New behaviors of α-particle preformation factors near doubly magic ¹⁰⁰Sn

$ \alpha $ decay is a fundamental nuclear decay mode. Research on $ \alpha $ decay has long been focused in the vicinities of doubly magic nuclei $ ^{208} $Pb ($ Z = 82,\; N = 126 $) and $ ^{298} $Fl ($ Z = 114, \;N = 184 $) because $ \alpha $ decay can be a probe to study unstable nucleus structures, and can be the only way to identify new synthesized superheavy nuclei [1–27]. Over the past two decades, $ \alpha $ emitters around the self-conjugate doubly magic nucleus $ ^{100} $Sn ($ Z = N = 50 $) at the opposite end of the mass table have also received a lot of attention and become a hot topic in nuclear physics [18, 28–35]. In particular, there is the fastest $ \alpha $ emitter $ ^{104} $Te near doubly magic nucleus $ ^{100} $Sn [34]. Since the $ \alpha $ emitters near self-conjugate doubly magic nucleus $ ^{100} $Sn are close to the $ N = Z $ line, the nuclear force is extremely sensitive to isospin. Therefore, it is a great chance to study the unique neutron-deficient nuclear structure and examine various $ \alpha $ decay theoretical models. Moreover, cluster radioactivity was also predicted as one of the decay modes of nuclei in the $ ^{100} $Sn region [36–39]. Further interest in the decay rates of nuclei around doubly magic nucleus $ ^{100} $Sn comes from research into astrophysical processes, for which this region has been considered as the end of the rapid proton capture process due to the Sn–Sb–Te cycle [33, 40, 41].

In addition, in the neutron-deficient Te, Xe, and Ba isotopes near $ ^{100} $Sn, one would expect that interactions between protons and neutrons occupying similar single-particle orbitals could enhance the $ \alpha $-particle preformation factors, and together with the significantly reduced $ \alpha $-widths compared to the analogous nuclei just above doubly magic nucleus $ ^{208} $Pb, can result in the so-called "superallowed" $ \alpha $ decay [42]. This effect would be expected to be strongest for $ N = Z $ self-conjugate nuclei [42]. Recently, $ \alpha $ radioactivity to a heavy self-conjugate nucleus was observed for the first time on the $ ^{108} $Xe$ \to $$ ^{104} $Te$ \to $$ ^{100} $Sn $ \alpha $ decay chain [34], including the measurements of the $ \alpha $-particle kinetic energy and $ \alpha $ decay half-lives of the $ \alpha $ emitters $ ^{108} $Xe [$ E_{\alpha} = 4.4(2) $ MeV, $ T_{1/2} = 58^{+106}_{-23} $ μs] and $ ^{104} $Te [$ E_{\alpha} = 4.9(2) $ MeV, $ T_{1/2}<18 $ ns]. The authors of this article suggested that the $ \alpha $-reduced width for $ ^{108} $Xe or $ ^{104} $Te is more than a factor of 5 larger than that for $ ^{212} $Po [34].

It is well known that $ ^{104} $Te, near the proton drip line, and $ ^{212} $Po, near the $ \mathcal{\beta}- $stability line, are the only two existing $ \alpha $ emitters decaying to doubly magic nuclei. In recent years, researchers adopted different models such as the density dependent cluster model (DDCM) [28, 43–45], the generalized liquid drop model (GLDM) [46, 47], the preformed cluster model (PCM) [48], etc. to study the $ \alpha $ decay of nuclei around doubly magic $ ^{208} $Pb and $ ^{100} $Sn, and provide some important insights on the $ \alpha $-particle preformation factor. In this letter, we focus on the $ \alpha $-particle preformation factors of nuclei near self-conjugate doubly magic nucleus $ ^{100} $Sn, and compare them to those of analogous nuclei just above the doubly magic nucleus $ ^{208} $Pb, based on the available experimental data of $ \alpha $ decay [34, 49–59] within the generalized liquid drop model (GLDM) [60–66]. These $ \alpha $ emitters have different isospins and mass numbers as well as different proton and neutron closed shells. We want to reveal some new behaviors of $ \alpha $-particle preformation factors for extremely neutron-deficient nuclei near self-conjugate doubly magic nucleus $ ^{100} $Sn, in order to understand the roles of proton–neutron correlation and the single-particle orbitals occupied by protons and neutrons in the preformation of $ \alpha $-clusters, as well as the physical mechanism of superallowed $ \alpha $ decay.

The GLDM can deal well with proton radioactivity [67], cluster radioactivity [68], fusion [69], fission [70], and the $ \alpha $ decay process [22, 60–66, 71] by introducing the quasimolecular shape mechanism [60], which can describe the complex deformation process from the parent nucleus through continuous transition to the appearance of a deep and narrow neck, finally resulting in two tangential fragments, and adding the proximity energy, including an accurate radius and mass asymmetry. In previous works [60–66], the GLDM has been introduced in detail. The $ \alpha $ decay half-life can be obtained by

with the $ \alpha $ decay constant $ \mathcal{\lambda} $ being expressed as

where the assault frequency $ \mathcal{\nu} $ is obtained using the classical method with the kinetic energy of the $ \alpha $-particle. The barrier-penetrating probability $ P $ is determined by tunneling the GLDM potential barriers [60–66] with the Wentzel–Kramers–Brillouin (WKB) approximation.

The experimental $ \alpha $-particle preformation factor $ P_{\alpha}^{\rm{Exp}} $ can be extracted from the ratios of the theoretical decay half-life $ T_{1/2}^{\rm{Cal1}} $, calculated by assuming the $ \alpha $-particle preformation factor is a constant $ P_{\alpha} = 1 $, to experimental decay half-life [65, 72–75]. This is expressed as

To examine the experimental $ \alpha $ decay half-life data, the analytic formula for estimating the $ \alpha $-particle preformation factor is also adopted, which was put forward in our previous works [66, 71]. It is expressed as

where

and

$ A $, $ Z $, and $ Q_{\alpha} $ represent mass number, proton number, and $ \alpha $ decay energy of the parent nucleus. $ A_1 $, $ Z_1 $, $ A_2 $, and $ Z_2 $ denote the mass and proton numbers of the $ \alpha $-particle and daughter nucleus. $ l $ is the angular momentum carried by the $ \alpha $-particle. The parameters values are listed in Ref. [66].

The calculated $ \alpha $ decay half-lives for nuclei above doubly magic nuclei $ ^{100} $Sn and $ ^{208} $Pb are presented in Tables 1 and 2, respectively. In these two tables, the first four columns represent the $ \alpha $ transition, the experimental kinetic energy of the $ \alpha $-particle, the experimental $ \alpha $ decay energy, and the minimum angular momentum carried by the $ \alpha $-particle. The fifth column is the experimental $ \alpha $ decay half-life. The sixth column denotes the calculated $ \alpha $ decay half-life $ T_{1/2}^{\rm{Cal1}} $ within the GLDM with $ P_{\alpha} = 1 $. The seventh column gives the calculated $ \alpha $ decay half-life $ T_{1/2}^{\rm{Cal2}} $ within the GLDM with the estimated $ \alpha $-particle preformation factor from Eq. (4). The eighth column shows the extracted experimental $ \alpha $-particle preformation factor by using Eq. (3) with $ T^{\rm{Cal1}}_{1/2} $ and $ T^{\rm{Exp}}_{1/2} $. The last two columns express the calculated proton–neutron correlation energy $ E_{p-n} $ and two protons–two neutrons correlation energy $ E_{2p-2n} $ determined by Eqs. (6) and (7). From these two tables, it can be seen immediately that the calculated $ \alpha $ decay half-lives $ T_{1/2}^{\rm{Cal2}} $ can accurately reproduce the experimental data including the newly observed self-conjugate nuclei $ ^{104} $Te and $ ^{108} $Xe [34]. Note that the calculations provide support for recent experimental observation data in Ref. [34]. In Table 1, for $ ^{110} $Te, the calculated $ \alpha $-particle preformation factor using Eq. (4) is an order of magnitude larger than the extracted value from the experimental $ \alpha $ decay half-life data. This is because the $ \alpha $ decay branch ratio of $ ^{110} $Te is very small, only 0.00067% [28], and the microscopic calculation of the $ \alpha $-particle preformation factor is very complicated, while Eq. (4) mainly considers the $ \alpha $ decay energy, proton number, and mass number when calculating the $ \alpha $-particle preformation factor. This leads to a deviation between the theoretically calculated $ \alpha $-particle preformation factors and the experimental data for nuclei with extremely low $ \alpha $ decay branch ratios such as $ ^{110} $Te.

$ \alpha $ transition	$ E_{\alpha} $ /MeV	$ Q_{\alpha} $ /MeV	$ l_{\rm min} $	$ T^{\rm{Exp}}_{1/2} $ /s	$ {T_{1/2}^{\rm{Cal1}}} $ /s	$ {T_{1/2}^{\rm{Cal2}}} $ /s	$ {P_{\alpha}^{\rm{Exp}}} $	$ E_{p-n} $ /MeV	$ E_{2p-2n} $ /MeV
$ ^{104} $Te$ \to^{100} $Sn	4.9 [34]	5.1	0	$<1.80\times10^{-8} $ [34]	$ 1.47\times10^{-8} $	$ 7.29\times10^{-8} $	$>0.81 $	1.26$ {^a} $	3.64$ {^a} $
$ ^{106} $Te$ \to^{102} $Sn	4.128 [51]	4.29	0	$ 7.00\times10^{-5} $ [50–52]	$ 2.52\times10^{-5} $	$ 8.38\times10^{-5} $	0.36	0.56$ {^b} $	1.84$ {^a} $
$ ^{108} $Te$ \to^{104} $Sn	3.314 [54]	3.44	0	$ 4.30\times10^{0} $ [51, 53, 54]	$ 9.30\times10^{-1} $	$ 1.68\times10^{0} $	0.22	1.06$ {^b} $	1.83$ {^b} $
$ ^{110} $Te$ \to^{106} $Sn	2.624 [55]	2.72	0	$ 2.78\times10^{6} $ [55, 56]	$ 3.32\times10^{5} $	$ 2.87\times10^{5} $	0.12	0.71$ {^b} $	1.73$ {^b} $
$ ^{108} $Xe$ \to^{104} $Te	4.32 [34, 49]	4.49	0	$ 5.80\times10^{-5} $ [34]	$ 3.70\times10^{-5} $	$ 1.10\times10^{-4} $	0.64	1.10$ {^a} $	3.35$ {^a} $
$ ^{110} $Xe$ \to^{106} $Te	3.72 [50]	3.86	0	$ 1.48\times10^{-1} $ [50]	$ 5.00\times10^{-2} $	$ 1.07\times10^{-1} $	0.34	0.73$ {^b} $	2.02$ {^a} $
$ ^{112} $Xe$ \to^{108} $Te	3.216 [51]	3.34	0	$ 3.38\times10^{2} $ [51, 53]	$ 1.01\times10^{2} $	$ 1.53\times10^{2} $	0.30	1.13$ {^b} $	1.66$ {^b} $
$ ^{114} $Ba$ \to^{110} $Xe	3.48 [50]	3.61	0	$ 4.20\times10^{1} $ [50]	$ 2.60\times10^{1} $	$ 4.19\times10^{1} $	0.62	0.66$ {^b} $	1.96$ {^a} $
a Calculated by using nuclear mass data in the WS4+ mass model [76]. b Calculated using nuclear mass data in the evaluated atomic mass table AME2016 [58, 59].

Table 1.  Calculations of $ \alpha $-particle preformation factor, $ \alpha $ decay half-lives, and the proton–neutron correlation energy $ E_{p-n} $ and two protons–two neutrons correlation energy $ E_{2p-2n} $ of even–even Te, Xe, and Ba isotopes near $ ^{100} $Sn.

$ \alpha $ transition	$ E_{\alpha} $ /MeV	$ Q_{\alpha} $ /MeV	$ l_{\rm min} $	$ T^{\rm{Exp}}_{1/2} $ /s	$ {T_{1/2}^{\rm{Cal1}}} $ /s	$ {T_{1/2}^{\rm{Cal2}}} $ /s	$ {P_{\alpha}^{\rm{Exp}}} $	$ E_{p-n} $ /MeV	$ E_{2p-2n} $ /MeV
$ ^{212} $Po$ \to^{208} $Pb	8.79	8.95	0	$ 2.95\times10^{-7} $	$ 1.00\times10^{-8} $	$ 1.48\times10^{-7} $	0.03	0.87	1.44
$ ^{214} $Po$ \to^{210} $Pb	7.69	7.83	0	$ 1.64\times10^{-4} $	$ 1.14\times10^{-5} $	$ 1.23\times10^{-4} $	0.07	0.70	1.28
$ ^{216} $Po$ \to^{212} $Pb	6.78	6.91	0	$ 1.45\times10^{-1} $	$ 1.39\times10^{-2} $	$ 1.09\times10^{-1} $	0.10	0.51	1.11
$ ^{218} $Po$ \to^{214} $Pb	6.00	6.12	0	$ 1.86\times10^{2} $	$ 1.93\times10^{1} $	$ 1.09\times10^{2} $	0.10	0.38	1.09
$ ^{212} $Rn$ \to^{208} $Po	6.26	6.38	0	$ 1.43\times10^{3} $	$ 1.40\times10^{1} $	$ 6.86\times10^{2} $	0.01	0.23	0.57
$ ^{214} $Rn$ \to^{210} $Po	9.04	9.21	0	$ 2.70\times10^{-7} $	$ 1.08\times10^{-8} $	$ 1.85\times10^{-7} $	0.04	0.67	1.24
$ ^{216} $Rn$ \to^{212} $Po	8.05	8.20	0	$ 4.50\times10^{-5} $	$ 4.99\times10^{-6} $	$ 6.56\times10^{-5} $	0.11	0.70	1.33
$ ^{218} $Rn$ \to^{214} $Po	7.13	7.26	0	$ 3.38\times10^{-2} $	$ 4.23\times10^{-3} $	$ 4.15\times10^{-2} $	0.13	0.58	1.29
$ ^{220} $Rn$ \to^{216} $Po	6.29	6.41	0	$ 5.56\times10^{1} $	$ 8.33\times10^{0} $	$ 5.92\times10^{1} $	0.15	0.52	1.18
$ ^{222} $Rn$ \to^{218} $Po	5.49	5.59	0	$ 3.30\times10^{5} $	$ 5.71\times10^{4} $	$ 2.79\times10^{5} $	0.17	0.51	1.15
$ ^{214} $Ra$ \to^{210} $Rn	7.14	7.27	0	$ 2.44\times10^{0} $	$ 3.25\times10^{-2} $	$ 1.53\times10^{0} $	0.01	0.22	0.65
$ ^{216} $Ra$ \to^{212} $Rn	9.35	9.53	0	$ 1.82\times10^{-7} $	$ 8.41\times10^{-9} $	$ 1.70\times10^{-7} $	0.05	0.52	1.14
$ ^{218} $Ra$ \to^{214} $Rn	8.39	8.55	0	$ 2.52\times10^{-5} $	$ 2.44\times10^{-6} $	$ 3.88\times10^{-5} $	0.10	0.58	1.21
$ ^{220} $Ra$ \to^{216} $Rn	7.45	7.59	0	$ 1.79\times10^{-2} $	$ 1.93\times10^{-3} $	$ 2.34\times10^{-2} $	0.11	0.68	1.35
$ ^{222} $Ra$ \to^{218} $Rn	6.56	6.68	0	$ 3.36\times10^{1} $	$ 4.46\times10^{0} $	$ 3.92\times10^{1} $	0.13	0.44	1.34
$ ^{224} $Ra$ \to^{220} $Rn	5.69	5.79	0	$ 3.14\times10^{5} $	$ 4.97\times10^{4} $	$ 2.99\times10^{5} $	0.16	0.41	1.25
$ ^{226} $Ra$ \to^{222} $Rn	4.78	4.87	0	$ 5.05\times10^{10} $	$ 1.10\times10^{10} $	$ 4.00\times10^{10} $	0.22	0.40	1.23

Table 2.  Same as Table 1, but for even–even Po, Rn, and Ra isotopes near $ ^{208} $Pb. The experimental $ \alpha $ decay half-lives are taken from the evaluated nuclear properties table NUBASE2016 [57]. The experimental $ \alpha $ decay energy are taken from the evaluated atomic mass table AME2016 [58, 59]. The $ E_{p-n} $ and $ E_{2p-2n} $ energies are calculated using the nuclear mass data in the AME2016 [58, 59].

To measure the agreement between the calculated $ \alpha $ decay half-lives $ T^{\rm{Cal2}}_{1/2} $ and experimental data $ T^{\rm{Exp}}_{1/2} $, the standard deviations are calculated by

For nuclei in Tables 1 and 2, the results of standard deviations $ \sigma_1 = 0.47 $ and $ \sigma_2 = 0.16 $ are satisfactory, showing that $ T^{\rm{Cal2}}_{1/2} $ can accurately reproduce $ T^{\rm{Exp}}_{1/2} $ within factors of $ 10^{0.47} = 2.95 $ and $ 10^{0.16} = 1.45 $, respectively. This demonstrates that the GLDM can be applied to extract the experimental $ \alpha $-particle preformation factors for studying the structure information of nuclei in these two regions.

Furthermore, Tables 1 and 2 show that the extracted experimental $ \alpha $-particle preformation factors $ P_{\alpha}^{\rm{Exp}} $ of nuclei near $ ^{100} $Sn are larger than $ P_{\alpha}^{\rm{Exp}} $ of nuclei near $ ^{208} $Pb, and in particular, larger than $ P_{\alpha}^{\rm{Exp}} $ of analogous nuclei just above $ ^{208} $Pb. The analogous nuclei refer to the two nuclei with the same valence proton and valence neutron located above doubly magic cores $ ^{100} $Sn and $ ^{208} $Pb, respectively. The valence protons $ N_p $ and valence neutrons $ N_n $ are defined as $ N_p = Z-Z_0 $ and $ N_n = N-N_0 $ with $ Z_0 = 50 $ and 82, $ N_0 = 50 $ and 126, being the magic numbers of protons and neutrons in the corresponding nuclear region. For example, $ ^{104} $Te is analogous to $ ^{212} $Po because they both have two valence protons and two valence neutrons outside of the doubly magic nuclei $ ^{100} $Sn and $ ^{208} $Pb, respectively.

The extracted experimental $ \alpha $-particle preformation factors $ P_{\alpha}^{\rm{Exp}} $ for nuclei above $ ^{100} $Sn and for analogous nuclei just above $ ^{208} $Pb are shown as functions of valence protons and valence neutrons in Fig. 1 (a), (b), and (c), respectively. In this figure, one can see that $ P_{\alpha}^{\rm{Exp}} $ for nuclei above $ ^{100} $Sn are significantly larger than those of analogous nuclei just above $ ^{208} $Pb. Furthermore, Fig. 1 (a) shows the variations of $ P_{\alpha}^{\rm{Exp}} $ for Te ($ Z = 52 $) and Po ($ Z = 84 $) isotopes, whose valence protons are $ N_p = $$ Z-Z_0 = 2 $, against valence neutrons $ N_n $. It is clear that for Te isotopes $ P_{\alpha}^{\rm{Exp}} $ exhibits an increasing trend when the nucleus moves towards the $ N = Z $ line, but Po isotopes do not show similar patterns due to the large asymmetry between neutrons and protons. Figure 1 (b) displays the variations of $ P_{\alpha}^{\rm{Exp}} $ for Xe ($ Z = 54 $) and Rn ($ Z = 86 $) isotopes, whose valence protons are $ N_p = Z-Z_0 = 4 $, against valence neutrons $ N_n $. We find that for Xe isotopes $ P_{\alpha}^{\rm{Exp}} $ also increases as the nucleus moves towards the $ N = Z $ line. However, Rn isotopes still do not show a similar trend of change. Figure 1 (c) plots $ P_{\alpha}^{\rm{Exp}} $ as a function of valence protons $ N_p $ for $ N = 58 $ and $ N = 134 $ isotones, whose valence neutrons are $ N_n = N-N_0 = 8 $. $ P_{\alpha}^{\rm{Exp}} $ for $ N = 58 $ isotopes also show an increasing tendency as the nuclei move towards the $ N = Z $ line, but this phenomenon does not occur in the analogous $ N = 134 $ isotopes just above $ ^{208} $Pb. This indicated that $ P_{\alpha}^{\rm{Exp}} $ is enhanced when a nucleus moves towards the $ N = Z $ line, resulting in the superallowed $ \alpha $ decay near doubly magic nucleus $ ^{100} $Sn. In recent work, Clark et al. adopted a very different model and studied the $ \alpha $-particle preformation factors of nuclei in these two regions [31]. A similar conclusion was obtained, though the $ \alpha $-particle preformation factors of nuclei near doubly magic nuclei $ ^{100} $Sn and $ ^{208} $Pb are in orders of $ 10^{-2} $ and $ 10^{-3} $, respectively.

Figure 1.  (color online) Variations of extracted experimental $\alpha$-particle preformation factors $P_{\alpha}^{\rm{Exp}}$ from Eq. (3) against the valence neutrons for $Z = 52$ and $Z = 84$ isotopes (left), and against the valence neutrons for $Z = 54$ and $Z = 86$ isotopes (middle), and against the valence protons for $N = 58$ and $N = 134$ isotones (right), respectively.

To investigate the effects of the proton–neutron interaction and the two protons–two neutrons interaction on $ \alpha $-particle preformation, we calculate the proton–neutron correlation energy $ E_{p-n} $ and two protons–two neutrons correlation energy $ E_{2p-2n} $ using

Equations (6) and (7) were proposed in Ref. [77] and used to determine the experimental pairing energy of the nucleons [78]. $ B(A, Z) $ is the binding energy of a nucleus with the mass number $ A $ and proton number $ Z $. The results of $ E_{p-n} $ energy and $ E_{2p-2n} $ energy are listed in the last two columns of Tables 1 and 2. In these two tables, it can be found that the $ E_{p-n} $ and $ E_{2p-2n} $ energies of nuclei above doubly magic nucleus $ ^{100} $Sn are larger than those of analogous nuclei just above $ ^{208} $Pb. This, in turn, leads to significant enhancement of $ P_{\alpha}^{\rm{Exp}} $ for nuclei near $ ^{100} $Sn. The results for the $ E_{p-n} $ and $ E_{2p-2n} $ energies are plotted in Fig. 2. In this figure, the $ E_{p-n} $ and $ E_{2p-2n} $ energies of nuclei above $ ^{100} $Sn are strengthened when compared to analogous nuclei just above $ ^{208} $Pb. For $ Z = 52 $ isotopes, the $ E_{p-n} $ and $ E_{2p-2n} $ energies increase rapidly in $ N_n = 2 $. Similarly, for $ Z = 54 $ isotopes, the $ E_{p-n} $ and $ E_{2p-2n} $ energies rise fast in $ N_n = 4 $. However, the $ E_{p-n} $ and $ E_{2p-2n} $ energies of analogous nuclei just above $ ^{208} $Pb change slowly. Therefore, it is demonstrated that the $ \alpha $-particle is more likely to form in self-conjugate nuclei, resulting in the superallowed $ \alpha $ decay. In addition, the $ E_{2p-2n} $ energy appears an increased tendency, the same as $ P_{\alpha}^{\rm{Exp}} $, when the nucleus moves towards the $ N = Z $ line, implying that the two protons–two neutrons interaction plays a more significant role than one proton–one neutron interaction in $ \alpha $-particle preformation.

Figure 2.  (color online) Proton–neutron $E_{p-n}$ correlation energy and two protons–two neutrons correlation energy $E_{2p-2n}$ for nuclei above $^{100}$Sn (denoted as solid symbols) and analogous nuclei above $^{208}$Pb (denoted as open symbols).

The extracted experimental $ \alpha $-particle preformation factors $ P_{\alpha}^{\rm{Exp}} $ for nuclei above $ ^{100} $Sn and $ ^{208} $Pb are shown as functions of $ ({N_pN_n})/({Z_0+N_0}) $ in Fig. 3 (a) and (b), respectively. In Fig. 3 (b), one can see that the closer $ ({N_pN_n})/({Z_0+N_0}) $ is to zero, representing the proton and/or neutron numbers approaching closed shells, the smaller is $ P_{\alpha}^{\rm{Exp}} $. When $ ({N_pN_n})/({Z_0+N_0}) $ is far from zero, $ P_{\alpha}^{\rm{Exp}} $ will increase. This indicates that the closer the proton and/or neutron number is to the magic number, the more difficult it is for an $ \alpha $-particle to form inside its parent nucleus. We find that $ P_{\alpha}^{\rm{Exp}} $ is linearly dependent on $ ({N_pN_n})/({Z_0+N_0}) $ for nuclei above $ ^{208} $Pb. This is consistent with the conclusions deduced by adopting different models, in which the $ \alpha $-particle preformation factors are extracted from the ratios between theoretical $ \alpha $ decay half-lives calculated by adopting the different models to experimental data [15, 79, 80], or calculated using the differences of binding energy between the $ \alpha $ decaying parent nucleus and its neighboring nuclei within the cluster-formation model [81]. It is shown that the nuclear shell effects and the nucleon configurations play key roles in $ \alpha $-cluster preformation for $ \alpha $-particle emitters around doubly magic $ ^{208} $Pb. However, in Fig. 3 (a) this phenomenon is broken in the $ ^{100} $Sn region. The values of $ P_{\alpha}^{\rm{Exp}} $ for nuclei above $ ^{100} $Sn are linearly independent of $ ({N_pN_n})/({Z_0+N_0} )$ and show a new behavior. When the nucleus is close to shell closure, $ P_{\alpha}^{\rm{Exp}} $ for nuclei near $ ^{100} $Sn does not decrease like that for nuclei near $ ^{208} $Pb, but rather increases. In addition, we can find that the maximum values of $ P_{\alpha}^{\rm{Exp}} $ in Fig. 3 (a) correspond to $ ^{104} $Te, $ ^{108} $Xe, and $ ^{114} $Ba. In particular, $ P_{\alpha}^{\rm{Exp}} $ is significantly enhanced along the N = Z line, which results in $ P_{\alpha}^{\rm{Exp}} $ for nuclei above $ ^{100} $Sn not being linearly dependent on $({N_pN_n})/({Z_0+N_0}) $.

Figure 3.  (color online) Variations of extracted experimental $\alpha$-particle preformation factors $P_{\alpha}^{\rm{Exp}}$ from Eq. (3) against $({N_pN_n})/({Z_0+N_0})$ for nuclei above $^{100}$Sn (left) and for nuclei above $^{208}$Pb (right).

In 2014 and 2017, Wang et al. and Seif et al. adopted the generalized liquid drop model (GLDM) [46, 47] and the preformed cluster model (PCM) [48], respectively, to study the $ \alpha $ decay of even–even, odd-$ A $, and doubly odd nuclei around doubly magic $ ^{100} $Sn. In these works [46–48], the calculated $ \alpha $ decay half-lives can accurately reproduce the experimental data, and the $ \alpha $-particle preformation factors evidently depend on the odd–even effect. In addition, in these works [46–48], the calculated $ \alpha $-particle preformation factors of nuclei near doubly magic nucleus $ ^{100} $Sn are of the order of $ 10^{-1} $, larger than ones of nuclei around doubly magic nucleus $ ^{208} $Pb. These works [46–48] also find that the proton–neutron interaction plays an important role in the formation of an $ \alpha $ cluster. These important insights support this present work. However, our research further shows that the $ \alpha $-particle preformation factors of nuclei near self-conjugate doubly magic $ ^{100} $Sn are significantly larger than those of analogous nuclei just above $ ^{208} $Pb, and they will be enhanced as the nuclei move towards the $ N = Z $ line. Furthermore, our research finds that the linear relationship between $ \alpha $-particle preformation factors and the product of valence protons and valence neutrons for nuclei around $ ^{208} $Pb is broken in the $ ^{100} $Sn region because the $ \alpha $-particle preformation factor is enhanced when the nucleus near $ ^{100} $Sn moves towards the $ N = Z $ line.

In general, $ \alpha $ decay is approximatively treated as a quasi-stationary-state problem that is characterized by the orbital angular momentum $ l $ and the global quantum number $ G $. Based on the Wildermuth rule, the global quantum number $ G = 2n_r+l $ denotes the principal quantum number with the radial quantum number $ n_r $ and angular momentum quantum number $ l $. For $ \alpha $ decay [82], $ G $ can be obtained by

In the different major proton and neutron closed shells, the different global quantum numbers $ G $ describe the different $ \alpha $-core relative motion for $ \alpha $ decay near $ ^{100} $Sn and $ ^{208} $Pb, which maybe can also influence the $ \alpha $-particle preformation. In the cluster model [7, 28, 43–45, 82], the quantum effect can be taken into account by the Bohr–Sommerfeld quasiclassical condition. However, the GLDM is a macroscopical model, which doesn't take into account the $ \alpha $-core relative motion. Therefore, the GLDM will need to be improved to introduce the $ \alpha $-core relative motion in the future.

In summary, we have systematically studied the $ \alpha $-particle preformation factors $ P_{\alpha}^{\rm{Exp}} $ of nuclei above doubly magic nuclei $ ^{100} $Sn and $ ^{208} $Pb, which are extracted from the ratios between the theoretical $ \alpha $-decay half-lives within the GLDM and the experimental data. The results show that $ P_{\alpha}^{\rm{Exp}} $ for nuclei near self-conjugate doubly magic $ ^{100} $Sn are significantly larger than those of analogous nuclei just above $ ^{208} $Pb, and they will be enhanced when the nucleus moves towards the $ N = Z $ line. The proton–neutron correlation energy $ E_{p-n} $ and two protons–two neutrons correlation energy $ E_{2p-2n} $ for nuclei near $ ^{100} $Sn are also larger than those of analogous nuclei just above $ ^{208} $Pb. This indicates that the interactions between protons and neutrons occupying similar single-particle orbitals could enhance $ P_{\alpha}^{\rm{Exp}} $ and result in the superallowed $ \alpha $ decay near doubly magic nucleus $ ^{100} $Sn. Furthermore, as the nucleus moves towards the $ N = Z $ line, the $ E_{2p-2n} $ energy shows an increased tendency which is the same as that of $ P_{\alpha}^{\rm{Exp}} $, while the $ E_{p-n} $ energy doesn't follow this pattern, indicating that the $ E_{2p-2n} $ energy plays a more important role than $ E_{p-n} $ energy in $ \alpha $-particle preformation of superallowed $ \alpha $ decay. The linear relationship between $ P_{\alpha}^{\rm{Exp}} $ and the product of valence protons and valence neutrons $ ({N_pN_n})/({Z_0+N_0}) $ for nuclei above $ ^{208} $Pb is broken in the $ ^{100} $Sn region because $ P_{\alpha}^{\rm{Exp}} $ is enhanced when the nucleus near $ ^{100} $Sn moves towards the $ N = Z $ line. Also, the calculated $ \alpha $ decay half-lives can accurately reproduce experimental data including the newly observed self-conjugate nuclei $ ^{108} $Xe and $ ^{104} $Te. This letter also provides evidence of the significant role of the proton–neutron interaction on $ \alpha $-particle preformation, which can shed some new light on $ \alpha $ decay and $ \alpha $-particle preformation factors in nuclear physics research in the future.