Two-proton radioactivity of the excited state within the Gamowlike and modified Gamow-like models

Two-proton radioactivity of the excited state within the Gamowlike and modified Gamow-like models 1 first-author 1 2 corresp † liuhongming13@126.com † liuhongming13@126.com liuhongming13@126.com 3 corresp ‡ wuxijun1980@yahoo.cn ‡ wuxijun1980@yahoo.cn wuxijun1980@yahoo.cn 4 1 5 6 7 corresp § lixiaohuaphysics@126.com § lixiaohuaphysics@126.com lixiaohuaphysics@126.com https://orcid.org/0000-0002-3399-6057 School of Nuclear Science and Technology, University of South China, Hengyang 421001, China School of Nuclear Science and Technology, University of South China, Hengyang 421001, China Institute of Modern Physics, Fudan University, Shanghai 200433, China Institute of Modern Physics, Fudan University, Shanghai 200433, China School of Math and Physics, University of South China, Hengyang 421001, China School of Math and Physics, University of South China, Hengyang 421001, China College of Physics and Electronics, Central South University, Changsha 410083, China College of Physics and Electronics, Central South University, Changsha 410083, China National Exemplary Base for International Sci & Tech. Collaboration of Nuclear Energy and Nuclear Safety, University of South China, Hengyang 421001, China National Exemplary Base for International Sci & Tech. Collaboration of Nuclear Energy and Nuclear Safety, University of South China, Hengyang 421001, China Cooperative Innovation Center for Nuclear Fuel Cycle Technology & Equipment, University of South China, Hengyang 421001, China Cooperative Innovation Center for Nuclear Fuel Cycle Technology & Equipment, University of South China, Hengyang 421001, China Key Laboratory of Low Dimensional Quantum Structures and Quantum Control, Hunan Normal University, Changsha 410081, China Key Laboratory of Low Dimensional Quantum Structures and Quantum Control, Hunan Normal University, Changsha 410081, China In this study, we systematically investigated the two-proton (2p) radioactivity half-lives from the excited state of nuclei near the proton drip line within the Gamow-like model (GLM) and modified Gamow-like model (MGLM). The calculated results were highly consistent with the theoretical values obtained using the unified fission model [Chin. Phys. C 45, 124105 (2021)], effective liquid drop model, and generalized liquid drop model [Acta Phys. Sin 71, 062301 (2022)] . Furthermore, utilizing the GLM and MGLM, we predicted the 2p radioactivity half-lives from the excited state for some nuclei that are not yet available experimentally. Simultaneously, by analyzing the calculated results from these theoretical models, it was found that the half-lives are strongly dependent on Q 2p and <math id="M1"><mi mathvariant="script">l</mi></math> . 2p radioactivity Gamow-like model Half-life Excited state Introduction 1 Since the discovery of radioactivity by Becquerel more than 100 years ago, different modes of nuclear decay and reactions have been researched, including alpha decay [ 1 1 - 3 3 1-3 ], beta decay [ 4 4 ], fragmentation reactions [ 5 5 , 6 6 ], and heavy-ion collisions [ 7 7 - 10 10 7-10 ]. In recent years, research on exotic decay around the proton drip line has attracted considerable attention. It is mainly investigated using proton and two-proton (2p) radioactivity processes [ 11 11 - 18 18 11-18 ], and the latter has been proposed as an extremely exotic decay mode for proton-rich nuclei far from the valley of beta stability. In 1960, Zel’dovich [ 19 19 ] reported that a pair of protons may be emitted from radioactive proton-rich nuclei. Subsequently, Goldansky and Jänecke attempted to determine candidates for 2p radioactivity, and Goldansky also coined the term ‘two-proton radioactivity.’ However, it was Galitsky and Cheltsov [ 20 20 ] who conducted the first theoretical attempt to describe the process of 2p radioactivity. Moreover, studying 2p radioactivity can extract abundant nuclear structure information, such as the sequences of particle energies, the wave function of two emitted protons, spin and parity, and the deformation effect [ 21 21 - 23 23 21-23 ]. More than 40 years after its theoretical proposal, the long-lived phenomenon of the decay of 2p radioactivity from the ground state to the ground state was first discovered, namely 45 Fe at GSI [ 24 24 ] and GANIL [ 25 25 ]. Subsequently, a series of the same phenomena was also detected, such as 54 Zn [ 26 26 , 27 27 ], 19 Mg [ 28 28 ], and 94 Ag [ 29 29 - 32 32 29-32 ]. In addition, short-lived radioactive nuclei processing the excited state can also produce the 2p phenomenon. Jänecke was the first to discuss β -delayed 2p ( β 2p) emission [ 33 33 ]. In 1983, the β 2p radioactivity of 22 Al was observed at the Lawrence Berkeley National Laboratory (LBL) for the first time [ 34 34 ], followed by further β 2p emitters, such as 23 Si [ 35 35 ], 26 P [ 36 36 ], 27 S [ 37 37 ], and 50 Ni [ 38 38 ]. For 2p radioactivity from the excited state, except for β 2p radioactivity, some 2p emitters may be fed by nuclear reactions, such as pick-up, transfer, or fragmentation, for example, 14 O [ 39 39 ], 17,18 Ne [ 40 40 - 44 44 40-44 ], 22 Mg [ 45 45 , 46 46 ], and 28,29 S [ 47 47 , 48 48 ]. The lifetimes of the excited 2p emissions are extremely short, approximately 10 -21 s, which is significantly shorter than the lifetimes of the ground-state 2p radioactivity originally predicted by the theory. From a theoretical point of view, during the last decades, several approaches have been applied to describe the emission mechanism and determine the typical half-life of 2p radioactivity. Whether the two protons emitted in this decay process are related to energy and angle is a question that has attracted attention for a long time. In general, there are three different mechanisms by which proton-rich nuclei emit two protons: (i) sequential emission, where two protons are successively emitted from the parent nucleus, and there is no relationship between them; (ii) three-body simultaneous emission, where two protons are emitted from the parent nucleus simultaneously, and the correlation is weak; and (iii) diproton emission (also called 2 He cluster emission). The cluster emission of 2 He is an extreme case with the emission of two strongly correlated protons that can only exist for a short period of time and then separate after penetrating the Coulomb barrier. The three-body simultaneous emission model treats radioactivity as a process in which the parent nucleus contains two protons and a remnant core. To date, a number of models and/or formulae have been proposed to handle the 2p radioactivity of the ground state [ 49 49 - 62 62 49-62 ]. In particular, the Gamow-like model (GLM) was proposed in 2013 by Zdeb et al. as a single-parameter model based on Wentzel-Kramers-Brillouin (WKB) theory to study α decay and cluster radioactivity [ 63 63 ]. Subsequently, the GLM proved to be successful in investigating the proton radioactivity and 2p radioactivity of the ground state [ 64 64 , 65 65 ]. Considering that two emitted protons form a 2 He cluster, the GLM assumed that the 2p radioactivity is due to the quantum mechanical tunneling of a charged two-proton particle through the nuclear Coulomb barrier. Under the assumption of a uniform charge distribution, the inner potential of the GLM is expressed as a square potential well, and the outer potential defaults to the Coulomb potential. As a result of the inhomogeneous charge distribution in the nucleus, superposition of the emitted particles, etc., the electrostatic shielding effect should be considered in the outer potential. Based on our previous studies, in which an exponential-type electrostatic potential, that is, the Hulthén potential, was introduced to describe the outer potential, Liu et al. modified the Gamow-like model, denoted as the MGLM, to calculate the half-lives of 2p radioactivity from the ground state [ 66 66 ]. The MGLM shows that the theoretical half-lives are in good agreement with the experimental data. It is certainly interesting to examine whether the GLM and its modified version can be extended to study the 2p radioactivity of the excited state. To this end, in the present study, we systematically analyzed the half-lives of 2p radioactive nuclei close to the proton drip line using the GLM and MGLM. The theoretical values were shown to be compatible with those of equivalent experiments. The remainder of this paper is organized as follows: In Sect. 2, the theoretical frameworks of the GLM and MGLM are briefly presented. Detailed results and discussion are presented in Sect. 3. Finally, a summary is given in Sect. 4. Theoretical framework 1 The 2p radioactivity half-life is typically calculated using <math display="block" id="M2"><mrow><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>ln</mi><mn>2</mn></mrow><mi>λ</mi></mfrac><mn>,</mn></mrow></math> where λ denotes the decay constant. It can be expressed as <math display="block" id="M3"><mrow><mi>λ</mi><mo>=</mo><mi>ν</mi><msub><mi>S</mi><mrow><mtext>2p</mtext></mrow></msub><mtext>P</mtext><mn>,</mn></mrow></math> where ν denotes the assault frequency associated with the harmonic oscillation frequency in the Nilsson potential [ 67 67 ]. It can be expressed as <math display="block" id="M4"><mrow><mi>h</mi><mi>ν</mi><mo>=</mo><mi>ℏ</mi><mi>ω</mi><mo>≃</mo><mfrac><mrow><mn>41</mn></mrow><mrow><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac><mn>,</mn></mrow></math> where h , ℏ , A , and ω are the Planck constant, reduced Plank constant, mass number of the parent nucleus, and angular frequency, respectively. S 2p = G 2 [ A /( A -2)] 2 n χ 2 represents the preformation probability of the two emitted protons in the parent nucleus, which is obtained using the cluster overlap approximation with G 2 = (2 n )!/[2 2 n ( n !) 2 ] [ 68 68 , 69 69 ]. Here, n  ≈(3 Z ) 1/3 -1 is the average principal proton oscillator quantum number, and χ 2 is the proton overlap function related to Ref. [ 70 70 ]. P is the Gamow penetration probability through the barrier, which can be calculated using the WKB approximation. It can be expressed as <math display="block" id="M5"><mrow><mi>P</mi><mo>=</mo><mtext>exp</mtext><mrow><mo>[</mo><mrow><mo>−</mo><mn>2</mn><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>R</mi><mrow><mi>i</mi><mi>n</mi></mrow></msub></mrow><mrow><msub><mi>R</mi><mrow><mi>o</mi><mi>u</mi><mi>t</mi></mrow></msub></mrow></msubsup></mrow></mstyle><msqrt><mrow><mfrac><mrow><mn>2</mn><mi>μ</mi></mrow><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>E</mi><mtext>k</mtext></msub><mo stretchy="false">)</mo></mrow></msqrt><mtext>d</mtext><mi>r</mi></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where <math id="M6"><mrow><mi>μ</mi><mo>=</mo><mfrac><mrow><msub><mi>m</mi><mrow><mtext>2p</mtext></mrow></msub><msub><mi>m</mi><mtext>d</mtext></msub></mrow><mrow><msub><mi>m</mi><mrow><mtext>2p</mtext></mrow></msub><mo>+</mo><msub><mi>m</mi><mtext>d</mtext></msub></mrow></mfrac></mrow></math> ≈ 938.3× 2 × A d / A MeV/ c 2 , with 2p , m d and A d as the mass of the two emitted protons, the residual daughter nucleus, and the mass number of the daughter nucleus, respectively. E k = Q 2p ( A -2)/ A denotes the kinetic energy of the two emitted protons, and <math id="M7"><mrow><msub><mi>R</mi><mrow><mtext>in</mtext></mrow></msub><mo>=</mo><msub><mi>r</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msubsup><mi>A</mi><mrow><mtext>2p</mtext></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mo>+</mo><msubsup><mi>A</mi><mtext>d</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></math> represents the spherical square well radius, where A 2p and r 0 are the mass number of the two emitted protons and the effective nuclear radius parameter, respectively. In this study, r 0 =1.28 fm, which is taken from Ref. [ 71 71 ]. R out is the outer turning point of the potential barrier, which satisfies the condition V ( r out ) = E k . In the framework of the GLM, V ( r ) is the total interaction potential between the two emitted protons and daughter nucleus, a square potential well represents the inner nuclear interaction potential, and a Coulomb potential, V C ( r ), is defaulted to represent the outer electrostatic potential. It can be expressed as <math display="block" id="M8"><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><msub><mi>V</mi><mn>0</mn></msub><mn>,</mn></mrow></mtd><mtd columnalign="left"><mrow><mn>0</mn><mo>⩽</mo><mi>r</mi><mo>⩽</mo><msub><mi>R</mi><mrow><mtext>in</mtext></mrow></msub><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mi mathvariant="script">l</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd><mtd columnalign="left"><mrow><mi>r</mi><mo>></mo><msub><mi>R</mi><mrow><mtext>in</mtext></mrow></msub><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></mrow></mrow></math> where V 0 denotes the depth of the potential well. Based on Blendowske et al. [ 72 72 ], we chose V 0 = 25 A 2p MeV. V C ( r ) = Z 2p Z d e 2 / r , where Z 2p and Z d are the proton numbers of the two emitted protons and daughter nucleus, respectively, and r is the mass center distance between the two emitted protons and daughter nucleus. To consider the electrostatic shielding effect, in the MGLM, we introduced an exponential-type electrostatic potential known as the Hulthén potential V H ( r ), which has been widely applied in the fields of atomic, molecular, and solid-state physics [ 73 73 - 77 77 73-77 ], to replace V C ( r ) for 2p radioactivity from the ground state. To display the difference between V H ( r ) and V C ( r ), we plotted a sketch of the total interaction potential between the two emitted protons and daughter nucleus versus the center-of-mass distance of the decay system in Fig. 1 Fig. 1 . From this figure, we can see that V H ( r ) has the same behavior as V C ( r ) within R in but drops quickly when <math id="M9"><mrow><mi>r</mi><mo>≫</mo><mn>0</mn></mrow></math> . Namely, the total interaction potential V ( r ) between the two emitted protons and daughter nucleus changes as follows: <math display="block" id="M10"><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><msub><mi>V</mi><mn>0</mn></msub><mn>,</mn></mrow></mtd><mtd columnalign="left"><mrow><mn>0</mn><mo>⩽</mo><mi>r</mi><mo>⩽</mo><msub><mi>R</mi><mrow><mtext>in</mtext></mrow></msub><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>V</mi><mtext>H</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mi mathvariant="script">l</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd><mtd columnalign="left"><mrow><mi>r</mi><mo>></mo><msub><mi>R</mi><mrow><mtext>in</mtext></mrow></msub><mn>.</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></mrow></mrow></math> V H ( r ) is the Hulthén potential and can be expressed as <math display="block" id="M37"><mrow><msub><mi>V</mi><mtext>H</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>a</mi><msub><mi>Z</mi><mrow><mtext>2p</mtext></mrow></msub><msub><mi>Z</mi><mtext>d</mtext></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><msup><mi>e</mi><mrow><mi>a</mi><mi>r</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mn>,</mn></mrow></math> where a =1.80810 -3 fm -1 denotes the screening parameter related to Ref. [ 66 66 ]. This can be used to determine the range of the potential, that is, the shortening of the exit radius. Additionally, we consider the effect of the centrifugal potential <math id="M38"><mrow><msub><mi>V</mi><mi mathvariant="script">l</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></math> on the half-life of 2p radioactivity in the GLM and MGLM. Because <math id="M39"><mrow><mi mathvariant="script">l</mi><mo stretchy="false">(</mo><mi mathvariant="script">l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="script">l</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math> is a necessary correction for one-dimensional problems [ 78 78 ], we chose the centrifugal potential <math id="M40"><mrow><msub><mi>V</mi><mi mathvariant="script">l</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></math> as the Langer modified form, which can be expressed as <math display="block" id="M41"><mrow><msub><mi>V</mi><mi mathvariant="script">l</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="script">l</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>μ</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M42"><mi mathvariant="script">l</mi></math> is the orbital angular momentum removed by the two emitted protons, which is calculated by parity and angular momentum conservation laws. Results and discussion 1 Based on our previous study [ 65 65 , 66 66 ], the main intention of this study is to extend the GLM and MGLM to the 2p radioactivity of the excited state. The selected two-proton emitters from the excited state were those with known experimentally released energies, which are both available and considerable. In this study, we calculated the half-lives of 2p radioactivity for 14 O * , 17 Ne * , 18 Ne * , 22 Mg * , 29 S * , and 94 Ag * ( * represents the excited state), and the results are listed in Table 1 Table 1 . The experimental data and theoretical results from the unified fission model (UFM) [ 79 79 ], effective liquid drop model (ELDM), and generalized liquid drop model (GLDM) [ 80 80 ] are also listed in this table. In Table 1 Table 1 , the first column represents the 2p decay process, and the second column represents the spin and parity of the initial and final states of the nucleus. The third column represents the angular momentum removed by the emitted two protons, which obeys spin-parity conservation laws, and the fourth column shows the experimental two-proton released energy, denoted as Q 2p . The sixth to tenth columns represent the logarithmic forms of 2p half-lives obtained using the GLM, MGLM, ELDM, GLDM, and UFM, respectively. In general, from this table, it is clear that the theoretical half-lives obtained using the GLM and MGLM are highly consistent with those of other theoretical models. Moreover, it is clear that the half-lives are sensitive to the released energy Q 2p and angular momentum <math id="M49"><mi mathvariant="script">l</mi></math> . 1 1 2p emission 1 1 <math id="M11"><mrow><msubsup><mi>j</mi><mi>i</mi><mi>π</mi></msubsup><mi>→</mi><msubsup><mi>j</mi><mi>f</mi><mi>π</mi></msubsup></mrow></math> 1 1 <math id="M12"><mi mathvariant="script">l</mi></math> 1 1 Q 2p (MeV) 1 1 <math id="M13"><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>Expt</mtext><mtext>.</mtext></mrow></msubsup></mrow></math>   1 1 <math id="M14"><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>GLM</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M15"><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>MGLM</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M16"><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>ELDM</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M17"><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>GLDM</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M18"><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>UFM</mtext></mrow></msubsup></mrow></math>   1 1 * 14 O * → 12 C 1 1 <math id="M19"><mrow><msup><mn>2</mn><mo>+</mo></msup><mtext>→</mtext><msup><mrow><mn>0</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 2 1 1 1.20 [ 39 39 ] 1 1 > -16.12 [ 39 39 ] 1 1 -15.68 1 1 -15.41 1 1 -15.49 1 1 -16.10 1 1 -16.02 1 1   1 1 <math id="M20"><mrow><msup><mn>2</mn><mo>+</mo></msup><mtext>→</mtext><msup><mrow><mn>0</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 2 1 1 3.15 [ 39 39 ] 1 1   1 1 -18.35 1 1 -18.07 1 1 -18.22 1 1 -19.58 1 1 -18.87 1 1   1 1 <math id="M21"><mrow><msup><mn>4</mn><mo>+</mo></msup><mi>→</mi><msup><mrow><mn>0</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 4 1 1 3.35 [ 39 39 ] 1 1   1 1 -16.36 1 1 -15.91 1 1 -16.25 1 1 -16.76 1 1 -15.96 1 1 * 17 Ne * → 15 O 1 1 <math id="M22"><mrow><msup><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mi>→</mi><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup></mrow></math> 1 1 2 1 1 0.35 [ 40 40 , 43 43 ] 1 1 > -10.59 [ 43 43 ] 1 1 -6.84 1 1 -6.98 1 1 -6.98 1 1 -6.79 1 1 -7.11 1 1   1 1 <math id="M23"><mrow><msup><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mi>→</mi><msup><mrow><mn>2192</mn><mo>;</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup></mrow></math> 1 1 2 1 1 0.82 [ 40 40 , 43 43 ] 1 1   1 1 -12.58 1 1 -12.36 1 1 -12.41 1 1 -12.68 1 1 -12.73 1 1   1 1 <math id="M24"><mrow><msup><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mi>→</mi><msup><mrow><mn>2192</mn><mo>;</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup></mrow></math> 1 1 1 1 1 0.97 [ 40 40 , 43 43 ] 1 1   1 1 -14.37 1 1 -14.18 1 1 -14.20 1 1 -14.68 1 1 -14.69 1 1 * 18 Ne * → 16 O 1 1 <math id="M25"><mrow><msup><mn>2</mn><mo>+</mo></msup><mi>→</mi><msup><mrow><mn>2192</mn><mo>;</mo><mn>0</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 2 1 1 0.59 [ 41 41 ] 1 1   1 1 -10.69 1 1 -10.56 1 1 -10.59 1 1 -10.96 1 1 -10.91 1 1   1 1 <math id="M26"><mrow><msup><mn>1</mn><mo>−</mo></msup><mi>→</mi><msup><mrow><mn>2192</mn><mo>;</mo><mn>0</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 1 1 1 1.63 [ 41 41 ] 1 1 <math id="M27"><mrow><mo>−</mo><msubsup><mrow><mn>16.15</mn></mrow><mrow><mo>−</mo><mn>0.06</mn></mrow><mrow><mo>+</mo><mn>0.06</mn></mrow></msubsup></mrow></math> [ 41 41 , 42 42 ] 1 1 -16.60 1 1 -16.36 1 1 -16.34 1 1 -17.20 1 1 -16.79 1 1 22 Mg * → 20 Ne 1 1 <math id="M28"><mrow><mo>−</mo><mi>→</mi><msup><mrow><mn>2192</mn><mo>;</mo><mn>0</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 0 1 1 6.11 [ 45 45 , 46 46 ] 1 1   1 1 -19.59 1 1 -19.59 1 1 -19.75 1 1 -19.58 1 1 -18.97 1 1 * 29 S * → 27 Si 1 1 <math id="M29"><mrow><mo>−</mo><mi>→</mi><msup><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 0 1 1 1.72-2.52 [ 47 47 ] 1 1   1 1 -16.01 <math id="M30"><mover accent="true"><mo>−</mo><mo>˜</mo></mover></math> 13.89 1 1 -15.72 <math id="M31"><mover accent="true"><mo>−</mo><mo>˜</mo></mover></math> 13.64 1 1 -15.5 <math id="M32"><mover accent="true"><mo>−</mo><mo>˜</mo></mover></math> 13.4 1 1 -17.2∼-14.7 1 1 -16.4 <math id="M33"><mover accent="true"><mo>−</mo><mo>˜</mo></mover></math> 14.3 1 1   1 1 <math id="M34"><mrow><mo>−</mo><mi>→</mi><msup><mrow><mn>2192</mn><mo>;</mo><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 0 1 1 4.32-5.12 [ 47 47 ] 1 1   1 1 -18.72∼-18.22 1 1 -18.47∼-17.94 1 1 -18.4∼-17.8 1 1 -19.2∼-18.8 1 1 -18.9∼-18.5 1 1 * 94 Ag * → 92 Rh 1 1 <math id="M35"><mrow><msup><mrow><mn>21</mn></mrow><mo>+</mo></msup><mi>→</mi><msup><mrow><mn>2192</mn><mo>;</mo><mn>11</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 6-10 1 1 1.90 [ 29 29 ] 1 1 1.90 <math id="M36"><mrow><msubsup><mrow/><mrow><mo>−</mo><mn>0.20</mn></mrow><mrow><mo>+</mo><mn>0.38</mn></mrow></msubsup></mrow></math> [ 29 29 ] 1 1 7.90 ∼ 12.91 1 1 7.81 ∼ 13.08 1 1 9.42 ∼ 14.63 1 1 8.22 ∼ 13.38 1 1 9.38 ∼ 15.21 1 1   1 1   1 1   1 1 1.98 [ 90 90 ] 1 1   1 1 7.09 ∼ 12.08 1 1 7.07 ∼ 12.31 1 1 8.61 ∼ 13.80 1 1 7.41 ∼ 12.55 1 1 8.56 ∼ 14.37 1 1   1 1   1 1   1 1 2.05 [ 90 90 ] 1 1   1 1 6.42 ∼ 11.39 1 1 6.45 ∼ 11.68 1 1 7.95 ∼ 13.11 1 1 6.74 ∼ 11.86 1 1 7.89 ∼ 13.68 1 1   1 1   1 1   1 1 3.45 [ 90 90 ] 1 1   1 1 -2.32 ∼ 2.32 1 1 -1.75 ∼ 3.15 1 1 -0.80 ∼ 4.04 1 1 -2.03 ∼ 2.75 1 1 -0.92 ∼ 4.56 To intuitively understand the effect of <math id="M50"><mi mathvariant="script">l</mi></math> and Q 2p on the half-lives of 2p radioactivity from the excited state, we selected the nuclei 14 O * and 94 Ag * to analyze the contribution of <math id="M51"><mi mathvariant="script">l</mi></math> and Q 2p to the corresponding theoretical half-lives. As shown in Table 1 Table 1 , the half-lives of 14 O * determined using both the GLM and MGLM differed by nearly three magnitudes for the same <math id="M52"><mi mathvariant="script">l</mi></math> value but different released energy Q 2p values ( Q 2p = 1.20 MeV and Q 2p = 3.15 MeV, respectively). In addition, for 94 Ag * , the half-lives obtained using the GLM and MGLM with identical Q 2p and different <math id="M53"><mi mathvariant="script">l</mi></math> differed by three magnitudes, whereas <math id="M54"><mi mathvariant="script">l</mi></math> varied from 6 to 10. It is clear that either Q 2p or <math id="M55"><mi mathvariant="script">l</mi></math> makes a non-negligible contribution to 14 O * and 94 Ag * for their corresponding theoretical half-lives within the GLM, MGLM, ELDM, and GLDM. In fact, the half-lives of 2p radioactivity are highly sensitive to proton-proton interactions owing to the pairing effect of valence protons. Most proton emitters considered in this study are deformed, and in some cases (for example, 67 Kr), both shape and structural changes occur [ 70 70 , 23 23 ]. The most remarkable effect of the deformed nuclear structure on proton-proton correlations is back-to-back emission, in which protons are emitted from opposite sides of the parent nucleus, yet still have strongly linked energies. The quasi-classical 2 He model cannot account for the experimentally observed proton-proton correlations, which indicate back-to-back proton emission. Moreover, to illustrate the agreement of the half-lives of the 2p radioactivity of the excited state, which was calculated using the GLM and MGLM, the theoretical results are shown in Fig. 2 Fig. 2 . It is evident from this figure that the calculated results are highly consistent with those of other theoretical models. In our previous study [ 71 71 ], the New Geiger-Nuttall law was applied to describe two-proton radioactivity within a two-parameter empirical formula. To further test the feasibility of our calculations, we plotted the quantity [log 10 T 1/2 +26.832]/( <math id="M56"><mrow><msubsup><mi>Z</mi><mtext>d</mtext><mrow><mn>0.8</mn></mrow></msubsup><mo>+</mo><msup><mi mathvariant="script">l</mi><mrow><mn>0.25</mn></mrow></msup></mrow></math> ) as a function of <math id="M57"><mrow><msubsup><mi>Q</mi><mrow><mtext>2p</mtext></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></math> , as shown in Fig. 3 Fig. 3 , which was classified with the value of angular momentum. When <math id="M58"><mi mathvariant="script">l</mi></math> =2, 6, and 10, there was a good linear relationship between the quantity [log 10 T 1/2 +26.832]/( <math id="M59"><mrow><msubsup><mi>Z</mi><mtext>d</mtext><mrow><mn>0.8</mn></mrow></msubsup><mo>+</mo><msup><mi mathvariant="script">l</mi><mrow><mn>0.25</mn></mrow></msup></mrow></math> ) and <math id="M60"><mrow><msubsup><mi>Q</mi><mrow><mtext>2p</mtext></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></math> , which is consistent with the results of our previous study. However, it should be noted that there was a poor linear relationship when <math id="M61"><mi mathvariant="script">l</mi></math> =0. To further explain this phenomenon, we checked the corresponding data in Fig. 3 Fig. 3 for <math id="M62"><mi mathvariant="script">l</mi></math> =0. Finally, we found that if 22 Mg * was deleted, a good linear relationship was displayed for <math id="M63"><mi mathvariant="script">l</mi></math> =2, 6, and 10. Based on this phenomenon, we suspect that the orbital angular momentum <math id="M64"><mi mathvariant="script">l</mi></math> removed by the two emitted protons for 22 Mg * is inappropriate because <math id="M65"><mrow><msubsup><mi>j</mi><mi>i</mi><mi>π</mi></msubsup></mrow></math> of 22 Mg * is uncertain. For verification, we modified the value of <math id="M66"><mi mathvariant="script">l</mi></math> for 22 Mg * to 1, 2, 6, and 10 and replotted the quantity [log 10 T 1/2 +26.832]/( <math id="M67"><mrow><msubsup><mi>Z</mi><mtext>d</mtext><mrow><mn>0.8</mn></mrow></msubsup><mo>+</mo><msup><mi mathvariant="script">l</mi><mrow><mn>0.25</mn></mrow></msup></mrow></math> ) as a function of <math id="M68"><mrow><msubsup><mi>Q</mi><mrow><mtext>2p</mtext></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></math> in Fig. 4 Fig. 4 . It was found that when <math id="M69"><mi mathvariant="script">l</mi></math> =1 and 2, there was a good linear relationship. Based on the parity and angular momentum conservation laws, we surmise that <math id="M70"><mrow><msubsup><mi>j</mi><mi>i</mi><mi>π</mi></msubsup></mrow></math> of 22 Mg * may be 1 - , 1 + , and 2 + . In recent years, excited 2p radioactivity experimental data have been rare because their extremely short half-lives make it difficult to observe the decay process. However, the abundance of information regarding the nuclear structure in this decay mode merits further investigation. Furthermore, apart from traditional methods that are applied to nuclear physics [ 81 81 - 84 84 81-84 ], the new approach of machining learning is widely applied to describe exotic decay processes [ 85 85 - 89 89 85-89 ], and it is worth extending it to 2p radioactivity in the future. Summary 1 In this study, we extended the GLM and MGLM to study the excited state 2p radioactivity of 14 O * , 17 Ne * , 18 Ne * , 22 Mg * , 29 S * , and 94 Ag * for the first time. The theoretical values obtained using the GLM and MGLM were found to be highly consistent with the corresponding experimental and theoretical values from the ELDM, GLDM, and UFM. 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All authors read and approved the final manuscript. 10.1007/s41365-022-01116-9 This work was supported by the National Natural Science Foundation of China (Nos. 12175100 and 11975132), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of the Education Bureau of Hunan Province, China (No. 18A237), the Natural Science Foundation of Hunan Province, China (No. 2018JJ2321), the Innovation Group of Nuclear and Particle Physics in USC, the Opening Project of the Cooperative Innovation Center for Nuclear Fuel Cycle Technology and Equipment, University of South China (No. 2019KFZ10). 2022-06-24 2022-08-29 2022-08-30 2022-10-08 2022-11-22 © 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. 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