1. Introduction
With the construction of a new generation of radioactive beam facilities and the development of the new detection technology, study on the exotic decay properties of unstable nuclei has been a hot subject in nuclear physics [1–9]. Nowadays, the proton radioactivity has been recognized as one of the exotic decay modes and paid attention by many researchers [4–7]. At the beginning of the 1960s, the proton radioactivity was proposed in very proton-rich nuclei by Goldansky, Zel'dovich and Karnaukhov [10–12]. In the region near the proton drip line, the one-proton (1p) and 2p radioactivity has been observed in recent decades [4–7]. For the 1p radioactivity, it was first discovered from the isomer state of 53Co (53Com) [13, 14], but the ground state 1p radioactivity was first observed in the decay of 151Lu and 147Tm [15, 16]. So far, about 25 ground state 1p emitters have been identified [7] and the 1p radioactivity has become a powerful tool to extract nuclear structure details, such as the sequences of single-particle energies, the wave function of an emitted proton, nuclear masses and deformations [17, 18].
For the 2p radioactivity, it is another new decay mode that two protons emit simultaneously from very proton-rich even-Z nuclei because of the pairing effect. In 1960, Zel'dovich predicted the possibility that a pair of protons might emit from a nucleus [11] and it was defined as ‘2p radioactivity' by Goldansky [10, 19]. More than 40 years later, the ground-state true 2p radioactivity (Q${}_{2p}\gt 0$ and Q${}_{p}\lt 0$, where Qp is the released energy of the 1p emission) [4] was firstly discovered from 45Fe [20, 21]. Later, the true 2p radioactivity was observed from the ground states of 54Zn [22], 48Ni [23], 19Mg [24] and 67Kr [25]. In addition to the ground-state true 2p radioactivity, in fact, before 2000, ones observed the 2p radioactivity from the very short-lived nuclear ground states and excited states. For the 2p radioactivity with a very short half-life, it was discovered from 6Be [26], 12O [27] and 16Ne [27]. In these systems, the 2p emitter states and the 1p daughter states overlap with each other because the decaying widths are rather large. This case was defined as the ‘democratic decay' in the subsequent work [4–6, 28]. For the 2p radioactivity from the excited states, it includes the β-delayed 2p decay [29] and the 2p emission from excited states populated in nuclear reactions [30]. Currently, the above mentioned three types of 2p emissions have been important frontiers of nuclear radioactivity studies.
To describe the true 2p radioactivity, various models have been proposed which can be roughly divided into two completely different cases [31–55]. One case considers the two emitted protons from a parent nucleus are correlated strongly due to the proton-proton attraction. This decay process is called as ‘2He' cluster emission [31–43]. The other one refers to the completely uncorrelated emission, which is usually named as a three-body radioactivity [43–55]. Within the two extreme pictures, the experimental 2p decay half-lives are reproduced by those models with different accuracies.
On the other hand, many candidates of the 2p radioactivity were predicted by various models [32, 40, 41, 43, 56–59]. As a matter of fact, the earliest determination for the candidates of the 2p radioactivity can date back to the pioneering work of Goldansky [10]. For the ground state 2p radioactivity, the predicted candidates are the nuclei with Z < 38. So, it is interesting to know whether or not the 2p radioactivity exists in heavier systems. Recently, Olsen et al delineated the full landscape of 2p radioactivity by the energy density functional theory with several Skyrme interactions to search for the new 2p-emitters of the heavier nuclides with Z > 38 [43]. This study shows that only in two mass regions the 2p-decay mode might occur and be close enough to be addressed by today's experiments. One region ranges from germanium to krypton and the second one is located just above tin [43].
However, the 2p radioactivity half-life is dependent strongly on the Q2p value. It is well known that the Q2p value can be extracted by the following expression
$\begin{eqnarray}{{Q}}_{2p}(N,Z)\approx -{{S}}_{2p}(N,Z)=B(N,Z-2)-B(N,Z),\end{eqnarray}$
where $B(N,Z)$ is the binding energy of the nuclide related to its mass $M(N,Z)$: $M(N,Z)={{ZM}}_{H}+{{Nm}}_{n}-B(N,Z)$ (MH and mn are the masses of the hydrogen atom and the neutron, respectively). Therefore, it is very important to get accurate Q2p values by the nuclear mass models with high precision and strong predictive ability.In recent years, various nuclear mass models have been developed phenomenologically and microscopically [60–77]. Relevant studies suggested that the WS4 and FRDM mass models have high accuracy and strong prediction ability and their accuracy is higher than that of the Hartree–Fock–Bogoliubov model with Skyrme or Gogny interactions [63, 78, 79]. Nowadays, the observed 2p radioactivity from 19Mg [24], 45Fe [20, 21], 54Zn [22], 48Ni [23] and 67Kr [25] is the true and simultaneous emission. The nucleon wave functions and nucleon–nucleon interaction are involved in the angular momentum and correlation of the two emitted protons. Therefore, this kind of 2p radioactivity is more significant for the study of the exotic nuclear structure. Thus, to search for the new 2p-emitters of the true and simultaneous case, in this article the WS4 [63] and FRDM [64] models will be used to extract the Q2p values. In calculations, to examine the model dependence of Q2p values, the HFB29 [67] and KTUY [74] models will also be applied. This is the first motivation of this article. In addition, for future measurements of the 2p radioactivity, based on our recent work on the 2p radioactivity [41], the half-lives of the 2p radioactivity will be calculated within the successful GLDM by inputting different kinds of Q2p values, which constitutes our second motivation. At last, the work of Olsen et al predicted that there exists a competition between 2p emission and α-decay for some nuclei of Z > 50 (nuclei around 103Te–110Ba) [43]. Thus, it is interesting to discuss whether the competition between 2p radioactivity and α-decay occurs for the nuclei of Z > 50 within the GLDM by inputting different types of Q2p values. This is the last motivation of our study. Driven by the above mentioned three motivations, we will predict the probable 2p radioactivity candidates and investigate the competition between 2p emission and α-decay using the GLDM and the WS4, FRDM, KTUY and HFB29 mass models.
This article is organized as follows. In section 2 , the theoretical framework is introduced. In section 3 , the calculated results are shown and relevant discussions are performed. The main conclusions are summarized in the last section.
2. Theoretical framework
2.1. Methods of extracting Q2p values
Various nuclear mass models, such as the macroscopic-microscopic models [60–66], the microscopic models based on the mean-field theory [67–73] and other kinds of models [74–77], have been proposed with root-mean-square deviations from several hundred keV to a few MeV with respect to all known nuclear masses. Within different nuclear mass tables combining equation (1 ), the Q2p values can be extracted. In the Q2p calculations, the WS4 [63], FRDM [64], HFB29 [67] and KTUY [74] mass tables are used.
2.2. GLDM
In the framework of the GLDM, the shape evolution process from one body to two separated fragments is described in a unified way. Its details can be found in [80–85]. For the 2p radioactivity, it is assumed that the 2p pair (2He cluster) with zero binding energy is preformed at the surface of a parent nucleus. Then, the two protons will separate quickly after the 2p pair penetrates the Coulomb barrier between the 2p cluster and daughter nucleus [41]. The half-life is defined as
$\begin{eqnarray}{T}_{1/2}^{2p}=\displaystyle \frac{\mathrm{ln}2}{{\lambda }_{2p}},\end{eqnarray}$
where ${\lambda }_{2p}$ is the decay constant and expressed as $\begin{eqnarray}{\lambda }_{2p}={S}_{2p}{\nu }_{0}P.\end{eqnarray}$
${S}_{2p}$ represents the spectroscopic factor of the 2p pair in the parent nucleus and can be calculated by the following cluster overlap approximation [32]: ${S}_{2p}={G}^{2}{\left[A/(A-2)\right]}^{2n}{\chi }^{2}$. Here, ${G}^{2}=(2n)!/[{2}^{2n}{\left(n!\right)}^{2}]$ [86], n is the average principal proton oscillator quantum number given by n ≈ ${(3Z)}^{1/3}-1$ [87]. A and Z are the mass number and the charge number of the parent nucleus, respectively. ${\chi }^{2}$ is the proton overlap function. Its value is taken as 0.0143, which is determined by fitting the experimental half-lives of 19Mg, 45Fe, 48Ni and 54Zn [41].${\nu }_{0}$ is the 2p pair frequency of assaults on the barrier. It can be obtained by the following classical approach
$\begin{eqnarray}{\nu }_{0}=\displaystyle \frac{1}{2{R}_{0}}\sqrt{\displaystyle \frac{2{E}_{2p}}{{M}_{2p}}},\end{eqnarray}$
where R0 is the charge radius of the parent nucleus. ${E}_{2p}$ and ${M}_{2p}$ represent the kinetic energy and the mass of the emitted 2p pair.The penetrability factor P is calculated by the WKB approximation, which is expressed as [80–85]
$\begin{eqnarray}P=\exp \left[-\displaystyle \frac{2}{{\rm{\hslash }}}{\int }_{{R}_{\mathrm{in}}}^{{R}_{\mathrm{out}}}\sqrt{2B(r)[E(r)-{E}_{\mathrm{sph}}]}{\rm{d}}{r}\right],\ \end{eqnarray}$
where Rin and Rout are the two turning points of the WKB action integral. Here, an approximation is used $B(r)=\mu $, and μ stands for the reduced mass of the 2p pair and the residual daughter nucleus.The nuclear shape of initial state (ground state) is assumed to be spherical. In equation (5 ) Esph refers to the energy of the initial state, which is composed of the volume energy EV, surface energy ES and Coulomb energy EC. That is6 ) can be expressed as [80–85]
$\begin{eqnarray}{E}_{\mathrm{sph}}={E}_{V}+{E}_{S}+{E}_{C}.\end{eqnarray}$
Each term of equation ( $\begin{eqnarray}{E}_{V}=-15.494(1-1.8{I}^{2})A,\end{eqnarray}$
$\begin{eqnarray}{E}_{S}=17.9439(1-2.6{I}^{2}){A}^{2/3}(S/4\pi {R}_{0}^{2}),\end{eqnarray}$
$\begin{eqnarray}{E}_{C}=0.6{e}^{2}({Z}^{2}/{R}_{0})\times 0.5\int (V(\theta )/{V}_{0}){\left(R(\theta )/{R}_{0}\right)}^{3}\sin \theta {\rm{d}}\theta ,\end{eqnarray}$
where S and I are the surface and relative neutron excess of the parent nucleus, respectively. $V(\theta )$ stands for the electrostatic potential at the surface. V0 is the sphere surface potential.When the two fragments are separated, the deformed energy E(r) is written as10 ) is written as [80–85]
$\begin{eqnarray}E(r)={E}_{V}+{E}_{S}+{E}_{C}+{E}_{\mathrm{Prox}}+{E}_{\mathrm{cen}}.\end{eqnarray}$
The specific form of each term in equation ( $\begin{eqnarray}{E}_{V}=-15.494\left[(1-1.8{I}_{1}^{2}){A}_{1}+(1-1.8{I}_{2}^{2}){A}_{2}\right],\end{eqnarray}$
$\begin{eqnarray}{E}_{S}=17.9439\left[(1-2.6{I}_{1}^{2}){A}_{1}^{2/3}+(1-2.6{I}_{2}^{2}){A}_{2}^{2/3}\right],\end{eqnarray}$
$\begin{eqnarray}{E}_{C}(r)=0.6{e}^{2}{Z}_{1}^{2}/{R}_{1}+0.6{e}^{2}{Z}_{2}^{2}/{R}_{2}+{e}^{2}{Z}_{1}{Z}_{2}/r,\end{eqnarray}$
$\begin{eqnarray}{E}_{\mathrm{Prox}}(r)=2\gamma {\int }_{{h}_{\min }}^{{h}_{\max }}{\rm{\Phi }}\left[D(r,h)/b\right]2\pi {h}{\rm{d}}{h},\end{eqnarray}$
$\begin{eqnarray}{E}_{\mathrm{cen}}(r)=\displaystyle \frac{{{\rm{\hslash }}}^{2}}{2\mu }\displaystyle \frac{l(l+1)}{{r}^{2}}.\end{eqnarray}$
Here A1(2), Z1(2) and I1(2) in equations (11 )–(13 ) are the mass numbers, charge numbers and relative neutron excesses of the two fragments, respectively. r means the distance between the two fragments.
In equation (13 ), R1 and R2 are the charge radii of the daughter nucleus and the emitted cluster. The values of R0, R1 and R2 are given by
$\begin{eqnarray}{R}_{i}=(1.28{A}_{i}^{1/3}-0.76+0.8{A}_{i}^{-1/3}){\rm{fm}},i=0,1,2.\end{eqnarray}$
E${}_{\mathrm{Prox}}(r)$ in equations (10 ), (14 ) represents the proximity energy, which is used to describe the effects of the nuclear forces between the close surfaces when there are nucleons in a neck or a gap between separated fragments. Under the influence of the proximity energy, the barrier top moves to an external position and the pure Coulomb barrier is strongly suppressed [80–85]. In equation (14 ), the surface parameter γ is the geometric mean for the surface parameters of the two fragments. Φ is the Feldmeier proximity function. D is the distance between the surfaces and the surface width $b=0.99$ fm. h is the distance varying from the neck radius or zero to the height of the neck border.
In equation (15 ), E${}_{\mathrm{cen}}(r)$ denotes the centrifugal potential energy. The symbol l represents the orbital angular momentum carried by the 2p pair. Due to the influence of E${}_{\mathrm{cen}}(r)$, ${R}_{\mathrm{out}}$ of equation (5 ) becomes the following form [83–85]
$\begin{eqnarray}{R}_{\mathrm{out}}=\displaystyle \frac{{Z}_{1}{Z}_{2}{e}^{2}}{2{Q}_{2p}}+\sqrt{{\left(\displaystyle \frac{{Z}_{1}{Z}_{2}{e}^{2}}{2{Q}_{2p}}\right)}^{2}+\displaystyle \frac{l(l+1){{\rm{\hslash }}}^{2}}{2\mu {Q}_{2p}}}.\end{eqnarray}$
The first turning point Rin is still expressed approximately as: ${R}_{\mathrm{in}}={R}_{1}+{R}_{2}$.
3. Results and discussions
Within equation (1 ) and the WS4, FRDM, KTUY and HFB29 mass tables, four types of Q2p values are extracted. Firstly, the experimental Q2p values of 19Mg, 45Fe, 48Ni, 54Zn and 67Kr and those from the four kinds of nuclear mass tables are listed in the left part of table 1. From table 1, we can see that the differences between the four kinds of Q2p values are very large. In addition, by comparing the experimental Q2p values and those calculated ones, it is found that the experimental Q2p values of 45Fe, 48Ni and 67Kr are reproduced best by the KTUY mass model. For 54Zn, the accuracy given by the HFB29 mass model is the highest. Here, it is worth mentioning that the binding energies of 19Mg and its daughter nucleus (17Ne) are not available in the WS4, FRDM and HFB29 mass tables, the Q2p value of 19Mg is estimated only by the KTUY mass table. However, the Q2p value from the KTUY mass model is 0.39 MeV larger than the experimental value. Therefore, it is difficult to determine which nuclear mass model has the strongest prediction power. This indicates that there exists some uncertainty about the masses of the nuclei near the proton drip-line predicted by the extant nuclear mass models.
Table 1. The comparison between the experimental Q2p values and those extracted from the WS4 [63], FRDM [64], KTUY [74] and HFB29 [67] nuclear mass models. log${}_{10}{T}_{1/2}^{\mathrm{cal}.}$ denotes the corresponding 2p radioactivity half-lives within the GLDM by inputting the experimental Q2p values and those extracted from the four kinds mass models. log${}_{10}{T}_{1/2}^{\mathrm{expt}.}$ stands for the experimental 2p radioactivity half-lives. All the Q2p and log${}_{10}{T}_{1/2}$ values are measured in MeV and second, respectively. The symbol ‘—' means the Q2p values or the log${}_{10}{T}_{1/2}^{\mathrm{cal}.}$ values are not available. |
| Nuclei | Q2p (MeV) | log${}_{10}{T}_{1/2}^{\mathrm{cal}.}$ (s) | log${}_{10}{T}_{1/2}^{\mathrm{expt}.}$ (s) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Expt. | WS4 | FRDM | KTUY | HFB29 | Expt. [41] | WS4 | FRDM | KTUY | HFB29 | ||
| ${}_{12}^{19}$Mg | 0.750 [24] | — | — | 1.14 | — | −11.79${}_{-0.42}^{+0.47}$ | — | — | −14.28 | — | −11.40${}_{-0.20}^{+0.14}$ [24] |
| ${}_{26}^{45}$Fe | 1.100 [20] | 2.06 | 1.89 | 1.19 | 1.92 | −2.23${}_{-1.17}^{+1.34}$ | −9.59 | −8.71 | −3.28 | −8.88 | −2.40${}_{-0.26}^{+0.26}$ [20] |
| 1.140 [21] | −2.71${}_{-0.57}^{+0.61}$ | −2.07${}_{-0.21}^{+0.24}$ [21] | |||||||||
| 1.154 [23] | −2.87${}_{-0.18}^{+0.19}$ | −2.55${}_{-0.12}^{+0.13}$ [23] | |||||||||
| 1.210 [88] | −3.50${}_{-0.52}^{+0.56}$ | −2.42${}_{-0.03}^{+0.03}$ [88] | |||||||||
| ${}_{28}^{48}$Ni | 1.350 [23] | 2.54 | 3.30 | 1.95 | 3.63 | −3.24${}_{-0.20}^{+0.20}$ | −10.47 | −12.84 | −7.73 | −13.62 | −2.08${}_{-0.78}^{+0.40}$ [23] |
| 1.290 [89] | −2.62${}_{-0.42}^{+0.44}$ | −2.52${}_{-0.22}^{+0.24}$ [89] | |||||||||
| 1.310 [90] | −2.83${}_{-0.41}^{+0.43}$ | −2.52${}_{-0.22}^{+0.24}$ [91] | |||||||||
| ${}_{30}^{54}$Zn | 1.480 [22] | 1.98 | 2.77 | 1.65 | 1.61 | −2.95${}_{-0.19}^{+0.19}$ | −6.67 | −10.33 | −4.40 | −4.08 | −2.43${}_{-0.14}^{+0.20}$ [22] |
| 1.280 [92] | −0.87${}_{-0.24}^{+0.25}$ | −2.76${}_{-0.14}^{+0.15}$ [92] | |||||||||
| ${}_{36}^{67}$Kr | 1.690 [25] | 3.06 | 1.33 | 1.52 | 1.94 | −1.25${}_{-0.16}^{+0.16}$ | −9.27 | 2.75 | 0.46 | −3.34 | −1.70${}_{-0.02}^{+0.02}$ [25] |
Then, the 2p radioactivity half-lives are calculated within the GLDM by inputting the experimental and the four kinds of Q2p values. In calculations, l is taken as zero, assuming the 2p decay process is the fastest. The corresponding half-lives are shown in columns 7–11 of table 1. Moreover, the experimental half-lives are given in the last column. By observing the half-lives of columns 7–12, it is easy to know that the 2p decay half-lives are extremely sensitive to the Q2p values. For example, for the case of 67Kr, the Q2p difference given by the WS4 and FRDM models is 1.73 MeV, but the corresponding order of magnitude difference of the half-life is as high as 1012 s. In addition, it is seen that the experimental half-lives can be reproduced best by inputting the experimental Q2p values. It means that the closer to the experimental Q2p values, the better the experimental half-life can be reproduced. Hence, the Q2p value is the most important for the 2p radioactivity and the nuclear mass models should be improved by taking into account more reasonable physical factors. However, doing so is a great challenge.
Next the 2p radioactivity half-lives are predicted within the GLDM by inputting the Q2p values extracted from the WS4, FRDM, KTUY and HFB29 mass models. To identify the true and simultaneous emission, an energy criterion was proposed in [43], which reads Q${}_{2p}\gt 0$ and Qp < 0.2Q2p. In addition, a condition on the 2p decay half-lives, −7 < log${}_{10}{T}_{1/2}^{2{\rm{p}}}$ < −1s (log${}_{10}{T}_{1/2}^{2{\rm{p}}}$ is the 2p decay decimal logarithm half-life, whose unit is second), was given to define the feasibility of experimental observation in [43]. The lower bound of log${}_{10}{T}_{1/2}^{2{\rm{p}}}={10}^{-7}$ s corresponds to the typical sensitivity limit of in-flight, projectile-fragmentation techniques. The upper bound of log${}_{10}{T}_{1/2}^{2{\rm{p}}}={10}^{-1}$ s ensures that the 2p radioactivity will not be dominated by β decay. However, the experimental 2p decay decimal logarithm half-life of the ${21}^{+}$ isomer state of 94Ag is 1.90 s [93]. Moreover, the 2p decay decimal logarithm half-life of 19Mg is measured as −11.40 s [24]. In addition, Goldansky pointed that one can possibly observe the true 2p radioactivity with the half-lives of log${}_{10}{T}_{1/2}^{2{\rm{p}}}$ > −12 s [10, 94]. Thus, the condition on the 2p decay half-lives of [43] should be extended. To avoid losing some 2p decay candidates, in this article an extended criterion on 2p decay half-lives is used, which is written as −12 < log${}_{10}{T}_{1/2}^{2{\rm{p}}}$ < 2s. According to the energy criterion and the new criterion on 2p decay half-lives, the Q2p values and half-lives of the true and simultaneous emission are listed in table 2.
Table 2. Same as table 1, but for the predicted 2p radioactivity half-lives by inputting the Q2p values extracted from the WS4 [63], FRDM [64], KTUY [74] and HFB29 [67] nuclear mass models. |
| Nuclei | Q2p (MeV) | log${}_{10}{T}_{1/2}^{\mathrm{cal}.}$ (s) | ||||||
|---|---|---|---|---|---|---|---|---|
| WS4 | FRDM | KTUY | HFB29 | WS4 | FRDM | KTUY | HFB29 | |
| ${}_{18}^{30}$Ar | — | 1.22 | — | — | — | −10.65 | — | — |
| ${}_{18}^{31}$Ar | 1.08 | — | — | 0.43 | −9.54 | — | — | 1.14 |
| ${}_{20}^{34}$Ca | — | 0.75 | 0.96 | — | — | −3.65 | −6.53 | — |
| ${}_{22}^{38}$Ti | — | 1.41 | 1.56 | — | — | −8.60 | −9.59 | — |
| ${}_{22}^{39}$Ti | 0.98 | — | — | — | −4.62 | — | — | — |
| ${}_{24}^{41}$Cr | — | 2.12 | 2.13 | — | — | −11.10 | −11.10 | — |
| ${}_{24}^{42}$Cr | 1.40 | 1.11 | — | — | −6.95 | −4.21 | — | — |
| ${}_{28}^{49}$Ni | 1.13 | 1.65 | — | 1.88 | −0.75 | −5.80 | — | −7.33 |
| ${}_{30}^{55}$Zn | — | 1.49 | — | — | — | −3.05 | — | — |
| ${}_{32}^{58}$Ge | 2.68 | 2.57 | 2.45 | 1.77 | −9.80 | −9.34 | −8.80 | −4.76 |
| ${}_{32}^{59}$Ge | 1.43 | 1.85 | — | — | −1.73 | −5.36 | — | — |
| ${}_{34}^{62}$Se | 3.64 | 2.93 | 3.27 | 3.04 | −12.00 | −9.75 | −10.90 | −10.20 |
| ${}_{34}^{63}$Se | 2.39 | 1.37 | 1.78 | 1.80 | −7.38 | 0.57 | −3.45 | −3.61 |
| ${}_{36}^{65}$Kr | — | — | 2.83 | 3.11 | — | — | −8.34 | −9.45 |
| ${}_{36}^{66}$Kr | — | 2.65 | — | — | — | −7.54 | — | — |
| ${}_{36}^{68}$Kr | 1.80 | — | — | — | −2.23 | — | — | — |
| ${}_{38}^{70}$Sr | — | 3.15 | 2.80 | 3.38 | — | −8.64 | −7.18 | −9.47 |
| ${}_{38}^{71}$Sr | 2.92 | 1.83 | — | 2.15 | −7.72 | −1.10 | — | −3.56 |
| ${}_{38}^{72}$Sr | 1.65 | — | — | — | 0.59 | — | — | — |
| ${}_{40}^{74}$Zr | — | 3.27 | 2.65 | 3.51 | — | −8.16 | −5.39 | −9.02 |
| ${}_{40}^{75}$Zr | 2.80 | 1.93 | — | 2.51 | −6.15 | −0.60 | — | −4.63 |
| ${}_{40}^{76}$Zr | 1.85 | — | — | — | 0.10 | — | — | — |
| ${}_{40}^{77}$Zr | — | — | — | 1.86 | — | — | — | −0.06 |
| ${}_{42}^{78}$Mo | — | — | 3.07 | 3.68 | — | — | −6.08 | −8.42 |
| ${}_{42}^{79}$Mo | 3.07 | — | 2.19 | — | −6.09 | — | −1.07 | — |
| ${}_{42}^{80}$Mo | 2.34 | 1.91 | — | — | −2.13 | 1.21 | — | — |
| ${}_{44}^{81}$Ru | — | — | — | 4.78 | — | — | — | −10.70 |
| ${}_{44}^{82}$Ru | — | — | — | 3.69 | — | — | — | −7.60 |
| ${}_{44}^{83}$Ru | 3.20 | 2.83 | — | 2.33 | −5.69 | −3.92 | — | −0.88 |
| ${}_{44}^{84}$Ru | 2.20 | — | — | 2.68 | 0.08 | — | — | −3.11 |
| ${}_{46}^{85}$Pd | — | — | — | 4.47 | — | — | — | −9.11 |
| ${}_{46}^{86}$Pd | 3.83 | — | 3.34 | 3.59 | −7.23 | — | −5.35 | −6.36 |
| ${}_{46}^{87}$Pd | 3.04 | — | — | 2.62 | −3.98 | — | — | −1.66 |
| ${}_{48}^{88}$Cd | — | — | — | 5.59 | — | — | — | −11.10 |
| ${}_{48}^{89}$Cd | — | — | — | 4.54 | — | — | — | −8.55 |
| ${}_{48}^{90}$Cd | 3.90 | — | 3.39 | 3.77 | −6.65 | — | −4.64 | −6.18 |
| ${}_{48}^{91}$Cd | 3.08 | — | — | 2.94 | −3.19 | — | — | −2.46 |
| ${}_{48}^{94}$Cd | — | — | 0.34 | — | — | — | 0.00 | — |
| ${}_{50}^{94}$Sn | — | — | 3.34 | — | — | — | −3.39 | — |
| ${}_{50}^{95}$Sn | — | — | 2.48 | — | — | — | 1.53 | — |
| ${}_{50}^{96}$Sn | 2.67 | — | — | — | 0.20 | — | — | — |
| ${}_{52}^{101}$Te | — | — | — | 5.45 | — | — | — | −9.54 |
| ${}_{54}^{107}$Xe | — | — | — | 3.08 | — | — | — | −0.24 |
| ${}_{56}^{111}$Ba | — | — | — | 3.46 | — | — | — | −2.28 |
| ${}_{58}^{114}$Ce | — | — | — | 4.94 | — | — | — | −7.12 |
| ${}_{58}^{116}$Ce | — | — | — | 3.22 | — | — | — | −0.01 |
From table 2, the probable 2p decay candidates can be found in the region of Z ≤ 50 by each mass model. However, the 2p decay modes are not observed in the Z = 20, 26, 30 nuclides, which are predicted by the WS4 model and in the Z = 26, 46, 48, 50 nuclides, which are predicted by the FRDM model. Similarly, in the cases of the KTUY and HFB29 mass models, the 2p radioactivity can not be detected by the current technique for the Z = 18, 26, 28, 30, 44 and Z = 20, 22, 24, 26, 30, 50 isotopes, respectively. In the region beyond Z = 50, the 2p decay candidates can be predicted only by the HFB29 mass model, which can be observed in the last five lines of table 2. These 2p decay candidates are in or very close to the N = Z line. Recently, the decays of 59Ge, 63Se and 67Kr were studied in an experiment with the BigRIPS separator at the RIKEN Nishina Center [25]. It was shown no evidence for 2p emission of 59Ge and 63Se except for 67Kr. However, 63Se is predicted as a probable 2p decay candidate by the four mass models. For 59Ge, its 2p radioactivity cannot be observed by the predictions of the KTUY and HFB29 models. By the comparison between our predictions and the measurements of Goigoux et al, it can be seen that the prediction power of the KTUY and HFB29 mass models seems stronger. Probably, the microscopic effective nucleon–nucleon interactions contained in the KTUY and HFB29 models enhance the prediction power of the two models. Now the number of the discovered 2p emitters is still small, more measurements on 2p radioactivity are expected with the new generation of radioactive beam facilities, for example, the High Intensity heavy-ion Accelerator Facility of China [95–98]. And we hope our predictions could be tested with them. In addition, in table 2, a general tendency for the log${}_{10}{T}_{1/2}^{2{\rm{p}}}$ can be seen: the log${}_{10}{T}_{1/2}^{2{\rm{p}}}$ half-lives of the light nuclei get shorter and the half-lives become longer for the heavy nuclei as long as the Q2p values of the light nuclei are not far away from those of the heavy nuclei. This is attributed to the following reason: for the light systems, the Coulomb barrier between the 2p cluster and daughter nucleus is low because of the smaller charge number so that the 2p cluster can penetrate the barrier more easily. However, the Coulomb barrier becomes higher and higher with the increase of Z. As a result, the 2p decay half-life gets longer in most cases for the heavy nuclei.
Relevant studies suggest that some neutron-deficient nuclei near the N = Z line, just above the N = Z = 50 shell closures, exhibit large α-decay branches [85, 99–103]. So, it is interesting to discuss the competition between the true 2p radioactivity and α-decay of the deficient-neutron nuclei beyond Z = 50. To compare the true 2p decay half-lives with the α-decay half-lives reasonably, the α-decay half-lives are also calculated by the GLDM. In the GLDM calculations, the preformation factor of an α-particle is used the analytic form of [104] and the angular momenta carried by the α-particle are selected as 0. The α-decay energies (Q${}_{\alpha }$), α-decay half-lives (log${}_{10}{T}_{1/2}^{\alpha }$), and competition between the two decay modes are shown in table 3. In the last column of table 3, 2p (α) represents that the 2p radioactivity (α-decay) is the dominant decay mode. As can be seen from table 3, 101Te, 111Ba and 114Ce are dominated by the 2p radioactivity. For 107Xe and 116Ce, α-decay is the dominant decay mode. Thus, the 2p radioactivity of the two nuclei is not easy to be observed due to the influence of so large α-decay branches. Note that the predicted decay energies of 101Te are Q2p = 5.45 MeV and Q${}_{\alpha }$ = −0.19 MeV, so, its main decay mode is the 2p radioactivity.
Table 3. The competition between the true 2p radioactivity and α-decay for the nuclei beyond the proton-drip line. Q${}_{\alpha }$ denotes the α-decay energy, which is measured in MeV. The α-decay half-lives (log${}_{10}{T}_{1/2}^{\alpha }$) are calculated within the GLDM and measured in second. |
| Nuclei | Mass Model | Q2p (MeV) | Q${}_{\alpha }$ (MeV) | log${}_{10}{T}_{1/2}^{2{\rm{p}}}$ (s) | log${}_{10}{T}_{1/2}^{\alpha }$ (s) | Decay mode |
|---|---|---|---|---|---|---|
| ${}_{52}^{101}$Te | HFB29 | 5.45 | −0.19 | −9.54 | — | 2p |
| ${}_{54}^{107}$Xe | HFB29 | 3.08 | 4.80 | −0.24 | −5.70 | α |
| ${}_{56}^{111}$Ba | HFB29 | 3.46 | 4.10 | −2.28 | −1.40 | 2p |
| ${}_{58}^{114}$Ce | HFB29 | 4.94 | 4.25 | −7.12 | −1.11 | 2p |
| ${}_{58}^{116}$Ce | HFB29 | 3.22 | 4.31 | −0.01 | −1.45 | α |
4. Conclusions
In this article, firstly the Q2p values of ${}_{12}^{19}$Mg, ${}_{26}^{45}$Fe, ${}_{28}^{48}$Ni, ${}_{30}^{54}$Zn and ${}_{36}^{67}$Kr have been extracted by the WS4, FRDM, KTUY and HFB29 nuclear mass models. By a comparison between the extracted Q2p values and the experimental ones, it is found that the experimental Q2p values can not be reproduced accurately by all the nuclear mass models. Meanwhile, the model dependence of the Q2p values is seen evidently. Then, the 2p radioactivity half-lives have been calculated in the framework of the GLDM by inputting different types of Q2p values. As a result, the uncertainties of the 2p decay half-lives are rather large due to the Q2p uncertainties. Meanwhile, the 2p radioactivity half-lives are reproduced well by inputting the Q2p values that are close to the experimental ones. In addition, to search for the new candidates of the true and simultaneous 2p radioactivity, the Q2p values of the even-Z nuclei beyond the 2p drip-line are extracted by the four kinds of nuclear mass models and the corresponding half-lives are predicted within the GLDM by inputting the four types of Q2p values. According to the energy and half-life constraint conditions, the probable 2p decay candidates are found in the region of Z ≤ 50 with all the mass models used in this article. In the region beyond Z = 50, the 2p-decaying candidates are predicted only by the HFB29 mass model. At last, the competition between the true 2p radioactivity and α-decay for the nuclei above N = Z = 50 shell closures has been investigated. It is shown that 101Te, 111Ba and 114Ce prefer to decay by the 2p radioactivity and α-decay is the dominant decay mode for 107Xe and 116Ce. Hence, the 2p radioactivity of 107Xe and 116Ce is difficult to be discovered due to their large α-decay channels. We hope our predictions and discussion are useful for searching for the new candidates of the true 2p radioactivity in future.