1 Introduction

With the efforts of the whole community, α-decay half-life T_1/2 of unstable nuclei is well known to be related to several quantities, such as the α-particle preformation factor [1, 2], the pre-exponential factor [3], and the penetration probability [4, 5]. The determining factor is the penetration probability which is extremely sensitive to α-decay energy, Q, as indicated by the Geiger-Nuttall empirical rule [6]. Besides the decay energy, Q, some other factors, such as the radius of daughter nucleus, R_d, are also important to calculate the penetration probability, but have been paid little attention to in α-decay half-life calculations. The sensitive correlation between R_d and the penetration probability, as well as T_1/2, has been pointed out in the nuclear textbook [6] that “the calculated half-lives are extremely sensitive to small changes in the assumed mean radius” of daughter nucleus.

The α-decay half-life T_1/2 is sensitive to R_d, which is closely related to the nuclear skin thickness, S, in the daughter nucleus, however, in many α-decay half-life calculations, S=0 fm is often assumed for simplification [4-11]. This is possibly due to the fact that little experimental information is known about the neutron density distribution, ρ_n(r), of most nuclei, as well as S. There are only a few experiments [12-21] on measuring the neutron density distributions and most of them attempt to extract ρ_n(r) and S_n in the double-magic nucleus ²⁰⁸Pb, whose neutron skin thickness has been recently pointed out to show a correlation with the density slope of symmetry energy in nuclear matter [22-25]. Oppositely, the proton density distribution, ρ_p(r), in many nuclei have been accurately investigated by electron scattering experiments and the experimental data is analyzed by different parameterized approaches, such as the Fourier-Bessel (FB) expansion and the two-parameter Fermi (2PF) model [26-34]. For instance, the ρ_p(r) of ²⁰⁸Pb is already known from electron scattering experiments with the widely-used 2PF model: ${\rho }_{\text{p/n}}(r)={\rho }_{\text{p/n}}^0/\{1+\text{exp}[(r-c_{\text{p/n}})/a_{\text{p/n}}]\}$, where the proton half-density radius, c_p, and diffuseness parameter, a_p, are extracted to be 6.680 fm and 0.447 fm [35]. From the very recent coherent pion photoproduction experiment [36], the ρ_n(r) of ²⁰⁸Pb is extracted with c_n and a_n to be 6.70±0.03(stat.) fm and $0.55\pm 0.01(\text{stat}{\mathrm{.})}_{-0.03}^{+0.02}(\text{sys}\mathrm{.})$ fm respectively, and $S_n=0.15\pm 0.03(\text{stat}{\mathrm{.})}_{-0.03}^{+0.01}(\text{sys}\mathrm{.})$ fm. In this paper, based on these newest experimental data of ²⁰⁸Pb, our main purpose is to explore the sensitivity of α-decay half-life, T_1/2, to nuclear skin thickness, S, for α emitters decaying to ²⁰⁸Pb and other daughter nuclei nearby, namely isotopes of Z_d=82 and isotones of N_d=126.

2 The correlation between α-decay half-life, T_1/2, and neutron skin thickness S

The close correlation between α-decay half-life, T_1/2, and neutron skin thickness, S, in the daughter nucleus can be analytically derived with a simplified α-decay model [6, 7], which assumes that only the point-charge Coulomb potential V_C(R)=Z_αZ_de²/R operates beyond mother nucleus surface, where Z_α and Z_d are the proton numbers of the α particle and daughter nucleus, respectively. Then α-decay half-life T_1/2 can be written as [4-7]

where P_α is the α-particle preformation factor and F is a normalization factor well-defined by the two-potential approach [3]. μ is the reduced mass of α-particle and daughter nucleus. R₂ and R₃ are the second and third classical turning points. Since we investigate favored α decays in this paper, the centrifugal potential is not involved. By inserting R₃=Z_αZ_de²/Q and R₂=R_d+Rα into Eq. (1), where Rα is the radius of α-particle, one can obtain [4-9]

where $x=\sqrt{(R_{\text{d}}+R_{\alpha })/R_3}\ll 1$. With arccos(x) and $x\sqrt{1-x^2}$ approximated by (π/2-x) and x, respectively, the following formula can be derived

where the parameters are $a=\frac{\sqrt{2}\pi e^2}{\hslash \ln 10}$, $b=-\frac{4e\sqrt{2}}{\hslash \ln 10}$ and $c={{\displaystyle \log }}_{10}\frac{4\mu \ln 2}{\hslash P_{\alpha }F}$. Since Pα and F are not always constants for different α decays and the point-charge Coulomb potential is assumed, the parameters a, b, and c can be determined by fitting experimental data. Besides, it is clearly shown that the α-decay half-life, T_1/2, is related to the radius of daughter nucleus, R_d, and ${{\displaystyle \log }}_{10}{T}_{1/2}\sim \sqrt{R_{\text{d}}+R_{\alpha }}$. The radius R_d can be obtained from the root-mean square radius, R_s, by applying the simple relation: $R_{\text{s}}=\sqrt{\frac{3}{5}}{R}_{\text{d}}$ [37]. The radius R_s is calculated from the proton and neutron density distributions ρ_p(r) and ρ_n(r) of daughter nucleus

where A_d is the mass number of daughter nucleus. The difference between ρ_p(r) and ρ_n(r) can be characterized by the so-called proton and neutron skin thicknesses

where R_p and R_n are the proton and neutron root-mean-square (RMS) radii, respectively. N_d=A_d-Z_d is the neutron number of the daughter nucleus. The correlation between T_1/2 and S can be directly shown by the α-decay half-life ratio derived from Eq. (3)

It is obvious that this ratio is irrelevant to the α-decay energy, Q, but exponentially dependent on the variation of $\sqrt{R_{\text{d}}^S+R_{\alpha }}$, while the uncertainties coming from Pα and other factors are all canceled out.

3 Numerical results and discussion

Based on Eq. (3), the sensitivity of α-decay half-life, T_1/2, to nuclear skin thickness, S, for α emitters decaying to ²⁰⁸Pb, isotopes of Z_d=82 and isotones of N_d=126 are explored. Here the ρ_p(r) for isotopes of Z_d=82 and ρ_n(r) for isotones of N_d=126 are calibrated by the measured proton and neutron density distributions of ²⁰⁸Pb [36]. The detailed parameters are given as follows

where the values 1.1308 and 1.1274 of the half-density radii are obtained by fitting the experimental ones in ²⁰⁸Pb [36].

By using Eqs.(5) and (7), the corresponding skin thicknesses in isotopes of Z_d=82 and isotones of N_d=126 are calculated, as shown in Fig. 1. It is clearly seen from Fig. 1(a) that the skin thickness in ¹⁹⁰Pb is approximately 0 fm. For isotopes ^191-214Pb, the neutron skin thickness, S_n, increases with the increase of neutron number, N, while for isotopes ^182-189Pb, the proton skin thickness, S_p, decreases with N. So the skin thickness varies smoothly with the neutron number either for the proton skin case or the neutron skin case. Similarly, as shown in Fig. 1(b) for isotones of N_d=126, S_n decreases with the proton number, Z, as the neutron number is fixed for nuclei from ²⁰⁹Bi to ²¹⁵Ac.

Figure 1:Full-screen View

The skin thickness, S, with the variation of neutron number, N, for isotopes of Z_d=82 (a) and of proton number, Z, for isotones of N_d=126 (b).

The radius, R_d, of each daughter nucleus is computed based on Eqs. (4) and (7). Then the corresponding α-decay half-lives, $T_{1/2}^S$, are calculated by using Eq. (3), where the parameters a=0.0136, b=-0.0165, and c=-14.0903 are fitted by the experimental data of α-decay energy, Q, and half-life $T_{1/2}^{\text{exp}}$ obtained from Refs. [38, 39]. The α-decay half-lives $T_{1/2}^{S=0}$ corresponding to the case of S=0 fm are also calculated with the same values of a, b, and c. The ratio $T_{1/2}^S/T_{1/2}^{S=0}$ with the variation of S is shown in Fig. 2. It is found in Fig. 2(a) that the ratio for ¹⁹⁰Pb is approximately 1.0 as its skin thickness is S≈0 fm. For α decays to ^191-214Pb, the ratio decreases with the increasing S_n, and they are all less than 1.0 because of the negative sign of b [see Eq. (6)]. For instance, the ratio is as small as 0.466 for ²¹⁴Pb. In contrast, for α decays to ^182-189Pb the ratio increases with S_p and the maximum one is 1.266 for ¹⁸²Pb. For α decays to isotones of N_d=126 in Fig. 2(b), the ratio increases with S_n and ranges from 1.279 to 1.415. Besides, it is noted that the nuclei ²⁰⁵Pb and ²⁰⁷Pb are not included in Fig. 2(a) since the involved α transitions are unfavored.

Figure 2:Full-screen View

The correlation between α-decay half-life ratio $T_{1/2}^S/T_{1/2}^{S=0}$ and nuclear skin thickness, S, for α decays to isotopes of Z_d=82 (a) and to isotones of N_d=126 (b).

The detailed values of the skin thicknesses, S, and the ratios $T_{1/2}^S/T_{1/2}^{S=0}$ are given in Table 1. The first and second columns of Table 1 are the α-decay mother nucleus and its isospin asymmetry δ=(N-Z)/A, respectively. The third column is the calculated radius, R_d, of the daughter nucleus and the corresponding skin thickness S_p/S_n is given in column four. It is clearly shown that the value of S_p decreases with the increasing isospin asymmetry, δ, while S_n increases with δ. It is found from the ratio $T_{1/2}^S/T_{1/2}^{S=0}$ of the fifth column that there exists a strong correlation between the α-decay half-life and the skin thickness S_p/S_n. In the last column, we give the comparison of calculated α-decay half-lives $T_{1/2}^S$ using Eq. (3) with the experimental ones $T_{1/2}^{\text{exp}}$ from Refs. [38, 39]. We found that the calculated α-decay half-lives $T_{1/2}^S$ agree with $T_{1/2}^{\text{exp}}$ well for both the Z=82 isotopes and the N=126 isotones. For the majority of cases, the experimental data is reproduced within a factor of two although a constant preformation probability of α cluster is used in Eq. (3). But for the ratio $T_{1/2}^S/T_{1/2}^{S=0}$, the shell effect and odd-even effect on the preformation probability are all canceled out and only the skin thickness effect is left. The skin thickness characterizes the difference between proton and neutron density distributions, especially the difference in the nuclear surface region. For the Z=82 isotopes, the difference in neutron density distributions shall only affect the nuclear part of the α-core potential. However, for the N=126 isotones, the change of proton density distribution will affect not only the nuclear potential but also the Coulomb potential. It is well known that the tunneling of the α particle in α decays takes place in the surface region, thus, the skin thickness effect should be included in α-decay half-life calculations.

4 Summary

In summary, with the analytical derived formula and newest experimental data of ²⁰⁸Pb, the sensitivity of α-decay half-life to nuclear skin thickness is explored for daughter nucleus around ²⁰⁸Pb, namely isotopes of Z_d=82 and isotones of N_d=126. From the numerical results, we find there is a close correlation between α-decay half-life and nuclear skin thickness, and the former one can be changed by a factor from 0.466 to 1.415. So it is necessary to consider the skin thickness for investigations on α-decay half-lives, and it could be a possible way to extract nuclear skin thickness from measured α-decay half-lives.