.3$, the calculated 2p ground-state half-life of ⁶⁷Kr is ${24}_{-7}^{+10}$ ms [170], which agrees with experimental results [49]. In future quantitative studies, a more microscopic description of the core might be needed for the three-body model. Furthermore, high-statistics data are expected to improve our understanding of 2p radioactivity.

Fig. 19Full-screen View

(Color online) Top: Nilsson levels Ω[Nn_zΛ] of the deformed core-p potential as functions of the oblate quadrupole deformation β₂ of the core. The dotted line indicates the valence level primarily occupied by the two valence protons. Bottom: Decay width (half-life) for the 2p ground-state radioactivity of ⁶⁷Kr. The solid and dashed lines denote the results within the rotational and vibrational coupling, respectively. The rotational-coupling calculations were performed by assuming that the 1/2[321] orbital is either occupied by the core (9/2[404]-valence) or valence (1/2[321]-valence) protons. See Ref. [170] for details.

Time-dependent framework. — An alternative strategy for tackling the decay process is the time-dependent formalism, which allows a broad range of questions (e.g., configuration evolution [185], decay rate [186], and fission [187]) to be addressed in a precise and transparent way. In the case of two-nucleon decay, the measured inter-particle correlations can be interpreted in terms of solutions propagating for long periods.

An approximate treatment of 2p emission was proposed in Ref. [188], in which the c.m. motion for the two valence protons was described classically. In a more realistic case, the early stage of the 2p emission from the ground state of ⁶Be was well demonstrated using a time-dependent method in Refs. [124, 165]. Initially, (t=0) when the wave function is relatively localized inside the nucleus, the density distribution shows two maxima for ⁶Be; these are associated with the diproton/cigarlike configuration characterized by small/large relative distances between valence protons (see Fig. 17). Meanwhile, the different time snapshots of the density distribution ρ and the corresponding density changes ρd are shown in Fig. 20. ρ and ρd are defined as $\begin{array}{l}\rho =8{\pi }^2{r}_1^2{r}_2^2\sin {\theta }_{12}{\left|\text{Ψ}\left(t\right)\right|}^2;\\ {\rho }_d=8{\pi }^2{r}_1^2{r}_2^2\sin {\theta }_{12}{\left|{\text{Ψ}}_d\left(t\right)\right|}^2/\langle {\text{Ψ}}_d(t)|{\text{Ψ}}_d(t)\rangle ;\\ \left|{\text{Ψ}}_d(t)\rangle =\right|\text{Ψ}(t)\rangle -\langle \text{Ψ}(0)|\text{Ψ}(t)\rangle |\text{Ψ}(0)\rangle \mathrm{.}\end{array}$ (8) During the early stage of decay, two strong branches are emitted from the inner nucleus, as shown in Fig. 20. The primary branch corresponds to the protons emitted at small opening angles; this indicates that a diproton structure is present during the tunneling phase. This can be understood in terms of the nucleonic pairing, which favors low angular momentum amplitudes and therefore lowers the centrifugal barrier and increases the 2p tunneling probability [51, 123, 124, 165]. The secondary branch corresponds to protons emitted in opposite directions. This accords with the flux current analysis shown in Fig. 17.

Fig. 20Full-screen View

(Color online) (a) Time-dependent 2p density distribution ρ of the ground state as a function of rp-p and rc-pp. (b) The corresponding decaying density distribution ρ_d. See Ref. [165] for details.

To capture the asymptotic dynamics and improve our understanding of the role of final-state interactions, the wave function of ⁶Be has also been propagated to large distances (over 500 fm) and long times (up to 30 pm/c); this facilitates analysis of asymptotic observables including nucleon-nucleon correlation [125]. After tunneling through the Coulomb barrier, the two emitted protons tend to gradually separate under Coulomb repulsion [125, 165]. Eventually, the 2p density becomes spatially diffused, which is consistent with the broad angular distribution measured in Ref. [98]. This corresponds to an opposite trend for the momentum distribution of the wave function, in which the emitted nucleons move with a well-defined total momentum, as indicated by a narrow resonance peak at long times [189]. Figure 21 illustrates the dramatic changes in the wave function and configurations of ⁶Be during the decay process. The gradual transition from the broad to narrow momentum distribution exhibits a pronounced interference pattern, which is universal for two-nucleon decays and is governed by Fermi’s golden rule. The interference frequencies, as indicated by dotted lines in Fig. 21, can be approximated by $\left(\frac{{\hslash }^2}{2m}{k}^2-Q_{2p}\right)t=\mathfrak{n}\pi \hslash$, where n = 1, 3, 5 ⋯ (i.e., they explicitly depend on the Q_2p energy).

Fig. 21Full-screen View

(Color online) Time evolution of the wave functions of ⁶Be. Configurations are labeled as $(K\mathrm{,}{\mathcal{l}}_{\text{x}}\mathrm{,}{\mathcal{l}}_y\mathrm{,}S)$ in Jacobi-T coordinates. K is the hyper-spherical quantum number, 𝒷 is the orbital angular momentum in Jacobi coordinate, and S is the total spin of valence protons. The projected contour map represents the sum of all configurations in momentum space; the interference frequencies are denoted by dotted lines corresponding to different n-values. See Ref. [125] for details.

Moreover, the configuration evolution also reveals a unique feature of three-body decay. As seen in Fig. 21, the initial ground state of ⁶Be is dominated by the p-wave (𝒷 = 1) components, and the small s-wave (𝒷 = 0) component originates from the non-resonant continuum. As the system evolves, the weight of the s-wave component—approximately corresponding to the Jacobi-T coordinate configuration $(K\mathrm{,}{\mathcal{l}}_{\text{x}}\mathrm{,}{\mathcal{l}}_y\mathrm{,}S)$ = (0,0,0,0)—gradually increases and eventually dominates, because it experiences no centrifugal barrier. Such transitions can also be revealed by comparing the internal and external configurations obtained via time-independent calculations [154]. This behavior can never occur in the single-nucleon decay, owing to the conservation of orbital angular momentum; however, it occurs in the two-nucleon decay because correlated di-nucleons involve components with different 𝒷-values [29, 30, 33, 35]. In addition, for 2p decays, the Coulomb potential and kinetic energy do not commute in the asymptotic region [122], leading to additional configuration mixing.

It has been also found that the 2p decay width is sensitive to the strength of the pairing interaction [125, 165, 191], because the diproton structure promotes the tunneling process. Moreover, Figure. 22 shows the asymptotic correlations of ⁶Be as a function of the nucleon-nucleon interaction strength Vpp/nn [125]. The obtained results have been also benchmarked with the Green’s function method, because the time evolution operator can be written as the Fourier transform of $\widehat{G}=(E-\widehat{H}+i\eta {)}^{-1}$: $e^{-i\frac{\widehat{H}}{\hslash }t}=\frac{e^{\frac{\eta }{\hslash }t}}{2\pi i}\mathcal{F}\left(\widehat{G}\mathrm{,}E\to \frac{t}{2\pi \hslash }\right)\mathrm{.}$ (9) As shown in Fig. 22, the attractive nuclear force is not only responsible for the presence of correlated di-nucleons in the initial state but also significantly influences the asymptotic energy correlations and angular correlations in the Jacobi-Y angle θk. This indicates that, even though the initial-state correlations are largely lost in the final state, certain fingerprints of the di-nucleon structure can still manifest themselves in the asymptotic observables.

Fig. 22Full-screen View

(Color online) (a) Asymptotic energy and (b) angular correlations of nucleons emitted from the ground state of ⁶Be (top) and ⁶He′ (bottom), as calculated at t=15 pm/c with different strengths of the Minnesota interaction [190]: standard (solid line), strong (increased by 50%; dashed line), and weak (decreased by 50%; dash-dotted line). Also shown are the benchmarking results obtained by the Green’s function method (GF; dotted line) using the standard interaction strength. θk is the opening angle between kx and ky in the Jacobi-Y coordinate system, and Epp/nn is the kinetic energy for the relative motions of the emitted nucleons. See Fig. 16 and Ref. [125] for definitions and details.

Connecting structure with asymptotic correlation

As shown above, the asymptotic nucleon-nucleon correlation and other decay properties are strongly determined by the nuclear structure. Therefore, deciphering these observables can help us to understand the nucleus interior and the properties of nuclear forces. In particular, the study of short-range correlations reveals the fundamental structure of nucleonic pairs at very high relative momenta [166, 192-194]. Regarding the 2p decay, this observed long-range correlation corresponds to a low-momentum phenomenon, which indicates the interplay between the diproton condensate inside the nucleus and the Cooper pairing during decay [10, 31].

Although these asymptotic observables can be evaluated using various kinds of theoretical models, the precise connection between nuclear structure (including pairing and continuum effects) is still largely unknown. One reason for this is the complicated open quantum nature of this three-body system, the other is the very different scales. The asymptotic nucleon-nucleon correlation and other observables are distorted during the decay process. This situation is exacerbated by 2p decay in the presence of long-range Coulomb interactions.

In one pioneering work, ⁴⁵Fe was—by analyzing the correlation of the emitted valence protons—found to contain a moderate amount of the p² component, [18, 148]. Later, the nucleon-nucleon correlations of psd-shell 2p emitters were systematically studied. As shown in Fig. 23, among these 2p emitters, the energy correlations of ¹²O and its isobaric analog ¹²N_IAS resemble those of ¹⁶Ne but differ notably from the energy correlations of ⁶Be and ⁸B_IAS. This indicates that the valence protons of ¹²O might have more sd-shell components, even though, in the naive shell-model picture, these valence protons fully occupy the p-wave orbitals. Consequently, these qualitative analyses provide us with useful information about the valence protons and corresponding 2p emitters. Moreover, to gain deep insight into the connection and extract structural information from these asymptotic observables, high-quality experimental data and comprehensive theoretical models are required.

Fig. 23Full-screen View

(Color online) Energy correlations of different 2p emitters in (a) Jacobi-T and (b) -Y coordinates. Ground-state 2p emitters are plotted as histograms whilst 2p emissions from isobaric analogs are plotted as data points. See Ref. [100] for details.

Excited 2p decay

As discussed, the phenomenon of 2p decay is not limited to the ground state: more and more excited nuclei have been found to exhibit 2p decay. Unlike ground-state 2p emitters, the 2p decay from excited states usually has a large decay energy Q_2p. This may lead to dense level density and more possible decay channels, which make it hard to determine the decay path and corresponding mechanism. Moreover, those high-lying states with large decay energies lie beyond the capabilities of many microscopic theories, in which the nuclear structure cannot be precisely described. So far, most tools in the market have dealt with this problem via assumptions and simulations. No self-consistent framework has yet been found that can be applied to comprehensively study these excited 2p emitters.

Meanwhile, nuclei can exhibit β⁺-delayed 2p emission. For instance, one such decay process was observed from the ground state of ²²Al [76]. The decay proceeds through the isobaric analogue state (IAS) of ²²Mg to the first excited state in ²⁰Ne, and it is confirmed by proton-gamma coincidences. This poses a big challenge for theoretical studies; this is because, even though protons and neutrons have tiny differences attributable to their inner structures and the Coulomb interaction, they are treated as identical particles in most theoretical models, by assuming isospin-symmetry conservation. However, this transition is considered to be forbidden by the isospins, because the 2p decay cannot change the isospin from T = 2 in the case of ²²Al and ²²Mg_IAS to T = 0 in the case of ²⁰Ne. This indicates that isospin mixing is crucial in such β⁺-delayed 2p emission processes; this requires further theoretical and experimental investigations.

Beyond the proton/neutron dripline, exotic decays may happen. 2p decay is just one of them. Recently, more and more exotic particle emissions (e.g., 3p or 4p decay) have been experimentally observed. The ground state of ³¹K has been found to exhibit a 3p decay [195]. Moreover, the newly discovered elements ⁸C and ¹⁸Mg are considered to exhibit 4p emissions [97, 99]. Owing to the large imbalance of the proton-neutron ratio, these 3p or 4p emitters are strongly coupled to the continuum and are more unstable compared to the 2p emitters. The decay mechanism of these multi-particle emissions is an interesting phenomena to investigate. As shown in Fig. 24(b), ⁸C and ¹⁸Mg arguably feature a 2p+2p decay mode, in which the intermediate states are the ground states of ⁶Be and ¹⁶Ne, respectively. However, owing to the low-statistic experimental data, their decay properties have not yet been fully determined. To address this problem, more physical observables are required.

Fig. 24Full-screen View

(Color online) (a) Subsection of the nuclear chart for multi-proton emission. These highlighted nuclei have been observed to decay via 1p (green), 2p (blue), 3p (purple), and 4p (pink) emissions. (b) The level diagrams for ⁸C and its isobaric analog ⁸B. The diagram for ⁸C is shifted up for display purposes. The full width at half maximum of the levels is indicated by the hatched regions. The isospin-permitted decays are presented in color. See Refs. [97, 99] for details.

Multi-proton radioactivity

Multi-neutron emission and mirror symmetry breaking

As the fermionic building blocks of a nucleus, the positively-charged proton and neutral neutron are almost identical in all respects except for their electric charge. This is a consequence of the isospin symmetry [196, 197], which is weakly broken in atomic nuclei, primarily by the Coulomb interaction. The interplay between the short-ranged nuclear force and long-range electromagnetic force can be studied by investigating 2p and two-neutron radioactivity [15, 17, 18, 125]. However, the neutron dripline is harder to reach and detecting neutrons is more challenging than detecting charged particles; hence, the experimental information regarding two-neutron (2n) decay is limited.

So far, the weakly neutron-unbound ¹⁶Be and ²⁶O are arguably the best current candidates for observing the phenomenon of 2n radioactivity [33, 35, 198-204]. Beryllium isotopes usually suffer large deformations, which might influence the 2n decay process of ¹⁶Be. As for ²⁶O, its ground state is very near to the threshold, which might result in some unique phenomena.

Inside the nucleus, the initial 2n density of ²⁶O also contains a dineutron, cigarlike, and triangular structure [33, 162]. These three configurations are characteristic of the d-wave component. The very small Q_2n=18± 5 keV value [199] in ²⁶O makes this nucleus a candidate for 2n radioactivity. When approaching the threshold, the presence of the centrifugal barrier is expected to lead to changes in the asymptotic correlations. As shown in Fig. 25, the angular correlation of ²⁶O becomes almost uniformly distributed. This is because, for small values of Q_2n, the 2n decay is dominated by the s-wave component, and the angular distribution becomes essentially isotropic [128, 154, 203]. This asymptotic behavior might represent an interesting avenue of research for further experimental studies.

Fig. 25Full-screen View

(Color online) The coordinate-space angular density of the ground state of ²⁶O as a functional two-neutron decay energy E. The upper panel is obtained by varying the energy (by changing the pairing interaction of the two valence neutrons). The lower panel is obtained by shifting the resonance energy of the d_3/2 state in ²⁵O, keeping the strength of the pairing interaction identical. See Ref. [158] for details.

When extending further beyond the neutron dripline, ²⁸O is predicted to be a 4n emitter [204-206]. It has been found recently that the four neutrons can exist transiently without any other matter [207]. Although it remains under debate whether the observed peak is a resonance [208-214], it does show several correlations in the presence of nuclear media, which constitutes an interesting feature to be further studied in ²⁸O and similarly proton-rich systems.