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Systematic study of two-proton radioactivity half-lives using the two-potential and Skyrme-Hartree-Fock approaches

  • In this work, we systematically study the two-proton (2p) radioactivity half-lives using the two-potential approach, and the nuclear potential is obtained using the Skyrme-Hartree-Fock approach and the Skyrme effective interaction of SLy8. For true 2p radioactivity (Q2p > 0 and Qp <0, where Qp and Q2p are the released energies of the one-proton and two-proton radioactivity, respectively), the standard deviation between the experimental half-lives and our theoretical calculations is 0.701. In addition, we extend this model to predict the half-lives of 15 possible 2p radioactivity candidates with Q2p > 0 obtained from the evaluated atomic mass table AME2016. The calculated results indicate a clear linear relationship between the logarithmic 2p radioactivity half-lives (log10T1/2) and coulomb parameters [(Z0.8d+l0.25)Q1/22p] considering the effect of orbital angular momentum proposed by Liu et al. [Chin. Phys. C 45, 024108 (2021)]. For comparison, the generalized liquid drop model (GLDM), effective liquid drop model (ELDM), and Gamow-like model are also used. Our predicted results are consistent with those obtained using other relevant models.
  • Recently, with the development of radioactive beam facilities, the study of exotic nuclei far from the β-stability line has became an interesting topic in nuclear physics [1-7]. Two-proton (2p) radioactivity, an important exotic decay mode, provides a new opportunity to obtain the nuclear structure information of rich-proton nuclei [8-10]. In the 1960s, this decay mode was first predicted independently by Zel’dovich [11] and Goldansky [12, 13]. However owing to the limitations in experiments, it was not until 2002 that true 2p radioactivity (Q2p > 0 and Qp <0, where Qp and Q2p are the released energies of the proton and two-proton radioactivity, respectively) was observed from the 45Fe ground state was observed at the Grand Accˊelˊerateur National d'lons Lourds (GANIL) [14] and Gesellschaft F¨ur Schwerionenforschung (GSI) [15], respectively. Later, the 2p radioactivity of 54Zn, 48Ni, 19Mg, and 67Kr were consecutively identified at different radioactive beam facilities [16-19].

    The 2p radioactivity process is considered as an isotropic emission with no angular correlation or a correlated emission forming a 2He-like cluster with a strong correlation from the even-Z nuclei either in the vicinity of or beyond the proton drip line [14, 20-22]. Based on these physical mechanisms, many theoretical models have been proposed to study 2p radioactivity, such as the direct decay model [23-29], simultaneous versus sequential decay model [30], diproton model [31, 32], and three-body model [33-36]. Moreover, some empirical formulas can successfully reproduce the half-lives of 2p radioactive nuclei, such as the four-parameter empirical formula proposed by Sreeja et al. [37] and the new Geiger-Nuttall law for two-proton radioactivity proposed by Liu et al [38]. The two-potential approach (TPA) [39, 40] proposed by Gurvitz, was initially used to address quasi-stationary problems and has been extended to studying α decay, cluster radioactivity, and proton radioactivity [41-55]. In our previous studies, we systematically investiagted proton radioactivity using the TPA approach, and the nuclear potential was calculated using the Skyrme-Hartree-Fock (SHF) approach [56, 57], and this is denoted as TPA-SHF. The calculated results can closely reproduce experimental data. Since 2p radioactivity process may share a similar theory of barrier penetration with proton radioactivity [58-61], whether TPA-SHF can be extended to study 2p radioactivity is an interesting question. Therefore, in this paper, considering the spectroscopic factor (S2p), we extend TPA-SHF to systematically study 2p radioactivity half-lives of nuclei with 4<Z<36. For comparison, the generalized liquid drop model (GLDM) [62], effective liquid drop model (ELDM) [20] and Gamow-like model [63] are also used.

    The remainder of this article is organized as follows. In Section II, the theoretical framework for the TPA-SHF is described in detail. The calculated results and discussion are given in Section III. In Section IV, a brief summary is provided.

    The 2p radioactivity half-life (T12) is an important indicator of nuclear stability, and it can be calculated using

    T12=ln2λ=ln2Γ.

    (1)

    Here λ, Γ, and are the two-proton radioactivity constant, decay width, and reduced Planck constant, respectively. In the TPA framework [39, 40], Γ can be represented by the normalized factor (F) and the penetration probability (P). It is expressed as

    Γ=2S2pFP4μ,

    (2)

    where S2p=G2[A/(A2)]2nχ2 denotes the spectroscopic factor of the 2p radioactivity. It can be obtained using cluster overlap approximation [64]. Here, G2=(2n)!/[22n(n!)2] where n(3Z)1/31 [65] is the average principal proton oscillator quantum number [66], χ2 is set as 0.0143 according to the study by Cui et al. [62], A and Z are the mass and proton number of parent nucleus, respectively, and F is the normalized factor, which can be calculated as

    Fr2r112k(r)dr=1,

    (3)

    where k(r)=2μ2|Q2pV(r)| is the wave number; μ=m2pmd/(m2p+md) is the reduced mass, where m2p and md are the masses of the two emitted protons and residual daughter nucleus, respectively; Q2p is the released energy of the two-proton radioactivity; V(r) is the total interaction potential between the two emitted protons and daughter nucleus, which are described in more detail in the following; r1, r2, and the subsequent r3 are the classical turning points. They satisfy the condition V(r1)=V(r2)=V(r3)=Q2p. Penetration probability P can be formulated as

    P=exp[2r3r2k(r)dr].

    (4)

    The total interaction potential (V(r)) is composed of the nuclear potential (VN(r)), Coulomb potential (VC(r)), and centrifugal potential (Vl(r)), and it can be expressed as

    V(r)=VN(r)+VC(r)+Vl(r).

    (5)

    In this work, based on the assumption that the two protons spontaneously emitted from parent nuclear share a momentum (p) on average, and the nuclear interaction potential between the two-proton emission and daughter nucleus is twice that between an emitted proton and daughter nucleus, we can obtain the nuclear potential of the two-proton emission VN(r)=2Uq(ρ,ρq,p2) using the SHF approach. In this model, the nuclear effective interaction is expressed as the standard Skyrme form [67]

    V12(r1,r2)=t0(1+x0Pσ)δ(r1r2)+12t1(1+x1Pσ)[P2δ(r1r2)+δ(r1r2)P2]+t2(1+x2Pσ)Pδ(r1r2)P+16t3(1+x3Pσ)[ρ(r1+r22)]αδ(r1r2)+iW0σ[P×δ(r1r2)P],

    (6)

    where t0t3, x0x3, W0, and α are the Skyrme parameters; ri (i = 1, 2) is the coordinate vector of i-th nucleon; P and P are the relative momentum operators acting on the left and right, respectively; and Pσ and σ are the spin exchange and Pauli spin operators, respectively. In the SHF model, the single-nucleon potential that depends on the momentum of the nucleon (p) can be calculated using [68]

    Uq(ρ,ρq,p2)=a(p2)2+b,

    (7)

    where subscript q is the proton/neutron ratio (q = p/n). The total nucleonic density (ρ) is the sum of the proton density (ρp) and neutron density (ρn). Coefficients a and b can be expressed as

    a=18[t1(x1+2)+t2(x2+2)]ρ+18[t1(2x1+1)+t2(2x2+1)]ρq,

    (8)

    b=18[t1(x1+2)+t2(x2+2)]k5f,n+k5f,p5π2+18[t2(2x2+1)t1(2x1+1)]k5f,q5π2+12t0(x0+2)ρ12t0(2x0+1)ρq+124t3(x3+2)(α+2)ρ(α+1)124t3(2x3+1)αρ(α1)(ρ2n+ρ2p)112t3(2x3+1)ραρq.

    (9)

    Here, kf,q=(3πρq)1/3 represents the Fermi momentum. The relationship among the total energy (E) of 2p emission in a nuclear medium, nuclear potential, and Coulomb potential can be expressed as

    E=2Uq(ρ,ρq,p2)+p22m2p+VC(r).

    (10)

    In this paper, E is obtained using the corresponding Q2p as E = [(A2)/A]Q2p. Based on the premise that the total energy remains constant when 2p emits from parent nuclei, using Eqs. (7) and (10), we can obtain the momentum of the two emitted protons |p|, expressed as

    |p|=2(E2bVc(r))a+1m2p.

    (11)

    The Coulomb potential VC(r) can be obtained from a uniformly charged sphere with radius R:

    VC(r)={ZdZ2pe22R[3(rR)2],r<R,ZdZ2pe2r,r>R,

    (12)

    where Z2p=2 is the proton number of the two emitted protons in 2p radioactivity. The radius R is given by [69]

    R=1.28A1/30.76+0.8A1/3.

    (13)

    For the last part of Eq. (5), centrifugal potential Vl(r), we select the Langer modified form since l(l+1)(l+1/2)2 is necessary in one-dimensional problems [70]. It can be expressed as

    Vl(r)=2(l+12)22μr2,

    (14)

    where l is the orbital angular momentum of the two emitted protons in 2p radioactivity.

    In this work, we first calculated the 2p radioactivity half-lives of nuclei with 4<Z<36 using the TPA, obtained the nuclear potential using the SHF apparoach, and compared our calculated results with the experimental data and theoretical results calculated usisng the GLDM [62], ELDM [20], and Gamow-like model [63]. The Skyrme effective interaction currently has approximately 120 sets of Skyrme parameters. The SLy series parameters have been widely used to describe the different nuclear reactions in various studies, and the α decay since spin-gradient term or a more refined two-body cent of mass correction is considered [29, 71-74]. These parameters are listed in Table 1. As an example, we select the Skyrme parameters of SLy8 in this paper. The detailed calculation results are listed in Table 2, in which the first two columns represent the two-proton emitter and texperimental released energy of 2p radioactivity (Q2p), respectively. The experimental data of 2p radioactivity half-lives and the theoretical ones obtained using the GLDM, ELDM, Gamow-like model, and our model are provided in logarithmic form in columns 3-7, respectively. As shown in Table 2, the theoretical 2p radioactivity half-lives calculated using our model can closely reproduce the experimental data. To intuitively survey their deviations, we plot the difference of 2p radioactivity logarithmic half-lives of the experimental data and the ones calculated busing the four models (our model, GLDM, ELDM, and Gamow-like) in Fig. 1. The figure clearly shows that all the points representing the difference are basically within ±1. For 48Ni of Q2p = 1.350 MeV and 54Zn of Q2p = 1.280 MeV in particualr, our calculated results can better reproduce the experimental data compared with the other models.

    Table 1

    Table 1.  Skyrme parameters of SLy series.
    model t0 t1 t2 t3 x0 x1 x2 x3 W0 α
    Sly0 [72] −2486.40 485.20 −440.50 13783.0 0.790 −0.500 −0.930 1.290 123.0 1/6
    Sly1 [72] −2487.60 488.30 −568.90 13791.0 0.800 −0.310 −1.000 1.290 125.0 1/6
    Sly2 [72] −2484.20 482.20 −290.00 13763.0 0.790 −0.730 −0.780 1.280 125.0 1/6
    Sly3 [72] −2481.10 481.00 −540.80 13731.0 0.840 −0.340 −1.000 1.360 125.0 1/6
    Sly4 [71] −2488.91 486.82 −546.39 13777.0 0.834 −0.344 −1.000 1.354 123.0 1/6
    Sly5 [71] −2484.88 483.13 −549.40 13763.0 0.778 −0.328 −1.000 1.267 126.0 1/6
    Sly6 [71] −2479.50 462.18 −448.61 13673.0 0.825 −0.465 −1.000 1.355 122.0 1/6
    Sly7 [71] −2482.41 457.97 −419.85 13677.0 0.846 −0.511 −1.000 1.391 126.0 1/6
    Sly8 [72] −2481.40 480.80 −538.30 13731.0 0.800 −0.340 −1.000 1.310 125.0 1/6
    Sly9 [72] −2511.10 510.60 −429.80 13716.0 0.800 −0.620 −1.000 1.370 125.0 1/6
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    Table 2

    Table 2.  Experimental and theoretical data of 2p radioactivity half-lives calculated using GLDM, ELDM, Gamow-like, and our model.
    Nucleus Q2p/MeV log10Texp1/2/s log10TGLDM1/2/s[62] log10TELDM1/2/s[20] log10TGamowlike1/2/s[63] log10Tourmodel1/2/s
    19Mg 0.750 [18] 11.40 [18] 11.79 11.72 11.46 11.00
    45Fe 1.100 [15] 2.40 [15] 2.23 2.09 2.31
    1.140 [14] 2.07 [14] 2.71 2.58 2.87
    1.210 [75] 2.42 [75] 3.50 3.37 3.53
    1.154 [17] 2.55 [17] 2.87 2.43 2.74 2.88
    48Ni 1.350 [17] 2.08 [17] 3.24 3.21 2.27
    1.290 [76] 2.52 [76] 2.62 2.59 2.23
    54Zn 1.480 [16] 2.43 [16] 2.95 1.32 3.01 2.08
    1.280 [77] 2.76 [77] 0.87 0.93 1.32
    67Kr 1.690 [19] 1.70 [19] 1.25 0.06 0.76 1.05
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    Figure 1

    Figure 1.  (color online) Difference between the experimental data of 2p radioactivity half-lives and theoretical data calculated using GLDM, ELDM, Gamow-like, and our model in logarithmic form.

    To obtain further insight into the agreement and systematics of the results, the standard deviation (σ) between the theoretical and experimental values is used to quantify the calculated capabilities of the above four models for 2p radioactivity half-lives. In this paper, it is defined as follows:

    σ=[ni=1[log10Ti1/2(expt.)log10Ti1/2(cal.)]2/n]1/2.

    (15)

    Here, log10Ti1/2(expt.) and log10Ti1/2(cal.) are the logarithmic forms of the experimental and calculated 2p radioactivity half-lives for the i-th nucleus, respectively. For comparison, the σ values of these four models are listed in Table 3, which clearly shows that the σ=0.701 of this paper is better than that of the GLDM and Gamow-like model with the same data. This indicates that our mode is suitable to studying 2p radioactivity half-lives.

    Table 3

    Table 3.  Standard deviation (σ) between the experimental and theoretical data calculated using our model, GLDM, ELDM, and Gamow-like model.
    ModelOur modelGLDMELDMGamow-like
    σ0.701(10)0.852(10)0.531(4)0.844(10)
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    In addition, as an application, we extend our model to predicting the half-lives of 15 possible 2p radioactivity candidates with Q2p>0 obtained from the evaluated atomic mass table AME2016 [78, 79]. For comparison, the GLDM, ELDM, and Gamow-like models are also used. The detailed results are shown in Table 4, where the first three columns represent the 2p radioactivity candidates, experimental 2p radioactivity released energy (Q2p) and orbital angular momentum (l), respectively. The last four columns are the theoretical values of 2p radioactivity half-lives calculated using GLDM, ELDM, Gamow-like model, and our model in logarithmic form, respectively. The table clearly shows that for short-lived 2p radioactivity nuclei, the orders of magnitude of most predicted results calculated using our model are consistent with the ones obtained using the other three models. However, for long-lived 2p radioactivity nuclei, such as 49Ni and 60Ge, the magnitude of our model is less than 2-3 orders of the other three models. To further clearly compare the evaluation capabilities of those four models, the relationship between the predicted results of those four models listed in Table 4 and coulomb parameters considering orbital angular momentum ((Z0.8d+l0.25)Q1/22p) i.e., new Geiger-Nuttall law for two-proton radioactivity proposed by Liu et al. [38] is plotted in Fig. 2. The figure shows that the predicted results of those four models are all linearly dependent on (Z0.8d+l0.25)Q1/22p and our model can better conform to the linear relationship.

    Figure 2

    Figure 2.  (color online) Relationship between the predicted results of these four models listed in Table 3 and coulomb parameters ((Z0.8d+l0.25)Q1/22p) considering the effect of the orbital angular momentum, i.e., new Geiger-Nuttall law for two-proton radioactivity proposed by Liu et al.

    Table 4

    Table 4.  Comparison of the predicted 2p radioactivity half-lives using GLDM, ELDM, Gamow-like model, and our model. The 2p radioactivity released energy (Q2p) and orbital angular momentum (l) of the two emitted protons were obtained from Ref. [20].
    Nucleus Q2p/MeV l log10TGLDM1/2/s[62] log10TELDM1/2/s[20] log10TGamowlike1/2/s[63] log10Tourmodel1/2/s
    22Si 1.283 0 13.30 13.32 13.25 11.78
    26S 1.755 0 14.59 13.86 13.92 12.93
    34Ca 1.474 0 10.71 9.91 10.10 9.51
    36Sc 1.993 0 11.74 12.00 11.12
    38Ti 2.743 0 14.27 13.56 13.84 11.77
    39Ti 0.758 0 1.34 0.81 0.91 1.62
    40V 1.842 0 9.85 10.15 9.34
    42Cr 1.002 0 2.88 2.43 2.65 2.83
    47Co 1.042 0 0.11 0.42 0.97
    49Ni 0.492 0 14.46 14.64 14.54 11.05
    56Ga 2.443 0 8.00 8.57 7.51
    58Ge 3.732 0 13.10 11.74 12.32 11.06
    59Ge 2.102 0 6.97 5.71 6.31 5.88
    60Ge 0.631 0 13.55 14.62 14.24 12.09
    61As 2.282 0 6.12 6.76 6.07
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    In this paper, using the two-potential approach comprising the Skyrme-Hartree-Fock to calculate the nuclear potential and the Skyrme effective interaction of SLy8, we systematically studied the 2p radioactivity half-lives of nuclei with 4<Z<36. The calculated results can closely reproduce experimental data. In addition, we extended our model to predict the half-lives of 15 possible 2p radioactivity candidates with Q2p>0 obtained from the evaluated atomic mass table AME2016 and compared our calculated results with the theoretical ones calculated using the GLDM, ELDM, and Gamow-like model. The predicted results of these four models are all linearly dependent on (Z0.8d+l0.25)Q1/22p, i.e., the new Geiger-Nuttall law for two-proton radioactivity proposed by Liu et al.

    We would like to thank X. -D. Sun, J. -G. Deng, J. -L. Chen and J. -H. Cheng for useful discussion.

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14. Zou, Y.-T., Pan, X., Li, X.-H. et al. Favored one proton radioactivity within a one-parameter model* * Supported by National Natural Science Foundation of China (Grant No.12175100 and No. 11705055), the construct program of the key discipline in Hunan province, the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 18A237), the Innovation Group of Nuclear and Particle Physics in USC, the Shandong Province Natural Science Foundation, China (Grant No. ZR2019YQ01), and the Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX20210942).[J]. Communications in Theoretical Physics, 2022, 74(11): 115302. doi: 10.1088/1572-9494/ac7e2c
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18. Santhosh, K.P.. Theoretical studies on two-proton radioactivity[J]. Physical Review C, 2021, 104(6): 064613. doi: 10.1103/PhysRevC.104.064613

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Xiao Pan, You-Tian Zou, Hong-Ming Liu, Biao He, Xiao-Hua Li, Xi-Jun Wu and Zhen Zhang. Systematic study of two-proton radioactivity half-lives within the two-potential approach with Skyrme-Hartree-Fock[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac2421
Xiao Pan, You-Tian Zou, Hong-Ming Liu, Biao He, Xiao-Hua Li, Xi-Jun Wu and Zhen Zhang. Systematic study of two-proton radioactivity half-lives within the two-potential approach with Skyrme-Hartree-Fock[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac2421 shu
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Systematic study of two-proton radioactivity half-lives using the two-potential and Skyrme-Hartree-Fock approaches

  • 1. School of Nuclear Science and Technology, University of South China, Hengyang 421001, China
  • 2. School of Math and Physics, University of South China, Hengyang 421001, China
  • 3. Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China
  • 4. College of Physics and Electronics, Central South University, Changsha 410083, China
  • 5. Cooperative Innovation Center for Nuclear Fuel Cycle Technology & Equipment, University of South China, Hengyang 421001, China
  • 6. Key Laboratory of Low Dimensional Quantum Structures and Quantum Control, Hunan Normal University, Changsha 410081, China

Abstract: In this work, we systematically study the two-proton (2p) radioactivity half-lives using the two-potential approach, and the nuclear potential is obtained using the Skyrme-Hartree-Fock approach and the Skyrme effective interaction of SLy8. For true 2p radioactivity (Q2p > 0 and Qp <0, where Qp and Q2p are the released energies of the one-proton and two-proton radioactivity, respectively), the standard deviation between the experimental half-lives and our theoretical calculations is 0.701. In addition, we extend this model to predict the half-lives of 15 possible 2p radioactivity candidates with Q2p > 0 obtained from the evaluated atomic mass table AME2016. The calculated results indicate a clear linear relationship between the logarithmic 2p radioactivity half-lives (log10T1/2) and coulomb parameters [(Z0.8d+l0.25)Q1/22p] considering the effect of orbital angular momentum proposed by Liu et al. [Chin. Phys. C 45, 024108 (2021)]. For comparison, the generalized liquid drop model (GLDM), effective liquid drop model (ELDM), and Gamow-like model are also used. Our predicted results are consistent with those obtained using other relevant models.

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    I.   INTRODUCTION
    • Recently, with the development of radioactive beam facilities, the study of exotic nuclei far from the β-stability line has became an interesting topic in nuclear physics [1-7]. Two-proton (2p) radioactivity, an important exotic decay mode, provides a new opportunity to obtain the nuclear structure information of rich-proton nuclei [8-10]. In the 1960s, this decay mode was first predicted independently by Zel’dovich [11] and Goldansky [12, 13]. However owing to the limitations in experiments, it was not until 2002 that true 2p radioactivity (Q2p > 0 and Qp <0, where Qp and Q2p are the released energies of the proton and two-proton radioactivity, respectively) was observed from the 45Fe ground state was observed at the Grand Accˊelˊerateur National d'lons Lourds (GANIL) [14] and Gesellschaft F¨ur Schwerionenforschung (GSI) [15], respectively. Later, the 2p radioactivity of 54Zn, 48Ni, 19Mg, and 67Kr were consecutively identified at different radioactive beam facilities [16-19].

      The 2p radioactivity process is considered as an isotropic emission with no angular correlation or a correlated emission forming a 2He-like cluster with a strong correlation from the even-Z nuclei either in the vicinity of or beyond the proton drip line [14, 20-22]. Based on these physical mechanisms, many theoretical models have been proposed to study 2p radioactivity, such as the direct decay model [23-29], simultaneous versus sequential decay model [30], diproton model [31, 32], and three-body model [33-36]. Moreover, some empirical formulas can successfully reproduce the half-lives of 2p radioactive nuclei, such as the four-parameter empirical formula proposed by Sreeja et al. [37] and the new Geiger-Nuttall law for two-proton radioactivity proposed by Liu et al [38]. The two-potential approach (TPA) [39, 40] proposed by Gurvitz, was initially used to address quasi-stationary problems and has been extended to studying α decay, cluster radioactivity, and proton radioactivity [41-55]. In our previous studies, we systematically investiagted proton radioactivity using the TPA approach, and the nuclear potential was calculated using the Skyrme-Hartree-Fock (SHF) approach [56, 57], and this is denoted as TPA-SHF. The calculated results can closely reproduce experimental data. Since 2p radioactivity process may share a similar theory of barrier penetration with proton radioactivity [58-61], whether TPA-SHF can be extended to study 2p radioactivity is an interesting question. Therefore, in this paper, considering the spectroscopic factor (S2p), we extend TPA-SHF to systematically study 2p radioactivity half-lives of nuclei with 4<Z<36. For comparison, the generalized liquid drop model (GLDM) [62], effective liquid drop model (ELDM) [20] and Gamow-like model [63] are also used.

      The remainder of this article is organized as follows. In Section II, the theoretical framework for the TPA-SHF is described in detail. The calculated results and discussion are given in Section III. In Section IV, a brief summary is provided.

    II.   THEORETICAL FRAMEWORK
    • The 2p radioactivity half-life (T12) is an important indicator of nuclear stability, and it can be calculated using

      T12=ln2λ=ln2Γ.

      (1)

      Here λ, Γ, and are the two-proton radioactivity constant, decay width, and reduced Planck constant, respectively. In the TPA framework [39, 40], Γ can be represented by the normalized factor (F) and the penetration probability (P). It is expressed as

      Γ=2S2pFP4μ,

      (2)

      where S2p=G2[A/(A2)]2nχ2 denotes the spectroscopic factor of the 2p radioactivity. It can be obtained using cluster overlap approximation [64]. Here, G2=(2n)!/[22n(n!)2] where n(3Z)1/31 [65] is the average principal proton oscillator quantum number [66], χ2 is set as 0.0143 according to the study by Cui et al. [62], A and Z are the mass and proton number of parent nucleus, respectively, and F is the normalized factor, which can be calculated as

      Fr2r112k(r)dr=1,

      (3)

      where k(r)=2μ2|Q2pV(r)| is the wave number; μ=m2pmd/(m2p+md) is the reduced mass, where m2p and md are the masses of the two emitted protons and residual daughter nucleus, respectively; Q2p is the released energy of the two-proton radioactivity; V(r) is the total interaction potential between the two emitted protons and daughter nucleus, which are described in more detail in the following; r1, r2, and the subsequent r3 are the classical turning points. They satisfy the condition V(r1)=V(r2)=V(r3)=Q2p. Penetration probability P can be formulated as

      P=exp[2r3r2k(r)dr].

      (4)

      The total interaction potential (V(r)) is composed of the nuclear potential (VN(r)), Coulomb potential (VC(r)), and centrifugal potential (Vl(r)), and it can be expressed as

      V(r)=VN(r)+VC(r)+Vl(r).

      (5)

      In this work, based on the assumption that the two protons spontaneously emitted from parent nuclear share a momentum (p) on average, and the nuclear interaction potential between the two-proton emission and daughter nucleus is twice that between an emitted proton and daughter nucleus, we can obtain the nuclear potential of the two-proton emission VN(r)=2Uq(ρ,ρq,p2) using the SHF approach. In this model, the nuclear effective interaction is expressed as the standard Skyrme form [67]

      V12(r1,r2)=t0(1+x0Pσ)δ(r1r2)+12t1(1+x1Pσ)[P2δ(r1r2)+δ(r1r2)P2]+t2(1+x2Pσ)Pδ(r1r2)P+16t3(1+x3Pσ)[ρ(r1+r22)]αδ(r1r2)+iW0σ[P×δ(r1r2)P],

      (6)

      where t0t3, x0x3, W0, and α are the Skyrme parameters; ri (i = 1, 2) is the coordinate vector of i-th nucleon; P and P are the relative momentum operators acting on the left and right, respectively; and Pσ and σ are the spin exchange and Pauli spin operators, respectively. In the SHF model, the single-nucleon potential that depends on the momentum of the nucleon (p) can be calculated using [68]

      Uq(ρ,ρq,p2)=a(p2)2+b,

      (7)

      where subscript q is the proton/neutron ratio (q = p/n). The total nucleonic density (ρ) is the sum of the proton density (ρp) and neutron density (ρn). Coefficients a and b can be expressed as

      a=18[t1(x1+2)+t2(x2+2)]ρ+18[t1(2x1+1)+t2(2x2+1)]ρq,

      (8)

      b=18[t1(x1+2)+t2(x2+2)]k5f,n+k5f,p5π2+18[t2(2x2+1)t1(2x1+1)]k5f,q5π2+12t0(x0+2)ρ12t0(2x0+1)ρq+124t3(x3+2)(α+2)ρ(α+1)124t3(2x3+1)αρ(α1)(ρ2n+ρ2p)112t3(2x3+1)ραρq.

      (9)

      Here, kf,q=(3πρq)1/3 represents the Fermi momentum. The relationship among the total energy (E) of 2p emission in a nuclear medium, nuclear potential, and Coulomb potential can be expressed as

      E=2Uq(ρ,ρq,p2)+p22m2p+VC(r).

      (10)

      In this paper, E is obtained using the corresponding Q2p as E = [(A2)/A]Q2p. Based on the premise that the total energy remains constant when 2p emits from parent nuclei, using Eqs. (7) and (10), we can obtain the momentum of the two emitted protons |p|, expressed as

      |p|=2(E2bVc(r))a+1m2p.

      (11)

      The Coulomb potential VC(r) can be obtained from a uniformly charged sphere with radius R:

      VC(r)={ZdZ2pe22R[3(rR)2],r<R,ZdZ2pe2r,r>R,

      (12)

      where Z2p=2 is the proton number of the two emitted protons in 2p radioactivity. The radius R is given by [69]

      R=1.28A1/30.76+0.8A1/3.

      (13)

      For the last part of Eq. (5), centrifugal potential Vl(r), we select the Langer modified form since l(l+1)(l+1/2)2 is necessary in one-dimensional problems [70]. It can be expressed as

      Vl(r)=2(l+12)22μr2,

      (14)

      where l is the orbital angular momentum of the two emitted protons in 2p radioactivity.

    III.   RESULTS AND DISCUSSION
    • In this work, we first calculated the 2p radioactivity half-lives of nuclei with 4<Z<36 using the TPA, obtained the nuclear potential using the SHF apparoach, and compared our calculated results with the experimental data and theoretical results calculated usisng the GLDM [62], ELDM [20], and Gamow-like model [63]. The Skyrme effective interaction currently has approximately 120 sets of Skyrme parameters. The SLy series parameters have been widely used to describe the different nuclear reactions in various studies, and the α decay since spin-gradient term or a more refined two-body cent of mass correction is considered [29, 71-74]. These parameters are listed in Table 1. As an example, we select the Skyrme parameters of SLy8 in this paper. The detailed calculation results are listed in Table 2, in which the first two columns represent the two-proton emitter and texperimental released energy of 2p radioactivity (Q2p), respectively. The experimental data of 2p radioactivity half-lives and the theoretical ones obtained using the GLDM, ELDM, Gamow-like model, and our model are provided in logarithmic form in columns 3-7, respectively. As shown in Table 2, the theoretical 2p radioactivity half-lives calculated using our model can closely reproduce the experimental data. To intuitively survey their deviations, we plot the difference of 2p radioactivity logarithmic half-lives of the experimental data and the ones calculated busing the four models (our model, GLDM, ELDM, and Gamow-like) in Fig. 1. The figure clearly shows that all the points representing the difference are basically within ±1. For 48Ni of Q2p = 1.350 MeV and 54Zn of Q2p = 1.280 MeV in particualr, our calculated results can better reproduce the experimental data compared with the other models.

      model t0 t1 t2 t3 x0 x1 x2 x3 W0 α
      Sly0 [72] −2486.40 485.20 −440.50 13783.0 0.790 −0.500 −0.930 1.290 123.0 1/6
      Sly1 [72] −2487.60 488.30 −568.90 13791.0 0.800 −0.310 −1.000 1.290 125.0 1/6
      Sly2 [72] −2484.20 482.20 −290.00 13763.0 0.790 −0.730 −0.780 1.280 125.0 1/6
      Sly3 [72] −2481.10 481.00 −540.80 13731.0 0.840 −0.340 −1.000 1.360 125.0 1/6
      Sly4 [71] −2488.91 486.82 −546.39 13777.0 0.834 −0.344 −1.000 1.354 123.0 1/6
      Sly5 [71] −2484.88 483.13 −549.40 13763.0 0.778 −0.328 −1.000 1.267 126.0 1/6
      Sly6 [71] −2479.50 462.18 −448.61 13673.0 0.825 −0.465 −1.000 1.355 122.0 1/6
      Sly7 [71] −2482.41 457.97 −419.85 13677.0 0.846 −0.511 −1.000 1.391 126.0 1/6
      Sly8 [72] −2481.40 480.80 −538.30 13731.0 0.800 −0.340 −1.000 1.310 125.0 1/6
      Sly9 [72] −2511.10 510.60 −429.80 13716.0 0.800 −0.620 −1.000 1.370 125.0 1/6

      Table 1.  Skyrme parameters of SLy series.

      Nucleus Q2p/MeV log10Texp1/2/s log10TGLDM1/2/s[62] log10TELDM1/2/s[20] log10TGamowlike1/2/s[63] log10Tourmodel1/2/s
      19Mg 0.750 [18] 11.40 [18] 11.79 11.72 11.46 11.00
      45Fe 1.100 [15] 2.40 [15] 2.23 2.09 2.31
      1.140 [14] 2.07 [14] 2.71 2.58 2.87
      1.210 [75] 2.42 [75] 3.50 3.37 3.53
      1.154 [17] 2.55 [17] 2.87 2.43 2.74 2.88
      48Ni 1.350 [17] 2.08 [17] 3.24 3.21 2.27
      1.290 [76] 2.52 [76] 2.62 2.59 2.23
      54Zn 1.480 [16] 2.43 [16] 2.95 1.32 3.01 2.08
      1.280 [77] 2.76 [77] 0.87 0.93 1.32
      67Kr 1.690 [19] 1.70 [19] 1.25 0.06 0.76 1.05

      Table 2.  Experimental and theoretical data of 2p radioactivity half-lives calculated using GLDM, ELDM, Gamow-like, and our model.

      Figure 1.  (color online) Difference between the experimental data of 2p radioactivity half-lives and theoretical data calculated using GLDM, ELDM, Gamow-like, and our model in logarithmic form.

      To obtain further insight into the agreement and systematics of the results, the standard deviation (σ) between the theoretical and experimental values is used to quantify the calculated capabilities of the above four models for 2p radioactivity half-lives. In this paper, it is defined as follows:

      σ=[ni=1[log10Ti1/2(expt.)log10Ti1/2(cal.)]2/n]1/2.

      (15)

      Here, log10Ti1/2(expt.) and log10Ti1/2(cal.) are the logarithmic forms of the experimental and calculated 2p radioactivity half-lives for the i-th nucleus, respectively. For comparison, the σ values of these four models are listed in Table 3, which clearly shows that the σ=0.701 of this paper is better than that of the GLDM and Gamow-like model with the same data. This indicates that our mode is suitable to studying 2p radioactivity half-lives.

      ModelOur modelGLDMELDMGamow-like
      σ0.701(10)0.852(10)0.531(4)0.844(10)

      Table 3.  Standard deviation (σ) between the experimental and theoretical data calculated using our model, GLDM, ELDM, and Gamow-like model.

      In addition, as an application, we extend our model to predicting the half-lives of 15 possible 2p radioactivity candidates with Q2p>0 obtained from the evaluated atomic mass table AME2016 [78, 79]. For comparison, the GLDM, ELDM, and Gamow-like models are also used. The detailed results are shown in Table 4, where the first three columns represent the 2p radioactivity candidates, experimental 2p radioactivity released energy (Q2p) and orbital angular momentum (l), respectively. The last four columns are the theoretical values of 2p radioactivity half-lives calculated using GLDM, ELDM, Gamow-like model, and our model in logarithmic form, respectively. The table clearly shows that for short-lived 2p radioactivity nuclei, the orders of magnitude of most predicted results calculated using our model are consistent with the ones obtained using the other three models. However, for long-lived 2p radioactivity nuclei, such as 49Ni and 60Ge, the magnitude of our model is less than 2-3 orders of the other three models. To further clearly compare the evaluation capabilities of those four models, the relationship between the predicted results of those four models listed in Table 4 and coulomb parameters considering orbital angular momentum ((Z0.8d+l0.25)Q1/22p) i.e., new Geiger-Nuttall law for two-proton radioactivity proposed by Liu et al. [38] is plotted in Fig. 2. The figure shows that the predicted results of those four models are all linearly dependent on (Z0.8d+l0.25)Q1/22p and our model can better conform to the linear relationship.

      Figure 2.  (color online) Relationship between the predicted results of these four models listed in Table 3 and coulomb parameters ((Z0.8d+l0.25)Q1/22p) considering the effect of the orbital angular momentum, i.e., new Geiger-Nuttall law for two-proton radioactivity proposed by Liu et al.

      Nucleus Q2p/MeV l log10TGLDM1/2/s[62] log10TELDM1/2/s[20] log10TGamowlike1/2/s[63] log10Tourmodel1/2/s
      22Si 1.283 0 13.30 13.32 13.25 11.78
      26S 1.755 0 14.59 13.86 13.92 12.93
      34Ca 1.474 0 10.71 9.91 10.10 9.51
      36Sc 1.993 0 11.74 12.00 11.12
      38Ti 2.743 0 14.27 13.56 13.84 11.77
      39Ti 0.758 0 1.34 0.81 0.91 1.62
      40V 1.842 0 9.85 10.15 9.34
      42Cr 1.002 0 2.88 2.43 2.65 2.83
      47Co 1.042 0 0.11 0.42 0.97
      49Ni 0.492 0 14.46 14.64 14.54 11.05
      56Ga 2.443 0 8.00 8.57 7.51
      58Ge 3.732 0 13.10 11.74 12.32 11.06
      59Ge 2.102 0 6.97 5.71 6.31 5.88
      60Ge 0.631 0 13.55 14.62 14.24 12.09
      61As 2.282 0 6.12 6.76 6.07

      Table 4.  Comparison of the predicted 2p radioactivity half-lives using GLDM, ELDM, Gamow-like model, and our model. The 2p radioactivity released energy (Q2p) and orbital angular momentum (l) of the two emitted protons were obtained from Ref. [20].

    IV.   SUMMARY
    • In this paper, using the two-potential approach comprising the Skyrme-Hartree-Fock to calculate the nuclear potential and the Skyrme effective interaction of SLy8, we systematically studied the 2p radioactivity half-lives of nuclei with 4<Z<36. The calculated results can closely reproduce experimental data. In addition, we extended our model to predict the half-lives of 15 possible 2p radioactivity candidates with Q2p>0 obtained from the evaluated atomic mass table AME2016 and compared our calculated results with the theoretical ones calculated using the GLDM, ELDM, and Gamow-like model. The predicted results of these four models are all linearly dependent on (Z0.8d+l0.25)Q1/22p, i.e., the new Geiger-Nuttall law for two-proton radioactivity proposed by Liu et al.

    ACKNOWLEDGEMENTS
    • We would like to thank X. -D. Sun, J. -G. Deng, J. -L. Chen and J. -H. Cheng for useful discussion.

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