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


 The α-particle preformation factors of nuclei above doubly magic nuclei 100Sn and 208Pb are investigated.The results show that the α-particle preformation factors of nuclei near self-conjugatedoubly magic 100Sn are larger significantly than those of analogous nuclei just above 208Pb, andthey will be enhanced as the nuclei move towards the N = Z line. The correlation energy of theproton-neutron Ep-n and two protons-two neutrons E2p-2n of nuclei near 100Sn also exhibit similarsituations indicating that the interactions between protons and neutrons occupying similar single-particleorbitals could enhance the α-particle preformation factors and result in the superallowedα decay. It also provides evidence of the significant role of proton-neutron interaction on α-particlepreformation. Besides, the linear relationship between α-particle preformation factors and the productof valence protons and valence neutrons for nuclei around 208Pb is broken in the 100Sn regionbecause the α-particle preformation factor is enhanced when the nucleus near 100Sn moves towardsthe N = Z line. Furthermore, the calculated decay half-lives can well reproduce the experimentaldata including the recent observed self-conjugate nuclei 104Te and 108Xe [Phys. Rev. Lett. 121,182501 (2018)].

α decay is a fundamental nuclear decay mode. The researches on α decay have long been focused on the vicinities of doubly magic nuclei 208 Pb (Z = 82, N = 126) and 298 Fl (Z = 114, N = 184) because α decay can be a probe to study the unstable nucleus structure, and can be the only way to identify the new synthesized superheavy nucleus . Over the past two decades, the α 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][29][30][31][32][33][34][35]. In particular, there is the fastest α emitter 104 Te near doubly magic nucleus 100 Sn [34]. Since the α 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 and obtain the unique neutron-deficient nuclear structure information and examine various α decay theoretical models. Moreover, the cluster radioactivity was also predicted as one of the decay modes of the nucleus in the 100 Sn region [36][37][38][39]. Further interest in the decay rates of nuclei around doubly magic nucleus 100 Sn comes from the research of 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 the interactions between protons and neutrons occupying similar single-particle orbitals could enhance the α-particle preformation factors and the reduced α-widths significantly when compared to the analogous nuclei just above doubly magic nucleus 208 Pb, and result in the so-called "superallowed" α decay [42]. And this effect would * zhanghongfei@lzu.edu.cn be expected to be the greatest for the N = Z selfconjugate nuclei [42]. Recently, the first time α radioactivity to a heavy self-conjugate nucleus was observed on the 108 Xe→ 104 Te→ 100 Sn α decay chain [34], including the measurements of the α-particle kinetic energy and α decay half-lives of the α emitters 108 Xe [E α = 4.4(2) MeV, T 1/2 = 58 +106 −23 µs] and 104 Te [E α = 4.9(2) MeV, T 1/2 < 18 ns]. The authors of this reference suggested that the α-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 β−stability line, are the only two existing α emitters decaying to the doubly magic nuclei. In this work, we focus on the α-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 α decay [34,[43][44][45][46][47][48][49][50][51][52][53] within the generalized liquid drop model (GLDM) [54][55][56][57][58][59][60]. These α emitters are in different isospins and mass numbers as well as around different protons and neutrons closed shells. We want to reveal some new behaviors of α-particle preformation factors of extremely neutron-deficient nuclei near self-conjugate doubly magic nucleus 100 Sn for understanding the roles of proton-neutron correlation and the single-particle orbitals occupied by protons and neutrons in the preformation of α-cluster as well as the physical mechanism of superallowed α decay.
The GLDM can well deal with proton radioactivity [61], cluster radioactivity [62], fusion [63], fission [64], and the α decay process [22,[54][55][56][57][58][59][60]65] because of introducing the quasimolecular shape mechanism [54], which can describe the complex deformation process from the parent nucleus 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 [54][55][56][57][58][59][60], the GLDM has been discussed in detail. The α decay half-life can be obtained by with the α decay constant λ being expressed as where the assault frequency ν is obtained by using the classical method with the kinetic energy of the α-particle.
The barrier penetrating probability P is determined by tunneling the GLDM potential barriers [54][55][56][57][58][59][60] with the Wentzel-Kramers-Brillouin (WKB) approximation. The experimental α-particle preformation factor P Exp α can be extracted from the ratios between the theoretical decay half-life T Cal1 1/2 calculated by assuming the αparticle preformation factor as a constant P α = 1 to experimental data [59,[66][67][68][69] and expressed as P Exp To examine the experimental α decay half-life data, the analytic formula for estimating the α-particle preformation factor is also adopted, which is put forward in our previous work [60,65]. It is expressed as where A, Z, and Q α represent mass number, proton number, and α decay energy of the parent nucleus. A 1 , Z 1 , A 2 , and Z 2 denote the mass and proton numbers of the α-particle and daughter nucleus. l is the angular momentum carried by the α-particle. The parameters values are listed in Ref. [60].
The calculated α decay half-lives for nuclei above doubly magic nuclei 100 Sn and 208 Pb are presented in Tables  I and II, respectively. In these two tables, the first four columns represent the α transition, the experimental kinetic energy of the α-particle, the experimental α decay energy, and the minimum angular momentum carried by the α-particle. The fifth column is the experimental α decay half-life. The sixth column denotes the calculated α decay half-life T Cal1 1/2 within the GLDM with P α = 1. The seventh column gives the calculated α decay half-life T Cal2 1/2 within the GLDM with the estimated α-particle preformation factor from Eq. (4). The eighth column shows the extracted experimental α-particle preformation factor by using Eq. (3) with T Cal1 1/2 and T Exp 1/2 . The last two columns express the calculated correlation energy of the proton-neutron E p−n and two protons-two neutrons E 2p−2n determined by Eqs. (6) and (7). From these two tables, it can be seen immediately that the calculated α decay half-lives T Cal2 1/2 can well reproduce the experimental data including the newly observed selfconjugate nuclei 104 Te and 108 Xe [34]. Note that the calculations provide supports for recent experimental observation data in Ref. [34]. To measure the agreements between the calculated α decay half-lives T Cal2 1/2 and experimental data T Exp 1/2 , the standard deviations are calculated by For nuclei in Tables I and II, the results of standard deviations σ 1 = 0.47 and σ 2 = 0.16 are satisfactory manifesting that T Cal2 1/2 can well reproduce T Exp 1/2 within factors of 10 0.47 = 2.95 and 10 0.16 = 1.45, respectively. It demonstrated that the GLDM can be applied to extract the experimental α-particle preformation factors for studying the structure information of nuclei in these two regions.
Furthermore, in Tables I and II, we can see that the extracted experimental α-particle preformation factors P Exp α of nuclei near 100 Sn are larger than P Exp α of nuclei near 208 Pb, and in particular, larger than P Exp α 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, as well as 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 α-particle preformation factors P Exp α 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 the P Exp α of 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 Exp α 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 clearly seen that for Te isotopes the P Exp α exhibits an increasing trend when the nucleus moves towards the N = Z line, but the P Exp α of Po isotopes do not show similar patterns due to the large asymmetry between neutrons and protons. Fig. 1 (b) displays the variations of P Exp α 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 can find that for Xe isotopes the P Exp α also increases as the nucleus moves towards the N = Z line. However, the P Exp  I. Calculations of α-particle preformation factor, α decay half-lives, and the correlation energy of the proton-neutron Ep−n and two protons-two neutrons E2p−2n of even-even Te, Xe, and Ba isotopes near 100 Sn.  The P Exp α of N = 58 isotones also show an increasing tendency as the nuclei move towards the N = Z line, but this phenomenon has not occurred in the analogous N = 134 isotopes just above 208 Pb. It is indicated that the P Exp α is enhanced when a nucleus moves towards the N = Z line, and result in the superallowed α decay near doubly magic nucleus 100 Sn. In recent work, Clark et al. adopted a very different model and studied the α-particle preformation factors of nuclei in these two regions [31]. A similar conclusion was obtained though the α-particle preformation factors of nuclei near doubly magic nuclei 100 Sn and 208 Pb are in orders of 10 −2 and 10 −3 , respectively.    For investigating the effects of proton-neutron interaction and two protons-two neutrons interaction on the α-particle preformation, we calculate the correlation energy of the proton-neutron E p−n and two protons-two neutrons E 2p−2n using Eqs. (6) and (7) were proposed in Ref. [71] and used to determine the experimental pairing energy of the nucleons [72]. 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 I and II. In these two tables, it can be found that the E p−n energy and E 2p−2n energy of nuclei above doubly magic nucleus 100 Sn are larger than those of analogous nuclei just above 208 Pb. This, in turn, leads to that the P Exp α of nuclei near 100 Sn are enhanced significantly. The results of E p−n energy and E 2p−2n energy are plotted in Fig. 2. In this figure, the E p−n energy and E 2p−2n energy of nuclei above 100 Sn are strengthened when compared to analogous nuclei just above 208 Pb. For Z = 52 isotopes, the E p−n energy and E 2p−2n energy increase rapidly in N n = 2. Similarly, for Z = 54 isotopes, the E p−n energy and E 2p−2n energy rise fast in N n = 4. However, the E p−n energy and E 2p−2n energy of analogous nuclei just above 208 Pb are changed slowly. Therefore, it demonstrated that the α-particle is more to form in self-conjugate nuclei and result in the superallowed α decay. In addition, the E 2p−2n energy appears an increased tendency, the same as P Exp α , when the nucleus moves towards the N = Z line implying that the two protons-two neutrons interaction play a more significant role than one proton-one neutron interaction in α-particle preformation.
The extracted experimental α-particle preformation factors P Exp α for nuclei above 100 Sn and 208 Pb are shown as functions of NpNn Z0+N0 in Fig. 3 (a) and (b), respectively. In Fig. 3 (b), one can see that the closer the NpNn Z0+N0 is to the zero, representing the proton and/or neutron num-bers approaches the closed shells, the smaller P Exp α is. When the NpNn Z0+N0 is far from zero, the P Exp α 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 α-particle to form inside its parent nucleus. And we can find that the P Exp α is linearly dependent on the NpNn Z0+N0 for nuclei above 208 Pb. It is consistent with the conclusions deduced by adopting the different models, in which the α-particle preformation factors are extracted from the ratios between theoretical α decay half-lives calculated by adopting the different models to experimental data [15,73,74], or calculated using the differences of binding energy between the α decaying parent nucleus and its neighboring nuclei within the cluster-formation model [75]. It is shown that the nuclear shell effects and the nucleons configuration play key roles in α-cluster preformation for α-particle emitters around doubly magic 208 Pb. However, in Fig. 3 (a) this phenomenon is broken in the 100 Sn region. The P Exp α of nuclei above 100 Sn are linearly independent of NpNn Z0+N0 and show a new behavior. When the nucleus is close to the shell closures, the P Exp α of the nucleus near 100 Sn does not decrease like that of the nucleus near 208 Pb, but it increases. In addition, we can find that the maximum values of the P Exp α in Fig. 3 (a) correspond to 104 Te, 108 Xe, and 114 Ba. In particular along the N = Z line, the P Exp α is significantly enhanced, which results in the P Exp α of nuclei above 100 Sn are not linearly dependent on NpNn Z0+N0 .
In summary, we systematically study the α-particle preformation factors P Exp α of nuclei above doubly magic nuclei 100 Sn and 208 Pb, which are extracted from the ra-tios between the theoretical α-decay half-lives within the GLDM to experimental data. The results show that the P Exp α of nuclei near self-conjugate doubly magic 100 Sn are larger significantly than those of analogous nuclei just above 208 Pb, and they will be enhanced when the nucleus moves towards the N = Z line. The correlation energy of proton-neutron E p−n and two protons-two neutrons E 2p−2n of nuclei near 100 Sn are also larger than those of analogous nuclei just above 208 Pb. It is indicated that the interactions between protons and neutrons occupying similar single-particle orbitals could enhance the P Exp α and result in the superallowed α 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 Exp α , while E p−n energy doesn't appear this pattern, indicating E 2p−2n energy plays a more important role than E p−n energy in α-particle preformation of superallowed α decay. The linear relationship between the P Exp α and the product of valence protons and valence neutrons NpNn Z0+N0 for nuclei above 208 Pb is broken in the 100 Sn region because the P Exp α is enhanced when the nucleus near 100 Sn moves towards the N = Z line. Besides, the calculated α decay half-lives can well reproduce experimental data including the newly observed self-conjugate nuclei 108 Xe and 104 Te. This work also provides evidence of the significant role of proton-neutron interaction on the α-particle preformation, which will shed some new light on α decay and α-particle preformation factors researches of nuclear physics in the future. This work is supported by National Natural Science Foundation of China (Grants No. 12175170, No. 11675066, No. 11665019).