Introduction

Proton radioactivity, the spontaneous process in which a nucleus with a large excess of protons transforms into the stable state by emitting an individual proton, was first observed by Jackson et al. [1, 2] from an isomeric state of ⁵³Co. In the early 1980s, Hofmann et al. [3] and Klepper et al. [4] further discovered proton emission from the nuclear ground states of ¹⁵¹Lu and ¹⁴⁷Tm, respectively. With the advancement of diverse infrastructures and radioactive beam installations, more than 40 proton emitters have been provenly illustrated in the proton number range 51≤ Z ≤ 83 from the ground state or low-lying isomeric state during the last decades [5-10]. Proton radioactivity can be used as an effective probe to identify nuclei near the proton drip line and explore diverse nuclear information, such as the released energy, half-lives, branching ratio, and wave function of the parent nucleus [11-14]. However, the study of proton-rich nuclei far from the β-stability line has become an extremely hot topic in nuclear physics.

Proton radioactivity shares the same decay mechanism as α decay [15-20], two-proton radioactivity [21-29], and cluster radioactivity [30, 31], that is, barrier penetration. These processes can be dealt with using the Wentzel–Kramers–Brillouin (WKB) approximation. To date, many models and/or formulae have been proposed to calculate the half-life of proton radioactivity, which can be divided into the following two categories: The first are theoretical models, including the effective interactions of density-dependent M3Y [32, 33], single-folding model [21], unified fission model (UFM) [34, 35], Gamow-like model [36], Coulomb and proximity potential model [37-39], generalized liquid-drop model [7, 40-42], and modified two-potential approach [11, 43, 44]. The second are empirical formulae proposed in the form of the Geiger–Nuttall (G-N) law [45], such as the universal decay law for proton emission (UDLP) [46], the new Geiger-Nuttall law (NG-N) [47], the four-parameter empirical formula of Zhang and Dong [12], and the screened decay law for proton emission [48]. Calculations using these theoretical methods provide excellent estimates of the lifetime of proton radioactivity and improve our understanding of the proton radioactivity phenomenon.

For proton radioactivity, it is well known that the emitted particle–nucleus interaction potential is extremely crucial to improving the accuracy of half-life calculations. In 2008, based on the Skyrme-energy-density-function approach, taking into account the isospin effect of the nuclear potential, Wang et al. [49, 50] proposed a modified Woods–Saxon potential to describe a large number of heavy-ion elastic or quasi-elastic scattering and fusion reactions. Later, it was widely used to study α decay and cluster radioactivity [51, 52]. Whether this modified potential can be extended to investigate proton radioactivity is an interesting topic. To this end, based on the two-potential approach within the modified Woods–Saxon potential (TPA-MWS), we systematically studied the half-lives of 32 spherical proton emitters. The results indicate that our calculations are consistent with the experimental data. In addition, we extended this model to predict the half-lives of 17 proton-emitting candidates whose radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020.

The remainder of this paper is organized as follows: In Sect. 2, the theoretical framework of the two-potential approach and two different empirical formulae are introduced in detail. The results and discussion are presented in Sect. 3. Finally, a brief summary is provided in Sect. 4.

Theoretical framework

2.1

Two potential approach

The proton radioactivity half-life, T_1/2, which is an important indicator of nuclear stability, is determined by $T_{1/2}=\frac{\hslash \ln 2}{\text{Γ}}\mathrm{,}$ (1) where ℏ is the reduced Planck constant, and Γ represents the proton radioactivity width. In the framework of the two-potential approach [53-55], Γ can be expressed as $\text{Γ}=S_p\frac{{\hslash }^{2}FP}{4\mu }\mathrm{,}$ (2) where $\mu =m_{\text{p}}{m}_{\text{d}}/(m_{\text{p}}+m_{\text{d}})\approx 938.272\times A_{\text{d}}/\left(A_{\text{d}}+A_{\text{p}}\right)\text{ M}eV/c^2$ represents the reduced mass of the emitted proton and daughter nucleus, with m_p, m_d, A_p, and A_d as the masses of the emitted proton and daughter nucleus, and the mass numbers of the emitted proton and daughter nucleus, respectively. S_p is the formation probability of the emitted proton–daughter system, which is related to the probability that the orbit of the emitted proton is empty in the daughter nucleus [7]. In this study, based on Ref. [34], we chose S_p=1. F is the normalized factor describing the collision frequency of the emitted proton in the potential barrier. This satisfies the condition $F{\displaystyle {\int }_{r_1}^{r_2}}\frac{1}{2k(r)}\text{d}r=1.$ (3) P, the penetration probability of the emitted proton penetrating the barrier, is calculated using the semiclassical WKB approximation. It can be expressed as $P=\text{exp}\left[-2{\displaystyle {\int }_{r_2}^{r_3}}k(r)\text{d}r\right]\mathrm{,}$ (4) where $k(r)=\sqrt{\frac{2\mu }{{\hslash }^2}\left|Q_{\text{p}}-V(r)\right|}$ is the wavenumber, and V(r) is the total interaction potential between the emitted proton and daughter nucleus, with r as the mass center distance between the emitter proton and daughter nucleus. In Eq. 3 and 4, r₁, r₂, and r₃ represent the classical turning points. These satisfy the condition $V(r_1)=V(r_2)=V(r_3)=Q_{\text{p}}$, where Q_p denotes the released energy. It can be obtained by $Q_{\text{p}}=\text{Δ}M-\left(\text{Δ}{M}_{\text{d}}+\text{Δ}{M}_{\text{p}}\right)+k\left(Z^{\epsilon }-Z_{\text{d}}^{\epsilon }\right)\mathrm{,}$ (5) where the experimental data of the mass excess Δ M, Δ M_d, and Δ M_p are taken from the latest atomic mass table NUBASE2020 [56], representing the mass excesses of the parent and daughter nuclei and emitted proton, respectively. The term $k\left(Z^{\epsilon }-Z_{\text{d}}^{\epsilon }\right)$ denotes the screening effect of atomic electrons, with Z and Z_d being the proton number of the parent and daughter nuclei, for Z lt;60, k=13.6 eV, ε=2.408, and for Z ≥ 60, k = 8.7 eV, ε=2.517 [57, 58].

In this study, the total interaction potential V(r) was composed of the nuclear potential V_N(r), Coulomb potential V_C(r), and centrifugal potential V_l(r) (see Fig. 1). It can be expressed as $V(r)=V_{\text{N}}(r)+V_{\text{C}}(r)+V_{\text{l}}(r)\mathrm{.}$ (6) For V_N(r), we chose the modified Woods–Saxon potential [49, 50], which can be parameterized as $V_{\text{N}}(r)=\frac{V_0}{1+\text{exp}\left[(r-R_0)/a\right]}\mathrm{.}$ (7) Here, R₀ is the effective nuclear radius, which can be expressed as $R_0=R_{\text{d}}+R_{\text{p}}-\mathrm{1.37,}$ (8) where $R_{\text{d}}=1.27A_{\text{d}}^{1/3}$ is the radius of the daughter nucleus. R_p=0.875 represents the proton radius [59]. a=0.4+0.33I_d is the diffuseness of the nuclear potential [60]. V₀ denotes the potential depth, which can be parameterized as $V_0=-44.16\left[1-0.4\left(I_{\text{d}}+I_{\text{p}}\right)\right]\frac{A_{\text{d}}^{1/3}{A}_{\text{p}}^{1/3}}{A_{\text{d}}^{1/3}+A_{\text{p}}^{1/3}}\mathrm{,}$ (9) where $I_{\text{d}}=\left(N_{\text{d}}-Z_{\text{d}}\right)/A_{\text{d}}$ and $I_{\text{p}}=\left(N_{\text{p}}-Z_{\text{p}}\right)/A_{\text{p}}$ are the isospin asymmetries of the daughter nucleus and emitted proton with Z_d, N_d and Z_p, N_p being the proton and neutron number of the daughter nucleus and emitted proton, respectively.

Fig. 1Full-screen View

(color online) Variation in the total V(r) and nuclear V_N(r) potentials for the parent nucleus ¹⁷⁰Au as a function of the nuclear radius.

V_C(r) is the Coulomb potential, which is taken as the potential of a uniformly charged sphere. It can be expressed as $V_{\text{C}}(r)=\{\begin{array}{ll}\frac{Z_{\text{d}}{Z}_{\text{p}}{e}^2}{2R}\left[3-\frac{r^2}{R^2}\right],\hfill & r\le R\mathrm{,}\hfill \\ \frac{Z_{\text{d}}{Z}_{\text{p}}{e}^2}{r},\hfill & r>R\mathrm{,}\hfill \end{array}$ (10) where e² ≈ MeV⋅fm denotes the square of the electronic elementary charge [61]. R denotes the sharp radius. For convenience, it can be taken as the effective nuclear radius R₀ of the modified Woods–Saxon potential [62].

For the centrifugal potential Vl(r), we chose the Langer modified form because $l(l+1)\to {(l+1/2)}^2$ is an essential correction for one-dimensional problems [63]. It can be expressed as $V_l(r)=\frac{{\hslash }^2{\left(l+\frac{1}{2}\right)}^2}{2\mu r^2}\mathrm{.}$ (11) Here, l is the angular momentum removed by the emitted proton, which satisfies the spin–parity conservation laws. It can be expressed as $l=\{\begin{array}{ll}{\text{Δ}}_j\hfill & \text{f}\text{o}\text{r}\text{e}\text{v}\text{e}\text{n}{\text{Δ}}_j\text{ and }\pi ={\pi }_{\text{d}}\mathrm{,}\hfill \\ {\text{Δ}}_j+1\hfill & \text{f}\text{o}\text{r}\text{e}\text{v}\text{e}\text{n}{\text{Δ}}_j\text{ and }\pi \ne {\pi }_{\text{d}}\mathrm{,}\hfill \\ {\text{Δ}}_j\hfill & \text{for odd }{\text{Δ}}_j\text{ and }\pi \ne {\pi }_{\text{d}}\mathrm{,}\hfill \\ {\text{Δ}}_j+1\hfill & \text{for odd }{\text{Δ}}_j\text{ and }\pi ={\pi }_{\text{d}}\mathrm{,}\hfill \end{array}$ (12) where Δ_j=|j-j_d-j_p|, with j, π, j_d, π_d, j_p, π_p denoting the spin and parity values of the parent and daughter nuclei and emitted proton, respectively.

Results and discussion

In this study, we systematically investigated the half-lives of 32 spherical proton emitters based on the TPA–MWS. For comparison, the UFM [35], Coulomb potential and proximity potential model with the Guo-2013 formalism (CPPM) [66], UDLP [46] and NG-N [47] were also used. The detailed numerical results are listed in Table 1. In this table, the first three columns contain the proton emitter, proton radioactivity released energy Q_p, and angular momentum l removed by the emitted proton. The last six columns contain the logarithmic form of the experimental half-lives and theoretical half-lives calculated using three models and two different empirical formulae denoted as Exp, TPA-MWS, UFM, CPPM, UDLP, and NG-N. As shown in Table 1, the calculated proton radioactivity half-lives obtained using our model can well reproduce the experimental data. To visually present their deviations, we plotted the differences between the experimental proton radioactivity half-lives and the calculated values using these five models and/or formulae in logarithmic form in Fig. 2. From this figure, we can see that the deviations are in the range of ±1. Overall, all of the calculated proton radioactivity half-lives were consistent with the experimental data. Nevertheless, it should be noted that the logarithmic deviation between the experimental and calculated half-lives was large, corresponding to the special case ¹⁷⁷Tl^m. Its deviations in logarithmic form were –0.889, -0.802, -1.176, -0.881, and -0.826 using TPA-MWS, UFM, CPPM, UDLP, and NG-N, respectively. Exploring the causes of such a large deviation is crucial for further investigation of proton radioactivity. As we know, the selection of the proton radioactivity released energy Q_p and angular momentum l is important in proton radioactivity. In our previous study, we discussed that the effect of angular momentum on the half-life is larger than the released energy [67]. The result indicated that the calculated half-life of proton radioactivity for ¹⁷⁷Tl^m was closer to the experimental data when l = 6 [67]. Therefore, we chose l =6 and calculated its logarithm proton radioactivity half-life as -2.779 with a deviation of 0.567, which has a higher consistency with the experimental data than the calculated value of -4.235 when the value of l is set as 5. In addition, the proton number of ¹⁷⁷Tl^m is close to the magic number Z = 82, and the proton shell effect may be another reason for this phenomenon.

Fig. 2Full-screen View

(color online) Deviations between the experimental proton radioactivity half-lives and the calculated values using three models and two different empirical formulae for spherical nuclei. The rhombi, circles, triangles, inverted triangles, and squares correspond to the results obtained using the UFM, CPPM, UDLP, NG-N, and TPA-MWS, respectively.

To obtain further insight into the agreement between the experimental and calculated data, the root-mean-square deviation σ was used to estimate the calculated capabilities of the above five models and/or formulae for the proton radioactivity half-lives. It can be defined as $\sigma =\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left({\text{log}}_{10}{T}_{1/2}^{\text{cal}\mathrm{.}i}-\text{l}o{g}_{10}{T}_{1/2}^{\text{exp}\mathrm{.}i}\right)}^2}}\mathrm{,}$ (15) where ${\text{log}}_{10}{T}_{1/2}^{\text{cal}\mathrm{.}i}$ and ${\text{log}}_{10}{T}_{1/2}^{\text{exp}\mathrm{.}i}$ denote the experimental proton radioactivity half-life and the calculated half-life in logarithmic form for the i-th nucleus, respectively. For comparison, the specific results of σ for the five theoretical models and/or formulae are listed in Table 2. From this table, we can clearly see that the standard deviation σ obtained using TPA-MWS (${\sigma }_{\text{TPA-MWS}}=0.341$) was smaller than that of the others. This means that the present model can better reproduce the experimental proton radioactivity half-lives than the others for spherical emitters.

A single line of the universal curve for α decay and cluster radioactivities could be manifested by plotting the decimal logarithm of the half-life versus the negative decimal logarithm of the penetration probability through the barrier [68, 69]. Whether this linear correlation can be extended to investigate proton radioactivity is an interesting topic [69]. Therefore, we plotted the correlation between the decimal logarithm of the experimental half-life against the negative decimal logarithm of penetrability in Fig 3. The penetrability was calculated theoretically in the framework of the semiclassical WKB approximation. As shown, the decimal logarithm of the experimental half-life had a clear linear dependence on the negative decimal logarithm of the penetrability, which implies that our calculations and the present methods are valid and reliable.

Fig. 3Full-screen View

(color online) Decimal logarithm of the experimental half-lives plotted against $-{\text{log}}_{10}P$ for proton radioactivity. The slope and intercept of the fitted solid line for each case are shown in the corresponding figure.

Considering the good agreement between the experimental half-lives and those calculated using the present model, we extended our model to predict the half-lives of proton radioactivity in the proton number region $51\le Z\le 83$. For comparison, the UFM, CPPM, UDLP, and NG-N were also used. The detailed predictions are listed in Table 3. In this table, the first three columns contain the possible proton radioactivity candidates, released energy Q_p, and orbital angular momentum l. The last five columns contain the predicted proton radioactivity half-lives obtained using the five models and/or formulae. From this table, we can clearly see that the magnitude of all results using our model is consistent with those calculated using the other four models and/or formulae, validating the predictive power of this method. Taking ¹¹⁶La as an example, the predictions were –10.256, –10.117, –9.963, –9.000, and –9.126. To further test the reliability of the predictions, we obtained a linear relationship between the quantities ${\text{log}}_{10}{T}_{1/2}^{\text{pre}}$ and $(Z_{\text{d}}^{0.8}+l)Q_{\text{p}}^{-1/2}$ based on the NG-N [47] in Fig. 4. The results show that the predictions are consistent with the NG-N, which indicates that our predictions of the proton radioactivity half-lives are valid and reliable. These predictions may provide valuable information for when searching for new nuclides with proton radioactivity in the future.

Fig. 4Full-screen View

(color online) Linear relationship between the quantities ${\text{log}}_{10}{T}_{1/2}^{\text{pre}}$ and $\left(Z_d^{0.8}+l\right)Q_p^{-1/2}$ based on the NG-N.

Furthermore, recent studies have shown that nuclear deformation plays an important role in proton radioactivity [37, 44, 48, 69]. Barrier penetration can be affected by the deformation of daughter nuclei during the proton emission process [70]. The deformation effect can result in the reduction of Coulomb and centrifugal barriers, which can influence the penetrability and reduce the theoretical half-lives [11]. In addition, some methods can be used to calculate the proton radioactivity half-lives of deformed proton emitters. In 2016, based on the WKB tunneling approximation, Qian and Ren et al. [44] proposed a simple formula including the deformation term to evaluate the proton radioactivity half-lives of the deformed nuclei. In 2022, the deformation dependence of the screened decay law for proton radioactivity was proposed by Budaca et al. [48] to systematically study proton radioactivity. Inspired by this, we will also discuss the influence of the deformation effect on the estimation of proton radioactivity half-lives in more detail in future work.

Summary

In summary, we present a systematic study of the proton radioactivity half-lives for 32 spherical proton emitters based on the TPA-MWS, which contains the isospin effect. It was found that the calculated results could reproduce the experimental data well. In addition, we extended this model to predict the proton radioactivity half-lives for 17 possible candidates. The predicted results are in reasonable agreement with those obtained using the UFM, CPPM, UDLP, and NG-N. This study may provide useful and reliable information for when predicting the existence of new proton emitters above the proton drip line, which may be detected in the future.