-
In recent years, there has been much work focused on nuclei beyond the proton drip line, due to the fact that they show new phenomena that cannot be found in stable nuclei [1-3]. This includes the two-proton (
2p ) radioactivity phenomenon, which was predicted by Zel'dovich [4] and Goldansky [5, 6] in the 1960s. From the perspective of pairing energy, the two protons emitted in a2p radioactivity process should be simultaneous emission from the ground state of a radioactive nucleus beyond the drip line. However, there is no agreement on whether the two protons are simultaneously emitted as two indepedent protons or as a "diproton emission" similar to the emission of a 2He-like cluster from the mother nucleus. Obviously, for odd-proton-number (odd-Z) nuclei, proton radioactivity is the predominant decay mode. For even-proton-number (even-Z) nuclei lying near the proton drip line, the2p radioactivity phenomenon may occur due to the effect of proton pairing [7]. Experimentally, the probability of the2p decay width of 16Ne and 12O was reported in 1978 [8]. Later, the ground-state true2p radioactivity of 45Fe was observed at the Grand Accˊe l ˊe rateur National d'Ions Lourds (GANIL) [9] and Gesellschaft f¨u r Schwerionenforschung (GSI) [10], respectively. In 2005, the2p radioactivity of 54Zn was detected at GANIL [11], followed by the2p radioactivity of 48Ni [12]. In 2007, the2p radioactivity of 19Mg was revealed by tracking the decay products [13]. Recently, the2p emission of 67Kr was observed in an experiment with the BigRIPS separator [14].Theoretically, various models have been proposed to investigate 2p radioactivity, including the direct decay model [15-21], the simultaneous versus sequential decay model [22], the diproton model [23, 24], and the three-body model [25-28]. Using an R-matrix formula, B. A. Brown et al. reproduced the
2p radioactivity half-lives of 45Fe [29]. Following this, using the continuum shell model, J. Rotureau et al. microcosmically described the2p radioactivity in 45Fe, 48Ni and 54Zn [30]. In 2017, M. Goncalves et al. used the effective liquid drop model (ELDM) to calculate the half-lives of2p radioactive nuclei [31]. Their calculated results can reproduce the experimental data well [32, 33]. Furthermore, based on the ELDM, they predicted the2p radioactivity half-lives of 33 nuclei with2p radioactivity released energyQ2p>0 , obtained from the latest evaluated atomic mass table AME2016 [34, 35]. In 2019, Sreeja et al. proposed a four-parameter empirical formula to study the2p radioactivity half-lives [36]. These parameters were obtained by fitting the predicted results from Goncalves et al. [33]. Their calculated results agree well with the known experimental data. Recently, Cui et al. studied the2p radioactivity of the ground state of nuclei based on a generalized liquid drop model (GLDM) [37], in which the2p radioactivity process is described as a pair particle preformed near the surface of the parent nucleus penetrating the barrier between the cluster and daughter nucleus. In this view,2p radioactivity shares a similar theory to barrier penetration with different kinds of charged particle radioactivity, such asα decay, cluster radioactivity, proton radioactivity and so on [38-43]. In our previous work [44], based on the Geiger-Nuttall (G-N) law [45], we proposed a two-parameter empirical formula for a new G-N law for proton radioactivity, which can be treated as an effective tool to study proton radioactivity. Therefore, whether the G-N law can be extended to study2p radioactivity or not is an interesting topic. In this work, a two-parameter analytic formula, which is related to the2p radioactivity half-lifeT1/2 ,2p radioactivity released energyQ2p , the charge of the daughter nucleusZd , and the orbital angular momentum l taken away by the two emitted protons, is proposed to study2p radioactivity.This article is organized as follows. In the next section, the theoretical framework for the new G-N law is described in detail. In Section III, the detailed calculations, discussion and predictions are provided. In Section IV, a brief summary is given.
-
In 1911, Geiger and Nuttall found there is a phenomenological relationship between the
α decay half-lifeT1/2 and the decay energyQα . This relationship is the so-called Geiger-Nuttall (G-N) law. It is expressed as:log10T1/2=aQα−1/2+b,
(1) where a and b represent the two isotopic chain–dependent parameters of this formula. Later, the G-N law was widely applied to study the half-lives of
α decay [38, 46-48], cluster radioactivity [49-51] and proton radioactivity [52-54]. However, relative toα decay and cluster radioactivity, the proton radioactivity half-life is more sensitive to the centrifugal barrier. This means that the linear relationship between the half-life of the proton radioactivity and the released energyQp only exists for proton-radioactive isotopes with the same orbital angular momentum l taken away by the emitted proton [44, 52, 54]. Similarly, the2p radioactivity half-life may also depend strongly on the2p radioactivity released energyQ2p and the orbital angular momentum l taken away by the two emitted protons. Recently, considering the contributions ofQ2p and the orbital angular momentum l to the2p radioactivity half-life, Sreeja et al. put forward a four-parameter empirical formula to study the2p radioactivity half-lives, which is expressed as [36]log10T1/2=((a×l)+b)Z0.8dQ2p−1/2+((c×l)+d),
(2) where a = 0.1578, b = 1.9474,
c=−1.8795 , andd=−24.847 denote the adjustable parameters, which are obtained by fitting the calculated results of the ELDM [33]. Their calculated results can reproduce the known experimental data well.In our previous work [44], considering the contributions of the daughter nuclear charge
Zd and the orbital angular momentum l taken away by the emitted proton to the proton radioactivity half-life, we proposed a two-parameter empirical formula for a new G-N law for proton radioactivity. This formula is written as:log10T1/2=aβ(Z0.8d+lβ)Qp−1/2+bβ,
(3) where
aβ=0.843 andbβ=−27.194 are the fitted parameters. The exponent on the orbital angular momentum l taken away by the emitted proton,β , is 1, which is obtained by fitting 44 experimental data points of proton radioactivity in the ground state and isomeric state. Combined with the work from Sreeja et al. [36, 54] and our previous work [44], it is interesting to examine whether or not a two-parameter form of the empirical formula is suitable to investigate2p radioactivity. In this work, because there are no experimental data for2p radioactive nuclei with orbital angular momentuml≠0 , we choose the experimental data for true2p radioactive nuclei (19Mg, 45Fe, 48Ni, 54Zn and 67Kr) with l = 0, and the predicted2p radioactivity half-lives of 7 nuclei withl≠0 (1 case with l = 1, 4 cases with l = 2 and 2 cases with l = 4) are extracted from Goncalves et al. [33].First, for the
β value describing the effect of l on the2p radioactivity half-life, we choose theβ value corresponding to the smallest standard deviationσ between the database and the calculated2p radioactivity half-lives as the optimal value, withβ varying from 0.1 to 0.5. The relationship between theσ andβ values is shown in Fig. 1. It is clear thatσ is smallest whenβ is equal to 0.25. Comparing with theβ value of Eq. (3) reflecting the effect of l on the proton radioactivity half-life, thisβ value is smaller. The reason may be that the reduced massμ of a proton-radioactive nucleus is smaller than that of a2p -radioactive nucleus, leading to the contribution of the centrifugal barrier to the half-life of the2p -radioactive nucleus being smaller. Correspondingly, the values of parameters a and b are given as:a=2.032,b=26.832,
(4) Then, we can obtain a final formula, which can be written as:
log10T1/2=2.032(Z0.8d+l0.25)Q2p−1/2−26.832.
(5) -
The primary aim of this work is to verify the feasibility of using Eq. (5) to investigate
2p radioactivity. The calculated logarithmic half-lives of2p -radioactive nuclei are listed in the seventh column of Table 1. Meanwhile, for comparison, the calculated results using GLDM, ELDM and a four-parameter empirical formula are shown in the fourth to sixth column of this table, respectively. In Table 1, the first three columns denote the2p -radioactive nucleus, the experimental2p radioactivity released energyQ2p and the logarithmic experimental2p radioactivity half-lifelog10Texp1/2 , respectively. For quantitative comparisons between the calculated2p radioactivity half-lives using our empirical formula and the experimental results, the last column gives the logarithm of errors between the experimental2p radioactivity half-lives and those calculated using our empirical formulalog10HF=log10Texp1/2−log10Tcal1/2 . From this table, it can be seen that for the true2p -radioactive nuclei 19Mg, 45Fe, 48Ni, 54Zn and 67Kr (Qp<0 ,Q2p>0 ), most values oflog10HF are between -1 and 1. Particularly, for the cases of 48Ni, withQ2p = 1.290, and 45Fe, withQ2p = 1.154, the values oflog10HF are 0.07 and 0.24, indicating our calculated results can reproduce the experimental data well. As for the sequential or pseudo-2p -radioactive nuclei 6Be, 12O and 16Ne (Qp>0 ,Q2p>0 ), the values oflog10HF for 6Be and 16Ne are relatively large. Likewise, the differences between the experimental data and the calculated2p radioactivity half-lives using GLDM, ELDM and the four-parameter empirical formula are more than three orders of magnitude. This may be due to the limitations of the early experimental equipment, resulting in the measured decay widths of these2p radioactivity nuclei not being accurate enough. It would be helpful to measure the experimental2p half-lives of these nuclei again in the future. In the case of 12O, the values oflog10HF are small, implying that our formula may also be suitable for studying pseudo-2p -radioactive nuclei which have relatively accurate experimental data.Nucleus Qexp2p /MeVlog10Texp1/2 /slog10TGLDM1/2 /s [37]log10TELDM1/2 /s [33]log10T1/2 /s [36]log10TThiswork1/2 /slog10 HF6Be 1.371 [55] −20.30 [55]−19.37 −19.97 −21.95 −23.81 3.51 12O 1.638 [56] >−20.20 [56]−19.17 −18.27 −18.47 −20.17 >−0.03 1.820 [8] −20.94 [8]−20.94 – −18.79 −20.52 −0.42 1.790 [57] −20.10 [57]−20.10 – −18.74 −20.46 0.36 1.800 [58] −20.12 [58]−20.12 – −18.76 −20.48 0.36 16Ne 1.33 [8] −20.64 [8]−16.45 – −15.94 −17.53 −3.11 1.400 [59] −20.38 [59]−16.63 −16.60 −16.16 −17.77 −2.61 19Mg 0.750 [13] −11.40 [13]−11.79 −11.72 −10.66 −12.03 0.63 45Fe 1.100 [10] −2.40 [10]−2.23 – −1.25 −2.21 −0.19 1.140 [9] −2.07 [9]−2.71 – −1.66 −2.64 0.57 1.210 [60] −2.42 [60]−3.50 – −2.34 −3.35 0.93 1.154 [12] −2.55 [12]−2.87 −2.43 −1.81 −2.79 0.24 48Ni 1.350 [12] −2.08 [12]−3.24 – −2.13 −3.13 1.05 1.290 [61] −2.52 [61]−2.62 – −1.61 −2.59 0.07 54Zn 1.480 [11] −2.43 [11]−2.95 −2.52 −1.83 −2.81 0.38 1.280 [62] −2.76 [62]−0.87 – −0.10 −1.01 −1.75 67Kr 1.690 [14] −1.70 [14]−1.25 −0.06 0.31 −0.58 −1.12 Table 1. Comparison of the experimental data for
2p -radioactive nuclei with different theoretical models (GLDM, ELDM, the four-parameter empirical formula of Ref. [36] and our empirical formula. Experimental data are taken from the corresponding references.To further test the feasibility of our empirical formula, we also use Eq. (5) to predict the
2p radioactivity half-lives of 22 nuclei with2p radioactivity released energyQ2p>0 . TheQ2p values are taken from the latest evaluated atomic mass table AME2016 and shown in the second column of Table 2. In this table, the first and third columns give the2p radioactivity candidates and the angular momentum l taken away by the two emitted protons, respectively. For a benchmark, the predicted results using GLDM, ELDM and the four-parameter empirical formula, extracted from Refs. [37], [33] and [36] respectively, are also listed in this table. We can clearly see that forl≠0 , the predicted results using our empirical formula are closer to those predicted using ELDM than those predicted using the four-parameter empirical formula. Most of the predicted results are of the same order of magnitude. As an example, in the cases of 28Cl (60As), the predicted2p radioactivity half-lives using ELDM, the four-parameter empirical formula and our empirical formula are−12.95 (−8.68 ),−14.52 (−10.84 ) and−12.46 (−8.33 ), respectively. This implies that our empirical formula is also suitable for studying nuclei with orbital angular momentuml≠0 . In the case ofl=0 , the predicted2p radioactivity half-lives using our empirical formula are in good agreement with those from GLDM and ELDM. To further demonstrate the significant correlation between the2p radioactivity half-livesT1/2 and the2p radioactivity released energiesQ2p , based on Eq. (5), we plot the quantity[log10T1/2+26.832]/(Z0.8d+l0.25) as a function ofQ−1/22p in Fig. 2. In this figure, there is an obvious linear dependence oflog10T1/2 onQ2p −1/2 , while the contributions of charge numberZd and orbital angular momentum l on the2p radioactivity half-lives are removed.Nucleus Q2p /MeVl log10TGLDM1/2 /s [37]log10TELDM1/2 /s [33]log10T1/2 /s [36]log10TThiswork1/2 /s22Si 1.283 0 −13.30 −13.32 −12.30 −13.74 26S 1.755 0 −14.59 −13.86 −12.71 −14.16 34Ca 1.474 0 −10.71 −9.91 −8.65 −9.93 36Sc 1.993 0 −11.74 −10.30 −11.66 38Ti 2.743 0 −14.27 −13.56 −11.93 −13.35 39Ti 0.758 0 −1.34 −0.81 −0.28 −1.19 40V 1.842 0 −9.85 −8.46 −9.73 42Cr 1.002 0 −2.88 −2.43 −1.78 −2.76 47Co 1.042 0 −0.11 0.21 −0.69 49Ni 0.492 0 14.46 14.64 12.78 12.43 56Ga 2.443 0 −8.00 −6.42 −7.61 58Ge 3.732 0 −13.10 −11.74 −9.53 −10.85 59Ge 2.102 0 −6.97 −5.71 −4.44 −5.54 60Ge 0.631 0 13.55 14.62 12.40 12.04 61As 2.282 0 −6.12 −4.74 −5.85 10N 1.3 1 −17.64 −20.04 −18.59 28Cl 1.965 2 −12.95 −14.52 −12.46 32K 2.077 2 −12.25 −13.46 −11.55 57Ga 2.047 2 −5.30 −5.22 −4.14 62As 0.692 2 14.52 13.83 14.18 52Cu 0.772 4 9.36 8.62 8.74 60As 3.492 4 −8.68 −10.84 −8.33 -
In this work, considering the contributions of the charge of the daughter nucleus
Zd and the orbital angular momentum l taken away by the two emitted protons, a two-parameter empirical formula of a new Geiger-Nuttall law is proposed for studying2p radioactivity. Using this formula, the experimental data of the true2p -radioactive nuclei can be reproduced well. Meanwhile, it is found that the calculated results using our empirical formula are agreement with those from GLDM, ELDM and the four-parameter empirical formula. Moreover, using our formula, the half-lives of possible2p radioactivity candidates are predicted. These predicted results may provide theoretical help for future experiments. -
We would like to thank X. -D. Sun, J. -G. Deng, and J. -H. Cheng for useful discussions.
