1. Introduction
The stability of a ground-state atomic nucleus is dependent on the complicated balance between the numbers of protons and neutrons. Especially, in the vicinities of proton and neutron drip lines, where the balance is strongly disturbed, the exotic nucleus with the extreme numbers of nucleons exhibits active radioactivity. The neutron-rich nuclei beyond the neutron drip line can become stable by β− decay, i.e. n → p + e− + $\bar{v}$. The proton-rich nuclei beyond the proton drip line could naturally emit excess protons to move toward stability via proton radioactivity. Generally, the modes of proton radioactivity include one proton radioactivity and two proton radioactivity. Experimentally, the one proton radioactivity phenomenon was firstly observed from the isomeric state of 53Co by Jackson et al in 1970 [1, 2]. Subsequently, Hofmann et al and Klepper et al also discovered the one proton radioactivity from nuclear ground states of 151Lu and 147Tm in 1982, independently [3, 4]. With the development of experimental facilities and radioactive beams, there are about 44 one proton emitters from the ground state or isomeric state being detected. Observing these one proton emitters, it is evidently found that the rich-proton odd-Z nuclei between Z = 51 and Z = 83 are more likely to emit a proton and form a new nucleus [5–10]. However, for two proton radioactivity, it was not until 2002 that this exotic phenomenon was firstly observed in experiments performed at GANIL [11] and GSI [12]. Up to now, there are 8 two proton radioactivity nuclei being discovered in different experiments [13–18]. It is found that the even-Z nuclei beyond the proton drip line are more likely to occur two proton radioactivity. The studies on these two kinds of proton radioactivity can provide important nuclear structure information for the exotic proton-rich nuclei, such as the shell structure and coupling between the bound and unbound nuclear states [19, 20]. In addition, one proton radioactivity can be treated as the inverse process of rapid proton capture, which is of great importance in the understanding of the origin of elements and the evolution of stars [21]. Theoretically, proton radioactivity shares the same decay mechanism as α decay, i.e. quantum tunneling through potential barriers [22–35], the half-lives can be evaluated by the one-dimensional WKB integral approximation. Based on the description, a lot of models and/or formulas were proposed to investigate proton radioactivity. For one proton radioactivity, there is the Woods-Saxon nuclear potential model [36–38], the effective interaction potential model of density-dependent M3Y [39–41], the generalized liquid drop model [42, 43], the single fold model [41, 44], the modified two-potential approach [8, 41] and others [45, 46]. For two proton radioactivity, there is the Coulomb and proximity potential model [47], the unified fission model [48], the electrostatic screened penetration model [49], the Gamow-like model [50], the two-potential and Skyrme-Hartree–Fock approaches [51], the generalized liquid drop model [52] and others [53]. These theoretical studies have improved our understanding of the proton radioactivity phenomenon [54–58]. Meanwhile, the new models and/or formulas are also awaiting proposal and/or employment to further research for proton radioactivity.
In 2005, based on the WKB theory, Tavares et al firstly proposed an OPM to calculate α decay half-lives of bismuth isotopes [59]. Their calculated results can reproduce the experimental data well, especially the predicted half-life for naturally occurring α emitter 209Bi has been confirmed by the experiment within a factor of ∼2. Moreover, in 2006, Medeiros et al used the OPM to systematically study α decay half-lives of 320 favored α decay nuclei [60]. The calculated results are in great agreement with the experimental data with the standard deviation σ = 0.317. Recently, using the OPM, we systematically investigate the α decay half-lives of neptunium isotopes [61]. The results indicate that the neutron number N = 126 shell effect is still robust in neptunium. Since one proton radioactivity shares the same mechanism of tunneling effect with α decay, whether the OPM can be extended to study the one proton radioactivity is an interesting issue. So in this article, using the OPM, we systematically investigate the favored one proton radioactivity half-lives of proton-rich nuclei with 51 ≤ Z ≤ 83. The calculated results show a good consistency with the experimental data.
This article is organized as follows. In the next section, the theoretical framework of OPM is briefly described. The detailed calculations and discussion are presented in section 3 . Finally, a summary is given in section 4 .
2. Theoretical framework
The one proton radioactivity half-life is generally calculated by
$\begin{eqnarray}{T}_{1/2}=\displaystyle \frac{\mathrm{ln}2}{{\nu }_{0}P},\end{eqnarray}$
where P is the penetration factor through the phenomenological potential barrier. ν0 is the assault frequency, which can be calculated by the oscillation frequency ω [62] and expressed as $\begin{eqnarray}{\nu }_{0}=\displaystyle \frac{\omega }{2\pi }=\displaystyle \frac{(2{n}_{{\rm{r}}}+l+\tfrac{3}{2}){\hslash }}{1.2\pi {\mu }_{0}{R}_{{\rm{n}}}^{2}}=\displaystyle \frac{(G+\tfrac{3}{2}){\hslash }}{1.2\pi {\mu }_{0}{R}_{0}^{2}}.\end{eqnarray}$
Here Rn is the nucleus root-mean-square (rms) radius, the relationship of ${R}_{{\rm{n}}}^{2}=\tfrac{3}{5}{R}_{0}^{2}$ is used in this work with ${R}_{0}=1.240{A}^{1/3}\left(1+\tfrac{1.646}{A}-0.191\tfrac{A-2Z}{A}\right)$ [63]. G = 2nr + l is the principal quantum number with nr and l being the radial and angular momentum quantum numbers, respectively. For one proton radioactivity, we choose G = 4 or G = 5 corresponding to the 4ℏω or 5ℏω oscillator shell depending on the individual proton emitter. ℏ is the reduced Planck constant. μ0 denotes the reduced mass between the emitted one proton and daughter nucleus in the final decaying nuclear system. It can be calculated by $\begin{eqnarray}\displaystyle \frac{1}{{\mu }_{0}}=\displaystyle \frac{1}{{m}_{{\rm{d}}}}+\displaystyle \frac{1}{{m}_{{\rm{z}}}},\end{eqnarray}$
$\begin{eqnarray}{m}_{{\rm{d}}}={A}_{{\rm{d}}}+\displaystyle \frac{{\rm{\Delta }}{M}_{{\rm{d}}}}{F}-\left({Z}_{{\rm{d}}}{m}_{{\rm{e}}}-\displaystyle \frac{{10}^{-6}{{kZ}}_{{\rm{d}}}^{\beta }}{F}\right)\ {\rm{u}},\end{eqnarray}$
where mz, md and me are the atomic mass of the emitted one proton, daughter nucleus and electron, respectively. Ad, Zd and ΔMd are the mass number, proton number and the mass excess of the daughter nucleus, respectively. F = 931.494009 MeV u−1 is the mass-energy conversion factor. The value of k = 8.7 eV, β = 2.517 for Z ≥ 60, and k = 13.6 eV, β = 2.408 for Z ≤ 60 [64].The penetration factor P given in equation (1 ) can be calculated by6 ) including the Coulomb VC and centrifugal Vl part, which are expressed as6 ) and also can be divided into two parts5 ), (6 ), (16 ) and (17 ), the barrier penetrability can be factorized as P = Pin Pout, where Pin = ${{\rm{e}}}^{-{\text{}}{G}_{\mathrm{in}}}$ and Pout = ${{\rm{e}}}^{-{\text{}}{G}_{\mathrm{out}}}$, with associated the Gamow factors as
$\begin{eqnarray}P={{\rm{e}}}^{-G},\end{eqnarray}$
$\begin{eqnarray}G=\displaystyle \frac{2}{{\hslash }}{\int }_{a}^{b}\sqrt{2\mu (s)[V(s)-{Q}_{{\rm{p}}}]}{\rm{d}}{s},\end{eqnarray}$
where G is the Gamow factor, s and μ(s) are the centre distance and reduced mass between the emitted one proton and daughter nucleus, respectively. V(s) is the emitted one proton-daughter nucleus interaction potential. a and b are the classical inner and outer turning points of a potential barrier satisfying the conditions V(a) = V(b) = Qp. Here Qp is the one proton radioactivity released energy. It can be expressed as [65] $\begin{eqnarray}\begin{array}{rcl}{Q}_{{\rm{p}}} & = & {\rm{\Delta }}{M}_{{\rm{p}}}-({\rm{\Delta }}{M}_{{\rm{d}}}+{\rm{\Delta }}{M}_{{\rm{z}}})\\ & & +{10}^{-6}{\text{}}k({Z}_{{\rm{p}}}^{\beta }-{Z}_{{\rm{d}}}^{\beta })\ \mathrm{MeV},\end{array}\end{eqnarray}$
where ΔMp and ΔMz are the mass excess of the parent nucleus and emitted one proton, respectively, which can be obtained from the evaluated nuclear properties table NUBASE2020 [66]. In the OPM, a = Rp − Rz is the difference between the radius of the parent nucleus and the emitted one proton. c = Rd + Rz is the centre distance between the daughter nucleus and emitted proton at the touching configuration point. Rz = 0.8409 fm is the radius of one proton in this work [67]. Rp and Rd are the radii of the parent nucleus and daughter nucleus, respectively. They are calculated by the droplet model of an atomic nucleus and expressed as [68, 69] $\begin{eqnarray}{R}_{{i}}=\displaystyle \frac{{Z}_{{i}}}{{A}_{{i}}}{R}_{{\rm{p}}i}+\left(1-\displaystyle \frac{{Z}_{{i}}}{{A}_{{i}}}\right){R}_{{\rm{n}}i},\qquad {i}={\rm{p}},{\rm{d}},\end{eqnarray}$
$\begin{eqnarray}{R}_{{ji}}={r}_{{ji}}\left[1+\displaystyle \frac{5}{2}{\left(\displaystyle \frac{1}{{r}_{{ji}}}\right)}^{2}\right],\qquad {j}={\rm{p}},{\rm{n}};{i}={\rm{p}},{\rm{d}},\end{eqnarray}$
where rji represent the equivalent sharp radius of a proton (j = p) or neutron (j = n) density distribution of a parent nucleus (i = p) or daughter nucleus (i = d), respectively. According to the finite-range droplet model theory of nuclei proposed by Möller et al [69], the equivalent sharp radius can be expressed as $\begin{eqnarray}\begin{array}{l}{r}_{{\rm{p}}i}=1.16(1+\bar{{\epsilon }_{{i}}})\\ \quad \times \left[1-\displaystyle \frac{2}{3}\left(1-\displaystyle \frac{{Z}_{{i}}}{{A}_{{i}}}\right)\left(1-\displaystyle \frac{2{Z}_{{i}}}{{A}_{{i}}}-\bar{{\delta }_{{i}}}\right)\right]{A}_{{i}}^{1/3},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{r}_{\mathrm{ni}}=1.16(1+\bar{{\epsilon }_{{i}}})\\ \quad \times \left[1+\displaystyle \frac{2}{3}\displaystyle \frac{{Z}_{{i}}}{{A}_{{i}}}\left(1-\displaystyle \frac{2{Z}_{{i}}}{{A}_{{i}}}-\bar{{\delta }_{{i}}}\right)\right]{A}_{{i}}^{1/3},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\bar{{\epsilon }_{{i}}}=\displaystyle \frac{1}{4{{\rm{e}}}^{0.831{A}_{{i}}^{1/3}}}-\displaystyle \frac{0.191}{{A}_{{i}}^{1/3}}+\displaystyle \frac{0.0031{Z}_{{i}}^{2}}{{A}_{{i}}^{4/3}},\end{eqnarray}$
$\begin{eqnarray}\bar{{\delta }_{{i}}}=\left(1-\displaystyle \frac{2{Z}_{{i}}}{{A}_{{i}}}+0.004781\displaystyle \frac{{Z}_{{i}}}{{A}_{{i}}^{2/3}}\right)/\left(1+\displaystyle \frac{2.52114}{{A}_{{i}}^{1/3}}\right).\end{eqnarray}$
The emitted one proton-daughter nucleus interaction potential V(s) appears in equation ( $\begin{eqnarray}{V}_{C}(s)=\displaystyle \frac{{Z}_{{\rm{z}}}{Z}_{{\rm{d}}}{{\rm{e}}}^{2}}{s},\end{eqnarray}$
$\begin{eqnarray}{V}_{l}(s)=\displaystyle \frac{{\left(l+\tfrac{1}{2}\right)}^{2}{{\hslash }}^{2}}{2{\mu }_{0}{s}^{2}},\end{eqnarray}$
where Zz = 1 is the charge number of one proton, e2 = 1.4399652 MeV fm is the square of the electronic elementary charge. l is the angular momentum taken by the emitted one proton. It can be obtained by spin-parity conservation laws [70]. In the process of emitting one proton preformed at the parent nucleus surface, the interaction potential between the emitted one proton and the daughter nucleus is denoted as inner potential Vin. Therefore, the total emitted one proton-daughter nucleus interaction potential V(s), shown in figure 1, is given by $\begin{eqnarray}V(s)=\left\{\begin{array}{ll}{V}_{\mathrm{in}}(s), & a\leqslant s\leqslant c,\\ {V}_{C}(s)+{V}_{l}(s), & c\leqslant s\leqslant b,\end{array}\right.\end{eqnarray}$
where Vin(s) = Qp + (V(c) − Qp) ${\left(\tfrac{s-a}{c-a}\right)}^{q}$ is used to describe inner potential barrier with an adjustable parameter q ≥ 1 [59, 71]. V(c) is the potential energy of critical point c between the overlapping region and separation region. The reduced mass μ(s) appears in equation ( $\begin{eqnarray}\mu (s)=\left\{\begin{array}{ll}{\mu }_{\mathrm{in}}(s), & a\leqslant s\leqslant c,\\ {\mu }_{0}, & c\leqslant s\leqslant b,\end{array}\right.\end{eqnarray}$
where μin(s) = ${\mu }_{0}{\left(\tfrac{s-a}{c-a}\right)}^{p}$ is used to describe inner reduced mass with an adjustable parameter p ≥ 0 [59, 72]. From equations ( $\begin{eqnarray}{G}_{\mathrm{in}}=\displaystyle \frac{2}{{\hslash }}{\int }_{a}^{c}\sqrt{2{\mu }_{\mathrm{in}}(s)[{V}_{\mathrm{in}}(s)-{Q}_{{\rm{p}}}]}{\rm{d}}{s},\end{eqnarray}$
$\begin{eqnarray}{G}_{\mathrm{out}}=\displaystyle \frac{2}{{\hslash }}{\int }_{c}^{b}\sqrt{2{\mu }_{0}[{V}_{\mathrm{out}}(s)-{Q}_{{\rm{p}}}]}{\rm{d}}{s}.\end{eqnarray}$
The inner Gamow factor Gin is given by a simple analytic form as [59] $\begin{eqnarray}\begin{array}{l}{G}_{\mathrm{in}}=0.4374702(a-c){\left(1+\displaystyle \frac{p+q}{2}\right)}^{-1}\\ \quad \times {\left\{{\mu }_{0}\left[\displaystyle \frac{{Z}_{{\rm{d}}}{{\rm{e}}}^{2}}{c}+\displaystyle \frac{20.9008{\left(l+\tfrac{1}{2}\right)}^{2}}{{\mu }_{0}{c}^{2}}-{Q}_{{\rm{p}}}\right]\right\}}^{1/2},\end{array}\end{eqnarray}$
where the adjustable one parameter is defined as ${\rm{g}}={\left(1+\tfrac{p+q}{2}\right)}^{-1}$ with 0 < g ≤ 2/3 in the model.
Figure 1. Shape of the emitted one proton-daughter nucleus interaction potential barrier for favored one proton radioactivity systems. a and b are the turning points of the potential energy, the barrier region of a to c and c to b is in the overlapping region and separation region, respectively. |
The outer Gamow factor Gout for the outer barrier region is analytically given by [73]
$\begin{eqnarray}{G}_{\mathrm{out}}=0.62994397{Z}_{{\rm{d}}}{\left(\displaystyle \frac{{\mu }_{0}}{{Q}_{{\rm{p}}}}\right)}^{1/2}F,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}F & = & \displaystyle \frac{{\text{}}{x}^{1/2}}{2y}\times \mathrm{ln}\displaystyle \frac{{\left[x(x+2y-1)\right]}^{1/2}+x+y}{\tfrac{x}{y}{\left[1+{\left(1+\tfrac{x}{{y}^{2}}\right)}^{1/2}\right]}^{-1}+y}\\ & & +\arccos {\left\{\displaystyle \frac{1}{2}\left[1-\displaystyle \frac{1-\tfrac{1}{y}}{{\left(1+{\tfrac{x}{y}}^{2}\right)}^{1/2}}\right]\right\}}^{1/2}\\ & & -{\left[\displaystyle \frac{1}{2y}\left(1+\displaystyle \frac{x}{2y}-\displaystyle \frac{1}{2y}\right)\right]}^{1/2},\end{array}\end{eqnarray}$
in which $\begin{eqnarray}x=\displaystyle \frac{20.9008{\left(l+\tfrac{1}{2}\right)}^{2}}{{\mu }_{0}{c}^{2}{Q}_{{\rm{p}}}},y=\displaystyle \frac{{Z}_{{\rm{d}}}{{\rm{e}}}^{2}}{2{{cQ}}_{{\rm{p}}}}.\end{eqnarray}$
So far, the one proton radioactivity half-lives can be evaluated by the OPM and expressed as $\begin{eqnarray}{T}_{1/2}=4.108054431\times {10}^{-23}\displaystyle \frac{{\mu }_{0}{R}_{0}^{2}}{\left(G+\tfrac{3}{2}\right)}{{\rm{e}}}^{({G}_{\mathrm{in}}+{G}_{\mathrm{out}})}\ {\rm{s}}.\end{eqnarray}$
3. Results and discussion
Firstly, we systematically investigate the favored one proton radioactivity half-lives within the OPM. From the experimentally discovered 44 one proton emitters, we select the experimental data of 9 favored one proton radioactivity nuclei as a database, and their experimental half-lives ${T}_{1/2}^{\mathrm{expt}.}$ are taken from the latest evaluated nuclear properties table NUBASE2020 [66] and the related [70]. The adjustable parameter g is the coupling term of the reduced mass parameter p and interaction potential parameter q between the preformed one proton and daughter nucleus in inner complex nuclear many-body systems. It is obtained by fitting the experimental half-lives of these favored one proton radioactivity nuclei by minimizing the root mean square difference (rms). The rms σ represents the difference between the experimental one proton radioactivity half-lives and the calculated ones using OPM. In this work, it is defined as
$\begin{eqnarray}\sigma =\sqrt{\displaystyle \frac{1}{9}\sum _{i=1}^{9}{\left({\mathrm{log}}_{10}{T}_{1/{2}_{{i}}}^{\mathrm{calc}}-{\mathrm{log}}_{10}{T}_{1/{2}_{{i}}}^{\mathrm{expt}.}\right)}^{2}}.\end{eqnarray}$
Here ${\mathrm{log}}_{10}{T}_{1/{2}_{{i}}}^{\mathrm{expt}.}$ and ${\mathrm{log}}_{10}{T}_{1/{2}_{{i}}}^{\mathrm{calc}}$ represent the logarithmic form of experimental and calculated one proton radioactivity half-life for the i-th nucleus, respectively. By fitting these 9 experimental favored one proton radioactivity half-lives, the adjustable parameter is determined to be 0.0326 when minimal σ = 0.470. The detailed relationship between σ and g is displayed in figure 2. The σ = 0.470 means that the calculated results can reproduce the experimental half-lives well and differ from the experiment data by a factor of 2.95 on average. Using the OPM and parameter g, we systematically calculate the favored one proton radioactivity half-lives of these 9 nuclei. For comparison, the Coulomb and proximity potential model (CPPM) with Guo2013 is analyzed from our previous work [74], and the Gamow-like model (GLM) with screened electrostatic barrier [75] and UDLP [76] are used. The detailed results are listed in table 1. In table 1, the first four columns present the experimental data of one proton radioactivity parent nucleus, corresponding to one proton radioactivity released energy Qp, spin-parity transformation (${j}_{{\rm{p}}}^{\pi }\to {j}_{{\rm{d}}}^{\pi }$) and the logarithmic form of experimental one proton radioactivity half-lives (Expt.), respectively. The last four columns represent the logarithmic form of theoretical one proton radioactivity half-lives ${\mathrm{log}}_{10}{T}_{1/2}$ (s), which are calculated using the CPPM [74], GLM [75], UDLP [76] and OPM, respectively. In order to intuitively give comparisons of the experimental half-lives with the calculated results, we present the individual decimal logarithmic deviation between experimental one proton radioactivity half-lives and calculated results in figure 3. In this figure, the blue regular triangle, green inverted triangle, red circle and purple rhombus denote the decimal logarithm deviations between the experimental one proton radioactivity half-lives and the calculated results using the CPPM, GLM, UDLP and OPM, respectively. From this figure, we can see that all the decimal logarithm deviations are basically within the range of ± 1, which indicates that OPM can be treated as a great tool to study the favored one proton radioactivity half-lives. Nevertheless, there are large decimal logarithm deviations between the experimental half-lives and calculated ones using the different models and/or formulas for the isomeric-state proton emitters. Taking 185Bim as an example, its experimental one proton radioactive half-life is −4.192, but the theoretical calculations using the CPPM, GLM, UDLP and OPM are −5.017, −4.971, −4.759 and −5.064, respectively. The decimal logarithm deviations between the experimental data and the calculated results using these models and formulas are almost an order of magnitude. In fact, the probability of one proton emission depends on the deformation of the system and the effects arising from the deformed shape, especially for the triaxial (see, for example 141Ho[77]) and oblate deformed (such as the Bi isotopes, see [78–80]) one proton emitters. Thus, the deformation effect can not be ignored for further study on one proton radioactivity half-lives of deformed proton emitters.
Figure 2. The relationship between the standard deviation σ and the value of adjustable parameter g. |
Figure 3. Decimal logarithm deviations between the favored experimental one proton radioactivity half-lives and the calculated ones using different theoretical models and/or formulas. |
Table 1. Comparison of favored experimental one proton radioactivity half-lives with the calculated ones using different theoretical models and/or formulas. The symbol m denotes the isomeric state. The Qp values are calculated by equation ( |
| ${\mathrm{log}}_{10}{T}_{1/2}({\rm{s}})$ | |||||||
|---|---|---|---|---|---|---|---|
| Nuclei | Qp (MeV) | ${j}_{{\rm{p}}}^{\pi }\to {j}_{{\rm{d}}}^{\pi }$ | Expt. | CPPM | GLM | UDLP | OPM |
| 141Hom | 1.264 | (1/2+) → 0+ | −5.137 | −5.769 | −5.865 | −5.331 | −5.979 |
| 146Tm | 0.904 | (1+) → 1/2+# | −0.810 | −0.315 | −0.773 | −0.610 | −0.711 |
| 157Ta | 0.956 | 1/2+ → 0+ | −0.529 | 0.122 | −0.305 | −0.188 | −0.154 |
| 161$\mathrm{Re}$ | 1.216 | 1/2+ → 0+ | −3.357 | −2.953 | −3.152 | −2.895 | −3.113 |
| 167Ir | 1.087 | 1/2+ → 0+ | −1.128 | −0.631 | −0.940 | −0.865 | −0.822 |
| 171Au | 1.464 | (1/2+) → 0+ | −4.652 | −4.527 | −4.569 | −4.298 | −4.579 |
| 176Tl | 1.278 | (3−, 4−, 5−) → (7/2−) | −2.208 | −1.909 | −2.133 | −2.059 | −2.067 |
| 177Tl | 1.172 | (1/2+) → 0+ | −1.178 | −0.610 | −0.855 | −0.863 | −0.740 |
| 185Bim | 1.625 | 1/2+ → 0+ | −4.192 | −5.017 | −4.971 | −4.759 | −5.064 |
To better demonstrate the reproducibility of OPM with other theoretical models and/or formulas for favored one proton radioactivity half-lives, we calculate the root mean square deviations σ by equation (25 ) and the results are listed in table 2. From table 2, we can see that the σ obtained by OPM is smaller than the ones calculated by the CPPM, which is reduced by 12.3%.
Table 2. The standard deviations σ between favored experimental half-lives and the calculated ones using different theoretical models and/or formulas. |
| Models | CPPM | GLM | UDLP | OPM |
|---|---|---|---|---|
| σ | 0.536 | 0.392 | 0.341 | 0.470 |
Finally, we use the OPM to predict the half-lives of possible favored one proton radioactivity nuclei whose decay is energetically allowed or observed but not quantified in NUBASE2020 [66] with the proton number region of 67 ≤ Z ≤ 85. The detailed results are listed in table 3. For comparison, the predicted results using UDLP are also listed in table 3, in which the first three columns represent the serial number (S.No.), one proton radioactivity parent nucleus and Qp value, respectively. The last two columns are the logarithmic form of one proton radioactivity half-lives calculated by the UDLP and OPM, respectively. Note from table 3 that the predicted one proton radioactivity half-lives using OPM are in reasonable agreement with the ones using UDLP. Moreover, we draw the predicted one proton radioactivity half-lives against the quantity of $({Z}_{{\rm{d}}}^{0.8}\,+\,$l $){Q}_{{\rm{p}}}^{-1/2}$ i.e. Geiger-Nuttall-like law for one proton radioactivity [70] in figure 4. Note from this figure that the predicted results depict an approximate straight line. These predictions are helpful for searching for the new nuclides with one proton radioactivity.
Figure 4. Predicted one proton radioactivity half-lives plotted against the quantity $({Z}_{{\rm{d}}}^{0.8}\,+\,$l $){Q}_{{\rm{p}}}^{-1/2}$, resembling a Geiger-Nuttall law for one proton radioactivity. |
Table 3. Predicted one proton radioactivity half-lives using OPM and UDLP for favored one proton radioactivity nuclei whose decay is energetically allowed or observed but not quantified in NUBASE2020 [66] and the related [81, 82]. The Qp values are calculated by equation ( |
| ${\mathrm{log}}_{10}{T}_{1/2}$(s) | ||||
|---|---|---|---|---|
| S.No. | Nuclei | Qp (MeV) | UDLP | OPM |
| 1 | 159$\mathrm{Re}$ [66] | 1.606 | −6.729 | −6.796 |
| 2 | 163$\mathrm{Re}$ [82] | 0.723 | 4.769 | 5.375 |
| 3 | 163Ir [66] | 1.917 | −8.175 | −8.434 |
| 4 | 165Ir [66] | 1.547 | −5.682 | −5.822 |
| 5 | 169Ir [81] | 0.628 | 8.294 | 8.971 |
| 6 | 168Au [66] | 2.007 | −8.148 | −8.534 |
| 7 | 169Au [66] | 1.947 | −7.812 | −8.186 |
| 8 | 170Au [82] | 1.587 | −5.396 | −5.652 |
| 9 | 173Au [81] | 1.002 | 1.030 | 1.134 |
| 10 | 176Tl [82] | 1.278 | −1.882 | −2.067 |
| 11 | 178Tl [82] | 0.898 | 3.603 | 3.745 |
| 12 | 179Tl [81] | 0.773 | 6.242 | 6.548 |
| 13 | 184Bi[82] | 1.559 | −3.973 | −4.486 |
| 14 | 193At [81] | 0.728 | 9.156 | 9.294 |
4. Summary
In summary, a phenomenological OPM based on the WKB theory is applied to systematically study the favored one proton radioactivity half-lives. The only adjustable parameter g is the coupling term of the reduced mass parameter p and the interaction potential parameter q between the preformed proton and daughter nucleus in inner complex nuclear many-body systems. By fitting the 9 favored experimental one proton radioactivity half-lives, we obtain g = 0.0326 with the ${\sigma }_{\min }=0.470$. Using the OPM and parameter g, we systematically calculate the one proton radioactivity half-lives of these nuclei. It is found that our results can reproduce the experimental data well. In addition, we extend the OPM to predict the half-lives of possible favored one proton radioactivity nuclei whose decay is energetically allowed or observed but not quantified in NUBASE2020. The predicted results are in reasonable agreement with the ones using the UDLP. These predictions may be useful for future experiments to explore the new possible one proton radioactivity.