Recent progress in two-proton radioactivity

Recent progress in two-proton radioactivity 1 2 first-author 1 2 corresp wangsimin@fudan.edu.cn wangsimin@fudan.edu.cn wangsimin@fudan.edu.cn https://orcid.org/0000-0002-8902-6842 1 2 corresp dqfang@fudan.edu.cn dqfang@fudan.edu.cn dqfang@fudan.edu.cn https://orcid.org/0000-0002-6123-3014 1 2 corresp mayugang@fudan.edu.cn mayugang@fudan.edu.cn mayugang@fudan.edu.cn https://orcid.org/0000-0002-0233-9900 Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai 200438, China Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai 200438, China During the last few decades, rare isotope beam facilities have provided unique data for studying the properties of nuclides located far from the beta-stability line. Such nuclei are often accompanied by exotic structures and radioactive modes, which represent the forefront of nuclear research. Among them, two-proton (2 p ) radioactivity is a rare decay mode found in a few highly proton-rich isotopes. The 2 p decay lifetimes and properties of emitted protons hold invaluable information regarding the nuclear structures in the presence of a low-lying proton continuum; as such, they have attracted considerable research attention. In this review, we present some of the recent experimental and theoretical progress regarding the 2 p decay, including technical innovations for measuring nucleon-nucleon correlations and developments in the models that connect their structural aspects with their decay properties. This impressive progress should play a significant role in elucidating the mechanism of these exotic decays, probing the corresponding components inside nuclei, and providing deep insights into the open quantum nature of dripline systems. Exotic decay Two-proton radioactivity Nucleon-nucleon correlation Experimental and theoretical development Introduction 1 A unique laboratory for many-body quantum physics, nuclei are composed of positively-charged protons and neutral neutrons; they offer invaluable data for testing fundamental theories and studying the universal properties of fermionic systems [ 1 1 - 5 5 1-5 ]. Of around the 7000 atomic nuclei thought to exist [ 6 6 , 7 7 ], less than half have been experimentally observed; these include the 286 primordial nuclides located in the β -stability valley. Meanwhile, the rare isotopes that inhabit remote regions of the nuclear landscape (around and beyond the particle driplines) are often accompanied by exotic structures and decay modes, owing to the weak binding and strong continuum coupling. Such nuclei are also referred to as open quantum systems, and they represent the forefront of nuclear structure and reaction research [ 8 8 - 13 13 8-13 ]. Among the most fundamental properties of such exotic nuclear systems, the first to be established are—in general—the possible radioactive modes and corresponding lifetimes. When approaching the edge of nuclear stability, new decay modes arise, including the exotic two-proton (2 p ) radioactivity. The concept of 2 p decay is relatively old; it was proposed by Goldansky in 1960 [ 15 15 ]. In such decay cases, the single-proton emission is energetically forbidden or strongly suppressed by the inner structure and angular momentum selection rule [ 15 15 - 22 22 15-22 ]. Experimental studies into 2 p -decay began in the 1970s and initially focused upon light nuclei such as 6 Be [ 23 23 , 24 24 ] or 12 O [ 25 25 , 26 26 ]. However, owing to their short lifetimes and broad intermediate states, the decay mechanisms of these nuclei are not fully understood. The first direct observation of 2 p emission from a long-lived ground state was achieved for 45 Fe via projectile fragmentation and in-flight identification [ 27 27 , 28 28 ]. So far, researchers have identified only a very small number of nuclei that can decay by emitting two protons from their ground and/or excited states. The 2 p decay lifetimes and properties of emitted protons carry invaluable information for nuclear physics research. In contrast to β and γ decays (which are mainly governed by weak and electromagnetic interactions, respectively), 2 p decays arise from the competition between strong nuclear forces and electromagnetic interactions. As a result, the corresponding lifetimes span a wide range of time scales; this offers a valuable testing ground for exploring unified theories and elucidating the interplay between these two types of fundamental interactions. Moreover, the 2 p emission process, which typically involves one core and two protons, is a three-body process. In the presence of a low-lying continuum, this feature produces difficulties and increases the investigation complexity compared to the one-proton emission. In general, the 2 p decay mechanisms can be roughly divided into three types (see Fig. 1 Fig. 1 and Ref. [ 17 17 ]): (i) Diproton emission, an extreme case in which two strongly correlated protons are emitted in almost identical directions; here, the diproton emission is essentially two protons constrained by pair correlation in a quasi-bound s -singlet (i.e., 1 S 0 ) configuration; because of the Coulomb barrier, such quasi-bound states can only exist for a short while before becoming separated after penetrating through the barrier. (ii) Three-body (large-angle) simultaneous emission, in which the core is separated from two protons simultaneously, and the valence protons are positioned remotely in coordinate space and emitted with large opening angles. (iii) Two-body sequential emission, in which the mother nucleus first emits a proton to an intermediate state of the neighboring nucleus, which itself then emits another proton to the final state, as shown in Fig. 1(c) Fig. 1(c) . The sequential decay can be treated as a two-step 1 p decay; hence, the first two emission mechanisms typically attract more interest, owing to their three-body character in the presence of low-lying continuum. In this three-body process, the relative momenta and opening angles between the two emitted protons contain the specific form of the nucleon wave function as well as the interactions between nucleons; hence, such processes are useful for studying the structures, decay properties, and nucleon-nucleon pairing correlations (in particular, the pp correlation) of nuclei around the proton dripline [ 29 29 - 35 35 29-35 ]. In addition, these processes offer a good method for investigating the astro-nuclear (2 p,γ ) and ( γ ,2 p ) processes, which are closely related to the waiting point nuclei [ 36 36 ]. Meanwhile, the structures and reactions of certain 2 p emitting nuclei are very important for understanding the cycle processes of element synthesis (e.g., NeNa and NaMg cycles) in nuclear astro-reactions [ 37 37 ]. Therefore, such processes pose a unique opportunity and daunting challenge for both experimental and theoretical studies. Experimentally, the utility of implantation into an Si array increases the efficiency of 2 p emission observations [ 17 17 , 18 18 ], and the development of time-projection chambers [ 38 38 ] has allowed researchers to trace the decay properties and asymptotic correlations of the emitted valence protons. In addition, from these properties and information regarding the intermediate state, the structure and decay mechanisms of the mother nucleus can be investigated. Meanwhile, these high-quality 2 p decay data obtained from exotic beam facilities also require the development of comprehensive theoretical approaches capable of simultaneously describing the structural and reaction aspects of this exotic decay problem [ 17 17 , 18 18 ]. Because only a few 2 p emitters have been identified, the mechanisms and properties of this exotic decay are largely unknown. To search for 2 p emitter candidates, systematical calculations have been performed using various theoretical models [ 21 21 , 22 22 , 39 39 - 48 48 39-48 ]. In these models, most predictions are made by treating the internal and asymptotic regions of the nucleus separately; this is efficient for estimating certain bulk proprieties (e.g., binding energy) and for spectroscopy. However, these 2 p emitters are located across vast regions of the nuclear landscape; thus, they have very different structures and decay properties. Meanwhile, this complicated three-body process is a consequence of the interplay between the internal nuclear structure, asymptotic behavior, and continuum effect. Therefore, to give a comprehensive description of a decay process (in particular, for the mechanism and asymptotic nucleon-nucleon correlations originating from the internal nuclear structure and distorted by the long-range Coulomb interaction), fine-tuned studies are required for each individual system. This becomes especially challenging for 2 p decays because the Coulomb barrier strongly suppresses the wave function at large distances; this also makes the 2 p lifetime relatively sensitive to the low-𝒷 wave function components inside the nucleus. Over the past few decades, impressive progress has been made (in both experimental and theoretical studies) in these directions; much of this has been well reviewed in Refs. [ 17 17 , 18 18 ]. In this review, we present some of the recent research, including technical innovations for measuring nucleon-nucleon correlations and model developments for connecting structural aspects and decay properties. Meanwhile, numerous open questions (e.g., the 2 p decay mechanism and impacts of internal structure) remain under debate. Further studies may help provide deep insights into the 2 p decay process as well as other open quantum systems. Brief History 1 In the early 1960s, Goldansky suggested a novel radioactive mode of 1 p (one-proton) or 2 p emission in proton-rich nuclei far from the stability line [ 15 15 , 17 17 , 52 52 ]. The earliest experimental discovery of such phenomena was the single-proton emission of excited state nuclei. In 1963, a β -delayed proton emission was believed to have been identified from 25 Si; however, this turned out to originate from the excited state of 25 Al [ 53 53 ]. Later, the proton emission was discovered in the isomer state of 53 Co [ 54 54 ] and the ground state of 151 Lu [ 55 55 ], in 1970 and 1982, respectively. Since then, more than 20 nuclides have been confirmed as the ground-state proton emitters; this offers invaluable information with which to study the nuclear structure. 2 p decay from excited states. — The ground-state 2 p emission occurs in a region even farther beyond the proton dripline. The candidates given in the early theories mainly included 39 Ti, 42 Cr, 45 Fe, and 48,49 Ni [ 56 56 - 59 59 56-59 ]. It is very difficult to produce these nuclei in experiments; hence, this exotic phenomenon has not been found, despite years of research. In 1980, Goldansky postulated a β -delayed 2 p emission from excited nuclei and proposed that the candidates could be proton-rich nuclei with charge numbers Z ranging from 10 to 20 [ 60 60 ]. The first observation of β -delayed 2 p emission from 22 Al was achieved at Lawrence Berkeley National Laboratory (LBNL) in 1983 [ 61 61 ]; here, the transition proceeded via the T =2 isobaric-analog state ( E x = 14.044 MeV) in 22 Mg. Inspired by this experiment, further β -delayed 2 p emission nuclei were investigated, including 23 Si, 26 P, 27 S, 31 Ar, 35 Ca, 39 Ti, 43 Cr, 46 Fe, and 50,51 Ni [ 62 62 - 71 71 62-71 ]. The excited 2 p emitters might not only be produced by β decay but also by nuclear reactions such as 14 O or 17,18 Ne [ 72 72 - 75 75 72-75 ] and may include fragmentation, transfer, and pick-up reactions. Owing to the large decay energy, the lifetime of a 2 p emission from the excited state is generally short, about the order of 10 -21 s, much shorter than those from the ground states. Consequently, the corresponding decay mechanism is harder to identify in these excited 2 p emission processes. In recent years, radioactive beam devices have been used to conduct a series of experimental measurements, including the β -delayed 2 p emission of 22 Al, 22,23 Si, and 27 S, and the 2 p emission of excited 22 Mg, 23 Al, 27,28 P, and 28,29 S nuclei [ 76 76 - 85 85 76-85 ]; this constitutes important progress in the characterization of excited 2 p emitters. 2 p decay from ground states. — The ground-state 2 p emitters are typically located beyond the proton dripline, and they are extremely difficult to produce. Therefore, researchers initially investigated the light-mass region of the nuclear landscape [ 23 23 , 26 26 , 50 50 , 86 86 , 87 87 ]. Owing to a large proton-neutron asymmetry, these short-lived 2 p emitters (e.g., 6 Be, 11,12 O, 16 Ne, and 19 Mg) tend to have short lifetimes and broad intermediate states, resulting in more complicated decay dynamics (see the discussion below for details). These very unstable systems are usually accompanied by exotic structures, which makes them harder to study experimentally and theoretically. For instance, as shown in Fig. 3 Fig. 3 , the mirror of the halo nucleus 11 Li, 11 O, was recently found to exhibit 2 p decay with a ground-state decay width of more than 1 MeV [ 50 50 , 88 88 ]. Many efforts—both theoretical and experimental—have been made to investigate their decay mechanisms and the impact of their internal nuclear structures; this has resulted in the excellent progress detailed below. Meanwhile, unlike the general situation for light-mass 2 p emitters, the criteria originally introduced by Goldansky [ 15 15 , 89 89 ] for the 2 p decay was Q 2 p > 0 and Q p < 0, where Q denotes the decay energy. In this case, the one-proton decay channel is energetically forbidden. The nuclei satisfying these requirements are typically in the mid-mass region, in which both mother and neighboring nuclei can be quasi-bound. Until 2002, the 2 p emission from the long-lived ground state was directly observed for the first time in 45 Fe [ 27 27 , 28 28 ]. The experiment was conducted at the Grand Accelerateur National d’Ions Lourds (GANIL) in France and the Gesellschaft für Schwerionenforschung (GSI) in Germany, in which 12 and 3 2 p decay events were observed, respectively. The extracted half-lives were <math id="M1"><mrow><msubsup><mrow><mn>4.7</mn></mrow><mrow><mo>−</mo><mn>1.4</mn></mrow><mrow><mo>+</mo><mn>3.4</mn></mrow></msubsup></mrow></math> and <math id="M2"><mrow><msubsup><mrow><mn>3.2</mn></mrow><mrow><mo>−</mo><mn>1.0</mn></mrow><mrow><mo>+</mo><mn>2.6</mn></mrow></msubsup></mrow></math> ms, respectively. Later, in 2005, a 2 p emission from the ground state of 54 Zn was identified in an experiment conducted by GANIL [ 90 90 ], which observed seven events and measured a half-life of <math id="M3"><mrow><msubsup><mrow><mn>3.7</mn></mrow><mrow><mo>−</mo><mn>1.0</mn></mrow><mrow><mo>+</mo><mn>2.2</mn></mrow></msubsup></mrow></math> ms. In the same year, GANIL studied the decay of the ground state of 48 Ni [ 91 91 ]: one of the four decay events was entirely compatible with 2 p radioactivity. This was later confirmed by Michigan State University (MSU). The corresponding experiment was performed at the National Superconducting Cyclotron Laboratory (NSCL/MSU), with four 2 p events being directly observed and a half-life of <math id="M4"><mrow><msubsup><mrow><mn>2.1</mn></mrow><mrow><mo>−</mo><mn>0.4</mn></mrow><mrow><mo>+</mo><mn>1.4</mn></mrow></msubsup></mrow></math> ms obtained [ 92 92 ]. Further experiments have been carried out or proposed, to more deeply probe the properties of this exotic decay mode in 45 Fe, 48 Ni, and 54 Zn [ 71 71 , 93 93 , 94 94 ]. In 2016, 67 Kr was measured with a half-life of 7.4(30) ms in an experiment performed at the Institute of Physical and Chemical Research (RIKEN) (see Fig. 2 Fig. 2 and Ref. [ 49 49 ]). By considering the dead-time correction of the acquisition system, 13 evens of 2 p decay were detected, in which the branching ratio was 37(14)%. This makes 67 Kr the heaviest 2 p emitter ever discovered. So far, only these four long-lived 2 p emitting ground-state nuclei have been identified. Other candidates are theoretically possible, including 59 Ge and 63 Se, which are predicted to be 2 p emitters. However, in the experimental measurements [ 49 49 ], 59 Ge and 63 Se are mainly observed via β -delayed proton(s) emission, and no 2 p decay phenomenon have been observed ( Table 1 Table 1 ). 1 1 Nuclei 1 1 Q 2 p 1 1 Γ / T 1/2 1 1 <math id="M5"><mrow><msubsup><mrow/><mn>4</mn><mn>6</mn></msubsup><mtext>Be</mtext></mrow></math> [ 23 23 , 24 24 , 98 98 ] 1 1 1.372(5) 1 1 92(6) keV 1 1 <math id="M6"><mrow><msubsup><mrow/><mn>6</mn><mn>8</mn></msubsup><mtext>C</mtext></mrow></math> [ 99 99 ] 1 1 2.111(19) 1 1 230(50) keV 1 1 <math id="M7"><mrow><msubsup><mrow/><mrow><mi> </mi><mn>8</mn></mrow><mrow><mn>11</mn></mrow></msubsup><mtext>O</mtext></mrow></math> [ 50 50 ] 1 1 4.25(6) 1 1 2.31(14) MeV 1 1 <math id="M8"><mrow><msubsup><mrow/><mrow><mi> </mi><mn>8</mn></mrow><mrow><mn>12</mn></mrow></msubsup><mtext>O</mtext></mrow></math> [ 25 25 , 26 26 , 100 100 ] 1 1 1.737(12) 1 1 0.40(25) MeV 1 1 <math id="M9"><mrow><msubsup><mrow/><mrow><mn>10</mn></mrow><mrow><mn>15</mn></mrow></msubsup><mtext>Ne</mtext></mrow></math> [ 96 96 ] 1 1 2.522(66) 1 1 0.59(23) MeV 1 1 <math id="M10"><mrow><msubsup><mrow/><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></msubsup><mtext>Ne</mtext></mrow></math> [ 25 25 , 86 86 , 101 101 - 103 103 101-103 ] 1 1 1.401(20) 1 1 <math id="M11"><mo>≤</mo></math> 80 keV 1 1 <math id="M12"><mrow><msubsup><mrow/><mrow><mn>12</mn></mrow><mrow><mn>18</mn></mrow></msubsup><mtext>Mg</mtext></mrow></math> [ 97 97 ] 1 1 3.44 1 1 115(100) keV 1 1 <math id="M13"><mrow><msubsup><mrow/><mrow><mn>12</mn></mrow><mrow><mn>19</mn></mrow></msubsup><mtext>Mg</mtext></mrow></math> [ 87 87 , 104 104 - 106 106 104-106 ] 1 1 0.76(6) 1 1 4.0(15) ps 1 1 <math id="M14"><mrow><msubsup><mrow/><mrow><mn>18</mn></mrow><mrow><mn>30</mn></mrow></msubsup><mtext>Ar</mtext></mrow></math> [ 106 106 , 107 107 ] 1 1 3.42(8) 1 1 <math id="M15"><mo>≤</mo></math> 10 ps 1 1 <math id="M16"><mrow><msubsup><mrow/><mrow><mn>26</mn></mrow><mrow><mn>45</mn></mrow></msubsup><mtext>Fe</mtext></mrow></math> [ 27 27 , 28 28 ] 1 1 1.80(20) 1 1 2.45(23) ms 1 1 <math id="M17"><mrow><msubsup><mrow/><mrow><mn>28</mn></mrow><mrow><mn>48</mn></mrow></msubsup><mtext>Ni</mtext></mrow></math> [ 69 69 , 71 71 , 91 91 , 92 92 ] 1 1 1.26(12) 1 1 2.1 <math id="M18"><mrow><msubsup><mrow/><mrow><mo>−</mo><mn>0.4</mn></mrow><mrow><mo>+</mo><mn>1.4</mn></mrow></msubsup></mrow></math> ms 1 1 <math id="M19"><mrow><msubsup><mrow/><mrow><mn>30</mn></mrow><mrow><mn>54</mn></mrow></msubsup><mtext>Zn</mtext></mrow></math> [ 90 90 , 94 94 ] 1 1 1.28(21) 1 1 1.59 <math id="M20"><mrow><msubsup><mrow/><mrow><mo>−</mo><mn>0.35</mn></mrow><mrow><mo>+</mo><mn>0.60</mn></mrow></msubsup></mrow></math> ms 1 1 <math id="M21"><mrow><msubsup><mrow/><mrow><mn>36</mn></mrow><mrow><mn>67</mn></mrow></msubsup><mtext>Kr</mtext></mrow></math> [ 49 49 ] 1 1 1.690(17) 1 1 7.4(30) ms Experimental Progress 1 The 2 p -emitting nuclei are likely to inhabit remote regions of the nuclear landscape, regardless of their energy state; thus, the production rate is always very low, which makes the experimental conditions difficult to realize. Consequently, only a few 2 p emitters have been identified, through rare events. Although the decay process can also be determined via other physical quantities (e.g., decay energy and half-life) to clearly identify the different mechanisms of 2 p emission, it is necessary to measure sufficient 2 p emission events, as well as to measure the correlation (including the relative momentum and opening angle) between the emitted core and two protons. To this end, many laboratories around the world are developing next-generation facilities and state-of-the-art detectors, to explore candidate 2 p emitters and their properties. In this section, we briefly introduce some of the recent experimental progress. Measurement in experiments 2 Generally, the half-lives of proton-rich nuclei decrease under the increase in proton-neutron asymmetry, and the span ranges from 10 1 to 10 -22 s. Therefore, different production mechanisms, detection methods, and technologies [ 108 108 - 112 112 108-112 ] are used for such nuclei in experiments, depending on the half-life. In early years, light particle-emission nuclei (e.g., 6 Be and 12 O) were produced by transfer reactions with protons or 3 He beams [ 23 23 , 26 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 [ 113 113 ]. 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 [ 114 114 ]. They bombarded a uranium target with a neutron beam, causing it to fragment and produce the neutron-rich nuclides 89,90,91 Kr and 89,90,91 Rb. 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 2 p 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 2 p 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 2 p emitters discovered thus far, 19 Mg was investigated using this technique [ 87 87 ]. Meanwhile, it also consistutes the main experimental technique for studying excited 2 p emissions. In the following, the 2 p emission of excited state 22 Mg [ 79 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 28 Si was used to produce secondary 23 Al and 22 Mg 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 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 Δ E - E detectors for the fragment. The three layers of plastic hodoscopes were located 3 m downstream of the target and were used as Δ E 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 2 p emission. These experimental studies indicate that the highly excited state of 22 Mg may produce a diproton emission [ 79 79 ]. Implantation decay method. — Implantation decay is one of the most popular methods used to experimentally measure long-lived nuclei. In terms of 2 p 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 2 p emission of 22 Al, performed at the Radioactive Ion Beam line in Lanzhou (RIBLL) [ 76 76 , 115 115 , 116 116 ]. A sketch of the detector setup for implantation decay is shown in Fig. 5 Fig. 5 . The emitted nucleus 22 Al 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 22 Al nuclei can be injected and halted at the decay position. Three silicon detectors with thicknesses of 300 μ m are installed after the aluminum degraders, to measure the energy loss (Δ E ) of the beam ions. The identification of incident ions is performed using a Δ E -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 1 p or 2 p 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 76 ]. 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 [ 117 117 ]. For the decay from proton-rich nuclei (e.g., 45 Fe), the traces of two valence protons are measured using an optical time projection chamber-charge coupled device (OTPC-CCD) method [ 93 93 ]. A sketch of the OTPC-CCD detector setup is shown in Fig. 6 Fig. 6 . The entire geometric volume of the OTPC is 20×20×15 cm 3 ; this is filled with a gas mixture of 49% He, 49% Ar, 1% N 2 , and 1% CH 4 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 45 Fe, the two protons emitted in the 2 p 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 2 p 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 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 1 p , 2 p , and 3 p emissions of 43 Cr, as measured by OTPC-CCD in the experiment at NSCL, are shown in Fig. 7 Fig. 7 ; here, the track of the emitted proton can be intuitively seen [ 118 118 ]. The abovementioned three experimental methods for measuring the 2 p 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 2 p 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 [ 119 119 , 120 120 ]. Similar to its development in the field of nuclear electronics, waveform sampling technology is becoming increasingly important in data acquisition for nuclear experiments [ 121 121 ]. If these new techniques are used in experiments, the original sig nal 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 2 As shown in Fig. 1 Fig. 1 , three mechanisms are available for 2 p 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 2 p emissions. For a 2 p 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 2 p . 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. 6 Be is one such case; it is thought to exhibit both diproton and large-angle (three-body) emissions compared with the theoretical calculations [ 122 122 - 125 125 122-125 ]; meanwhile, the observed pp correlation has a wide energy distribution [ 18 18 , 24 24 , 98 98 ]. Similar situations occur in 11,12 O [ 50 50 , 51 51 , 126 126 - 128 128 126-128 ]. However, one can still extract useful information from these observables. We take the ground-state 2 p emitter 16 Ne 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 Fig. 8 and Refs. [ 102 102 , 103 103 ]). This suggests that interference between these processes is responsible for the observed features, which can be described in terms of a “tethered decay” mechanism. For the 2 p 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 18 Ne show a clear diproton emission mechanism in the 6.15 MeV excited state [ 75 75 ]. In the study of 2 p emissions for excited states of 27,28 P and 28,29 S, conducted at RIBLL, a diproton emission mechanism was also identified using the same method [ 81 81 - 83 83 81-83 ]. In the following, we present in detail a method for identifying the 2 p decay mechanism. As described above, in 1983, a β -delayed decay experiment performed at the LBNL identified a 2 p emission in the 14.044 MeV excited state of 22 Mg; however, the emission mechanism could not be clearly identified because the statistics for the relative angle distribution were too low [ 129 129 ]. To elucidate this decay mechanism, the 2 p emissions of two excited proton-rich nuclei, 23 Al and 22 Mg, were recently measured using the RIPS at the RIKEN RI Beam Factory [ 79 79 ]. The in-flight decay introduced above was adopted, and a sketch of the detector setup is shown in Fig. 4 Fig. 4 . The relative momenta and opening angle between the two emitted protons decayed from 23 Al and 22 Mg and the excitation energies of the decay nuclei were measured. Figure 9(a),(b) 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 23 Al were dominated by the sequential emissions or three-body (large-angle) emissions in different energy ranges. However, as shown in Fig. 9(c),(d) Fig. 9(c),(d) , around the 14.044 MeV excited state of 22 Mg, 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 22 Mg were basically governed by three-body (large-angle) or sequential emission. Consequently, the experiment quantitatively determined the 2 p decay mechanism for the 14.044 MeV excited state of 22 Mg [ 79 79 ]; this is useful for further studying the properties of excited 2 p emissions. As a different experimental method, the implantation decay was used to measure the β -delayed 2 p emission of 22 Al in RIBLL [ 76 76 ]; this confirms the diproton emission from the 14.044 MeV excited state of 22 Mg. To further analyze the 2 p decay mechanism, simulation tools are required. The two-particle intensity interferometry (HBT) [ 130 130 ] method is widely used to obtain the emission source size and emission time for middle- and high-energy nuclear reactions [ 131 131 ]. 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 seq uential emission, the HBT method was used in the above study investigating the 2 p emission of 23 Al and 22 Mg [ 80 80 ]. Assuming that the first proton is emitted at time t 1 =0 and the second proton is emitted at time t 2 = t , the space and time profile of the Gaussian source can be simulated according to the function S ( r,t ) ∼ <math id="M22"><mrow><mtext>exp</mtext><mo stretchy="false">(</mo><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn><msubsup><mi>r</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><mi>t</mi><mo>/</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></math> in the CRAB code [ 80 80 ]. r 0 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 r 0 and τ values was evaluated by determining the value of the reduced χ 2 , as shown in Fig. 10 Fig. 10 . The results show that the ranges of the source size parameters were obtained as r 0 = 1.2 ~ 2.8 fm and r 0 = 2.2 ~ 2.4 fm for 23 Al and 22 Mg, 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 23 Al and 22 Mg, respectively. The large difference in emission time between the two nuclei means that the 2 p emission mechanism of 23 Al is primarily sequential emission, owing to the long emission-time difference. Meanwhile, the emission mechanism for 22 Mg 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 2 p emission can be clearly identified [ 80 80 , 132 132 ]. Meanwhile, the HBT method has been used to analyze experimental data from the two-neutron (2 n ) decay, which has also come to represent a frontier of radioactive nuclear beam physics in recent years [ 133 133 ]. The 2 n emission of 18 C and 20 O have been measured at GSI, and the measured momentum correlation for the emitted neutrons was analyzed using the HBT method [ 134 134 ]. The results also indicate that the time scales substantially differ between simultaneous and sequential decay. So far, only a few ground-state 2 p emitters have been observed with relatively low statistics; hence, the primary goal is to explore more 2 p emitter candidates. Meanwhile, the study of the 2 p 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 22 , 21 21 , 46 46 ] for details), ground-state 2 p 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 2 p 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 2 p and 2 n decay. Theoretical Developments 1 In recent years, many related theoretical developments have been made in various directions. In this section, we briefly introduce certain innovative and important ones. Each focuses upon certain aspects of 2 p decays, such as their candidates, inner structures, or asymptotic observables, which reveals the impacts of the low-lying continuum and the interplay among various of interactions. These different approaches form a comprehensive description of the 2 p decay process, which is useful for better understanding its open quantum nature and exotic properties. Towards quantified prediction of the 2 p landscape 2 Exploring the candidate 2 p emitters using density functional theory. — The ground-state 2 p decay usually occurs beyond the 2 p 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 17 - 19 19 17-19 , 135 135 - 137 137 135-137 ]. Meanwhile, the presence of the Coulomb barrier has a confining effect upon the nucleonic density; hence, relatively long-lived 2 p emitters should be expected. So far, the 2 p 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 39 , 40 40 , 43 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 6 , 138 138 , 139 139 ]. The candidate ground-state 2 p emitters have been explored using the self-consistent Hartree-Fock-Bogoliubov (HFB) equations [ 21 21 , 22 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 21 , 22 22 ] produced a very consistent prediction for the 2 p dripline. The corresponding mean value and uncertainty are shown in Fig. 11 Fig. 11 . In Ref. [ 21 21 , 22 22 ], the 2 p decay candidates were selected according to the decay energy criteria Q 2 p > 0 and Q p < 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 Fig. 11 ). Meanwhile, the decay half-lives are estimated via both direct [ 19 19 ] and diproton decay [ 56 56 , 57 57 ] models and compared with those of alpha decay. As a result, this 2 p 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 2 p -decaying isotopes (see Fig. 12 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. 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 46 ]. In the framework of Ref. [ 46 46 ], predictions were obtained for each model 𝒨 k using Bayesian Gaussian processes trained upon separation-energy residuals with respect to the experimental data [ 141 141 , 142 142 ] and combined via Bayesian model averaging. The corresponding posterior weights conditional to the data y are given by [ 143 143 - 145 145 143-145 ] <math display="block" id="M23"><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi mathvariant="script">M</mi><mi>k</mi></msub><mo>|</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mo>|</mo><msub><mi mathvariant="script">M</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi mathvariant="script">M</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi mathvariant="script">l</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mo>|</mo><msub><mi mathvariant="script">M</mi><mi mathvariant="script">l</mi></msub><mo stretchy="false">)</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi mathvariant="script">M</mi><mi mathvariant="script">l</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mrow></mfrac><mo>,</mo></mrow></math> where π ( <math id="M24"><mi mathvariant="script">M</mi></math> k ) are the prior weights and p ( y | <math id="M25"><mi mathvariant="script">M</mi></math> 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 7 ]. In particular, for this 2 p decay topic, Ref. [ 46 46 ] provides the quantified model results for 1 p and 2 p separation energies and the derived probabilities for proton emission (see Fig. 13 Fig. 13 ); this indicates promising candidates for 2 p decay and will be tested using experimental data from rare-isotope facilities. Towards a comprehensive description of three-body decay 2 One of the most fundamental problems of 2 p 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) 2 p 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 2 p decay, one commonly used selection method (also introduced by Goldansky [ 15 15 ]) is to use the decay energy criteria Q 2 p > 0 and Q p < 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 17 , 18 18 , 123 123 , 146 146 ] (see Fig. 14 Fig. 14 ). The energy criterion might not be very strict for 2 p decays originating from the excited states or in the light-mass region. For the former case [as shown in Fig. 14(a) 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) 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) Fig. 14(d,e) ]. Consequently, both direct and sequential decay processes are allowed; this is often referred to as “democratic decay” [ 18 18 , 24 24 ]. 6 Be and the recently discovered 11,12 O are two such cases [ 50 50 , 100 100 , 147 147 ]. 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 18 ]. All of these results indicate that the 2 p 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 2 p mechanism. This will also introduce large uncertainties in the decay property studies. This complicated 2 p 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 45 Fe [ 148 148 ], 19 Mg [ 101 101 ], and 48 Ni [ 71 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 2 p wave function. The high-quality 2 p 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 17 , 18 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 2 p decay can help us to better understand the internal structure information and its impact upon the 2 p 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) [ 149 149 , 150 150 ], which has been successfully applied to describe the 2 p decay from the <math id="M26"><mrow><msubsup><mn>1</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> state in 18 Ne and other ground-state 2 p emitters (see Fig. 15 Fig. 15 and Refs. [ 151 151 , 152 152 ]). Regarding the latter phenomena, the contribution of the decay width from the three-body dynamics has been estimated using a hybrid model [ 153 153 ]. In this hybrid framework (as shown in Ref. [ 153 153 ]), we can investigate the spectroscopic amplitude via the two-nucleon decay amplitudes (TNAs) for the removal of two protons from the initial state <math id="M27"><mrow><mo>|</mo><mi>A</mi><msup><mi>ω</mi><mo>'</mo></msup><msup><mi>J</mi><mo>'</mo></msup><mo>〉</mo></mrow></math> , leaving it in the final state <math id="M28"><mrow><mo>〈</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mi>ω</mi><mi>J</mi><mo>|</mo></mrow></math> given by the reduced matrix element, as <math display="block" id="M29"><mrow><mtext>TNA</mtext><mrow><mo>(</mo><mrow><msub><mi>q</mi><mi>a</mi></msub><mn>,</mn><msub><mi>q</mi><mi>b</mi></msub></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>〈</mo><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mi>ω</mi><mi>J</mi><mo>|</mo><mo>|</mo><msup><mrow><mrow><mo>[</mo><mrow><msub><mover accent="true"><mi>a</mi><mo>˜</mo></mover><mrow><msub><mi>q</mi><mi>a</mi></msub></mrow></msub><mo>⊗</mo><msub><mover accent="true"><mi>a</mi><mo>˜</mo></mover><mrow><msub><mi>q</mi><mi>b</mi></msub></mrow></msub></mrow><mo>]</mo></mrow></mrow><mrow><msub><mi>J</mi><mi>o</mi></msub></mrow></msup><mo>|</mo><mo>|</mo><mi>A</mi><msup><mi>ω</mi><mo>'</mo></msup><msup><mi>J</mi><mo>'</mo></msup></mrow><mo>〉</mo></mrow></mrow><mrow><msqrt><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>δ</mi><mrow><msub><mi>q</mi><mi>a</mi></msub><msub><mi>q</mi><mi>b</mi></msub></mrow></msub></mrow><mo>)</mo></mrow><mo stretchy="false">(</mo><mn>2</mn><mi>J</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msqrt></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M30"><mrow><msub><mover accent="true"><mi>a</mi><mo>˜</mo></mover><mi>q</mi></msub></mrow></math> 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 = J o = 0 and q a = q b = q has been considered. Notably, this spin-singlet nucleon pair might not be spatially close. The corresponding 2 p 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 2 p decay width can be calculated from an incoherent sum: <math display="block" id="M31"><mrow><msub><mtext>Γ</mtext><mi>i</mi></msub><mrow><mo>(</mo><mrow><msub><mi>Q</mi><mrow><mn>2</mn><mi>p</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>q</mi></munder></mstyle><msub><mtext>Γ</mtext><mi>s</mi></msub><mrow><mo>(</mo><mrow><msub><mi>Q</mi><mrow><mn>2</mn><mi>p</mi></mrow></msub><mn>,</mn><msup><mi mathvariant="script">l</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><msup><mrow><mrow><mo>[</mo><mrow><mtext>TNA</mtext><mrow><mo>(</mo><mrow><msup><mi>q</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup><mn>,</mn></mrow></math> where Γ s 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 <math display="block" id="M32"><mrow><msub><mtext>Γ</mtext><mi>c</mi></msub><mrow><mo>(</mo><mrow><msub><mi>Q</mi><mrow><mn>2</mn><mi>p</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>[</mo><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>q</mi></munder></mstyle><msqrt><mrow><msub><mtext>Γ</mtext><mi>s</mi></msub><mrow><mo>(</mo><mrow><msub><mi>Q</mi><mrow><mn>2</mn><mi>p</mi></mrow></msub><mn>,</mn><msup><mi mathvariant="script">l</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></msqrt><mtext>TNA</mtext><mrow><mo>(</mo><mrow><msup><mi>q</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup><mn>,</mn></mrow></math> in which the phase is taken as positive for all terms. As shown in Table 2 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. [ 153 153 ], it is important to include small s 2 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 67 Kr retain a large discrepancy between the experimental data and theoretical predictions [ 153 153 ]. As a recently observed 2 p emitter, the half-life of 67 Kr is systematically over-estimated by various theoretical models [ 43 43 , 154 154 ]. It seems that such a short lifetime can only be reproduced by using a large amount of s - or p -wave components [ 153 153 , 155 155 ], which indicates that deformation effects or certain other factors might be missing from the current theoretical or experimental frameworks [ 49 49 , 153 153 ]. 1 1 Nucleus 1 1 T 1/2 2 1 T 1/2 without s 2 2 1 T 1/2 with s 2 1 1 J π 1 1 Expt. 1 1 Incoherent 1 1 Coherent 1 1 Incoherent 1 1 Coherent 1 1 19 Mg 1/2 - 1 1 4.0(15) 1 1   1 1   1 1 0.73 <math id="M33"><mrow><msubsup><mrow/><mrow><mo>−</mo><mn>0.17</mn></mrow><mrow><mo>+</mo><mn>1.5</mn></mrow></msubsup></mrow></math> 1 1 0.20 <math id="M34"><mrow><msubsup><mrow/><mrow><mo>−</mo><mn>0.05</mn></mrow><mrow><mo>+</mo><mn>0.40</mn></mrow></msubsup></mrow></math> 1 1 45 Fe 3/2 + 1 1 3.6(4) 1 1 20(8) 1 1 6.6(26) 1 1 5.9(24) 1 1 1.8(7) 1 1 48 Ni 0 + 1 1 4.1(20) 1 1 5.1(29) 1 1 1.8(11) 1 1 1.3(6) 1 1 0.43(22) 1 1 54 Zn 0 + 1 1 1.9(6) 1 1 1.8(8) 1 1 0.9(4) 1 1 1.7(8) 1 1 0.6(3) 1 1 67 Kr 3/2 - 1 1 20(11) 1 1 850(390) 1 1 320(140) 1 1 820(380) 1 1 250(110) 1 1 67 Kr 1/2 - 1 1 20(11) 1 1 904(420) 1 1 290(130) 1 1 940(430) 1 1 360(160) 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 2 p 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 ( n 1 and n 2 ). The i -th cluster ( i = c,n 1 , n 2 ) contains the position vector r i and linear momentum k i [ 31 31 , 123 123 , 126 126 , 156 156 - 159 159 156-159 ]. Two types of coordinates are commonly used to construct the three-body framework. As shown in Fig. 16 Fig. 16 , one is the cluster-orbital shell model (COSM) [ 160 160 ] and the other is Jacobi coordinates [ 126 126 ]. The former uses single-particle coordinates ( r n - r c ) with a recoil term to handle the extra energy introduced by center-of-mass (c.m.) motion [ 160 160 , 161 161 ]. In this framework, it is easy to calculate matrix elements and extend them t o 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 <math display="block" id="M35"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold"><mtext>x</mtext></mstyle><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msqrt><mrow><msub><mi>μ</mi><mstyle mathvariant="bold"><mtext>x</mtext></mstyle></msub></mrow></msqrt><mo stretchy="false">(</mo><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold"><mtext>r</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>−</mo><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold"><mtext>r</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold"><mtext>y</mtext></mstyle><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msqrt><mrow><msub><mi>μ</mi><mi>y</mi></msub></mrow></msqrt><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold"><mtext>r</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold"><mtext>r</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub></mrow></mfrac><mo>−</mo><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold"><mtext>r</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>3</mn></msub></mrow></msub></mrow><mo>)</mo></mrow><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> where i 1 = n 1 , i 2 = n 2 , i 3 = c for T-coordinates and i 1 = n 2 , i 2 = c , i 3 = n 1 for Y-coordinates, see Fig. 16 Fig. 16 . In Eq. (5) A i is the i -th cluster mass number, and <math id="M36"><mrow><msub><mi>μ</mi><mstyle mathvariant="bold"><mtext>x</mtext></mstyle></msub><mo>=</mo><mfrac><mrow><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub></mrow></mfrac></mrow></math> and <math id="M37"><mrow><msub><mi>μ</mi><mi>y</mi></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>3</mn></msub></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>3</mn></msub></mrow></msub></mrow></mfrac></mrow></math> 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. The benchmarking between COSM and Jacobi coordinates has been performed in Ref. [ 162 162 ], in which both weakly bound and unbound systems ( 6 He, 6 Li, 6 Be, and 26 O) were investigated using the Berggren ensemble technique [ 163 163 ]. 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 [ 162 162 ]. 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 2 p emitter, 6 Be has been considered to feature a “democratic” decay mode, attributable to the large width of the ground state of its neighboring nucleus 5 Li ( Q p = 1.97 MeV, Γ = 1.23 MeV). The density distribution of 6 Be has been studied in various few-body models; it shows two maxima associated with diproton and cigarlike configurations (see Fig. 17 Fig. 17 ). This accords with the angular densities obtained by the three-body model [ 31 31 , 162 162 ] and GSM [ 164 164 ]. This angular density can be calculated using <math display="block" id="M38"><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mrow><mo>〈</mo><mrow><mi>Ψ</mi><mrow><mo>|</mo><mrow><mi>δ</mi><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><mi>r</mi></mrow><mo>)</mo></mrow><mi>δ</mi><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>2</mn></msub><mo>−</mo><msup><mi>r</mi><mo>'</mo></msup></mrow><mo>)</mo></mrow><mi>δ</mi><mrow><mo>(</mo><mrow><msub><mi>θ</mi><mrow><mn>12</mn></mrow></msub><mo>−</mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mi>Ψ</mi></mrow><mo>〉</mo></mrow><mtext>d</mtext><mi>r</mi><mtext>d</mtext><msup><mi>r</mi><mo>'</mo></msup><mn>,</mn></mrow></math> 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. Instead, we can look at the configuration evolution of the internal structure during decay by using the flux current <math id="M39"><mrow><mtext> </mtext><mtext> </mtext><mi>j</mi><mtext> </mtext><mo>=</mo><mtext>I</mtext><mi>m</mi><mo stretchy="false">(</mo><msup><mi>Ψ</mi><mo>†</mo></msup><mo>∇</mo><mi>Ψ</mi><mo stretchy="false">)</mo><mi>ℏ</mi><mo>/</mo><mi>m</mi></mrow></math> , which shows how the two valence protons evolve within a given state wave function Ψ . In the case of 6 Be (as shown in Fig. 17 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 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-2 p 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 [ 123 123 , 124 124 , 162 162 , 165 165 ]. 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: <math display="block" id="M40"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold-italic"><mtext>k</mtext></mstyle><msub><mtext> </mtext><mstyle mathvariant="bold-italic"><mtext>x</mtext></mstyle></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mi>μ</mi><mstyle mathvariant="bold-italic"><mtext>x</mtext></mstyle></msub><mrow><mo>(</mo><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold-italic"><mtext>k</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold-italic"><mtext>k</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold-italic"><mtext>k</mtext></mstyle><msub><mtext> </mtext><mi>y</mi></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mi>μ</mi><mi>y</mi></msub><mrow><mo>(</mo><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold-italic"><mtext>k</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>+</mo><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold-italic"><mtext>k</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>2</mn></msub></mrow></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold-italic"><mtext>k</mtext></mstyle><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>3</mn></msub></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msub><mi>i</mi><mn>3</mn></msub></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> No c.m. motion occurs; hence, it is easy to see that <math id="M41"><mrow><mstyle displaystyle="true"><msub><mo>∑</mo><mi>i</mi></msub></mstyle><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold"><mtext>k</mtext></mstyle><msub><mtext> </mtext><mi>i</mi></msub><mo>=</mo><mn>0</mn></mrow></math> , and k y is aligned in the opposite direction to <math id="M42"><mrow><mtext> </mtext><mtext> </mtext><mi>k</mi><msub><mtext> </mtext><mrow><msub><mi>i</mi><mn>3</mn></msub></mrow></msub></mrow></math> . θ k and <math id="M43"><mrow><msub><msup><mi>θ</mi><mo>′</mo></msup><mi>k</mi></msub></mrow></math> denote the opening angles of ( k x , k y ) in Jacobi-T and -Y coordinates, respectively (see Fig. 16 Fig. 16 ). The kinetic energy of the relative motions of the emitted nucleons is given by <math id="M44"><mrow><msub><mi>E</mi><mrow><mi>p</mi><mi>p</mi><mo>/</mo><mi>n</mi><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup><msubsup><mi>k</mi><mstyle mathvariant="bold"><mtext>x</mtext></mstyle><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><msub><mi>μ</mi><mstyle mathvariant="bold"><mtext>x</mtext></mstyle></msub></mrow></mfrac></mrow></math> , and E core- p/n denotes that of the core-nucleon pair. Therefore, the T -type ( θ k , E pp/nn ) and Y -type ( <math id="M45"><mrow><msub><msup><mi>θ</mi><mo>′</mo></msup><mi>k</mi></msub></mrow></math> , 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 <math id="M46"><mrow><msqrt><mrow><mfrac><mrow><msubsup><mi>k</mi><mstyle mathvariant="bold"><mtext>x</mtext></mstyle><mn>2</mn></msubsup></mrow><mrow><msub><mi>μ</mi><mstyle mathvariant="bold"><mtext>x</mtext></mstyle></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msubsup><mi>k</mi><mi>y</mi><mn>2</mn></msubsup></mrow><mrow><msub><mi>μ</mi><mi>y</mi></msub></mrow></mfrac></mrow></msqrt></mrow></math> , which at later times approaches the limit <math id="M47"><mrow><mfrac><mrow><msqrt><mrow><mn>2</mn><mi>m</mi><msub><mi>Q</mi><mrow><mn>2</mn><mi>p</mi><mo>/</mo><mn>2</mn><mi>n</mi></mrow></msub></mrow></msqrt></mrow><mi>ℏ</mi></mfrac></mrow></math> , where Q 2 p /2 n 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 [ 166 166 ]. Meanwhile, the long-range correlations of 2 p 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 [ 167 167 - 169 169 167-169 ]; 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 Fig. 18 , the asymptotic nucleon-nucleon correlations of 6 Be, 45 Fe, and several other 2 p emitters have been reproduced or predicted using the three-body model [ 18 18 , 122 122 ]. 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 45 Fe, the small-angle emission dominates in the asymptotic region, which may correspond to a diproton decay. Meanwhile, the situation in 6 Be 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. [ 124 124 , 125 125 ] 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 [ 171 171 - 174 174 171-174 ]. For 2 p emission, the GCC method has been extended to a deformed case [ 162 162 , 170 170 ], 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 <math id="M48"><mrow><msup><mi>Ψ</mi><mrow><mi>J</mi><mi>π</mi></mrow></msup><mo>=</mo><mstyle displaystyle="true"><msub><mo>∑</mo><mrow><msub><mi>J</mi><mi>p</mi></msub><msub><mi>π</mi><mi>p</mi></msub><msub><mi>j</mi><mi>c</mi></msub><msub><mi>π</mi><mi>c</mi></msub></mrow></msub></mstyle><msup><mrow><mrow><mo>[</mo><mrow><msup><mi>Φ</mi><mrow><msub><mi>J</mi><mi>p</mi></msub><msub><mi>π</mi><mi>p</mi></msub></mrow></msup><mo>⊗</mo><msup><mi>ϕ</mi><mrow><msub><mi>j</mi><mi>c</mi></msub><msub><mi>π</mi><mi>c</mi></msub></mrow></msup></mrow><mo>]</mo></mrow></mrow><mrow><mi>J</mi><mi>π</mi></mrow></msup></mrow></math> , where <math id="M49"><mrow><msup><mi>Φ</mi><mrow><msub><mi>J</mi><mi>p</mi></msub><msub><mi>π</mi><mi>p</mi></msub></mrow></msup></mrow></math> and <math id="M50"><mrow><msup><mi>ϕ</mi><mrow><msub><mi>j</mi><mi>c</mi></msub><msub><mi>π</mi><mi>c</mi></msub></mrow></msup></mrow></math> are the wave functions of the two valence protons and core, respectively. The wave function of the valence protons <math id="M51"><mrow><msup><mi>Φ</mi><mrow><msub><mi>J</mi><mi>p</mi></msub><msub><mi>π</mi><mi>p</mi></msub></mrow></msup></mrow></math> is expressed in Jacobi coordinates and expanded using the Berggren basis [ 163 163 , 175 175 ], 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 2 p decay, and it effectively allows nuclear structures and reactions to be treated on the same footing. As discussed above, the lifetime of 67 Kr can be influenced by deformation effects [ 49 49 ]. Indeed, studies of single-proton (1 p ) emitters [ 172 172 - 174 174 172-174 , 176 176 - 181 181 176-181 ] have demonstrated the impact of rotational and vibrational couplings on 1 p half-lives. The corresponding influence in the 2 p decay of 67 Kr has been studied using the deformed GCC method [ 170 170 ]. Figure 19(a) Figure 19(a) shows the proton Nilsson levels (labeled with asymptotic quantum numbers Ω [Nn z Λ]) 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 0 g 9/2 shell. This gap is responsible for the oblate ground-state shapes of proton-deficient Kr isotopes [ 182 182 - 184 184 182-184 ]. 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 67 Kr. Figure 19(b) Figure 19(b) shows the 2 p 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) Fig. 19(b) . The decreasing 𝒷 content of the 2 p wave function results in a dramatic increase in the decay width. At the deformation <math id="M52"><mrow><msub><mi>β</mi><mn>2</mn></msub><mo>≈</mo><mo>−</mo><mn>0.3</mn></mrow></math> , the calculated 2 p ground-state half-life of 67 Kr is <math id="M53"><mrow><msubsup><mrow><mn>24</mn></mrow><mrow><mo>−</mo><mn>7</mn></mrow><mrow><mo>+</mo><mn>10</mn></mrow></msubsup></mrow></math> ms [ 170 170 ], which agrees with experimental results [ 49 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 2 p radioactivity. 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 [ 185 185 ], decay rate [ 186 186 ], and fission [ 187 187 ]) 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 2 p emission was proposed in Ref. [ 188 188 ], in which the c.m. motion for the two valence protons was described classically. In a more realistic case, the early stage of the 2 p emission from the ground state of 6 Be was well demonstrated using a time-dependent method in Refs. [ 124 124 , 165 165 ]. Initially, ( t =0) when the wave function is relatively localized inside the nucleus, the density distribution shows two maxima for 6 Be; these are associated with the diproton/cigarlike configuration characterized by small/large relative distances between valence protons (see Fig. 17 Fig. 17 ). Meanwhile, the different time snapshots of the density distribution ρ and the corresponding density changes ρ d are shown in Fig. 20 Fig. 20 . ρ and ρ d are defined as <math display="block" id="M54"><mtable columnalign="left"><mtr><mtd><mi>ρ</mi><mo>=</mo><mn>8</mn><msup><mi>π</mi><mn>2</mn></msup><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup><msubsup><mi>r</mi><mn>2</mn><mn>2</mn></msubsup><mi>sin</mi><msub><mi>θ</mi><mrow><mn>12</mn></mrow></msub><msup><mrow><mo>|</mo><mrow><mtext>Ψ</mtext><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>;</mo></mtd></mtr><mtr><mtd><msub><mi>ρ</mi><mi>d</mi></msub><mo>=</mo><mn>8</mn><msup><mi>π</mi><mn>2</mn></msup><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup><msubsup><mi>r</mi><mn>2</mn><mn>2</mn></msubsup><mi>sin</mi><msub><mi>θ</mi><mrow><mn>12</mn></mrow></msub><msup><mrow><mo>|</mo><mrow><msub><mtext>Ψ</mtext><mi>d</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>/</mo><mo>〈</mo><msub><mtext>Ψ</mtext><mi>d</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>|</mo><msub><mtext>Ψ</mtext><mi>d</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>〉</mo><mo>;</mo></mtd></mtr><mtr><mtd><mrow><mo>|</mo><mrow><msub><mtext>Ψ</mtext><mi>d</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>〉</mo><mo>=</mo></mrow><mo>|</mo></mrow><mtext>Ψ</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>〉</mo><mo>−</mo><mo>〈</mo><mtext>Ψ</mtext><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>|</mo><mtext>Ψ</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>〉</mo><mo>|</mo><mtext>Ψ</mtext><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>〉</mo><mn>.</mn></mtd></mtr></mtable></math> During the early stage of decay, two strong branches are emitted from the inner nucleus, as shown in Fig. 20 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 2 p tunneling probability [ 51 51 , 123 123 , 124 124 , 165 165 ]. The secondary branch corresponds to protons emitted in opposite directions. This accords with the flux current analysis shown in Fig. 17 Fig. 17 . To capture the asymptotic dynamics and improve our understanding of the role of final-state interactions, the wave function of 6 Be 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 [ 125 125 ]. After tunneling through the Coulomb barrier, the two emitted protons tend to gradually separate under Coulomb repulsion [ 125 125 , 165 165 ]. Eventually, the 2 p density becomes spatially diffused, which is consistent with the broad angular distribution measured in Ref. [ 98 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 [ 189 189 ]. Figure 21 Figure 21 illustrates the dramatic changes in the wave function and configurations of 6 Be 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 Fig. 21 , can be approximated by <math id="M55"><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><msub><mi>Q</mi><mrow><mn>2</mn><mi>p</mi></mrow></msub></mrow><mo>)</mo></mrow><mi>t</mi><mo>=</mo><mi mathvariant="fraktur">n</mi><mi>π</mi><mi>ℏ</mi></mrow></math> , where n = 1, 3, 5 ⋯ (i.e., they explicitly depend on the Q 2 p energy). Moreover, the configuration evolution also reveals a unique feature of three-body decay. As seen in Fig. 21 Fig. 21 , the initial ground state of 6 Be 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 <math id="M57"><mrow><mo stretchy="false">(</mo><mi>K</mi><mn>,</mn><msub><mi mathvariant="script">l</mi><mstyle mathvariant="bold"><mtext>x</mtext></mstyle></msub><mn>,</mn><msub><mi mathvariant="script">l</mi><mi>y</mi></msub><mn>,</mn><mi>S</mi><mo stretchy="false">)</mo></mrow></math> = (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 [ 154 154 ]. 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 29 , 30 30 , 33 33 , 35 35 ]. In addition, for 2 p decays, the Coulomb potential and kinetic energy do not commute in the asymptotic region [ 122 122 ], leading to additional configuration mixing. It has been also found that the 2 p decay width is sensitive to the strength of the pairing interaction [ 125 125 , 165 165 , 191 191 ], because the diproton structure promotes the tunneling process. Moreover, Figure. 22 Figure. 22 shows the asymptotic correlations of 6 Be as a function of the nucleon-nucleon interaction strength V pp/nn [ 125 125 ]. 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 <math id="M58"><mrow><mover accent="true"><mi>G</mi><mo>^</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mo>−</mo><mover accent="true"><mi>H</mi><mo>^</mo></mover><mo>+</mo><mi>i</mi><mi>η</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> : <math display="block" id="M59"><mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mfrac><mover accent="true"><mi>H</mi><mo>^</mo></mover><mi>ℏ</mi></mfrac><mi>t</mi></mrow></msup><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mrow><mfrac><mi>η</mi><mi>ℏ</mi></mfrac><mi>t</mi></mrow></msup></mrow><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow></mfrac><mi mathvariant="script">F</mi><mrow><mo>(</mo><mrow><mover accent="true"><mi>G</mi><mo>^</mo></mover><mn>,</mn><mi>E</mi><mo>→</mo><mfrac><mi>t</mi><mrow><mn>2</mn><mi>π</mi><mi>ℏ</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> As shown in Fig. 22 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 θ k . 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. Open Problems 1 Because the exotic phenomenon of direct 2 p decay was discovered not long ago, only a few nuclei have thus far been found to exhibit it, which precludes systematic studies. Meanwhile, the corresponding microscopic theories remain under development. Although impressive progress has been achieved in both experiment and theory, many open problems remain under debate. Here, we briefly introduce some of them that are crucial for a better understanding of the 2 p decay process; this might help us gain a deeper insight into the properties of open quantum systems. Connecting structure with asymptotic correlation 2 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 [ 166 166 , 192 192 - 194 194 192-194 ]. Regarding the 2 p 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 10 , 31 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 2 p decay in the presence of long-range Coulomb interactions. In one pioneering work, 45 Fe was—by analyzing the correlation of the emitted valence protons—found to contain a moderate amount of the p 2 component, [ 18 18 , 148 148 ]. Later, the nucleon-nucleon correlations of psd -shell 2 p emitters were systematically studied. As shown in Fig. 23 Fig. 23 , among these 2 p emitters, the energy correlations of 12 O and its isobaric analog 12 N IAS resemble those of 16 Ne but differ notably from the energy correlations of 6 Be and 8 B IAS . This indicates that the valence protons of 12 O 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 2 p 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. Excited 2p decay 2 As discussed, the phenomenon of 2 p decay is not limited to the ground state: more and more excited nuclei have been found to exhibit 2 p decay. Unlike ground-state 2 p emitters, the 2 p decay from excited states usually has a large decay energy Q 2 p . 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 2 p emitters. Meanwhile, nuclei can exhibit β + -delayed 2 p emission. For instance, one such decay process was observed from the ground state of 22 Al [ 76 76 ]. The decay proceeds through the isobaric analogue state (IAS) of 22 Mg to the first excited state in 20 Ne, 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 2 p decay cannot change the isospin from T = 2 in the case of 22 Al and 22 Mg IAS to T = 0 in the case of 20 Ne. This indicates that isospin mixing is crucial in such β + -delayed 2 p emission processes; this requires further theoretical and experimental investigations. Beyond the proton/neutron dripline, exotic decays may happen. 2 p decay is just one of them. Recently, more and more exotic particle emissions (e.g., 3 p or 4 p decay) have been experimentally observed. The ground state of 31 K has been found to exhibit a 3 p decay [ 195 195 ]. Moreover, the newly discovered elements 8 C and 18 Mg are considered to exhibit 4 p emissions [ 97 97 , 99 99 ]. Owing to the large imbalance of the proton-neutron ratio, these 3 p or 4 p emitters are strongly coupled to the continuum and are more unstable compared to the 2 p emitters. The decay mechanism of these multi-particle emissions is an interesting phenomena to investigate. As shown in Fig. 24(b) Fig. 24(b) , 8 C and 18 Mg arguably feature a 2 p +2 p decay mode, in which the intermediate states are the ground states of 6 Be and 16 Ne, 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. Multi-proton radioactivity 2 Multi-neutron emission and mirror symmetry breaking 2 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 [ 196 196 , 197 197 ], 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 2 p and two-neutron radioactivity [ 15 15 , 17 17 , 18 18 , 125 125 ]. 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 (2 n ) decay is limited. So far, the weakly neutron-unbound 16 Be and 26 O are arguably the best current candidates for observing the phenomenon of 2 n radioactivity [ 33 33 , 35 35 , 198 198 - 204 204 198-204 ]. Beryllium isotopes usually suffer large deformations, which might influence the 2 n decay process of 16 Be. As for 26 O, its ground state is very near to the threshold, which might result in some unique phenomena. Inside the nucleus, the initial 2 n density of 26 O also contains a dineutron, cigarlike, and triangular structure [ 33 33 , 162 162 ]. These three configurations are characteristic of the d -wave component. The very small Q 2 n =18± 5 keV value [ 199 199 ] in 26 O makes this nucleus a candidate for 2 n 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 Fig. 25 , the angular correlation of 26 O becomes almost uniformly distributed. This is because, for small values of Q 2 n , the 2 n decay is dominated by the s -wave component, and the angular distribution becomes essentially isotropic [ 128 128 , 154 154 , 203 203 ]. This asymptotic behavior might represent an interesting avenue of research for further experimental studies. When extending further beyond the neutron dripline, 28 O is predicted to be a 4 n emitter [ 204 204 - 206 206 204-206 ]. It has been found recently that the four neutrons can exist transiently without any other matter [ 207 207 ]. Although it remains under debate whether the observed peak is a resonance [ 208 208 - 214 214 208-214 ], it does show several correlations in the presence of nuclear media, which constitutes an interesting feature to be further studied in 28 O and similarly proton-rich systems. Summary 1 As one of the most recently discovered exotic decay modes, 2 p radioactivity has attracted considerable theoretical and experimental attention. The massive processes have been desgined to help us gain insight into the 2 p decay mechanism and its corresponding asymptotic observables. In this review, we briefly introduced a small fraction of the recent studies. In terms of experiments, by using the decay dynamics and lifetime, different methods and detectors (e.g., including in-flight decay techniques, implantation decay, and time projection chambers) have been developed. Moreover, new techniques such as waveform sampling are under development as an extension of the traditional methods. Meanwhile, the theoretical developments (starting from the configuration interaction and three-body model) are aiming to construct a unified framework in which the interplay of structural information and decay properties can be more fully understood. With the establishment of next-generation rare isotope beam facilities, more and more 2 p emitters (as well as other exotic decays) will be discovered. 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All authors read and approved the final manuscript. 10.1007/s41365-022-01091-1 This work is partially supported by the National Natural Science Foundation of China (Nos. 12147101, 11925502, 11935001, 11961141003, 11890714), the National Key R&D Program of China (No. 2018YFA0404404), the Strategic Priority Research Program of Chinese Academy of Sciences (No. XDB34030000), and the Shanghai Development Foundation for Science and Technology (No. 19ZR1403100). 2022-04-25 2022-06-30 2022-07-01 2022-08-22 2022-09-26 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2022 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01091-1 http://dx.doi.org/10.1007/s41365-022-01091-1 . 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