-
The proton radioactivity half-life
T12 can be express-ed using the decay width Γ as follows:T12=ln2ℏΓ.
(1) In the framework of the TPA [18], Γ depends on the formation probability of the proton radioactivity
Sp , the normalized factor F, and the penetration probability of the emitted proton crossing the barrier P. It is given byΓ=ℏ2SpFP4μ,
(2) where
ℏ is the reduced Planck constant, andμ=MdMp(Md+Mp) is the reduced mass, withMp andMd as the masses of the emitted proton and daughter nuclei, respectively.In the classical WKB approximation, the penetration probability P and normalized factor F are given by
P=exp[−2∫r3r2k(r)dr],
(3) F∫r2r112k(r)dr=1.
(4) Here,
r1 ,r2 , andr3 represent the classical turning points, which satisfy the conditionsV(r1)=V(r2)=V(r3)=Qp .k(r) is the wave number, which can be written ask(r)=√2μℏ2|Qp−V(r)|,
(5) where
Qp is the proton radioactivity energy. The total potentialV(r) is given byV(r)=VN(r)+VC(r)+Vl(r),
(6) where
VN(r) ,VC(r) , andVl(r) represent the nuclear, Coulomb, and centrifugal potentials, respectively. For the centrifugal potentialVl(r) ,l(l+1)→(l+1/2)2 is an essential correction [26]. In this study, the centrifugal potentialVl(r) is chosen to be in the Langer-modified form, which can be written asVl(r)=ℏ2(l+12)22μr2,
(7) where l is the orbital angular momentum taken by the emitted proton [27]. This can be obtained by the parity and angular momentum conservation laws. The Coulomb potential
VC(r) is taken as the potential of a uniformly charged sphere connected to the sharp radiusR=1.28A1/3−0.76+0.8A−1/3 , where A is the mass number of the parent nucleus [28]. It can be expressed asVC(r)={ZpZde22R[3−(rR)2],r<R,ZpZde2r,r>R,
(8) where
Zp andZd are the proton numbers of the emitted proton and daughter nuclei, respectively. -
The emitted proton–daughter nucleus nuclear potential
VN(r)=Uq(ρ,ρp,p) is calculated using SHF. The nuclear effective interactions with zero-range, momentum, and density dependent forms is given by [15]VSkyrme12(r1,r2)=t0(1+x0Pσ)δ(r)+12t1(1+x1Pσ)[P′2δ(r)+δ(r)P2]+t2(1+x2Pσ)P′⋅δ(r)P+16t3(1+x3Pσ)[ρ(R)]αδ(r)
+iW0σ⋅[P′×δ(r)P],
(9) where
r=r1−r2 ,R=(r1+r2)/2 ,ri (i = 1, 2) is the coordinate vector of the i-th nucleon,Pσ is the spin exchange operator,P′ andP are the relative momentum operators acting on the left and right, respectively,t0 ,t1 ,t2 ,t3 ,x0 ,x1 ,x2 ,x3 ,W0 , and α are the Skyrme parameters.In local density approximation, the single-nucleon potential from the SHF model can be expressed as [29]
Uq(ρ,ρq,p)=ap2+b,
(10) where
p is the momentum of the nucleon. The coefficient a and b are given bya=18[t1(x1+2)+t2(x2+2)]ρ+18[−t1(2x1+1)+t2(2x2+1)]ρq,
(11) b=18[t1(x1+2)+t2(x2+2)]k5f,n+k5f,p5π2+18[t2(2x2+1)−t1(2x1+1)]k5f,q5π2+12t0(x0+2)ρ−12t0(2x0+1)ρq+124t3(x3+2)(α+2)ρ(α+1)−124t3(2x3+1)αρ(α−1)(ρ2n+ρ2p)−112t3(2x3+1)ραρq,
(12) where
kf,q=(3πρq)1/3 is the Fermi momentum of the nucleon,ρq is the proton (neutron) density withq=p(n) , andρ=ρp+ρn represents the total nucleon density.The total energy E of a nucleon in a nuclear medium can be written as
E=Uq(ρ,ρq,p)+p22m=p22m∗+b,
(13) where
m∗ denotes the effective mass defined by12m∗=12m+a.
(14) We assume that the total energy of the emitted protons remains constant during proton radioactivity.
|p| can be derived from the total energy, nucleon density, and isospin asymmetry and is given by|p|=√2m∗(E−b).
(15) The potential energy
Uq(ρ,ρp,p)=E−p22m .By comparing the expressions in the SHF with the corresponding expressions in the modified Skyrme-like (MSL) model [30], the nine parameters σ, β, γ, C, D, y,
Elocsym(ρ0) , the gradientGs , and the symmetry-gradient coefficientGv in the MSL model can be related to the nine Skyrme interaction parameters via the following analytic relations [25]:t0=4σ/3ρ0,
(16) t1=20C/(9ρ0(k0F)2)+8Gs/3,
(17) t2=4(25C−18D)9ρ0(k0F)2−8(Gs+2Gv)3,
(18) t3=16β/(ργ0(γ+1)),
(19) x0=3(y−1)Elocsym(ρ0)/σ−1/2,
(20) x1=12Gv−4Gs−6D/(ρ0(k0F)2)3t1,
(21) x2=20Gv+4Gs−5(16C−18D)/(3ρ0(k0F)2)3t2,
(22) x3=−3y(γ+1)Elocsym(ρ0)/(2β)−1/2,
(23) α=γ−1,
(24) with
k0F=(1.5π2ρ0)1/3 . The seven parameters σ, β, γ, C, D, y, andElocsym(ρ0) in the MSL model can be expressed analytically in terms of the seven macroscopic quantities, that is, the normal densityρ0 , the equation of state atρ0 E0 (ρ0) , the incompressibilityK0 , the isoscalar effective massm∗s,0 , the isovector effective massm∗v,0 , the nuclear symmetry energy atρ0 Esym (ρ0) , and the extracted density slope L [30]. The nine Skyrme interaction parameterst0 ,t1 ,t2 ,t3 ,x0 ,x1 ,x2 ,x3 , and α can also be expressed analytically in terms of the nine macroscopic quantitiesρ0 ,E0(ρ0) ,K0 ,m∗s,0 ,m∗v,0 ,Esym(ρ0) , L,GS , andGV via the above relations. -
The aim of this study is to investigate the proton radioactivity half-life based on the relationship between macroscopic quantities and Skyrme parameters [25]. To study the sensitivity of proton radioactivity to macroscopic quantities [25], we adopt the set of Skyrme parameters MSL0 obtained using the empirical values of the macroscopic quantities to ensure that the adjustment of the macroscopic quantities is not out of line with the actual values. The macroscopic quantities and corresponding Skyrme parameters are listed in Table 1.
G′0(ρ0) is the Landau parameter, whose value can vary from approximately 0 to 1.6, depending on the method and model [31–33]. In this study, experimental data on spin, parity, the proton radioactivity energyQp , and the proton radioactivity half-lives are taken from the latest evaluated nuclear property table NUBASE2020 [34] and the latest evaluated atomic mass table AME2020 [35, 36], except for those of 140Ho, 144Tm, 151Lu, 159Re, and 164Ir, which are taken from Ref. [37]. The standard deviation Δ indicates the divergence between experimental data and the the proton radioactivity half-lives calculated using TPA-SHF, which can be expressed asΔ=√∑(lgTexp1/2(s)−lgTcal1/2(s))2/n .Quantity MSL0 Quantity MSL0 ρ0 0.16 fm−3 t0 −2118.06 MeV fm5 E0(ρ0) −16 MeV t1 395.196 MeV fm5 K0 230 MeV t2 −63.953 MeV fm5 m∗s,0/m 0.8 t3 12857.7 MeV fm3+3α m∗v,0/m 0.7 x0 −0.0709496 Esym(ρ0) 30 MeV x1 −0.332282 L 60 MeV x2 1.35830 GS 132 MeV fm5 x3 −0.0228181 GV 5 MeV fm5 α 0.235879 G′0(ρ0) 0.42 W0 133.3 MeV fm5 Table 1. Macroscopic quantities and corresponding Skyrme parameters in MSL0.
Using the MSL0 parameter set, we systematically calculate the proton radioactivity half-lives of
69≤Z≤81 nuclei withSp=1 [24]. The detailed results are plotted in Fig. 1. The blue circles and red triangles represent the logarithmic forms of the experimental data and the logarithmic forms of the calculated proton radioactivity half-life, respectively. As shown in Fig. 1, the proton radioactivity half-lives vary over a wide range from10−6 s to102 s. Although the proton radioactivity half-life variation is up to eight orders of magnitude, the theoretical points follow the experimental points well, almost coinciding with each other. Furthermore, we obtain the standard deviation Δ between the experimental data and the calculated proton radioactivity half-life as equal to 0.405, which means that the theoretical proton radioactivity half-lives calculated using TAP-SHF with MSL0 (TPA-SHF-MSL0) can reproduce the experimental data well.Figure 1. (color online) Comparison of experimental proton radioactivity half-lives and theoretical proton radioactivity half-lives calculated using TPA-SHF-MSL0 (
69≤Z≤81 ).To clearly reveal the dependence of the theoretical proton radioactivity half-lives on each macroscopic quantity, we vary one aspect at a time while keeping all other macroscopic quantities at their default values in MSL0. In this study, the variation in each macroscopic quantity is controlled within the experimentally obtained values to ensure that the dependencies we obtain are meaningful. To intuitively reflect the relationship between the theoretical proton radioactivity half-life change and the evolution of each macroscopic quantity, we plot the standard deviations between experimental data and the calculated proton radioactivity half-lives corresponding to the change in each macroscopic quantity in Fig. 2. From this figure, we can clearly see that the variety of different macroscopic quantities causes different effects on the standard deviations between experimental data and the calculated proton radioactivity half-lives. The standard deviation exhibits a robust correlation with
ρ0 and a relatively weak correlation withGS . For the other seven macroscopic quantities, their changes have little effect on the standard deviations between experimental data and the calculated proton radioactivity half-lives. The discovery of this relationship means that we can constrain the Skyrme parameters using two macroscopic quantities that significantly affect the standard deviation, which have a higher weight in the standard deviation calculation. Thus, a new set of Skyrme parameters suitable for calculating the proton radioactivity half-life is obtained with as few adjustable parameters as possible. Similarly, such a relationship, in turn, limits the range of these two macroscopic parameters. This means that the actual value ofρ0 is approximately 0.156fm−3 , and the real value ofGS is approximately 187 MeVfm5 .Figure 2. (color online) Standard deviations Δ between experimental data and theoretical proton radioactivity half-lives calculated by individually varying
ρ0 ,E0(ρ0) ,K0 ,m∗s,0 ,m∗v,0 ,Esym(ρ0) , L,GS , andGV .In this study, a new Skyrme parameter MQSP is given by fitting the two highest weight macroscopic quantities,
ρ0 andGS . The detailed results are listed in Table 2. The new theoretical value of the proton radioactivity half-life can be calculated using TAP-SHF with MQSP (TPA-SHF-MQSP). To verify the accuracy of our new Skyrme parameters in calculating the proton radioactivity half-life, the universal decay law for proton radioactivity (UDLP) is chosen as a comparison. The UDLP is given by Qi et al. [38], which is an extension of the universal decay law (UDL) [39, 40]. In the UDLP, the logarithm of proton radioactivity half-life can be expressed asQuantity MQSP Quantity MQSP ρ0 0.155 fm−3 t0 −2134.31 MeV fm5 E0(ρ0) −16 MeV t1 529.92 MeV fm5 K0 230 MeV t2 −187.13 MeV fm5 m∗s,0/m 0.8 t3 13130.28 MeV fm3+3α m∗v,0/m 0.7 x0 −0.0561957 Esym(ρ0) 30 MeV x1 −0.3721052 L 60 MeV x2 0.159008 GS 182 MeV fm5 x3 −0.21680 GV 5 MeV fm5 α 0.24205 G′0(ρ0) 0.42 W0 133.3 MeV fm5 Table 2. Macroscopic quantities and corresponding Skyrme parameters obtained using MQSP.
log10T12=aχ′+bρ′+c+dl(l+1),
(25) where
χ′=ZpZd√μQp , andρ′=√μZpZd(Ad13+Ap13) . Here, the parametersa=0.386,b=−0.502 ,c=−17.8 , andd= 2.386 are determined by fitting to the experimental data of proton radioactivity taken from Ref. [38]. As a comparison, we use TPA-SHF-MSL0, TPA-SHF-MQSP, and the UDLP to calculate the proton radioactivity half-life, and the results are listed in Table 3. In this table, columns one to four represent the parent nuclei, the orbital angular momentum l taken by the emitted proton, the proton radioactivity energyQp , and the logarithmic form of the experimental proton radioactivity half-lives, respectively. The following three columns represent the logarithmic forms of the theoretical proton radioactivity half-lives calculated using TPA-SHF with the Skyrme effective interaction of MSL0 (TPA-SHF-MSL0), TPA-SHF with the Skyrme effective interaction of MQSP (TPA-SHF-MQSP), and the UDLP, denoted as MSL0, MQSP, and UDLP, respectively. From this table, we can clearly see that the theoretical proton radioactivity half-lives calculated using TPA-SHF-MQSP can reproduce the experimental data well.Nucleus l Qp/MeV Measured lgTMSL01/2/s lgTMQSP1/2/s lgTUDLP1/2/s 144Tm 5 1.725 −5.569 −5.243 −5.42 −4.691 145Tm 5 1.736 −5.499 −5.339 −5.496 −4.767 146Tm m 5 1.206 −1.137 −0.788 −0.958 −0.814 146Tm 0 0.896 −0.81 −0.583 −0.746 −0.482 147Tm 5 1.059 0.587 1.063 0.849 0.772 147Tm m 2 1.12 −3.444 −2.89 −3.207 −2.713 150Lu m 2 1.29 −4.398 −4.341 −4.52 −3.91 150Lu 5 1.27 −1.347 −0.954 −1.116 −0.988 151Lu m 2 1.301 −4.796 −4.473 −4.647 −4.025 151Lu 5 1.255 −0.896 −0.81 −0.98 −0.863 155Ta 5 1.453 −2.495 −2.248 −2.39 −2.17 156Ta 2 1.02 −0.826 −0.401 −0.498 −0.417 156Ta m 5 1.11 0.933 1.522 1.361 1.129 157Ta 0 0.935 −0.527 0.137 −0.005 0.123 159Re m 5 1.801 −4.665 −4.531 −4.706 −4.181 159Re 5 1.816 −4.678 −4.62 −4.806 −4.268 160Re 0 1.267 −3.163 −3.778 −3.942 −3.408 161Re m 5 1.317 −0.678 −0.455 −0.633 −0.611 161Re 0 1.197 −3.306 −2.992 −3.126 −2.689 164Ir 5 1.844 −3.959 −4.418 −4.594 −4.114 165Ir m 5 1.711 −3.433 −3.496 −3.68 −3.306 166Ir 2 1.152 −0.824 −1.034 −1.169 −1.014 166Ir m 5 1.332 −0.076 −0.118 −0.283 −0.335 167Ir 0 1.07 −1.12 −0.682 −0.804 −0.645 167Ir m 5 1.245 0.842 0.855 0.688 0.523 170Au 2 1.472 −3.487 −3.891 −4.186 −3.716 170Au m 5 1.752 −3.975 −3.394 −3.569 −3.234 171Au m 5 1.702 −2.587 −3.037 −3.205 −2.915 171Au 0 1.448 −4.652 −4.578 −4.728 −4.157 176Tl 0 1.265 −2.208 −2.041 −2.162 −1.919 177Tl m 5 1.963 −3.346 −4.473 −4.27 −4.206 177Tl 0 1.172 −1.178 −0.836 −0.982 −0.863 Table 3. Calculation of the spherical proton radioactivity half-lives. Measured,
lgTMSL01/2 ,lgTMQSP1/2 , andlgTUDLP1/2 are the logarithmic forms of the experimental proton radioactivity half-life and the calculations using STPA-SHF-MSL0, TPA-SHF-MQSP, and the UDLP.To intuitively compare TPA-SHF-MSL0, TPA-SHF-MQSP, and UDLP with experimental data, the logarithmic deviations between the experimental data of the proton radioactivity half-lives and the calculated values are shown in Fig. 3. In this figure, the X-axis represents the mass number of the proton radioactivity parent nucleus, and the Y-axis represents the logarithmic deviations. The three different colors and symbols refer to the calculations obtained using three different models. From Fig. 3, we can clearly see that the values of
lg(Tcal1/2−Texp1/2) obtained using TPA-SHF-MQSP are mainly near zero, indicating that the theoretical half-life of proton radioactivity calculated using our model is in good agreement with the experimental data. Furthermore, this figure shows that the calculations using TPA-SHF-MQSP can better reproduce the experimental data than TPA-SHF-MSL0 and UDLP for most nuclei. To intuitively compare the deviations between the half-lives of proton radioactivity obtained using different models and experimental data, we calculate the standard deviation.ΔMSL0= 0.405,ΔMQSP= 0.362, andΔUDLP= 0.459 represent the standard deviations betweenlgTMSL01/2 ,lgTMQSP1/2 ,lgTUDLP1/2 , and the values oflgTexp1/2 , respectively. These results show that the TPA-SHF-MQSP method is better than the other models in calculating the spherical proton radioactivity half-lives. Therefore, it is credible to use TPA-SHF-MQSP to study the proton radioactivity half-lives.
Quantity | MSL0 | MSL0 | |
0.16 | −2118.06 MeV | ||
−16 MeV | 395.196 MeV | ||
230 MeV | −63.953 MeV | ||
0.8 | 12857.7 MeV | ||
0.7 | −0.0709496 | ||
30 MeV | −0.332282 | ||
L | 60 MeV | 1.35830 | |
132 MeV | −0.0228181 | ||
α | 0.235879 | ||
0.42 | 133.3 MeV |