Introduction

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 ⁵³Co^m [1, 2] and independently confirmed by Hofmann et al. and Klepper et al. from ground states of ¹⁵¹Lu [3] and ¹⁴⁷Tm [4] in 1982. Since then, proton radioactivity has attracted wide attention in the nuclear physics community [5-19] because it can provide crucial information of neutron-deficient nuclei, such as their shell structure [20] and the coupling between bound and unbound nuclear states [21]. Moreover, as the inverse process of rapid proton capture, it can contribute significantly to the understanding of element origin and star evolution [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 [23-26], 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], distorted-wave Born approximation (DWBA) [28], Lejeune and Mahaux (JLM) [29], coupled-channels approach (CCA) [30-32], unified fission model (UFM) [33, 34], two-potential approach (TPA) [35], generalized liquid drop model (GLDM) [36-38], single fold model (SFM) [39], Coulomb and proximity potential model (CPPM) [40], universal decay law of proton radioactivity (UDLP) [41], and new Geiger-Nuttall law of proton radioactivity (NG-N) [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]. 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 [44-46]. Recently, Zou et al. successfully generalized OPM to favored proton radioactivity based on the same mechanism of tunneling effect [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], the screened effect shifts the outer turning point to the left [49, 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].

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

The proton radioactivity half-life is generally defined as $T_{1/2}=\frac{\text{ln2}}{\lambda }=\frac{\text{ln2}}{{\nu }_0{S}_{\text{p}}{P}_{\text{se}}}\mathrm{,}$ (1) where λ is the proton radioactivity constant depending on the collision frequency of the emitted proton in the potential barrier ν₀, 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); ν₀ can be calculated as ${\nu }_0=\frac{\omega }{2\pi }=\frac{(2n_{\text{r}}+l+3/2)\hslash }{2\pi {\mu }_0{R}_{\text{n}}{}^2}=\frac{(G+3/2)\hslash }{1.2\pi {\mu }_0{R}_0{}^2}\mathrm{.}$ (2) Here, ω is the oscillation frequency [52] and μ₀ 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 $R_{\text{n}}{}^2$=3/5 $R_0{}^2$ [53], where R₀=1.240 $A_p^{1/3}(1+1.616/A_{\text{p}}-0.191(A_{\text{p}}-2Z_{\text{p}})/A_{\text{p}})$, with A_p and Z_p being the mass and proton number of the parent nucleus, respectively. G=2n_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: $\begin{array}{c}{S}_{\text{p}}=e^{-G_{\text{ov}}}\mathrm{,}{G}_{\text{ov}}=\frac{2}{\hslash }{\displaystyle {\int }_a^b}\sqrt{2{\mu }_{\text{ov}}[V_{\text{ov}}(r)-Q_{\text{p}}]}\text{d}r\mathrm{,}\\ P_{\text{se}}=e^{-G_{\text{se}}}\mathrm{,}{G}_{\text{se}}=\frac{2}{\hslash }{\displaystyle {\int }_b^c}\sqrt{2{\mu }_{\text{se}}[V_{\text{se}}(r)-Q_{\text{p}}]}\text{d}r\mathrm{.}\end{array}$ (3) 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; 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, 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]. 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]. Finally, c is the outer turning point of the potential barrier that satisfies the condition V(c)=Q_p. μ₀ and Q_p can be calculated as $\frac{1}{{\mu }_0}=\frac{1}{m_{\text{d}}}+\frac{1}{m_{\text{proton}}}\mathrm{,}$ (4) $m_{\text{d}}=A_{\text{d}}+\frac{\Delta M_{\text{d}}}{F}-(Z_{\text{d}}{m}_{\text{e}}-\frac{{10}^{-6}k{Z}_{\text{d}}{}^{\beta }}{F})\mathrm{,}$ (5) $\begin{array}{c}{Q}_{\text{p}}=\Delta M_{\text{p}}-\left(\Delta M_{\text{d}}+\Delta M_{\text{proton}}\right)\\ +{10}^{-6}\text{k}\left(Z_{\text{p}}{}^{\beta }-Z_{\text{d}}{}^{\beta }\right)\text{MeV}\mathrm{.}\end{array}$ (6) Here, m, A, Z, and ΔM 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 $\text{k}{Z}^{\beta }$ is the total binding energy of the Z electrons in the atom, whereas the term $\text{k}\left(Z_{\text{p}}{}^{\beta }-Z_{\text{d}}{}^{\beta }\right)$ represents the screened effect of the atomic electrons. For Z≥60, $\text{k}$ = 8.7 eV and β=2.517, while for Z<60, $\text{k}$=13.6 eV and β=2.408; these values were derived from data reported by Huang et al. [58]. In addition, V(r) denotes the total interaction potential between the emitted proton and daughter nucleus, which is sketched in Fig. 1.

Fig. 1Full-screen View

(Color online) Schematic diagram of proton-daughter nucleus interaction potential V(r).

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 $V(r)=V_{\text{n}}(r)+V_{\text{c}}(r)+V_l(r)\mathrm{.}$ (7) Here, the centrifugal potential Vl(r) is written as $V_l(r)=\frac{{(l+1/2)}^2{\hslash }^2}{2{\mu }_0{r}^2}\mathrm{,}$ (8) where the minimum angular momentum l_min removed by the emitted proton is calculated as $l_{\text{min}}=\{\begin{array}{ll}{\Delta }_j\mathrm{,}\hfill & \text{for even }{\Delta }_j\text{ and }{\pi }_{\text{p}}={\pi }_{\text{d}}\mathrm{,}\hfill \\ {\Delta }_j+\mathrm{1,}\hfill & \text{for even }{\Delta }_j\text{ and }{\pi }_{\text{p}}\ne {\pi }_{\text{d}}\mathrm{,}\hfill \\ {\Delta }_j\mathrm{,}\hfill & \text{for odd }{\Delta }_j\text{ and }{\pi }_{\text{p}}\ne {\pi }_{\text{d}}\mathrm{,}\hfill \\ {\Delta }_j+\mathrm{1,}\hfill & \text{for odd }{\Delta }_j\text{ and }{\pi }_{\text{p}}={\pi }_{\text{d}}\mathrm{.}\hfill \end{array}$ (9) Here, ${\Delta }_j=\left|j_{\text{p}}-j_{\text{d}}-j_{\text{proton}}\right|$, 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 $V_{\text{c}}(r)=\frac{Z_{\text{proton}}{Z}_{\text{d}}{e}^2}{r}\mathrm{.}$ (10) 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 $V_{\text{h}}(r)=\frac{t{Z}_{\text{proton}}{Z}_{\text{d}}{e}^2}{e^{tr}-1}\mathrm{,}$ (11) where e²=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 displays a schematic diagram of interaction potential in a separating region, taking the process of proton radioactivity of nucleus ¹⁴⁶Tm 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.

Fig. 2Full-screen View

(Color online) Schematic diagram of proton-¹⁴⁵Er nucleus Coulomb potential and Hulthen potential in the separating region.

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, 60], they can be expressed as ${\mu }_{\text{ov}}={\mu }_0{\left(\frac{r-a}{b-a}\right)}^p\mathrm{,}p\ge 0\mathrm{,}$ (12) $V_{\text{ov}}(r)=Q_{\text{p}}+\left(V(b)-Q_{\text{p}}\right){\left(\frac{r-a}{b-a}\right)}^q\mathrm{,}q\ge \mathrm{1,}$ (13) with $\begin{array}{c}V(b)=V_{\text{h}}(b)+V_{\text{l}}(b)\\ =\frac{t{Z}_{\text{proton}}{Z}_{\text{d}}{e}^2}{e^{tb}-1}+\frac{{(l+1/2)}^2{\hslash }^2}{2{\mu }_0{b}^2}\mathrm{.}\end{array}$ (14) According to Eqs.(3), (12), and (13), the expression for G_ov can be rewritten as $\begin{array}{c}{G}_{\text{ov}}=0.4374702\left(b-a\right){\left(1+\frac{p+q}{2}\right)}^{-1}\\ {\left\{{\mu }_0\left[\frac{t{Z}_{\text{proton}}{Z}_{\text{d}}{e}^2}{e^{tb}-1}+\frac{20.9008{\left(l+1/2\right)}^2}{{\mu }_0{b}^2}-Q_p\right]\right\}}^{1/2}\mathrm{,}\end{array}$ (15) where ${\left(1+\frac{p+q}{2}\right)}^{-1}$ is denoted by g, with 0≤g≤ $\frac{2}{3}$. 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., μ₀; meanwhile, the potential energy V_se(r) includes the Hulthen-type and centrifugal potentials. Therefore, G_se can be expressed as $G_{\text{se}}=0.62994397Z_{\text{d}}{\left(\frac{{\mu }_0}{Q_{\text{p}}}\right)}^{1/2}\times F\mathrm{,}$ (16) where $\begin{array}{c}F=\frac{x^{1/2}}{2y}\times \ln \left\{\frac{{\left[x(x+2y-1)\right]}^{1/2}+x+y}{(x/y){\left[1+{(1+x/y^2)}^{1/2}\right]}^{-1}+y}\right\}\\ +\arccos {\left\{\frac{1}{2}\left[1-\frac{1-1/y}{{(1+x/y^2)}^{1/2}}\right]\right\}}^{1/2}\\ -{\left[\frac{1}{2y}\left(1+x/2y-1/2y\right)\right]}^{1/2}\mathrm{,}\end{array}$ (17) with $\begin{array}{c}x=\frac{20.9008(l+{1/2)}^2}{{\mu }_0{b}^2{Q}_{\text{p}}}\mathrm{,}\\ y=\frac{\ln \left(t{Z}_{\text{proton}}{Z}_{\text{d}}{e}^2/Q_{\text{p}}+1\right)}{2tb}\mathrm{.}\end{array}$ (18) Then, the proton radioactivity half-life can be calculated by $T_{1/2}=4.108054431\times {10}^{-23}\frac{{\mu }_0{R}_0{}^2}{G+3/2}{S}_{\text{p}}{}^{-1}{P}_{\text{se}}{}^{-1}\mathrm{.}$ (19)

Results and discussion

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 $\sigma =\sqrt{{\displaystyle \sum }{\left({\log }_{10}{T}_{1/2}^{\text{cal}}-{\log }_{10}{T}_{1/2}^{\text{exp}}\right)}^2/n}\mathrm{,}$ (20) where ${\log }_{10}{T}_{1/2}^{\text{cal}}$ and ${\log }_{10}{T}_{1/2}^{\text{exp}}$ 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 shows the difference between the outer turning points before and after considering the screened effect versus $Z_{\text{d}}/Q_{\text{p}}$. $R_{\text{out}}^{\text{H}}$ and $R_{\text{out}}^{\text{C}}$ 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 $Z_{\text{d}}/Q_{\text{p}}$, the more evident this effect is for the penetrability factor.

Fig. 3Full-screen View

(Color online) Difference between $R_{\text{out}}^{\text{C}}$ and $R_{\text{out}}^{\text{H}}$ obtained by the proposed improved Gamow model, where the electrostatic barrier is dismissed and considered, respectively. The turning point radii are defined as Vi($R_{\text{out}}^i$)=Q_p (i=C, H).

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. 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 ${\text{log}}_{10}{T}_{1/2}^{\text{exp}}$, ${\text{log}}_{10}{T}_{1/2}^{\text{cal}}$, ${\text{log}}_{10}{T}_{1/2}^{\text{UDLP}}$, and ${\text{log}}_{10}{T}_{1/2}^{\text{NG}-\text{N}}$, 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 ¹⁷⁷Tl^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 ¹⁷⁷Tl^m is not considered. For comparison, Table 2 lists the standard deviations σ calculated within 31 spherical nuclei (except for ¹⁷⁷Tl^m), NG-N, and UDLP. The results show that the calculated proton radioactivity half-lives of spherical nuclei are reliable.

To intuitively demonstrate the consistency between our results and the experimental data, 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 ¹⁷⁷Tl^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 presents the proton radioactivity half-lives of ¹⁷⁷Tl^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 ¹⁷⁷Tl^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]. They analyzed the single-particle energies from microscopic calculations using the Hartree Fock-Bogolubow model with the Gogny-type force D1S parameter set of ¹⁷⁷Tl^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 ¹⁷⁷Tl^m, which has an excitation energy of only 807 keV. However, the experimental data of spin and parity for ¹⁷⁷Tl^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 ¹⁷⁷Tl^m may be l = 6.

Fig. 4Full-screen View

(Color online) Deviations between the experimental proton radioactivity half-lives and corresponding calculated half-lives in logarithmic form for spherical (a) and deformed (b) nuclei. The red triangle, green square, and blue circle represent the deviations calculated by the proposed improved Gamow model, NG-N, and UDLP, respectively.

Fig. 5Full-screen View

(Color online) Q_p dependence of the proton radioactivity half-lives for ¹⁷⁷Tl^m with various angular momenta (l_exp=5).

Recently, the study of deformed nuclei has attracted extensive attention [55, 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 and subfigure (b) in 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 Sp 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. For a more visual comparison, the predictions of the three models are plotted in 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 $Z_{\text{d}}^{0.8}/Q_{\text{p}}^{1/2}$, i.e., the new Geiger-Nuttall law [42] for proton radioactivity, for l=0, 2, 3, and 5; the results are depicted in 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.

Fig. 6Full-screen View

(Color online) Comparison of the predicted proton radioactivity half-lives using our model, NG-N, and UDLP. They are denoted by red triangle, green square, and blue circle, respectively.

Fig. 7Full-screen View

(Color online) Logarithmic values of the experimental half-lives and predicted half-lives versus $Z_{\text{d}}^{0.8}/Q_{\text{p}}^{1/2}$ for l=0, 2, 3, 5. The blue triangle, green square, and red sphere denote the experimental proton radioactivity half-lives of spherical and deformed nuclei, and predicted half-lives, respectively.

Summary

In summary, based on the Gamow model and considering the screened electrostatic effect, we systematically studied proton radioactivity half-lives. The calculated results show that the experimental data for both spherical and deformed nuclei can be reproduced well with the corresponding parameters. We also analyzed the relationship between the half-life and l of ¹⁷⁷Tl^m, and propose a possible reference value: l=6. Moreover, we extended this 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 and compare them with the predictions of UDLP and NG-N. The results predicted by our model and by these two formulas were consistent with each other. In addition, we verified the reliability of our predictions using the new Geiger-Nuttall law. This study will prompt inquiries regarding nuclear structures and provide information for future experiments.