Alpha-Cluster formation in heavy alpha-emitters within a multistep model
๐ผ๐ผ -cluster formation in heavy ๐ผ -emitters within a multistep model J. M. Dong a , โ , Q. Zhao b , L. J. Wang c , W. Zuo a and J. Z. Gu d a Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China a School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China b School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China c School of Physical Science and Technology, Southwest University, Chongqing 400715, China d China Institute of Atomic Energy, P. O. Box 275(10), Beijing 102413, China
A R T I C L E I N F O
Keywords :alpha-decayFormation probabilityMultistep modelSuperheavy nucleiisland of stability
A B S T R A C T ๐ผ -decay always has enormous impetuses to the development of physics and chemistry, in particulardue to its indispensable role in the research of new elements. Although it has been observed in lab-oratories for more than a century, it remains a di๏ฌcult problem to calculate accurately the formationprobability ๐ ๐ผ microscopically. To this end, we establish a new model, i.e., multistep model, and thecorresponding formation probability ๐ ๐ผ values of some typical ๐ผ -emitters are calculated without ad-justable parameters. The experimental half-lives, in particular their irregular behavior around a shellclosure, are remarkably well reproduced by half-life laws combined with these ๐ ๐ผ . In our strategy,the cluster formation is a gradual process in heavy nuclei, di๏ฌerent from the situation that cluster pre-exists in light nuclei. The present study may pave the way to a fully understanding of ๐ผ -decay fromthe perspective of nuclear structure. ๐ผ -decay is a typical radioactive phenomenon in which anatomic nucleus emits a helium nucleus spontaneously. Asone of the most important decay modes for heavy and super-heavy nuclei, it was regarded as a quantum-tunneling e๏ฌect๏ฌrstly in the pioneering works of Gamov, Condon and Gur-ney in 1928 [1, 2], which provided an extremely signi๏ฌcantevidence supporting the probability interpretation of quan-tum mechanics in the early stage of nuclear physics. How-ever, a full understanding of ๐ผ -decay mechanism and hencean accurate description of the half-life, have not been set-tled yet. The critical problem lies in how to understand themechanism of ๐ผ -cluster formation and compute the forma-tion probability, known as a long-standing problem for nu-clear physics for more than eighty years that has attractedconsiderable interest continuously [3]. The ๐ผ -decay is reallyunderstood only if the formation probability ๐ ๐ผ can be welldetermined microscopically.The investigation of the ๐ผ -formation probability also pro-motes the exploration of cluster structures in nuclei. Actu-ally, numerous experimental observations have already re-vealed clustering phenomena in some light nuclei, such asthe famous Hoyle state in stellar nucleosynthesis that ex-hibits a structure composed of three ๐ผ -particles [4]. Thetheoretical exploration of the mechanism of cluster forma-tion has been a hot topic in nuclear physics [5, 6, 7]. Forheavy nuclei, a novel manifestation of ๐ผ -clustering structure,namely, " ๐ผ + Pb" states in
Po was revealed experimen-tally by their enhanced ๐ธ decays [8]. Yet, it is still an openquestion that whether or not the light and heavy nuclei sharethe same mechanism of cluster formation.Importantly, ๐ผ -decay has far-reaching implications in theresearch of superheavy nuclei (SHN) [9, 10]. Since an "is-land of stability" of SHN was predicted in the 1960s, ex-perimental e๏ฌorts worldwide have continuously embarked โ [email protected] ORCID (s): on such hugely expensive programs since it is always at theexciting forefront in both chemistry and physics [11, 10].However, the exact locations of the corresponding nuclearmagic numbers remain unknown, and theoretical approachesto date do not yield consistent predictions. The direct mea-surement of nuclear binding energies and detailed spectro-scopic studies of SHN with
๐ > have been beyondexperimental capabilities [12, 13, 14], therefore, to uncovertheir underlying structural information, one has to resort tothe mere knowledge about measured ๐ผ -decay energies andhalf-lives [14]. The formation probability, if available mi-croscopically, is of enormous importance to change this em-barrassing situation in combination with accurately measured ๐ผ -decay properties.Because of its fundamental importance, theoretically, theexploration of the ๐ผ -particle formation can be traced backto 1960 [15], which triggered extensive investigations withshell models [3, 16, 17, 18], Bardeen-Cooper-Schri๏ฌer (BCS)models [3, 19, 20] and Skyrme energy density functionals [21]later. The formation amplitude is regarded as the overlapbetween the con๏ฌguration of a parent nucleus and the onedescribed by an ๐ผ -particle coupled to the daughter nucleus.In particular, Po as a typical ๐ผ -emitter with two protonsand two neutrons outside the doubly magic core Pb, wasdiscussed extensively. Nevertheless, these calculations dis-agree on the decay width, and underestimate it substantially [3,20, 22]. To improve the calculations, the shell model com-bined with a cluster con๏ฌguration, was proposed with a treat-ment of all correlations between nucleons on the same foot-ing [23]. Yet, these calculations tend to be di๏ฌcult to gen-eralize for nuclei more complex than
Po. Over the pasttwo decades, several new approaches have been put forwardto calculate the formation probability ๐ ๐ผ in di๏ฌerent frame-works, including the pairing approach [24], ๐ ๐ ๐ ๐ scheme [25],quantum-mechanical fragmentation theory [26], cluster for-mation models [27, 28, 29], a quartetting wave function ap- : Preprint submitted to Elsevier
Page 1 of 7 a r X i v : . [ nu c l - t h ] J a n C (Parent, v i =1) Daughter+2n+2p ฮฑ Parent |< ฮจ P | ฮจ IC >| |< ฮจ IC | ฮจ ฮจ D >| P ( ฮจ โฮจ ฮฑ )=1 Figure 1: (Color online) Schematic illustration of the physical picture for an ๐ผ -cluster formation in the multistep model throughone of many pathways. The intermediate con๏ฌguration (IC) is the parent state but with a neutron level ๐ ๐ and a proton level ๐ ๐ fully occupied ( ๐ฃ ๐ = 1 ). These four nucleons ๏ฌlling the two single-particle levels will jump to the unoccupied ๐ ๐ - and ๐ ๐ -levels ofthe daughter nucleus, and then automatically assemble into an ๐ผ -particle ๏ฌnally. proach [30, 31, 32], internal barrier penetrability approach [33],statistical method [34], some empirical relations [35, 36, 37,38], and extraction combined with experimental data [39,40, 41, 42, 43, 44]. Although great e๏ฌorts have been madeand considerable progress has been achieved, no fully satis-factory approach has yet been found until now.To explore the ๐ผ -particle formation probability ๐ ๐ผ , wepropose a multistep model, and calculate the ๐ ๐ผ explicitlywithout introducing any adjustable parameter with the helpof a self-consistent energy density functional theory.Before we explore the ๐ผ -cluster formation probability,we ๏ฌrst discuss brie๏ฌy the proton spectroscopic factor of pro-ton radioactivity. The initial state is proton quasiparticle ex-citations of parent BCS vacuum ๐ผ โ ๐ | BCS โฉ P and the ๏ฌnal stateis ๐ โ ๐ | BCS โฉ D with ๐ โ ๐ | BCS โฉ D = [ ๐ข ( D ) ๐ ๐ผ โ ๐ + ๐ฃ ( D ) ๐ ๐ผ ๐ ] | BCS โฉ D [45].The spectroscopic factor (for spherical nuclei) is then givenby ๐ ๐ = | D โจ BCS | ๐ ๐ ๐ผ โ ๐ | BCS โฉ P | โ ( ๐ข ( D ) ๐ ) [45, 46], where ( ๐ข ( ๐ท ) ๐ ) = 1 โ ( ๐ฃ ( ๐ท ) ๐ ) is the probability that the spherical or-bit of the emitted proton is empty in the daughter nucleus.Accordingly, the ๐ ๐ can be iconically interpreted as a proba-bility that the blocked odd-proton in the ๐ ๐ -orbit of the parentnucleus jumps to the unoccupied ๐ ๐ -orbit of the daughter nu-cleus. With the inclusion of the calculated ๐ ๐ from nuclearmany-body approaches, the partial half-lives for sphericalproton emitters can be quite well reproduced [47, 48], indi-cating the success of the strategy for proton spectroscopicfactor.Inspired by the proton radioactivity, our multistep modelis proposed, and a sketch is exhibited in Fig. 1 to show schemat-ically the physical picture of our model. However, di๏ฌer-ent from the proton radioactivity where the emitted protoncomes from the blocked odd-proton orbit in the parent nu-cleus, the neutrons (protons) inside the emitted ๐ผ -particlecould come from any single-neutron (proton) level in princi-ple. Therefore, we introduce the intermediate con๏ฌguration(IC) with mass number ๐ด driven perhaps by quantum ๏ฌuc-tuation and residual interactions, to characterize which lev-els donate the complete four nucleons for the ๐ผ -formation.The IC is a state that a single-neutron level ( ๐ ๐ ) and a single-proton level ( ๐ ๐ ) in the parent nucleus are fully-occupied,namely, their occupation probabilities ๐ฃ ๐ = 1 for ๐ = ๐ ๐ , ๐ ๐ (which makes sure there are exactly four nucleons from thesetwo levels to generate an ๐ผ -particle), being analogous to theblocked odd-proton in proton radioactivity. In fact, it is acomponent of the initial parent state according to the in-terpretation of quantum mechanics, for example, the four-quasiparticle states in the picture of angular-momentum pro-jected models [49]. And there are many IC states and hencemany pathways to form the ๐ผ -particle, where Fig. 1 just il-lustrates one of the pathways, and hence our strategy is obvi-ously distinguished from other models. These four quasipar-ticles in the ๐ ๐ - and ๐ ๐ -levels are going to form an ๐ผ -particleand the remaining ๐ด โ4 nucleons accordingly form a daugh-ter nucleus, and the pathway via this IC is marked as ( ๐ ๐ , ๐ ๐ )for the sake of the following discussion. Accordingly, theprobability to ๏ฌnd a ๏ฌnal con๏ฌguration ฮจ ฮจ D in the wave-function of the parent nucleus ฮจ P through a given interme-diate con๏ฌguration ฮจ IC is ๐ ( ๐ ๐ ,๐ ๐ ) D = |โจ ฮจ P | ฮจ IC โฉ| โ |โจ ฮจ IC | ฮจ ฮจ D โฉ| , (1)which is the formation probability of the daughter nucleusthrough this pathway.For a fully microscopic treatment, in principle, many-body wave functions in the laboratory frame are need for | ฮจ P โฉ , | ฮจ IC โฉ , | ฮจ ฮจ D โฉ . While such treatments are chal-lenging and in this work a practical approximation is adopted,i.e., we neglected beyond-mean-๏ฌeld e๏ฌects and employ thecorresponding mean-๏ฌeld wave functions in the intrinsic frame.For a deformed super๏ฌuid nucleus with nucleons paired byup and down spins, within the BCS formulation, the overlapintegral of โจ ฮจ P | ฮจ IC โฉ is written as โจ ฮจ P | ฮจ IC โฉ = ฮ ๐ ( ๐ข ( P ) ๐ ๐ข ( IC ) ๐ + ๐ฃ ( P ) ๐ ๐ฃ ( IC ) ๐ ) , (2)in which ๐ฃ ๐ ( ๐ข ๐ ) represents the probability that the two-folddegenerate ๐ -th single-particle level is occupied (unoccu-pied). The ๏ฌnal state is written as ฮจ ฮจ D = ๐ โ ๐ ๐ ๐ โ ๐ ๐ | BCS โฉ ๐ท ,where ๐ โ ๐ ๐ ( ๐ โ ๐ ๐ ) creates two neutrons (protons). As an ap-proximation, we assume the daughter even-even core can bedescribed within the BCS approach, just like that in protonradioactivity [45]. Therefore, โจ ฮจ IC | ฮจ ฮจ D โฉ expressed interms of single-particle properties is given by โจ ฮจ IC | ฮจ ฮจ D โฉ : Preprint submitted to Elsevier
Page 2 of 7 ( ฮ ๐ = ๐ ๐ ,๐ ๐ ๐ข ( D ) ๐ ) ฮ ๐ โ ๐ [ ๐ข ( D ) ๐ ๐ข ( IC ) ๐ + ๐ฃ ( D ) ๐ ๐ฃ ( IC ) ๐ โจ ๐ ( D ) ๐ | ๐ ( IC ) ๐ โฉ ] , (3)where ๐ ๐ denotes the normalized single-particle wavefunc-tion. Iconically, the four nucleons escaping from the IC jumpsinto the unoccupied ๐ ๐ - and ๐ ๐ -levels of the daughter nucleus,with a probability ( ๐ข ( D ) ๐ ๐ ๐ข ( D ) ๐ ๐ ) .We make the following two assumptions: 1) The forma-tion probability of the ๐ผ -cluster is identical to that of thedaughter nucleus, i.e., ๐ ( ๐ ๐ ,๐ ๐ ) ๐ผ = ๐ ( ๐ ๐ ,๐ ๐ ) D , that is, the for-mation of the ๐ผ -particle is achieved accordingly once thedaughter nucleus is generated. This means the four nucleonsescaping from the IC jumping into the unoccupied ๐ ๐ - and ๐ ๐ -levels of the daughter nucleus are expected to automati-cally assemble into an ๐ผ -particle at nuclear surface sponta-neously. Namely, the transition probability from ฮจ toan actual ๐ผ -cluster state ฮจ ๐ผ is ๐ (ฮจ โ ฮจ ๐ผ ) = 1 near thelow-density nuclear surface. It is generally believed that an ๐ผ -cluster is formed only in regions that the nuclear matterdensity is low, and there is a similar case that an ๐ผ -like stategenerates automatically below ๐ Mott โ 0 . fm โ3 in quar-tetting wave function approach [30, 31, 32]. Intuitively, thatthe nucleons spontaneously organize into an doubly magic ๐ผ -particle, makes the system more stable. 2) Each pathwayfor the formation process is expected to be independent ofthe others . We sum over all pathways (i.e., through di๏ฌerentICs) to eventually achieve the ๐ผ -particle formation probabil-ity via ๐ ๐ผ = โ ( ๐ ๐ ,๐ ๐ ) ๐ ( ๐ ๐ ,๐ ๐ ) D = โ ( ๐ ๐ ,๐ ๐ ) { ฮ ๐ โฒ ( ๐ข ( P ) ๐ โฒ ๐ข ( IC ) ๐ โฒ + ๐ฃ ( P ) ๐ โฒ ๐ฃ ( IC ) ๐ โฒ ) โ ฮ ๐ = ๐ ๐ ,๐ ๐ ( ๐ข ( D ) ๐ ) โ ฮ ๐ โ ๐ [ ๐ข ( D ) ๐ ๐ข ( IC ) ๐ + ๐ฃ ( D ) ๐ ๐ฃ ( IC ) ๐ โจ ๐ ( D ) ๐ | ๐ ( IC ) ๐ โฉ ] } , (4)with Eqs. (1-3). The dimensionless formation probabilityhere is the expectation value of the ๐ผ -cluster number thatcan be emitted. The stationary-state description of a time-dependent cluster formation process is a quite good approxi-mation and simpli๏ฌes the problem enormously [3], which iswidely used at present for ๐ผ -decay. It is valid because half-lives of ๐ผ -emitters are very long ( โ6 โ 10 s) comparedwith the โperiods" of nuclear motion ( โ21 s) and hencein the time evolution of a decaying state the nucleons has alarge number of opportunities to get clustered and to get theclusters dissolved before it can actually escaped from the nu-cleus [3].Our approach involves the structure of both parent anddaughter nuclei, but does not involve an intrinsic state ora localized density distribution of the ๐ผ -cluster, being sig-ni๏ฌcantly di๏ฌerent from the standard shell or BCS modelswhere a Gauss-shaped intrinsic ๐ผ -cluster wavefunction is in-troduced. The quartetting wave function approach is a suc-cessful method proving a reasonable behavior for the ๐ ๐ผ ofeven-even Po isotopes [31], which does not involve such an Table 1
The rms deviations โโจ ๐ โฉ and average deviations โจ ๐ โฉ for theVSF and UDL with ๐ ๐ผ = 1 and ๐ ๐ผ โ .Formulas โโจ ๐ โฉ โจ ๐ โฉ VSF ( ๐ ๐ผ = 1 ) 0.360 0.296VSF ( ๐ ๐ผ โ ) 0.109 0.0861UDL ( ๐ ๐ผ = 1 ) 0.316 0.268UDL ( ๐ ๐ผ โ ) 0.0888 0.0812 intrinsic ๐ผ -cluster wavefunction to calculate ๐ ๐ผ either, anddoes not employ the overlap of the wavefunctions betweenthe initial and ๏ฌnal states. The wavefunction of the boundstate for the center-of-mass motion of four correlated nucle-ons is obtained by solving the corresponding Schrรถdingerequation, and then the ๐ ๐ผ is calculated by integrating themodular square of this wavefunction in the region below Mottdensity since an ๐ผ -like state generates automatically at suchlow densities [30, 31, 32]. Di๏ฌerent substantially from thisquartetting wave function approach, the ๐ ๐ผ in our work isstill based on the concept of overlap integrals, and ๏ฌnallycan be calculated with the compact expression of Eq. (4)with the help of existing many-body approaches without in-troducing any adjustable parameter.The single particle properties in Eq. (4) are determinedwithin the framework of a covariant density functional (CDF)approach starting from an interacting Lagrangian density [50,51, 52, 53]. The nuclear CDF employed in self-consistentcalculations is parameterized by means of about ten couplingconstants that are calibrated to basic properties of nuclearmatter and ๏ฌnite nuclei, which enables one to perform an ac-curate description of ground state properties and collectiveexcitations over the whole nuclear chart [51, 52, 53], and hasbecome a standard tool in low energy nuclear structure. Theexplicit calculations are carried out based on a standard codeDIZ [54] for deformed nuclei, with the NL3 interaction [55]for the mean-๏ฌeld and the calibrated D1S Gogny force [57]for the pairing channel. The NL3 parameter set has beenused with enormous success in the description of a varietyof ground-state properties of spherical, deformed and exoticnuclei [55, 56], and the calibrated D1S Gogny force enablesone to well reproduce the odd-even staggerings on nuclearbinding energies [57]. We concentrate on the even-even Po,Rn and Ra isotopes with spherical or near-spherical shapes,because their ๐ผ -decays tend to have large branching ratios(100% in most cases) and their corresponding half-lives werebest measured experimentally [58]. On the other hand, these ๐ผ -decay cases usually do not involve excited states and angu-lar momentum transfers, and thus serve as an optimal testingground to examine our model. Moreover, the values of over-lap integrals ฮ ๐ โจ ๐ ( D ) ๐ | ๐ ( IC ) ๐ โฉ for these nuclei can be taken asunity. The products in Eqs. (2,3) along with the summa-tion in Eq. (4) are truncated at 5 MeV for the single-nucleonspectra to achieve convergence.To assess the validity of our multistep model, we explorethe role of the formation probability ๐ ๐ผ in half-life calcula-tions. The widely accepted formulas, i.e., the semi-empirical : Preprint submitted to Elsevier
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10 120 130 140283032343638404244 110 120 130 140182022242628303234 r = 0.9998S 1 l og T / E xp . ( s )- b โ + l og S โ (MeV -1/2 )S = 1r = 0.9975 โ (MeV -1/2 ) Figure 2: log ๐ Exp. โ ๐๐ โฒ + log ๐ ๐ผ as a function of ๐ โฒ ob-tained with the UDL for ๐ ๐ผ = 1 (left) and ๐ ๐ผ โ (right).The coe๏ฌcient ๐ is ๏ฌxed at the ๏ฌtted value in the two cases,respectively. The straight lines are given by ๐๐ โฒ + ๐ . Here ๐ isthe corresponding correlation coe๏ฌcient. Viola-Seaborg formula (VSF) [59] and the universal decaylaw (UDL) based on the ๐ -matrix expression [60] are em-ployed, which are respectively given as log ๐ = ๐๐ + ๐ โ ๐ ๐ผ + ๐๐ + ๐ โ log ๐ ๐ผ , (5) log ๐ = ๐๐ โฒ + ๐๐ โฒ + ๐ โ log ๐ ๐ผ , (6)with ๐ โฒ = 2( ๐ โ 2) โ ๐ด ๐ผ๐ ๐ ๐ผ , ๐ด ๐ผ๐ = 4( ๐ด โ 4)โ ๐ด,๐ โฒ = โ ๐ด ๐ผ๐ ( ๐ โ 2) [ ( ๐ด โ 4) + 4 ] .๐ ( ๐ด ) is the proton (mass) number of a given parent nucleus.The decay energy ๐ ๐ผ and half-life ๐ are in units of MeVand second, respectively. ๐ ๐ผ = 1 ( ๐ ๐ผ โ ) correspondsto the results without (with) the inclusion of the formationprobability.The ๏ฌtting procedures are performed in the cases of ๐ ๐ผ =1 and ๐ ๐ผ โ respectively to test whether or not the pre-dicted ๐ ๐ผ could improve substantially the accuracy of thetwo formulas. It is worth pointing out that the UDL has al-ready included the logarithmic formation amplitude whichis assumed to be linearly dependent upon ๐ โฒ . Therefore, inEq. (6), the ๐ โฒ -dependent formation probability is replacedby the presently calculated ๐ ๐ผ . The root-mean-square (rms)deviations โโจ ๐ โฉ and average deviations โจ ๐ โฉ for the two for-mulas with ๐ ๐ผ = 1 and ๐ ๐ผ โ are summarized in Table 1.The inclusion of the ๐ ๐ผ indeed greatly improves the accu-racy of both the VSF and UDL. The UDL with a solid phys-ical ground but less parameters, works better than the VSF,and reproduces the available experimental half-lives withina factor of 2 in the case of ๐ ๐ผ = 1 . Yet, when the micro-scopically calculated ๐ ๐ผ is included, the deviation of the re-๏ฌtting is reduced down to around . The good agreementbetween the calculated half-lives and the experimental data RaRnPo
Exp. + CM Present Cal. S Parent neutron number
Figure 3: (Color online) Calculated formation probability ๐ ๐ผ values within the multistep model for the Po, Rn and Ra iso-topes, compared with those extracted by using the experimen-tal half-lives in combination with the CM calculated penetra-tion probabilities. is quite encouraging, indicating the reliability of the forma-tion probability ๐ ๐ผ given by the multistep model. In Fig. 2,we plot the UDL ๏ฌttings but replace the half-lives with theexperimental values to more visually reveal the role of ๐ ๐ผ ,and that the inclusion of the ๐ ๐ผ systematically improves theagreement with data is exhibited. The highly linear correla-tion is displayed for ๐ ๐ผ โ , with a correlation coe๏ฌcient ashigh as ๐ = 0 . , suggesting the success of our formationprobability and the validity of the two assumptions.Furthermore, ๐ ๐ผ is extracted in turn by using the ratio ofthe theoretical half-life to the experimentally observed value.The barrier penetrability of ๐ผ -particle, is achieved theoreti-cally by the WKB approximation which turns out to work ex-cellently [61], where the potential barrier is constructed bya simple "Cosh" potential plus the Coulomb barrier model(CM) [62]. Here the extracted ๐ ๐ผ should be considered as arelative value. By selecting an optimal constant assault fre-quency, the extracted ๐ ๐ผ values with varying neutron num-ber ๐ are compared with the results given by the multistepmodel in Fig. 3. In sharp contrast with half-lives, the ๐ ๐ผ values are located in a relatively narrow range, leading tothe success of the empirical half-life laws even when ๐ ๐ผ isnot included. The ๐ ๐ผ values follow the similar behavior withregard to the Po, Rn and Ra isotopic chainsโthat is, gradu-ally drop with increasing neutron number up to the spheri-cal magic number ๐ = 126 , attributed to the increased sta-bility of isotopes when approaching the magic number, andthen they increase drastically with neutron number. Such ageneral trend of the extracted ๐ ๐ผ is successfully reproduced : Preprint submitted to Elsevier
Page 4 of 7 Fn p n -8-7-6-5-4-3-2-1 Fp Figure 4:
Contour plot of log ๐ ( ๐ ๐ ,๐ ๐ ) ๐ผ versus the single-particleenergies (in units of MeV) of the ๐ ๐ - and the ๐ ๐ -levels for theillustrative example of Po to show the role of each pathway.The dashed lines denote the corresponding Fermi energies ๐ Fn and ๐ Fp for neutrons and protons, respectively. within our method. Typically, the ๐ ๐ผ of Po, being ex-pected to be large owing to its two protons and two neutronsoutside the shell closure core
Pb, is about six times largerthan that of its neighbor
Po. The weight of the clustercomponent in
Po is large, which is exactly what one needsto simultaneously describe the B(E2) [63] and the absolute ๐ผ -decay width [3, 23] within the shell model plus a clustercomponent. The distinct behavior that ๐ ๐ผ varies abruptlywhen the magic number is crossed, con๏ฌrms that the par-ticularly signi๏ฌcant role of shell e๏ฌects on ๐ ๐ผ is reasonablyaccounted for in Eq. (4) via the single-particle properties.As one expects, ๐ ๐ผ reaches its minimum at the shell closure ๐ = 126 as the result of the well-known shell stability thatstrongly enhances the nuclear binding.In order to analyze the contribution of each pathway inEq. (4) to ๐ ๐ผ , Fig. 4 illustrates ๐ ( ๐ ๐ ,๐ ๐ ) ๐ผ = |โจ ฮจ IC | ฮจ P โฉ| โ |โจ ฮจ ฮจ D | ฮจ IC โฉ| versus the single neutron and single pro-ton energies ( ๐ n , ๐ p ) by taking the typical nucleus Po asan example. The formation probability in Eq. (4) is predom-inantly determined by the pathways belonging, in the IC, tothe fully-occupied neutron and proton levels ๐ ๐ and ๐ ๐ slightlyabove the Fermi surfaces, i.e., these levels are the major nu-cleon donors to constitute the emitted ๐ผ -particle. The con-tributions from other pathways drop sharply when the fully-occupied levels ( ๐ ๐ , ๐ ๐ ) gradually go away from these leadingones.It is well-known that the concept of ๐ผ -clustering is essen-tial for understanding the structure of light nuclei. In somecases, light nuclei behave like molecules composed of clus-ters of protons and neutrons, such as the de๏ฌnite ๐ผ clusterstructure of Be [64] re๏ฌected in a localized density distri-butions. But whether such type of cluster structure existsor not in heavy nuclei is uncertain. In our strategy, how-ever, a localized ๐ผ -cluster does not pre-exist inside a par-ent nucleus, but is generated during the decay process in ourmodel. Therefore, the scenario that a continuous formation and breaking of the ๐ผ -cluster until it escapes randomly fromthe parent nucleus [3], is supported. As a result, the mecha-nism of ๐ผ formation in heavy ๐ผ -emitters is di๏ฌerent markedlyfrom that in light nuclei.Since the formation probability ๐ ๐ผ is highly relevant tothe quantum-mechanical shell e๏ฌects, the extracted ๐ ๐ผ witha high-precision UDL combined with experimentally mea-sured ๐ผ -decay properties, is of great importance for diggingup valuable structural information of SHN and exotic nuclei.The correlations between the ๐ ๐ผ values of SHN have sug-gested that the heaviest isotopes reported in Dubna do not liein a region of rapidly changing shapes [65]. Therefore, the ๐ ๐ผ versus proton number exhibits a behavior analogous toisotopic chains shown in Fig. 3, attributed to the nearby shellclosure. By employing the NL3 interaction, the heaviest nu-cleus Og ( ๐ = 118 ) and its isotonic neighbor Lv ( ๐ =116 ), are predicted to be nearly spherical. The calculated ๐ ๐ผ is 0.66 for Lv while it reduces to 0.45 for
Og with theratio of ๐ ๐ผ ( Lv )โ ๐ ๐ผ ( Og ) = 1 . , being indeed similar tothe trend shown in Fig. 3, which is consistent with the factthat the NL3 interaction itself predicts the adjacent ๐ = 120 as a magic number. On the other hand, with the UDL ofEq. (6) along with experimental data [10], the extracted ratioof ๐ ๐ผ ( Lv )โ ๐ ๐ผ ( Og ) is as high as . +4 . . ( . +5 . . ) withhalf-life data of Lv from Ref. [66] (Ref. [67]) where thesigni๏ฌcant uncertainties are due to the low-statistics data forthe measurements, and agrees marginally with the above the-oretical value. Hence the probable magic nature of ๐ = 120 is suggested. The future measurements with a much higheraccuracy for these nuclei together with their isotopes are en-couraged, which would pin down the proton magic numbereventually.Intriguingly, the superallowed ๐ผ -decay to doubly magic Sn was observed recently, which indicates a much larger ๐ผ -formation probability than Po counterpart [68]. Withinthe framework of the quartetting wave function approach,an enhanced ๐ผ -cluster formation probability for Te wasfound because the bound state wavefunction of the four nu-cleons has a large component at the nuclear surface [32].Yet, a large ๐ ๐ผ for Te is inconsistent with our predictionof ๐ ๐ผ = 0 . , suggesting the onset of an unusual type of nu-clear super๏ฌuidity for self-conjugate nuclei, i.e., the proton-neutron pairing which is not well-con๏ฌrmed at present. Thisisoscalar pairing would considerably impact on the single-particle spectra and hence the ๐ ๐ผ , and its absence in our CDFcalculations leads to the underestimated ๐ ๐ผ . Therefore, as anew way independent of Ref. [69], combined with precise ๐ผ -decay measurements for ๐ โ ๐ nuclei, our approach for the ๐ ๐ผ could enable us to clarify this abnormal pairing interac-tion in turn by employing CDF approaches with the inclusionof a tentative isoscalar super๏ฌuidity.In general, computing the formation probability ๐ ๐ผ de-๏ฌned in Eq. (4) by nuclear density functionals with a veryhigh accuracy, is out of reach at present because of the well-known fact that the single-particle levels are not well-de๏ฌnedin the concept of the mean-๏ฌeld approximation especiallyfor well-deformed nuclei. The approximate single-particle : Preprint submitted to Elsevier
Page 5 of 7 pectrum is perhaps the primary cause which leads to the de-viation of the theoretical calculations. Nevertheless, basedon two intuitive assumptions and without any phenomeno-logical adjustment, our strategy opens a new perspective toaccount for the formation mechanism. This is highly impor-tant to help one to uncover the underlying knowledge aboutsuperheavy nuclei and isoscalar pairing, in combination withexperimental observations. For example, the experimentallymeasured ๐ผ -decay properties of heaviest nuclei combinedwith our calculated ๐ ๐ผ suggest the probable proton-magicnature of ๐ = 120 . To treat the many-body wave functionsin a laboratory frame beyond mean-๏ฌeld approximation, isplaned as a future work.J. M. Dong thanks B. Zhou, W. Scheid, and C. Qi forhelpful comments and suggestions. This work is supportedby the Strategic Priority Research Program of Chinese Academyof Sciences (Grant No. XDB34000000), by the NationalNatural Science Foundation of China (Grants Nos. 11775276,11905175, 11975282, and 11675265), by the National KeyProgram for S&T Research and Development (Grant No.2016YFA0400502), by the Youth Innovation Promotion As-sociation of Chinese Academy of Sciences under Grant No.Y201871, by the Continuous Basic Scienti๏ฌc Research Project(Grant No. WDJC-2019-13), by the Leading Innovation Project(Grant No. LC 192209000701), and by the Continuous Ba-sic Scienti๏ฌc Research Project (Grant No. WDJC-2019-13). References [1] G. Gamov, Z. Phys. (1928) 204.[2] E. U. Condon, R. W. Gurney, Nature (London) 122, (1928) 439.[3] R. G. Lovas, R. J. Liotta, A. Insolia, K. Varga, D. S. Delion, Phys.Rep. 294 (1998) 265.[4] A. Tohsaki, H. Horiuchi, P. Schuck, G. Rรถpke, Phys. Rev. Lett.87 (2001) 192501; Rev. Mod. Phys. 89 (2017) 011002.[5] J.-P. Ebran, E. Khan, T. Nikลกiฤ, D. 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