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The stability of nuclei in the ground state depends on a complicated balance between the number of protons and neutrons. In the vicinity of the proton drop line, where this balance is strongly disturbed, the nuclear force no longer binds the additional nucleon. However, in the case of proton-rich nuclei, the Coulomb potential barrier resulting from the charged protons can keep one (for odd-Z nuclei) or two (for even-Z nuclei) additional protons inside the nucleus for a finite time [1]. These protons will eventually be emitted through the Coulomb barrier, leading to proton or two-proton (
2p ) radioactivity. These two decay modes were predicted by Goldansky et al. using the isobaric invariance principle and the isotopic invariance principle in the 1960s [2-4]. Proton radioactivity was initially discovered by Jacksonet al. from the isomeric state of53Co in 1970 [5, 6]. In terms of2p radioactivity, in theory, it can be divided into two cases depending on the single-proton emission released energy. One is not true2p radioactivity, whereQ2p> 0 andQp> 0 (Q2p andQp are the released energies of2p radioactivity and the single-proton emission, respectively), meaning the two emitted protons actually sequentially decay. The other is true2p radioactivity, whereQ2p> 0 andQp< 0 [7], meaning the two protons are emitted simultaneously. Not true2p radioactivity was initially observed by a series of extremely short-lived ground-state2p radioactivity emitters before 2002, such as6Be [8],12O [9], and16Ne [9]. With the development of experimental facilities and radioactive beams, true2p radioactivity was first reported in 2002 via the decay of45Fe in experiments performed at GANIL [10] and GSI [11], respectively. Later, true2p radioactivity of19Mg ,48Ni ,54Zn , and67Kr was also observed in different experiments [12-15].For the true
2p radioactivity process, the protons can emit from two kinds of states of the parent nucleus. One state involves an isotropic emission with no angular correlation, while the other has a strong correlation occurring as a2He -like emission from the parent nuclei. Based on these two descriptions and Wentzel-Kramers-Brillouin (WKB) theory, a number of theoretical models and/or empirical formulas were proposed for investigation of2p radioactivity, such as the direct decay model [16-18], the diproton model [17, 19-24], the three-body model [25-28], the empirical formulas of Sreeja [29] and Liu [30], and others [31-34]. These theoretical approaches have partially improved our understanding of the2p radioactivity phenomenon. However, the calculations in these theoretical approaches do not take into account the electrostatic screening effect caused by the superposition of the involved charges, the magnetic field generated by the movement of the two emitted protons, or the inhomogeneous charge distribution of the nucleus [35-37]. This effect was first considered by Hulthenet al in 1942 [38]. In their study, they proposed an analytic form of electrostatic interaction to describe this effect, named Hulthen potential. In recent years, this potential has been extensively employed to study the half-lives of α decay and proton radioactivity [36, 37, 39, 40]. The calculated results reproduce the experimental data well. The2p radioactivity process shares the same theory, i.e., barrier penetration as α decay and proton radioactivity [41-50]. Whether the Hulthen potential can be extended to study2p radioactive half-lives is an interesting topic. In this study, based on the WKB theory and employing the Hulthen potential as a replacement for Coulomb potential to consider the electrostatic screening effect, we systematically investigate the half-lives of2p radioactive nuclei near or beyond the proton drip line.The article is arranged as follows. In the next section, the theoretical framework is briefly presented. The detailed calculations and discussion are presented in Section III. Finally, a summary is given in Section IV.
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The
2p radioactive half-lifeT1/2 can be calculated using the decay constant λ and expressed asT1/2=ln2λ
(1) with
λ=S2pνP.
(2) Here,
S2p=G20[A/(A−2)]2nρ2 denotes the spectroscopic factor of2p radioactivity estimated using the cluster overlap approximation [20] withG20=(2n)!/[22n(n!)2] [51], where n≈(3Z)1/3−1 is the average principal proton oscillator quantum number [52], A and Z denote the mass number and proton number of the parent nucleus, respectively,ρ2=0.015 denotes the proton overlap function, which is determined by aχ2 optimization from the experimental half-lives of19Mg ,45Fe ,48Ni , and54Zn [16], and ν is the assault frequency, which can be calculated by the oscillation frequency ω and expressed as [53]ν=ω2π=(2nr+l+32)ℏ1.2πμR2n=(G+32)ℏ1.2πμR200,
(3) where
Rn=√35R00 is the nucleus root-mean-square (rms) radius withR00=1.24A1/3(1+1.646A−0.191A−2ZA) [54], andG=2nr+l is the principal quantum number, withnr and l being the radial and angular momentum quantum numbers, respectively. For2p radioactivity, we set G = 4 or 5 corresponding to the4ℏω or5ℏω oscillator shells, depending on the individual two-proton emitters. The reduced mass of the decaying nuclear system is given byμ=mdm2p/(md+m2p)≃938.3×2×(A−2) /A MeV/c2 , withmd andm2p being the mass of daughter nucleus and the two emitted protons, respectively. Finally,ℏ is the reduced Planck constant.The quantity P given in Eq. (2) is the penetration probability of the two emitted protons crossing the barrier. It can be calculated by the semi-classical WKB approximation [55, 56] and written as
P=exp{−2ℏ∫RoutRt√2μ[V(r)−Q2p]dr} ,
(4) where
V(r) is the diproton-daughter nucleus interaction potential,Rout is the outer turning point of potential barrier satisfying the conditionV(Rout)=Q2p , andRt=Rd+R2p is the distance of the touching configuration, whereRd andR2p are the radii of daughter nucleus and the two emitted protons, respectively. They can be calculated using [57]R=1.28A1/3−0.76+0.8A−1/3.
(5) In general, the diproton-daughter nucleus electrostatic potential is by default Coulomb type, given as
VC=ZdZ2pe2/r,
(6) where
Zd andZ2p are the proton numbers of the daughter nucleus and the two emitted protons, respectively, ande2 is the square of the electronic elementary charge. However, in the2p radioactivity process, for the superposition of the involved charges, due to the magnetic field generated by the movement of the two emitted protons and the inhomogeneous charge distribution of the nucleus, the diproton-daughter nucleus electrostatic potential behaves as a Coulomb potential at a short distance and drops exponentially at large distance, i.e., the screened electrostatic effect [38, 58]. This behavior of electrostatic potential can be described using the Hulthen type potential and is defined asVH=aZdZ2pe2ear−1,
(7) where a is the screening parameter. In this framework, the total diproton-daughter nucleus interaction potential
V(r) , shown in Fig. 1, is given byFigure 1. (color online) Schematic plot of the potential energy
V(r) for67Kr as a function of the distance between the centers of the daughter nucleus and the two emitted protons.V(r)={−V0,0⩽r⩽Rt,VH(r)+Vl(r),r⩾Rt,
(8) where
V0 is the depth of square well andVl (r) is the centrifugal potential.As
l(l+1)→(l+12)2 is a necessary correction for one-dimensional problems [59], the centrifugal potential is chosen as the Langer modified centrifugal potential in this study. It can be expressed asVl(r)=ℏ2(l+12)22μr2,
(9) where l is the orbital angular momentum taken away by the two emitted protons. It can be obtained by the spin and parity conservation laws.
Obviously, the outer turning point of potential barrier
Rout given in Eq. (4) is an important quantity in evaluating the half-life of2p radioactivity. To obtain an analytic expression forRout , the centrifugal term should be approximated as [60]1r2≈a2(ear−1)2
(10) with the value of a being small. Under this assumption, the outer turning point of the potential barrier
Rout can be expressed asRout=1aln[2V1√V02+4V1Q2p−V0+1]
(11) with
V0=2ae2Z1, V1=a2ℏ2(l+12)22μ.
(12) We define the Gamow factor as
Gse = -logP for the outer potential barrier region. It can be given by the definite integral as [36]Gse=1a[I1(r)+I2(r)]|RoutRt.
(13) These two terms,
I1(r) andI2(r) , have the following expressions: [36, 37]I1(r)=−√V1x2+V0x−Q2p+√Q2parcsin[xV0−2Q2px√4Q2pV1+V02]−V02√V1ln[2√(V1x2+V0x−Q2p)+V0+2V1x],
(14) I2(r)=√V1y2+U0y−U1−√U1arctanyU0−2U12√U1(V1y2+U0y−U1)+U02√V1ln[2√V1(V1y2+U0y−U1)+U0+2V1y],
(15) where the following notations are employed:
x=(ear−1)−1, y=1+(ear−1)−1,U0=V0−2V1, U1=Q2p+V0−V1.
(16) -
In this study, we systematically investigate the
2p radioactive half-lives of nuclei near or beyond the proton drip line. The experimental2p radioactive half-lives and corresponding released energies are taken from the corresponding references of2p radioactivity. With the Hulthen potential for the electrostatic barrier being considered, this model contains one adjustable parameter, i.e., the screening parameter a, which is obtained by fitting the true2p radioactive nuclei of19Mg ,45Fe ,48Ni ,54Zn , and67Kr , amounting to 10 experimental datasets. The standard deviation σ, describing the difference between the experimental data and calculated data, can be expressed asσ=√11010∑i=1[log(Tcacl1/2iTexpt1/2i)]2.
(17) Here,
Texpt1/2i andTcacl1/2i represent the experimental and calculated2p radioactive half-lives for the i-th nucleus, respectively. Through minimizing σ, the screening parameter is determined asa=2.9647×10−2fm−1 . The value of a is small, though it observably impacts the classical outer turning pointRout obtained by Eq. (11), with the2p radioactive half-life being sensitive toRout . For intuitively displaying the screening effect, in Fig. 2, we illustrate the difference betweenRCout -RHout against the proton number of the daughter nucleus for different2p radioactivity released energies,Q2p . Here,RCout andRHout represent theRout value calculated using pure Coulomb and Hulthen potentials, respectively, while the centrifugal potential contribution is not considered. From this figure, we can clearly see that the smaller2p radioactivity released energyQ2p and larger proton number of the daughter nucleusZd increase the difference betweenRCout andRHout .Figure 2. (color online) Difference between
RCout andRHout obtained by V(r) =Q2p when only considering the Coulomb potential. For calculatingRCout , the Coulomb potential is taken as the potential of a uniformly charged sphere, i.e.,VC (r) =ZdZ2pe2/r , while forRHout , the Coulomb potential is taken as the Hulthen potential, i.e.,VH(r) =aZdZ2pe2ear−1 , with a =2.9647×10−2fm−1 .For a more intuitive illustration of the relationship between the screening effect and
Q2p , taking the true2p radioactive nucleus45Fe as an example, we plot the difference ofRCout -RHout againstQ2p in Fig. 3. From this figure, we can see an approximately linear relation betweenRCout -RHout andQ2p , indicating a strong correlation between the screening effect and2p radioactivity released energy.Figure 3. (color online) Difference between
RCout andRHout plotted against the 2p radioactivity released energyQ2p for45Fe .In the following, using this model and the value of parameter a, we systematically calculate the
2p radioactive half-lives of nuclei using the experimental data, including true and not true2p radioactive nuclei. The detailed results are listed in Table 1. For comparison, the Gamow-like model [21], generalized liquid drop model [24], Sreeja formula [29], and Liu formula [30] are also employed. In this table, the first three columns represent the2p radioactive parent nucleus,2p radioactivity released energy, and the logarithmic form of experimental2p radioactive half-lives (Expt), respectively. The last five columns represent the logarithmic form of theoretical2p radioactive half-lives, which are calculated in this study and with the Gamow-like model (GLM) [21], generalized liquid drop model (GLDM) [24], Sreeja formula (Sreeja) [29], and Liu formula (Liu) [30], respectively. In order to intutively provide comparisons of the experimental2p radioactive half-lives with the calculated values, we present the decimal logarithm deviations between the experimental2p radioactive half-lives and the calculated values in Fig. 4. From this figure, it can be seen that the decimal logarithm deviations between the experimental2p radioactive half-lives and the values calculated in this study are within±1 . However, for54Zn withQ2p = 1.280 MeV, the decimal logarithm deviations between the experimental2p radioactive half-lives and the values calculated using the theoretical approaches of the Gamow-like model, generalized liquid drop model, Sreeja formula, Liu formula, and this study approach approximately an order of magnitude. The result suggests that this experimental data may not be accurate enough for theoretical comparisons. Furthermore, in order to compare the reproducibility of our model with other theoretical models and/or formulas for2p radioactive half-lives, we calculate the standard deviation σ using Eq. (17), with the results listed in Table 2. From this table, we find that the minimum σ obtained by our study is 0.736. Compared to the best result of previous studies, the value of σ is reduced by0.846−0.7360.846=13.0 %. It means that our study can be treated as a new and effective tool to investigate2p radioactivity.Nuclei Q2p /MeVlog10T1/2 /sExpt This study GLM GLDM Sreeja Liu 6Be 1.371 [8] −20.30 [8] −19.86 −19.70 −19.37 −21.95 −23.81 12O 1.638 [61] >−20.20 [61] −17.70 −18.04 −19.71 −18.47 −20.17 1.820 [3] −20.94 [3] −18.03 −18.30 −19.46 −18.79 −20.52 1.790 [62] −20.10 [62] −17.98 −18.26 −19.43 −18.74 −20.46 1.800 [63] −20.12 [63] −18.00 −18.27 −19.44 −18.76 −20.48 16Ne 1.330 [3] −20.64 [3] −15.47 −16.23 −16.45 −15.94 −17.53 1.400 [64] −20.38 [64] −15.71 −16.43 −16.63 −16.16 −17.77 19Mg 0.750 [12] −11.40 [12] −10.58 −11.46 −11.79 −10.66 −12.03 45Fe 1.100 [11] −2.40 [11] −2.32 −2.09 −2.23 −1.25 −2.21 1.140 [10] −2.07 [10] −2.67 −2.58 −2.71 −1.66 −2.64 1.154 [15] −2.55 [15] −2.78 −2.74 −2.87 −1.80 −2.79 1.210 [65] −2.42 [65] −3.24 −3.37 −3.50 −2.34 −3.35 48Ni 1.290 [66] −2.52 [66] −2.55 −2.59 −2.62 −1.61 −2.59 1.350 [15] −2.08 [15] −3.00 −3.21 −3.24 −2.13 −3.13 54Zn 1.280 [67] −2.76 [67] −1.31 −0.93 −0.87 −0.10 −1.01 1.480 [13] −2.43 [13] −2.81 −3.01 −2.95 −1.83 −2.81 67Kr 1.690 [14] −1.70 [14] −0.95 −0.76 −1.25 0.31 −0.58 Table 1. Comparisons between the experimental
2p radioactive half-lives and the calculated values using five different theoretical models and/or formulas. The experimental2p radioactive half-liveslog10Texpt1/2 and corresponding2p released energiesQ2p are extracted from the different references.Figure 4. (color online) Deviations between the experimental
2p radioactive half-lives and the calculated values using different theoretical models and/or formulas for the true and not true2p radioactive nuclei.Model This study Gamow GLDM Sreeja Liu σ 0.736 0.846 0.852 1.221 0.851 Table 2. Standard deviation σ between the experimental data and the calculated data using different theoretical models and/or formulas for true
2p radioactivity.Finally, we extend this model to predict the half-lives of possible
2p radioactive nuclei withQ2p>0 , which are extracted from the evaluated atomic mass table AME2016 [68, 69]. The predicted results with other predictions using the Gamow-like model, GLDM, and two empirical formulas of Sreeja and Liu are all listed in Table 3. This table is similar to Table 1, except the experimental2p radioactive half-lives denoted as "Expt" are replaced by the orbital angular momentum denoted by l in the third column. Based on Table 3, we plot the logarithmic form of the predicted2p radioactive half-lives for these possible2p radioactive nuclei in Fig. 5. From this figure, we observe that our predicted results are in agreement with the results from the Gamow-like model, generalized liquid drop model, Sreeja formulas, and Liu formulas. In addition, we plot the predicted2p radioactive half-liveslog10TPre1/2 against(Z0.8d+l0.25)Q−1/22p , i.e., the new Geiger-Nuttall law for2p radioactivity, [30] in Fig. 6. The figure depicts an approximately straight line, which indicates that our predicted results are reliable.Nuclei Q2p /MeVl log10T1/2 /sThis study GLM GLDM Sreeja Liu 22Si 1.283 0 −12.17 −13.25 −13.30 −12.30 −13.74 26S 1.755 0 −12.82 −13.92 −14.59 −12.71 −14.16 34Ca 1.474 0 −8.99 −10.10 −10.71 −8.65 −9.93 36Sc 1.993 0 −10.79 −12.00 − −10.30 −11.66 38Ti 2.743 0 −12.70 −13.84 −14.27 −11.93 −13.35 39Ti 0.758 0 −1.91 −0.91 −1.34 −0.28 −1.19 40V 1.842 0 −8.97 −10.15 − −8.46 −9.73 42Cr 1.002 0 −2.87 −2.65 −2.88 −1.78 −2.76 47Co 1.042 0 −1.13 −0.42 − 0.21 −0.69 56Ga 2.443 0 −7.41 −8.57 − −6.42 −7.61 58Ge 3.732 0 −11.10 −12.32 −13.10 −9.53 −10.85 59Ge 2.102 0 −5.41 −6.31 −6.97 −4.44 −5.54 61As 2.282 0 −5.78 −6.76 − −4.74 −5.58 10N 1.300 1 −16.76 −17.36 − −20.04 −18.59 28Cl 1.965 2 −11.78 −13.11 − −14.52 −12.46 32K 2.077 2 −11.13 −12.49 − −13.46 −11.55 57Ga 2.047 2 −4.94 −5.91 − −5.22 −4.14 60As 3.492 4 −7.88 −9.40 − −10.84 −8.33 Table 3. Comparison of the predicted half-lives for possible
2p radioactive nuclei whose2p radioactivity is energetically allowed or observed but not yet quantified in NUBASE2016 [70].Figure 5. (color online) Predicted
2p radioactive half-lives using different theoretical models and/or formulas for possible2p radioactive nuclei whose2p radioactivity is energetically allowed or observed but not yet quantified in NUBASE2016 [70].Figure 6. Predicted
2p radioactive half-liveslog10TPre1/2 plotted against(Z0.8d+l0.25)Q−1/22p , i.e., the new Geiger-Nuttall law for2p radioactivity [30]. -
In summary, based on the WKB theory considering the electrostatic screening effect and using a Hulthen potential to replace the Coulomb potential, we systematically investigate the
2p radioactive half-lives of nuclei near or beyond the proton drip line. The screening parameter a is obtained by fitting the experimental half-lives of true2p radioactive nuclei according to the smallest standard deviation. The calculated results are found to be in agreement with the corresponding experimental data. In addition, we extend this model to predict the half-lives of possible2p radioactive nuclei whose2p radioactivity is energetically allowed or observed but not yet quantified in NUBASE2016. The predicted results are in agreement with the results calculated using the Gamow-like model, generalized liquid drop model, Sreeja formula, and Liu formula. Furthermore, there is an approximate linear trend between our predicted2p radioactive half-liveslog10TPre1/2 and(Z0.8d+l0.25)Q−1/22p , i.e., the new Geiger-Nuttall law for2p radioactivity. It indicates that our predictions are reliable. -
We would like to thank X. -D. Sun, J. -G. Deng, J. -H. Cheng, and J. -L. Chen for their useful discussions and input.
Systematic study of two-proton radioactivity with a screened electrostatic barrier
- Received Date: 2021-05-31
- Available Online: 2021-10-15
Abstract: In this study, a phenomenological model is proposed based on Wentzel-Kramers-Brillouin (WKB) theory and applied to investigate the two-proton (