Alpha-Cluster formation in heavy alpha-emitters within a multistep model

$\alpha$-decay always has enormous impetuses to the development of physics and chemistry, in particular due to its indispensable role in the research of new elements. Although it has been observed in laboratories for more than a century, it remains a difficult problem to calculate accurately the formation probability $S_\alpha$ microscopically. To this end, we establish a new model, i.e., multistep model, and the corresponding formation probability $S_\alpha$ values of some typical $\alpha$-emitters are calculated without adjustable parameters. The experimental half-lives, in particular their irregular behavior around a shell closure, are remarkably well reproduced by half-life laws combined with these $S_\alpha$. In our strategy, the cluster formation is a gradual process in heavy nuclei, different from the situation that cluster pre-exists in light nuclei. The present study may pave the way to a fully understanding of $\alpha$-decay from the perspective of nuclear structure.

-decay is a typical radioactive phenomenon in which an atomic nucleus emits a helium nucleus spontaneously. As one of the most important decay modes for heavy and superheavy nuclei, it was regarded as a quantum-tunneling effect firstly in the pioneering works of Gamov, Condon and Gurney in 1928 [1,2], which provided an extremely significant evidence supporting the probability interpretation of quantum mechanics in the early stage of nuclear physics. However, a full understanding of -decay mechanism and hence an accurate description of the half-life, have not been settled yet. The critical problem lies in how to understand the mechanism of -cluster formation and compute the formation probability, known as a long-standing problem for nuclear physics for more than eighty years that has attracted considerable interest continuously [3]. The -decay is really understood only if the formation probability can be well determined microscopically.
The investigation of the -formation probability also promotes the exploration of cluster structures in nuclei. Actually, numerous experimental observations have already revealed clustering phenomena in some light nuclei, such as the famous Hoyle state in stellar nucleosynthesis that exhibits a structure composed of three -particles [4]. The theoretical exploration of the mechanism of cluster formation has been a hot topic in nuclear physics [5,6,7]. For heavy nuclei, a novel manifestation of -clustering structure, namely, " + 208 Pb" states in 212 Po was revealed experimentally by their enhanced 1 decays [8]. Yet, it is still an open question that whether or not the light and heavy nuclei share the same mechanism of cluster formation.
Importantly, -decay has far-reaching implications in the research of superheavy nuclei (SHN) [9,10]. Since an "island of stability" of SHN was predicted in the 1960s, experimental efforts worldwide have continuously embarked * dongjm07@impcas.ac.cn ORCID(s): on such hugely expensive programs since it is always at the exciting forefront in both chemistry and physics [11,10]. However, the exact locations of the corresponding nuclear magic numbers remain unknown, and theoretical approaches to date do not yield consistent predictions. The direct measurement of nuclear binding energies and detailed spectroscopic studies of SHN with > 110 have been beyond experimental capabilities [12,13,14], therefore, to uncover their underlying structural information, one has to resort to the mere knowledge about measured -decay energies and half-lives [14]. The formation probability, if available microscopically, is of enormous importance to change this embarrassing situation in combination with accurately measured -decay properties. Because of its fundamental importance, theoretically, the exploration of the -particle formation can be traced back to 1960 [15], which triggered extensive investigations with shell models [3,16,17,18], Bardeen-Cooper-Schriffer (BCS) models [3,19,20] and Skyrme energy density functionals [21] later. The formation amplitude is regarded as the overlap between the configuration of a parent nucleus and the one described by an -particle coupled to the daughter nucleus. In particular, 212 Po as a typical -emitter with two protons and two neutrons outside the doubly magic core 208 Pb, was discussed extensively. Nevertheless, these calculations disagree on the decay width, and underestimate it substantially [3,20,22]. To improve the calculations, the shell model combined with a cluster configuration, was proposed with a treatment of all correlations between nucleons on the same footing [23]. Yet, these calculations tend to be difficult to generalize for nuclei more complex than 212 Po. Over the past two decades, several new approaches have been put forward to calculate the formation probability in different frameworks, including the pairing approach [24], scheme [25], quantum-mechanical fragmentation theory [26], cluster formation models [27,28,29], a quartetting wave function ap- Figure 1: (Color online) Schematic illustration of the physical picture for an -cluster formation in the multistep model through one of many pathways. The intermediate configuration (IC) is the parent state but with a neutron level and a proton level fully occupied ( 2 = 1). These four nucleons filling the two single-particle levels will jump to the unoccupied -and -levels of the daughter nucleus, and then automatically assemble into an -particle finally.
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 efforts have been made and considerable progress has been achieved, no fully satisfactory approach has yet been found until now. To explore the -particle formation probability , we propose a multistep model, and calculate the explicitly without introducing any adjustable parameter with the help of a self-consistent energy density functional theory.
Before we explore the -cluster formation probability, we first discuss briefly the proton spectroscopic factor of proton radioactivity. The initial state is proton quasiparticle excitations of parent BCS vacuum † |BCS⟩ P and the final state is † |BCS⟩ D with † |BCS⟩ D = (D) † + (D) |BCS⟩ D [45]. The spectroscopic factor (for spherical nuclei) is then given by = | D ⟨BCS| † |BCS⟩ P | 2 ≈ ( (D) ) 2 [45,46], where ( ( ) ) 2 = 1 − ( ( ) ) 2 is the probability that the spherical orbit of the emitted proton is empty in the daughter nucleus. Accordingly, the can be iconically interpreted as a probability that the blocked odd-proton in the -orbit of the parent nucleus jumps to the unoccupied -orbit of the daughter nucleus. With the inclusion of the calculated from nuclear many-body approaches, the partial half-lives for spherical proton emitters can be quite well reproduced [47,48], indicating the success of the strategy for proton spectroscopic factor.
Inspired by the proton radioactivity, our multistep model is proposed, and a sketch is exhibited in Fig. 1 to show schematically the physical picture of our model. However, different from the proton radioactivity where the emitted proton comes from the blocked odd-proton orbit in the parent nucleus, the neutrons (protons) inside the emitted -particle could come from any single-neutron (proton) level in principle. Therefore, we introduce the intermediate configuration (IC) with mass number driven perhaps by quantum fluctuation and residual interactions, to characterize which levels donate the complete four nucleons for the -formation. The IC is a state that a single-neutron level ( ) and a singleproton level ( ) in the parent nucleus are fully-occupied, namely, their occupation probabilities 2 = 1 for = , (which makes sure there are exactly four nucleons from these two levels to generate an -particle), being analogous to the blocked odd-proton in proton radioactivity. In fact, it is a component of the initial parent state according to the interpretation of quantum mechanics, for example, the fourquasiparticle states in the picture of angular-momentum projected models [49]. And there are many IC states and hence many pathways to form the -particle, where Fig. 1 just illustrates one of the pathways, and hence our strategy is obviously distinguished from other models. These four quasiparticles in the -and -levels are going to form an -particle and the remaining −4 nucleons accordingly form a daughter nucleus, and the pathway via this IC is marked as ( , ) for the sake of the following discussion. Accordingly, the probability to find a final configuration Ψ 2n-2p Ψ D in the wavefunction of the parent nucleus Ψ P through a given intermediate configuration Ψ IC is which is the formation probability of the daughter nucleus through this pathway. For a fully microscopic treatment, in principle, manybody wave functions in the laboratory frame are need for |Ψ P ⟩, |Ψ IC ⟩, |Ψ 2n2p Ψ D ⟩. While such treatments are challenging and in this work a practical approximation is adopted, i.e., we neglected beyond-mean-field effects and employ the corresponding mean-field wave functions in the intrinsic frame. For a deformed superfluid nucleus with nucleons paired by up and down spins, within the BCS formulation, the overlap integral of ⟨Ψ P |Ψ IC ⟩ is written as in which 2 ( 2 ) represents the probability that the two-fold degenerate -th single-particle level is occupied (unoccupied). The final state is written as Ψ 2n-2p Ψ D = † † |BCS⟩ , where † ( † ) creates two neutrons (protons). As an approximation, we assume the daughter even-even core can be described within the BCS approach, just like that in proton radioactivity [45]. Therefore, ⟨Ψ IC |Ψ 2n-2p Ψ D ⟩ expressed in terms of single-particle properties is given by where denotes the normalized single-particle wavefunction. Iconically, the four nucleons escaping from the IC jumps into the unoccupied -and -levels of the daughter nucleus, with a probability ( (D) (D) ) 4 .
We make the following two assumptions: 1) The formation probability of the -cluster is identical to that of the daughter nucleus, i.e., , that is, the formation of the -particle is achieved accordingly once the daughter nucleus is generated. This means the four nucleons escaping from the IC jumping into the unoccupied -and -levels of the daughter nucleus are expected to automatically assemble into an -particle at nuclear surface spontaneously. Namely, the transition probability from Ψ 2n-2p to an actual -cluster state Ψ is (Ψ 2n-2p → Ψ ) = 1 near the low-density nuclear surface. It is generally believed that an -cluster is formed only in regions that the nuclear matter density is low, and there is a similar case that an -like state generates automatically below Mott ≃ 0.03 fm −3 in quartetting wave function approach [30,31,32]. Intuitively, that the nucleons spontaneously organize into an doubly magic -particle, makes the system more stable. 2) Each pathway for the formation process is expected to be independent of the others. We sum over all pathways (i.e., through different ICs) to eventually achieve the -particle formation probability via with Eqs. (1)(2)(3). The dimensionless formation probability here is the expectation value of the -cluster number that can be emitted. The stationary-state description of a timedependent cluster formation process is a quite good approximation and simplifies the problem enormously [3], which is widely used at present for -decay. It is valid because halflives of -emitters are very long (10 −6 − 10 17 s) compared with the "periods" of nuclear motion (10 −21 s) and hence in the time evolution of a decaying state the nucleons has a large number of opportunities to get clustered and to get the clusters dissolved before it can actually escaped from the nucleus [3]. Our approach involves the structure of both parent and daughter nuclei, but does not involve an intrinsic state or a localized density distribution of the -cluster, being significantly different from the standard shell or BCS models where a Gauss-shaped intrinsic -cluster wavefunction is introduced. The quartetting wave function approach is a successful method proving a reasonable behavior for the of even-even Po isotopes [31], which does not involve such an intrinsic -cluster wavefunction to calculate either, and does not employ the overlap of the wavefunctions between the initial and final states. The wavefunction of the bound state for the center-of-mass motion of four correlated nucleons is obtained by solving the corresponding Schrödinger equation, and then the is calculated by integrating the modular square of this wavefunction in the region below Mott density since an -like state generates automatically at such low densities [30,31,32]. Different substantially from this quartetting wave function approach, the in our work is still based on the concept of overlap integrals, and finally can be calculated with the compact expression of Eq. (4) with the help of existing many-body approaches without introducing any adjustable parameter.
The single particle properties in Eq. (4) are determined within 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-consistent calculations is parameterized by means of about ten coupling constants that are calibrated to basic properties of nuclear matter and finite nuclei, which enables one to perform an accurate description of ground state properties and collective excitations over the whole nuclear chart [51,52,53], and has become a standard tool in low energy nuclear structure. The explicit calculations are carried out based on a standard code DIZ [54] for deformed nuclei, with the NL3 interaction [55] for the mean-field and the calibrated D1S Gogny force [57] for the pairing channel. The NL3 parameter set has been used with enormous success in the description of a variety of ground-state properties of spherical, deformed and exotic nuclei [55,56], and the calibrated D1S Gogny force enables one to well reproduce the odd-even staggerings on nuclear binding energies [57]. We concentrate on the even-even Po, Rn and Ra isotopes with spherical or near-spherical shapes, because their -decays tend to have large branching ratios (100% in most cases) and their corresponding half-lives were best measured experimentally [58]. On the other hand, these -decay cases usually do not involve excited states and angular momentum transfers, and thus serve as an optimal testing ground to examine our model. Moreover, the values of overlap integrals Π ⟨ (D) | (IC) ⟩ for these nuclei can be taken as unity. The products in Eqs. (2,3) along with the summation in Eq. (4) are truncated at 5 MeV for the single-nucleon spectra to achieve convergence.
To assess the validity of our multistep model, we explore the role of the formation probability in half-life calculations. The widely accepted formulas, i.e., the semi-empirical Exp.
1∕2 − ′ + log 10 as a function of ′ obtained with the UDL for = 1 (left) and ≠ 1 (right). The coefficient is fixed at the fitted value in the two cases, respectively. The straight lines are given by ′ + . Here is the corresponding correlation coefficient.
Viola-Seaborg formula (VSF) [59] and the universal decay law (UDL) based on the -matrix expression [60] are employed, which are respectively given as log 10 1∕2 = + √ + + − log 10 , (5) with ( ) is the proton (mass) number of a given parent nucleus. The decay energy and half-life 1∕2 are in units of MeV and second, respectively. = 1 ( ≠ 1) corresponds to the results without (with) the inclusion of the formation probability.
The fitting procedures are performed in the cases of = 1 and ≠ 1 respectively to test whether or not the predicted could improve substantially the accuracy of the two formulas. It is worth pointing out that the UDL has already included the logarithmic formation amplitude which is assumed to be linearly dependent upon ′ . Therefore, in Eq. (6), the ′ -dependent formation probability is replaced by the presently calculated . The root-mean-square (rms) deviations √ ⟨ 2 ⟩ and average deviations ⟨ ⟩ for the two formulas with = 1 and ≠ 1 are summarized in Table 1. The inclusion of the indeed greatly improves the accuracy of both the VSF and UDL. The UDL with a solid physical ground but less parameters, works better than the VSF, and reproduces the available experimental half-lives within a factor of 2 in the case of = 1. Yet, when the microscopically calculated is included, the deviation of the refitting is reduced down to around 20%. The good agreement between the calculated half-lives and the experimental data  is quite encouraging, indicating the reliability of the formation probability given by the multistep model. In Fig. 2, we plot the UDL fittings but replace the half-lives with the experimental values to more visually reveal the role of , and that the inclusion of the systematically improves the agreement with data is exhibited. The highly linear correlation is displayed for ≠ 1, with a correlation coefficient as high as = 0.9998, suggesting the success of our formation probability and the validity of the two assumptions.
Furthermore, is extracted in turn by using the ratio of the theoretical half-life to the experimentally observed value. The barrier penetrability of -particle, is achieved theoretically by the WKB approximation which turns out to work excellently [61], where the potential barrier is constructed by a simple "Cosh" potential plus the Coulomb barrier model (CM) [62]. Here the extracted should be considered as a relative value. By selecting an optimal constant assault frequency, the extracted values with varying neutron number are compared with the results given by the multistep model in Fig. 3. In sharp contrast with half-lives, the values are located in a relatively narrow range, leading to the success of the empirical half-life laws even when is not included. The values follow the similar behavior with regard to the Po, Rn and Ra isotopic chains-that is, gradually drop with increasing neutron number up to the spherical magic number = 126, attributed to the increased stability of isotopes when approaching the magic number, and then they increase drastically with neutron number. Such a general trend of the extracted is successfully reproduced within our method. Typically, the of 212 Po, being expected to be large owing to its two protons and two neutrons outside the shell closure core 208 Pb, is about six times larger than that of its neighbor 210 Po. The weight of the cluster component in 212 Po is large, which is exactly what one needs to simultaneously describe the B(E2) [63] and the absolute -decay width [3,23] within the shell model plus a cluster component. The distinct behavior that varies abruptly when the magic number is crossed, confirms that the particularly significant role of shell effects on is reasonably accounted for in Eq. (4) via the single-particle properties. As one expects, reaches its minimum at the shell closure = 126 as the result of the well-known shell stability that strongly enhances the nuclear binding.
In order to analyze the contribution of each pathway in Eq. (4) to , Fig. 4 illustrates ( , ) = |⟨Ψ IC |Ψ P ⟩| 2 ⋅ |⟨Ψ 2n-2p Ψ D |Ψ IC ⟩| 2 versus the single neutron and single proton energies ( n , p ) by taking the typical nucleus 212 Po as an example. The formation probability in Eq. (4) is predominantly determined by the pathways belonging, in the IC, to the fully-occupied neutron and proton levels and slightly above the Fermi surfaces, i.e., these levels are the major nucleon donors to constitute the emitted -particle. The contributions from other pathways drop sharply when the fullyoccupied levels ( , ) gradually go away from these leading ones.
It is well-known that the concept of -clustering is essential for understanding the structure of light nuclei. In some cases, light nuclei behave like molecules composed of clusters of protons and neutrons, such as the definite 2 cluster structure of 8 Be [64] reflected in a localized density distributions. But whether such type of cluster structure exists or not in heavy nuclei is uncertain. In our strategy, however, a localized -cluster does not pre-exist inside a parent nucleus, but is generated during the decay process in our model. Therefore, the scenario that a continuous formation and breaking of the -cluster until it escapes randomly from the parent nucleus [3], is supported. As a result, the mechanism of formation in heavy -emitters is different markedly from that in light nuclei.
Since the formation probability is highly relevant to the quantum-mechanical shell effects, the extracted with a high-precision UDL combined with experimentally measured -decay properties, is of great importance for digging up valuable structural information of SHN and exotic nuclei. The correlations between the values of SHN have suggested that the heaviest isotopes reported in Dubna do not lie in a region of rapidly changing shapes [65]. Therefore, the versus proton number exhibits a behavior analogous to isotopic chains shown in Fig. 3, attributed to the nearby shell closure. By employing the NL3 interaction, the heaviest nucleus 294 Og ( = 118) and its isotonic neighbor 292 Lv ( = 116), are predicted to be nearly spherical. The calculated is 0.66 for 292 Lv while it reduces to 0.45 for 294 Og with the ratio of ( 292 Lv)∕ ( 294 Og) = 1.5, being indeed similar to the trend shown in Fig. 3, which is consistent with the fact that the NL3 interaction itself predicts the adjacent = 120 as a magic number. On the other hand, with the UDL of Eq. (6) along with experimental data [10], the extracted ratio of ( 292 Lv)∕ ( 294 Og) is as high as 3.2 +4.3 −1.8 (4.5 +5.8 −2.2 ) with half-life data of 292 Lv from Ref. [66] (Ref. [67]) where the significant uncertainties are due to the low-statistics data for the measurements, and agrees marginally with the above theoretical value. Hence the probable magic nature of = 120 is suggested. The future measurements with a much higher accuracy for these nuclei together with their isotopes are encouraged, which would pin down the proton magic number eventually.
Intriguingly, the superallowed -decay to doubly magic 100 Sn was observed recently, which indicates a much larger -formation probability than 212 Po counterpart [68]. Within the framework of the quartetting wave function approach, an enhanced -cluster formation probability for 104 Te was found because the bound state wavefunction of the four nucleons has a large component at the nuclear surface [32]. Yet, a large for 104 Te is inconsistent with our prediction of = 0.24, suggesting the onset of an unusual type of nuclear superfluidity for self-conjugate nuclei, i.e., the protonneutron pairing which is not well-confirmed at present. This isoscalar pairing would considerably impact on the singleparticle spectra and hence the , and its absence in our CDF calculations leads to the underestimated . Therefore, as a new way independent of Ref. [69], combined with precisedecay measurements for ≃ nuclei, our approach for the could enable us to clarify this abnormal pairing interaction in turn by employing CDF approaches with the inclusion of a tentative isoscalar superfluidity.
In general, computing the formation probability defined in Eq. (4) by nuclear density functionals with a very high accuracy, is out of reach at present because of the wellknown fact that the single-particle levels are not well-defined in the concept of the mean-field approximation especially for well-deformed nuclei. The approximate single-particle spectrum is perhaps the primary cause which leads to the deviation of the theoretical calculations. Nevertheless, based on two intuitive assumptions and without any phenomenological adjustment, our strategy opens a new perspective to account for the formation mechanism. This is highly important to help one to uncover the underlying knowledge about superheavy nuclei and isoscalar pairing, in combination with experimental observations. For example, the experimentally measured -decay properties of heaviest nuclei combined with our calculated suggest the probable proton-magic nature of = 120. To treat the many-body wave functions in a laboratory frame beyond mean-field approximation, is planed as a future work.