Systematic study of proton radioactivity half-lives

Systematic study of proton radioactivity half-lives 1 first-author 1 1 2 corresp † wuxijun1980@yahoo.cn † wuxijun1980@yahoo.cn wuxijun1980@yahoo.cn 3 corresp ‡ kyois@126.com ‡ kyois@126.com kyois@126.com 1 4 5 6 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 School of Math and Physics, University of South China, Hengyang 421001, China School of Math and Physics, University of South China, Hengyang 421001, China The Research Center for Theoretical Physics, Science School, Qingdao University of Technology, Qingdao 266033, China The Research Center for Theoretical Physics, Science School, Qingdao University of Technology, Qingdao 266033, 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 the present study, on the basis of the screened electrostatic effect of the Coulomb potential, we propose an improved Gamow model within the centrifugal potential in which there are only two adjustable parameters, i.e., the screened parameters t and g , which represent the combined effect of the interaction pot ential and reduced mass of the emitted proton-daughter nucleus on the half-life of proton radioactivity in the overlapping region. Using this model, we systematically calculated the proton radioactivity half-lives of 31 spherical nuclei and 13 deformed nuclei and obtained corresponding root-mean-square deviations of 0.274 and 0.367, respectively. The relationship between the proton radioactivity half-life of 177 Tl m and the corresponding angular momentum l removed by the emitted proton is also discussed. In addition, we used the proposed model to predict the proton radioactivity half-lives of 18 nuclei whose proton radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020. For comparison, we used the universal decay law of proton radioactivity proposed by Qi et al. (Phys Rev C 85:011303, 2012. https://doi.org/10.1103/PhysRevC.85.011303), and the new Geiger–Nuttall law of proton radioactivity proposed by Chen et al. (Eur Phys J 55:214, 2019. https:// doi.org/10.1140/epja/i2019-12927-7). proton radioactivity Gamow model half-lives Introduction 1 Proton radioactivity, i.e., the disintegration of nuclei by emitting a proton and the formation of a new nuclide, is a typical decay mode for odd- Z emitters beyond the proton drip line, which represents a fundamental limit of nuclear existence where the nuclei spontaneously shed off excess protons to stabilize. This phenomenon was first discovered in 1970 by Jackson et al. from a high-spin isomer 53 Co m [ 1 1 , 2 2 ] and independently confirmed by Hofmann et al. and Klepper et al. from ground states of 151 Lu [ 3 3 ] and 147 Tm [ 4 4 ] in 1982. Since then, proton radioactivity has attracted wide attention in the nuclear physics community [ 5 5 - 19 19 5-19 ] because it can provide crucial information of neutron-deficient nuclei, such as their shell structure [ 20 20 ] and the coupling between bound and unbound nuclear states [ 21 21 ]. Moreover, as the inverse process of rapid proton capture, it can contribute significantly to the understanding of element origin and star evolution [ 22 22 ]. With the development of experimental facilities and technology, new nuclei undergoing proton radioactivity have been discovered. So far, there are approximately 45 proton emitters with 51 ≤ Z ≤ 83 detected [ 23 23 - 26 26 23-26 ], 15 of which are in isomeric state and the remaining are in ground state. We can also divide them into approximately 32 spherical nuclei and 13 deformed nuclei, according to the degree of deformation. Theoretically, proton radioactivity obeys the quantum tunnel theory, i.e., a proton tunnels through a potential barrier between the emitted proton and daughter nucleus, which is the same decay mechanism as α decay. Based on this description, a great number of models, microscopic approaches, and empirical formulas have been proposed to analyze this process, such as the effective interaction potential model of density dependent M3Y (DDM3Y) [ 27 27 ], distorted-wave Born approximation (DWBA) [ 28 28 ], Lejeune and Mahaux (JLM) [ 29 29 ], coupled-channels approach (CCA) [ 30 30 - 32 32 30-32 ], unified fission model (UFM) [ 33 33 , 34 34 ], two-potential approach (TPA) [ 35 35 ], generalized liquid drop model (GLDM) [ 36 36 - 38 38 36-38 ], single fold model (SFM) [ 39 39 ], Coulomb and proximity potential model (CPPM) [ 40 40 ], universal decay law of proton radioactivity (UDLP) [ 41 41 ], and new Geiger-Nuttall law of proton radioactivity (NG-N) [ 42 42 ], among others. These studies have greatly advanced our understanding of proton radioactivity and are still evolving. In 2005, based on the Gamow model and considering the overlapping effect, Tavares et al. first proposed the one-parameter model (OPM) to study the α decay of bismuth isotopes with an angular momentum l =5 removed by the emitted α particle [ 43 43 ]. Subsequently, OPM was applied to evaluate the α decay half-lives of platinum, neptunium, and uranium isotopes, with the calculated results being in good agreement with the experimental data [ 44 44 - 46 46 44-46 ]. Recently, Zou et al. successfully generalized OPM to favored proton radioactivity based on the same mechanism of tunneling effect [ 47 47 ]. For proton radioactivity, an odd proton must penetrate a barrier containing nuclear, Coulomb, and centrifugal potentials. Compared with α and cluster decays, the centrifugal potential plays a more important role in proton radioactivity because protons have less mass than α particles and clusters. In addition, the probability of protons penetrating the barrier is sensitive to the value of the outer turning point, i.e., the right-most intersection of the decay energy and proton-daughter nucleus interaction potential. When the Coulomb potential is replaced by the Hulthen potential [ 48 48 ], the screened effect shifts the outer turning point to the left [ 49 49 , 50 50 ]. Thus, it is crucial to consider the contribution of the centrifugal potential and screened electrostatic effect when analyzing proton radioactivity. To this end, based on the Gamow model, we systematically studied proton radioactivity by jointly considering the screened electrostatic effect and centrifugal potential together with experimental data from the latest table of evaluated nuclear properties, i.e., NUBASE2020 [ 51 51 ]. The remainder of this paper is organized as. In Sect. 2, the theoretical frameworks for the calculation of proton radioactivity half-life and screened electrostatic barrier are described in detail. The calculations and discussion are presented in Sect. 3. Finally, Sect. 4 provides a brief summary. Theoretical frameworks 1 The proton radioactivity half-life is generally defined as <math display="block" id="M1"><mrow><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mtext>ln2</mtext></mrow><mi>λ</mi></mfrac><mo>=</mo><mfrac><mrow><mtext>ln2</mtext></mrow><mrow><msub><mi>ν</mi><mn>0</mn></msub><msub><mi>S</mi><mtext>p</mtext></msub><msub><mi>P</mi><mrow><mtext>se</mtext></mrow></msub></mrow></mfrac><mn>,</mn></mrow></math> where λ is the proton radioactivity constant depending on the collision frequency of the emitted proton in the potential barrier ν 0 , the spectroscopic factor S p (probability of finding the daughter nucleus with a certain state J π in the mother nucleus), and the penetrability factor P se (probability of a proton penetrating through the external barrier); ν 0 can be calculated as <math display="block" id="M2"><mrow><msub><mi>ν</mi><mn>0</mn></msub><mo>=</mo><mfrac><mi>ω</mi><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>n</mi><mtext>r</mtext></msub><mo>+</mo><mi>l</mi><mo>+</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mi>ℏ</mi></mrow><mrow><mn>2</mn><mi>π</mi><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>R</mi><mtext>n</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>+</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mi>ℏ</mi></mrow><mrow><mn>1.2</mn><mi>π</mi><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>R</mi><mn>0</mn></msub><msup><mrow/><mn>2</mn></msup></mrow></mfrac><mn>.</mn></mrow></math> Here, ω is the oscillation frequency [ 52 52 ] and μ 0 denotes the reduced mass between the emitted proton and daughter nucleus in the final decaying nuclear system. In this study, the nucleus root-mean-square (rms) radius R n was estimated as <math id="M3"><mrow><msub><mi>R</mi><mtext>n</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow></math> =3/5 <math id="M4"><mrow><msub><mi>R</mi><mn>0</mn></msub><msup><mrow/><mn>2</mn></msup></mrow></math> [ 53 53 ], where R 0 =1.240 <math id="M5"><mrow><msubsup><mi>A</mi><mi>p</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>1.616</mn><mo>/</mo><msub><mi>A</mi><mtext>p</mtext></msub><mo>−</mo><mn>0.191</mn><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>p</mtext></msub><mo>−</mo><mn>2</mn><msub><mi>Z</mi><mtext>p</mtext></msub><mo stretchy="false">)</mo><mo>/</mo><msub><mi>A</mi><mtext>p</mtext></msub><mo stretchy="false">)</mo></mrow></math> , with A p and Z p being the mass and proton number of the parent nucleus, respectively. G =2 n r + l is the principal quantum number, where n r and l are the radial and angular momentum quantum numbers, respectively; ℏ denotes the reduced Planck constant. Spectroscopic factor S p and penetrability factor P se are defined as follows: <math display="block" id="M6"><mtable><mtr><mtd><msub><mi>S</mi><mtext>p</mtext></msub><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>G</mi><mrow><mtext>ov</mtext></mrow></msub></mrow></msup><mn>,</mn><msub><mi>G</mi><mrow><mtext>ov</mtext></mrow></msub><mo>=</mo><mfrac><mn>2</mn><mi>ℏ</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup></mrow></mstyle><msqrt><mrow><mn>2</mn><msub><mi>μ</mi><mrow><mtext>ov</mtext></mrow></msub><mo stretchy="false">[</mo><msub><mi>V</mi><mrow><mtext>ov</mtext></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Q</mi><mtext>p</mtext></msub><mo stretchy="false">]</mo></mrow></msqrt><mtext>d</mtext><mi>r</mi><mn>,</mn></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><mtext>se</mtext></mrow></msub><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>G</mi><mrow><mtext>se</mtext></mrow></msub></mrow></msup><mn>,</mn><msub><mi>G</mi><mrow><mtext>se</mtext></mrow></msub><mo>=</mo><mfrac><mn>2</mn><mi>ℏ</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mi>b</mi><mi>c</mi></msubsup></mrow></mstyle><msqrt><mrow><mn>2</mn><msub><mi>μ</mi><mrow><mtext>se</mtext></mrow></msub><mo stretchy="false">[</mo><msub><mi>V</mi><mrow><mtext>se</mtext></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Q</mi><mtext>p</mtext></msub><mo stretchy="false">]</mo></mrow></msqrt><mtext>d</mtext><mi>r</mi><mn>.</mn></mtd></mtr></mtable></math> Here, G ov and G se are the Gamow factors in the overlapping ( a ∼ b ) and separating ( b ∼ c ) regions, as shown in Fig. 1 Fig. 1 ; μ ov , μ se , V ov , and V se are the reduced mass and interaction potential in the overlapping and separating regions, respectively; Q p is the energy released by proton radioactivity. In Fig. 1 Fig. 1 , a and b denote the inner turning point and separating point, respectively; their values are equal to R p - R proton and R d + R proton . R proton = 0.8409 fm is the radius of the proton, obtained from Ref. [ 54 54 ]. R p and R d are the radii of the parent and daughter nuclei, respectively. They were calculated using the droplet model of an atomic nucleus; detailed calculations can be found in Ref. [ 57 57 ]. Finally, c is the outer turning point of the potential barrier that satisfies the condition V ( c )= Q p . μ 0 and Q p can be calculated as <math display="block" id="M7"><mrow><mfrac><mn>1</mn><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>m</mi><mtext>d</mtext></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>m</mi><mrow><mtext>proton</mtext></mrow></msub></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M8"><mrow><msub><mi>m</mi><mtext>d</mtext></msub><mo>=</mo><msub><mi>A</mi><mtext>d</mtext></msub><mo>+</mo><mfrac><mrow><mi>Δ</mi><msub><mi>M</mi><mtext>d</mtext></msub></mrow><mi>F</mi></mfrac><mo>−</mo><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>d</mtext></msub><msub><mi>m</mi><mtext>e</mtext></msub><mo>−</mo><mfrac><mrow><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>6</mn></mrow></msup><mi>k</mi><msub><mi>Z</mi><mtext>d</mtext></msub><msup><mrow/><mi>β</mi></msup></mrow><mi>F</mi></mfrac><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M9"><mtable><mtr><mtd><msub><mi>Q</mi><mtext>p</mtext></msub><mo>=</mo><mi>Δ</mi><msub><mi>M</mi><mtext>p</mtext></msub><mo>−</mo><mrow><mo>(</mo><mrow><mi>Δ</mi><msub><mi>M</mi><mtext>d</mtext></msub><mo>+</mo><mi>Δ</mi><msub><mi>M</mi><mrow><mtext>proton</mtext></mrow></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup><mtext>k</mtext><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mtext>p</mtext></msub><msup><mrow/><mi>β</mi></msup><mo>−</mo><msub><mi>Z</mi><mtext>d</mtext></msub><msup><mrow/><mi>β</mi></msup></mrow><mo>)</mo></mrow><mtext> </mtext><mtext>MeV</mtext><mn>.</mn></mtd></mtr></mtable></math> Here, m , A , Z , and Δ M are the atomic mass, mass number, proton number, and mass excess, respectively, and the subscripts p, d, e, and proton represent the parent nucleus, daughter nucleus, electron, and proton, respectively. F =931.494009 MeV/u is the mass-energy conversion factor. The quantity <math id="M10"><mrow><mtext>k</mtext><msup><mi>Z</mi><mi>β</mi></msup></mrow></math> is the total binding energy of the Z electrons in the atom, whereas the term <math id="M11"><mrow><mtext>k</mtext><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mtext>p</mtext></msub><msup><mrow/><mi>β</mi></msup><mo>−</mo><msub><mi>Z</mi><mtext>d</mtext></msub><msup><mrow/><mi>β</mi></msup></mrow><mo>)</mo></mrow></mrow></math> represents the screened effect of the atomic electrons. For Z ≥ 60, <math id="M12"><mtext>k</mtext></math> = 8.7 eV and β =2.517, while for Z < 60, <math id="M13"><mtext>k</mtext></math> =13.6 eV and β =2.408; these values were derived from data reported by Huang et al. [ 58 58 ]. In addition, V ( r ) denotes the total interaction potential between the emitted proton and daughter nucleus, which is sketched in Fig. 1 Fig. 1 . Generally, in the process of proton radioactivity, the total interaction potential V ( r ) between the emitted proton and daughter nucleus consists of the nuclear potential V n ( r ), Coulomb potential V c ( r ), and centrifugal potential V l ( r ). It is expressed as <math display="block" id="M14"><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>n</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>+</mo><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>l</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Here, the centrifugal potential V l ( r ) is written as <math display="block" id="M15"><mrow><msub><mi>V</mi><mi>l</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msub><mi>μ</mi><mn>0</mn></msub><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> where the minimum angular momentum l min removed by the emitted proton is calculated as <math display="block" id="M16"><mrow><msub><mi>l</mi><mrow><mtext>min</mtext></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>Δ</mi><mi>j</mi></msub><mn>,</mn></mrow></mtd><mtd columnalign="left"><mrow><mtext>for even </mtext><msub><mi>Δ</mi><mi>j</mi></msub><mtext> and </mtext><msub><mi>π</mi><mtext>p</mtext></msub><mo>=</mo><msub><mi>π</mi><mtext>d</mtext></msub><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>Δ</mi><mi>j</mi></msub><mo>+</mo><mn>1,</mn></mrow></mtd><mtd columnalign="left"><mrow><mtext>for even </mtext><msub><mi>Δ</mi><mi>j</mi></msub><mtext> and </mtext><msub><mi>π</mi><mtext>p</mtext></msub><mo>≠</mo><msub><mi>π</mi><mtext>d</mtext></msub><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>Δ</mi><mi>j</mi></msub><mn>,</mn></mrow></mtd><mtd columnalign="left"><mrow><mtext>for odd </mtext><msub><mi>Δ</mi><mi>j</mi></msub><mtext> and </mtext><msub><mi>π</mi><mtext>p</mtext></msub><mo>≠</mo><msub><mi>π</mi><mtext>d</mtext></msub><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>Δ</mi><mi>j</mi></msub><mo>+</mo><mn>1,</mn></mrow></mtd><mtd columnalign="left"><mrow><mtext>for odd </mtext><msub><mi>Δ</mi><mi>j</mi></msub><mtext> and </mtext><msub><mi>π</mi><mtext>p</mtext></msub><mo>=</mo><msub><mi>π</mi><mtext>d</mtext></msub><mn>.</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> Here, <math id="M17"><mrow><msub><mi>Δ</mi><mi>j</mi></msub><mo>=</mo><mrow><mo>|</mo><mrow><msub><mi>j</mi><mtext>p</mtext></msub><mo>−</mo><msub><mi>j</mi><mtext>d</mtext></msub><mo>−</mo><msub><mi>j</mi><mrow><mtext>proton</mtext></mrow></msub></mrow><mo>|</mo></mrow></mrow></math> , where j p , π p , j d , π d , j proton , and π proton are the spin and parity values for parent, daughter, and proton, respectively. In addition, when a proton is separated from the parent nucleus, the Coulomb potential V c ( r ) is expressed as <math display="block" id="M18"><mrow><msub><mi>V</mi><mtext>c</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mrow><mtext>proton</mtext></mrow></msub><msub><mi>Z</mi><mtext>d</mtext></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac><mn>.</mn></mrow></math> To describe the process of proton radioactivity, we employed the Hulthen-type potential instead of the Coulomb potential, which causes the superposition of the involved charges, movement of the emitted proton (which generates a magnetic field), and an inhomogeneous charge distribution in the nucleus. It can be defined as <math display="block" id="M19"><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>t</mi><msub><mi>Z</mi><mrow><mtext>proton</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>t</mi><mi>r</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mn>,</mn></mrow></math> where e 2 =1.4399652 MeV ⋅ fm and t is the screened parameter. In fact, the Hulthen-type potential is a generation of Coulomb potential. At short distances, its behavior is very similar to that of the Coulomb potential; however, at large distances, it drops exponentially. To intuitively show the difference between the Coulomb potential and Hulthen potential at large distances, Fig. 2 Fig. 2 displays a schematic diagram of interaction potential in a separating region, taking the process of proton radioactivity of nucleus 146 Tm as an example. It can be concluded from this figure that with an increase in r , the error between the Coulomb potential and Hulthen potential increases. Meanwhile, the Hulthen potential shifts the outer turning point to the left. In the overlapping region, the reduced mass and interaction potential cannot be treated as a free two-body system, because the proton is not completely separated from the parent nucleus. Here, we use μ ov and V ( r ) ov to represent the reduced mass and interaction potential in this region, respectively. Inspired by Refs. [ 59 59 , 60 60 ], they can be expressed as <math display="block" id="M20"><mrow><msub><mi>μ</mi><mrow><mtext>ov</mtext></mrow></msub><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>r</mi><mo>−</mo><mi>a</mi></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mi>p</mi></msup><mn>,</mn><mtext> </mtext><mi>p</mi><mo>≥</mo><mn>0</mn><mtext> </mtext><mn>,</mn></mrow></math> <math display="block" id="M21"><mrow><msub><mi>V</mi><mrow><mtext>ov</mtext></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Q</mi><mtext>p</mtext></msub><mo>+</mo><mrow><mo>(</mo><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Q</mi><mtext>p</mtext></msub></mrow><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>r</mi><mo>−</mo><mi>a</mi></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mi>q</mi></msup><mn>,</mn><mtext> </mtext><mi>q</mi><mo>≥</mo><mn>1,</mn></mrow></math> with <math display="block" id="M22"><mtable><mtr><mtd><mi>V</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>h</mtext></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mtext>l</mtext></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>t</mi><msub><mi>Z</mi><mrow><mtext>proton</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>t</mi><mi>b</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msub><mi>μ</mi><mn>0</mn></msub><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mn>.</mn></mtd></mtr></mtable></math> According to Eqs.(3), (12), and (13), the expression for G ov can be rewritten as <math display="block" id="M23"><mtable><mtr><mtd><msub><mi>G</mi><mrow><mtext>ov</mtext></mrow></msub><mo>=</mo><mn>0.4374702</mn><mrow><mo>(</mo><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mo>)</mo></mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>{</mo><mrow><msub><mi>μ</mi><mn>0</mn></msub><mrow><mo>[</mo><mrow><mfrac><mrow><mi>t</mi><msub><mi>Z</mi><mrow><mtext>proton</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>t</mi><mi>b</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>20.9008</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>l</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><msub><mi>Q</mi><mi>p</mi></msub></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mn>,</mn></mtd></mtr></mtable></math> where <math id="M24"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> is denoted by g , with 0≤ g≤ <math id="M25"><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></math> . The parameter g encodes an adjustable coupling of the mass power parameter p and potential energy power parameter q . It represents the combined effect of the interaction potential and reduced mass of the proton-daughter nucleus on the half-life of proton radioactivity in the overlapping region. Once a proton is separated from the parent nucleus, the proton radioactivity system becomes a simple two-body problem. The reduced mass μ se can be obtained using Eq.(4), i.e., μ 0 ; meanwhile, the potential energy V se ( r ) includes the Hulthen-type and centrifugal potentials. Therefore, G se can be expressed as <math display="block" id="M26"><mrow><msub><mi>G</mi><mrow><mtext>se</mtext></mrow></msub><mo>=</mo><mn>0.62994397</mn><msub><mi>Z</mi><mtext>d</mtext></msub><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow><mrow><msub><mi>Q</mi><mtext>p</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>×</mo><mi>F</mi><mn>,</mn></mrow></math> where <math display="block" id="M27"><mtable><mtr><mtd><mi>F</mi><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><mi>y</mi></mrow></mfrac><mo>×</mo><mi>ln</mi><mrow><mo>{</mo><mrow><mfrac><mrow><msup><mrow><mrow><mo>[</mo><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>/</mo><mi>y</mi><mo stretchy="false">)</mo><msup><mrow><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>/</mo><msup><mi>y</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>y</mi></mrow></mfrac></mrow><mo>}</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mi>arccos</mi><msup><mrow><mo>{</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>y</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>/</mo><msup><mi>y</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd><mo>−</mo><msup><mrow><mo>[</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mi>y</mi></mrow></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>/</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mn>,</mn></mtd></mtr></mtable></math> with <math display="block" id="M28"><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mn>20.9008</mn><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub><msup><mi>b</mi><mn>2</mn></msup><msub><mi>Q</mi><mtext>p</mtext></msub></mrow></mfrac><mn>,</mn></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mrow><mo>(</mo><mrow><mi>t</mi><msub><mi>Z</mi><mrow><mtext>proton</mtext></mrow></msub><msub><mi>Z</mi><mtext>d</mtext></msub><msup><mi>e</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Q</mi><mtext>p</mtext></msub><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mi>t</mi><mi>b</mi></mrow></mfrac><mn>.</mn></mtd></mtr></mtable></math> Then, the proton radioactivity half-life can be calculated by <math display="block" id="M29"><mrow><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mn>4.108054431</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>23</mn></mrow></msup><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>R</mi><mn>0</mn></msub><msup><mrow/><mn>2</mn></msup></mrow><mrow><mi>G</mi><mo>+</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></mfrac><msub><mi>S</mi><mtext>p</mtext></msub><msup><mrow/><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>P</mi><mrow><mtext>se</mtext></mrow></msub><msup><mrow/><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>.</mn></mrow></math> Results and discussion 1 On the basis of the Gamow model, replacing the Coulomb potential with the Hulthen-type potential, an improved model is proposed to investigate the half-lives of proton radioactivity. We selected 45 proton emitters with 51 ≤ Z ≤ 83 as the database and divided them into 32 spherical nuclei and 13 deformed nuclei. For spherical nuclei, using a genetic algorithm with the optimal solution of σ as the objective function, we obtained the values of the adjustable parameters: t =9.186×10 -4 and g =4.313×10 -3 . In this study, σ , i.e., the deviation between the experimental and calculated data, is defined as <math display="block" id="M30"><mrow><mi>σ</mi><mo>=</mo><msqrt><mrow><mstyle displaystyle="true"><mo>∑</mo></mstyle><msup><mrow><mrow><mo>(</mo><mrow><msub><mrow><mi>log</mi></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>cal</mtext></mrow></msubsup><mo>−</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>exp</mtext></mrow></msubsup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>/</mo><mi>n</mi></mrow></msqrt><mn>,</mn></mrow></math> where <math id="M31"><mrow><msub><mrow><mi>log</mi></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>cal</mtext></mrow></msubsup></mrow></math> and <math id="M32"><mrow><msub><mrow><mi>log</mi></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>exp</mtext></mrow></msubsup></mrow></math> are the experimental and calculated proton radioactivity half-lives in logarithmic form, and n denotes the number of nuclei involved in each case. To visualize the screened electrostatic effect, Fig. 3 Fig. 3 shows the difference between the outer turning points before and after considering the screened effect versus <math id="M33"><mrow><msub><mi>Z</mi><mtext>d</mtext></msub><mo>/</mo><msub><mi>Q</mi><mtext>p</mtext></msub></mrow></math> . <math id="M34"><mrow><msubsup><mi>R</mi><mrow><mtext>out</mtext></mrow><mtext>H</mtext></msubsup></mrow></math> and <math id="M35"><mrow><msubsup><mi>R</mi><mrow><mtext>out</mtext></mrow><mtext>C</mtext></msubsup></mrow></math> represent the outer turning points obtained by the Gamow model with Hulthen-type and Coulomb potentials, respectively. According to this figure, it can be concluded that the screening of electrostatic repulsion shortens the outer turning point by several percentage points, but we also conclude that the larger the ratio <math id="M36"><mrow><msub><mi>Z</mi><mtext>d</mtext></msub><mo>/</mo><msub><mi>Q</mi><mtext>p</mtext></msub></mrow></math> , the more evident this effect is for the penetrability factor. Using the proposed improved Gamow model and the obtained values of the parameters t and g , we calculated the proton radioactivity half-lives of 32 spherical nuclei. Detailed calculations are presented in Table 1 Table 1 . In this table, the first two columns present the proton emitter and corresponding energy released by proton radioactivity, Q p , respectively. The next two columns denote the spin and parity transition and the minimum angular momentum removed by the emitted proton l min , respectively. The last four columns are the half-lives of the experimental proton radioactivity, half-lives calculated using the proposed improved model, UDLP, and NG-N, all expressed in logarithmic form as <math id="M40"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>exp</mtext></mrow></msubsup></mrow></math> , <math id="M41"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>cal</mtext></mrow></msubsup></mrow></math> , <math id="M42"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>UDLP</mtext></mrow></msubsup></mrow></math> , and <math id="M43"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>NG</mtext><mo>−</mo><mtext>N</mtext></mrow></msubsup></mrow></math> , respectively. It can be easily seen from this table that the calculations from the proposed model are very close to the experimental values for all nuclei except for 177 Tl m , whose calculated half-life differs by approximately one order of magnitude from the experimental value. Furthermore, the total rms deviation for spherical nuclei calculated by our model is 0.331 orders of magnitude and decreases to 0.274 when 177 Tl m is not considered. For comparison, Table 2 Table 2 lists the standard deviations σ calculated within 31 spherical nuclei (except for 177 Tl m ), NG-N, and UDLP. The results show that the calculated proton radioactivity half-lives of spherical nuclei are reliable. 1 1 Nucleus 1 1 Q p 1 1 <math id="M44"><mrow><msubsup><mi>j</mi><mtext>p</mtext><mi>π</mi></msubsup><mo>→</mo><msub><mi>j</mi><mtext>d</mtext></msub><msup><mrow/><mi>π</mi></msup></mrow></math> 1 1 l 1 1 <math id="M45"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>exp</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M46"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>cal</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M47"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>NG</mtext><mo>−</mo><mtext>N</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M48"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mtext>T</mtext><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mi>�</mi><mtext>#x2009;UDLP</mtext></mrow></msubsup></mrow></math> 3 1 Part I: Spherical nuclei 1 1 144 Tm 1 1 1.724 1 1 <math id="M49"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>10</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>9</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mtext>#</mtext></mrow></math> 1 1 5 1 1 -5.569 1 1 -5.263 1 1 -5.212 1 1 -4.687 1 1 145 Tm 1 1 1.754 1 1 <math id="M50"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 -5.499 1 1 -5.467 1 1 -5.401 1 1 -4.871 1 1 146 Tm 1 1 0.904 1 1 <math id="M51"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 0 1 1 -0.810 1 1 -0.854 1 1 -1.272 1 1 -0.610 1 1 146 Tm m 1 1 1.214 1 1 <math id="M52"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>5</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 5 1 1 -1.137 1 1 -0.959 1 1 -0.999 1 1 -0.896 1 1 147 Tm m 1 1 1.133 1 1 <math id="M53"><mrow><msup><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 -3.444 1 1 -3.181 1 1 -2.455 1 1 -2.859 1 1 147 Tm 1 1 1.072 1 1 <math id="M54"><mrow><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 0.587 1 1 0.752 1 1 0.681 1 1 0.614 1 1 150 Lu m 1 1 1.305 1 1 <math id="M55"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mrow><mn>,2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 2 1 1 -4.398 1 1 -4.454 1 1 -3.633 1 1 -4.050 1 1 150 Lu 1 1 1.285 1 1 <math id="M56"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>5</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 5 1 1 -1.347 1 1 -1.186 1 1 -1.219 1 1 -1.132 1 1 151 Lu m 1 1 1.315 1 1 <math id="M57"><mrow><msup><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 -4.796 1 1 -4.561 1 1 -3.722 1 1 -4.150 1 1 151 Lu 1 1 1.255 1 1 <math id="M58"><mrow><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 -0.896 1 1 -0.877 1 1 -0.910 1 1 -0.862 1 1 155 Ta 1 1 1.466 1 1 <math id="M59"><mrow><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 -2.495 1 1 -2.427 1 1 -2.397 1 1 -2.269 1 1 156 Ta 1 1 1.036 1 1 <math id="M60"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mtext>#</mtext></mrow></math> 1 1 2 1 1 -0.826 1 1 -0.654 1 1 -0.180 1 1 -0.624 1 1 156 Ta m 1 1 1.126 1 1 <math id="M61"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>9</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mtext>#</mtext></mrow></math> 1 1 5 1 1 0.933 1 1 1.205 1 1 1.101 1 1 0.947 1 1 157 Ta 1 1 0.946 1 1 <math id="M62"><mrow><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 -0.527 1 1 -0.145 1 1 -0.657 1 1 -0.038 1 1 159 Re m 1 1 1.816 1 1 <math id="M63"><mrow><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 -4.665 1 1 -4.646 1 1 -4.494 1 1 -4.269 1 1 159 Re 1 1 1.816 1 1 <math id="M64"><mrow><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 -4.678 1 1 -4.645 1 1 -4.493 1 1 -4.268 1 1 160 Re 1 1 1.267 1 1 <math id="M65"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>4</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mtext>#</mtext></mrow></math> 1 1 0 1 1 -3.163 1 1 -3.786 1 1 -3.761 1 1 -3.408 1 1 161 Re 1 1 1.216 1 1 <math id="M66"><mrow><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 -3.306 1 1 -3.223 1 1 -3.277 1 1 -2.895 1 1 161 Re m 1 1 1.336 1 1 <math id="M67"><mrow><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 -0.678 1 1 -0.712 1 1 -0.729 1 1 -0.789 1 1 164 Ir 1 1 1.844 1 1 <math id="M68"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>9</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup></mrow></math> 1 1 5 1 1 -3.959 1 1 -4.426 1 1 -4.247 1 1 -4.114 1 1 165 Ir m 1 1 1.727 1 1 <math id="M69"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 -3.433 1 1 -3.626 1 1 -3.482 1 1 -3.408 1 1 166 Ir 1 1 1.167 1 1 <math id="M70"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mo>−</mo></msup><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 2 1 1 -0.824 1 1 -1.198 1 1 -0.688 1 1 -1.188 1 1 166 Ir m 1 1 1.347 1 1 <math id="M71"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo></mrow><mo>+</mo></msup><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 5 1 1 -0.076 1 1 -0.318 1 1 -0.344 1 1 -0.475 1 1 167 Ir 1 1 1.087 1 1 <math id="M72"><mrow><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 -1.120 1 1 -0.967 1 1 -1.347 1 1 -0.865 1 1 167 Ir m 1 1 1.262 1 1 <math id="M73"><mrow><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 0.842 1 1 0.611 1 1 0.546 1 1 0.348 1 1 170 Au 1 1 1.487 1 1 <math id="M74"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mo>−</mo></msup><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 2 1 1 -3.487 1 1 -4.074 1 1 -3.254 1 1 -3.845 1 1 170 Au m 1 1 1.767 1 1 <math id="M75"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo></mrow><mo>+</mo></msup><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 5 1 1 -3.975 1 1 -3.499 1 1 -3.330 1 1 -3.333 1 1 171 Au 1 1 1.464 1 1 <math id="M76"><mrow><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 -4.652 1 1 -4.669 1 1 -4.460 1 1 -4.298 1 1 171 Au m 1 1 1.702 1 1 <math id="M77"><mrow><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 -2.587 1 1 -3.025 1 1 -2.876 1 1 -2.915 1 1 176 Tl 1 1 1.278 1 1 <math id="M78"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn></mrow><mo>−</mo></msup><msup><mrow><mn>,4</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 0 1 1 -2.208 1 1 -2.194 1 1 -2.361 1 1 -2.059 1 1 177 Tl 1 1 1.173 1 1 <math id="M79"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 -1.178 1 1 -0.901 1 1 -1.274 1 1 -0.875 1 1 177 Tl m 1 1 1.963 1 1 <math id="M80"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 -3.346 1 1 -4.431 1 1 -4.148 1 1 -4.205 3 1 Part II: Deformed nuclei 1 1 108 I 1 1 0.610 1 1 <math id="M81"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mtext>#</mtext><mo>→</mo><msup><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext></mrow></math> 1 1 2 1 1 0.723 1 1 0.502 1 1 0.438 1 1 -0.019 1 1 109 I 1 1 0.829 1 1 <math id="M82"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 -4.032 1 1 -3.558 1 1 -3.493 1 1 -3.671 1 1 112 Cs 1 1 0.820 1 1 <math id="M83"><mrow><msup><mn>1</mn><mo>+</mo></msup><mtext>#</mtext><mo>→</mo><msup><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext></mrow></math> 1 1 2 1 1 -3.310 1 1 -2.681 1 1 -2.697 1 1 -2.923 1 1 113 Cs 1 1 0.978 1 1 <math id="M84"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 -4.771 1 1 -4.836 1 1 -4.760 1 1 -4.865 1 1 117 La 1 1 0.823 1 1 <math id="M85"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 -1.602 1 1 -1.991 1 1 -2.072 1 1 -2.350 1 1 121 Pr 1 1 0.901 1 1 <math id="M86"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 -1.921 1 1 -2.456 1 1 -2.552 1 1 -2.811 1 1 130 Eu 1 1 1.028 1 1 <math id="M87"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><msup><mrow><mn>,3</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 2 1 1 -3.000 1 1 -2.828 1 1 -2.950 1 1 -3.233 1 1 131 Eu 1 1 0.951 1 1 <math id="M88"><mrow><msup><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 -1.699 1 1 -1.800 1 1 -1.988 1 1 -2.310 1 1 135 Tb 1 1 1.203 1 1 <math id="M89"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 3 1 1 -2.996 1 1 -3.408 1 1 -3.479 1 1 -3.806 1 1 140 Ho 1 1 1.104 1 1 <math id="M90"><mrow><msup><mn>6</mn><mo>−</mo></msup><msup><mrow><mn>,0</mn></mrow><mo>+</mo></msup><msup><mrow><mn>,8</mn></mrow><mo>+</mo></msup><mo>→</mo><msup><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></math> 1 1 3 1 1 -2.222 1 1 -1.702 1 1 -1.877 1 1 -2.317 1 1 141 Ho m 1 1 1.256 1 1 <math id="M91"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 -5.180 1 1 -5.115 1 1 -5.717 1 1 -5.261 1 1 141 Ho 1 1 1.194 1 1 <math id="M92"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 3 1 1 -2.387 1 1 -2.747 1 1 -2.850 1 1 -3.257 1 1 185 Bi m 1 1 1.607 1 1 <math id="M93"><mrow><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 -4.191 1 1 -4.011 1 1 -4.606 1 1 -4.623 1 1 Nuclei 4 1 σ 1 1   1 1 Cases 1 1 σ 1 1 1 σ 2 1 1 σ 3 1 1 Spherical nuclei 1 1 31 1 1 0.274 1 1 0.399 1 1 0.385 1 1 Deformed nuclei 1 1 13 1 1 0.367 1 1 0.437 1 1 0.571 To intuitively demonstrate the consistency between our results and the experimental data, Fig. 4 (a) Fig. 4 (a) shows the differences between the experimental half-lives of proton radioactivity and the calculated half-lives in logarithmic form for spherical proton emitters using the proposed improved Gamow model, NG-N, and UDLP. They are represented by red triangle, green square, and blue circle, respectively. As can be seen from this figure, compared with the calculations using the other two formulas, our results are generally closer to the experimental values. The deviations of most of the spherical nuclei are within 0.5. This indicates that our model can reproduce the experimental half-lives accurately. The exception is 177 Tl m , for which the agreement with the experimental data is worse for the improved Gamow model, NG-N, and UDLP, with corresponding deviations of -1.084, -0.802, and -0.859, respectively. The reason for this large discrepancy is worth investigating. Given that our calculations are determined by two experimental quantities, i.e., the energy released by proton radioactivity Q p and the angular momentum l , Fig. 5 Fig. 5 presents the proton radioactivity half-lives of 177 Tl m calculated by our model in logarithmic form versus Q p . The dotted lines with different colors correspond to the proton radioactivity half-lives with different angular momenta l whereas the red spheres represent the experimental data of 177 Tl m . According to this figure, we can conclude that the results are sensitive to l whereas the dependence on Q p is not so pronounced. For the same decay energy, the half-lives increase by an order of magnitude or more for each increase in angular momentum. Moreover, the experimental data perfectly fit on the line with l =6, which is larger than that of experimental l =5. The same phenomenon was reported by A. Zdeb et al. [ 54 54 ]. They analyzed the single-particle energies from microscopic calculations using the Hartree Fock-Bogolubow model with the Gogny-type force D1S parameter set of 177 Tl m . It can be concluded that the first l = 6 state 13/2 + is approximately 9.5 MeV above the 1/2 + ground state, and that the angular momentum l = 6 cannot be associated with 177 Tl m , which has an excitation energy of only 807 keV. However, the experimental data of spin and parity for 177 Tl m in NUBASE2020 are uncertain. According to the proton radioactivity satisfying the conservation law of spin and parity, this leads to an uncertain value of l removed by the emitted proton. Thus, in our opinion, the orbital angular momentum of 177 Tl m may be l = 6. Recently, the study of deformed nuclei has attracted extensive attention [ 55 55 , 56 56 ]. Therefore, it would be interesting to extend our model to deformed nuclei. Theoretically, the probability of proton formation depends on deformation of the decaying nucleus. In a well-deformed nucleus, decay can proceed through one of the spherical components of the deformed orbit, which can be very small in case of large deformations. Thus, the probability of formation is small. Using our model with the parameters fitted from the spherical nuclei to calculate 13 deformed nuclei, we found that the calculated results were not in satisfactory agreement with the experimental values. The error of several nuclei reached an order of magnitude. This phenomenon is probably due to the deformation effect that affects the total potential energy and some other factors, such that the probabilities of finding a proton at the nuclear surface S p and a proton penetrating through the external barrier P se vary with respect to those of spherical nuclei. Using the experimental data of 13 deformed nuclei as a database, we obtained a new set of parameters, i.e., t =1.831×10 -4 and g =0.666, by refitting. It is worth noting that the values of t for the deformed and spherical nuclei are the same in magnitude, whereas g for the spherical nuclei is close to zero and for the deformed nuclei is greater than 0.5. This difference in the g value directly affects S p ; thus, the corresponding S p values became 0.98 and 0.17 on average for spherical and deformed nuclei, respectively. This suggests that spherical nuclei have narrower overlapping regions than deformed nuclei, and that protons are more likely to escape the spherical parent nucleus. The relevant results and deviations are displayed in Part II of Table 1 Table 1 and subfigure (b) in Fig. 4 Fig. 4 , respectively; note that the calculations agree well with the experimental data. Generally, the advantage of our model is that it avoids direct consideration of the complex nuclear potential and Coulomb potential in the overlapping region. Besides, it uses a single parameter to characterize the influence of the physical quantity of the overlapping region on the proton radioactivity half-life. We fitted different parameter values for deformed and spherical nuclei and systematically studied the effect of deformation on the spectroscopic factor S p and half-life of the proton radioactivity. Nevertheless, by fitting the adjustable parameters through experimental data, the physical meaning of our model is not as clear as that of microscopic models when considering the effect of deformation on the half-life of proton radioactivity. In the following, we extend our improved Gamow model to predict the proton radioactivity half-lives of 18 nuclei, whose proton radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020. For comparison, UDLP and NG-N were also used. The predicted proton radioactivity half-lives are listed in Table 3 Table 3 . For a more visual comparison, the predictions of the three models are plotted in Fig. 6 Fig. 6 . The results show that the values predicted by UDLP, NG-N, and our model are consistent. To further verify the reliability of our predictions, we plotted the relationships between the logarithm of experimental and predicted half-lives and <math id="M94"><mrow><msubsup><mi>Z</mi><mtext>d</mtext><mrow><mn>0.8</mn></mrow></msubsup><mo>/</mo><msubsup><mi>Q</mi><mtext>p</mtext><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup></mrow></math> , i.e., the new Geiger-Nuttall law [ 42 42 ] for proton radioactivity, for l =0, 2, 3, and 5; the results are depicted in Fig. 7 Fig. 7 . The dashed lines in this figure were fitted to the experimental data. From this figure, we can clearly see that our predicted proton radioactivity half-lives fit the linear relevance well. This indicates that our predicted results may be useful for future studies on proton emission half-lives in newly synthesized isotopes. 1 1 Nucleus 1 1 Q p 1 1 <math id="M95"><mrow><msubsup><mi>j</mi><mtext>p</mtext><mi>π</mi></msubsup><mo>→</mo><msub><mi>j</mi><mtext>d</mtext></msub><msup><mrow/><mi>π</mi></msup></mrow></math> 1 1 l 1 1 <math id="M96"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>exp</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M97"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>cal</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M98"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>NG</mtext><mo>−</mo><mtext>N</mtext></mrow></msubsup></mrow></math> 1 1 <math id="M99"><mrow><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><msubsup><mtext>T</mtext><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mi>�</mi><mtext>#x2009;UDLP</mtext></mrow></msubsup></mrow></math> 3 1 Part I: Spherical nuclei 1 1 146 Tm n 1 1 1.144 1 1 <math id="M100"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>10</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mtext>#</mtext></mrow></math> 1 1 5 1 1 - 1 1 -0.145 1 1 -0.206 1 1 -0.177 1 1 159 Re 1 1 1.606 1 1 <math id="M101"><mrow><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 - 1 1 -6.854 1 1 -6.381 1 1 -6.227 1 1 165 Ir 1 1 1.547 1 1 <math id="M102"><mrow><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 - 1 1 -5.897 1 1 -5.530 1 1 -5.387 1 1 169 Ir m 1 1 0.782 1 1 <math id="M103"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 - 1 1 8.499 1 1 8.043 1 1 7.362 1 1 171 Ir m 1 1 0.403 1 1 <math id="M104"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 - 1 1 23.122 1 1 21.891 1 1 20.337 1 1 169 Au 1 1 1.947 1 1 <math id="M105"><mrow><msup><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 0 1 1 - 1 1 -8.227 1 1 -7.478 1 1 -7.572 1 1 172 Au 1 1 0.877 1 1 <math id="M106"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn></mrow><mo>−</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>7</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup></mrow></math> 1 1 2 1 1 >0.146 1 1 4.070 1 1 3.983 1 1 3.578 1 1 172 Au m 1 1 0.627 1 1 <math id="M107"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>9</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>13</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup></mrow></math> 1 1 2 1 1 >-0.260 1 1 10.497 1 1 9.678 1 1 9.433 3 1 Part II: Deformed nuclei 1 1 103 Sb 1 1 0.979 1 1 <math id="M108"><mrow><msup><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 - 1 1 -6.148 1 1 -6.009 1 1 -5.948 1 1 104 Sb 1 1 0.509 1 1 <math id="M109"><mrow><mo stretchy="false">(</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext></mrow></math> 1 1 2 1 1 >0.827 1 1 2.200 1 1 2.187 1 1 1.550 1 1 105 Sb 1 1 0.331 1 1 <math id="M110"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 >3.049 1 1 9.388 1 1 9.240 1 1 8.002 1 1 111 Cs 1 1 1.740 1 1 <math id="M111"><mrow><msup><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 - 1 1 -10.640 1 1 -10.375 1 1 -10.094 1 1 116 La 1 1 1.591 1 1 <math id="M112"><mrow><mo stretchy="false">(</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext></mrow></math> 1 1 2 1 1 - 1 1 -9.375 1 1 -9.126 1 1 -9.000 1 1 127 Pm 1 1 0.792 1 1 <math id="M113"><mrow><msup><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 2 1 1 - 1 1 0.033 1 1 -0.239 1 1 -0.620 1 1 137 Tb 1 1 0.843 1 1 <math id="M114"><mrow><msup><mrow><mn>11</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 - 1 1 4.038 1 1 2.974 1 1 2.714 1 1 185 Bi 1 1 1.541 1 1 <math id="M115"><mrow><msup><mrow><mn>9</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 - 1 1 0.330 1 1 -0.732 1 1 -1.030 1 1 185 Bi n 1 1 1.721 1 1 <math id="M116"><mrow><msup><mrow><mn>13</mn><mo>/</mo><mn>2</mn></mrow><mo>+</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 6 1 1 - 1 1 -0.020 1 1 -1.512 1 1 -1.171 1 1 211 Pa 1 1 0.721 1 1 <math id="M117"><mrow><msup><mrow><mn>9</mn><mo>/</mo><mn>2</mn></mrow><mo>−</mo></msup><mtext>#</mtext><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></math> 1 1 5 1 1 - 1 1 16.535 1 1 14.099 1 1 13.366 Summary 1 In summary, based on the Gamow model and considering the screened electrostatic effect, we systematically studied proton radioactivity half-lives. 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Material preparation, data collection and analysis were performed by Xiao-Yuan Hu, De-Xing Zhu, Xi-Jun Wu, Peng-Cheng Chu and Xiao-Hua Li. The first draft of the manuscript was written by Yang-Yang Xu, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-023-01178-3 This work was supported in part 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 Education Bureau of Hunan Province, China (Nos. 21B0402 and 18A237), the Natural Science Foundation of Hunan Province, China (No. 2018JJ2321), the Innovation Group of Nuclear and Particle Physics in USC, the Shandong Province Natural Science Foundation, China (No. ZR2022JQ04), the Hunan Provincial Innovation Foundation For Postgraduate (No. CX20220993), and the Opening Project of Cooperative Innovation Center for Nuclear Fuel Cycle Technology and Equipment, University of South China (No. 2019KFZ10). 2022-10-14 2022-12-20 2022-12-21 2023-02-28 2023-03-20 © 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. 2023 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/s41365-023-01178-3 http://dx.doi.org/10.1007/s41365-023-01178-3 . Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. 2023 Nuclear Science and Techniques 2 34