Cluster-Daughter Overlap as a New Probe of Alpha-Cluster Formation in Medium-Mass and Heavy Even-Even Nuclei

We study the possibility to use the cluster-daughter overlap as a new probe of alpha-cluster formation in medium-mass and heavy even-even nuclei. We introduce a dimensionless parameter $\mathfrak{O}$, which is the ratio between the root-mean-square (rms) intercluster separation and the sum of rms point radii of the daughter nucleus and the alpha particle, to measure the degree of the cluster-daughter overlap quantitatively. By using this parameter, a large (small) cluster-daughter overlap between the alpha cluster and daughter nucleus corresponds to a small (large) $\mathfrak{O}$ value. The alpha-cluster formation is shown explicitly, in the framework of the quartetting wave function approach, to be suppressed when the $\mathfrak{O}$ parameter is small, and be favored when the $\mathfrak{O}$ parameter is large. We then use this $\mathfrak{O}$ parameter to explore systematically the landscape of alpha-cluster formation probabilities in medium-mass and heavy even-even nuclei, with $\mathfrak{O}$ being calculated from experimentally measured charge radii. The trends of alpha-cluster formation probabilities are found to be generally consistent with previous studies. The effects of various shell closures on the alpha-cluster formation are identified, along with some hints on a possible subshell structure at $N=106$ along the Hg and Pb isotopic chains. The study here could be a useful complement to the traditional route to probe alpha-cluster formation in medium-mass and heavy even-even nuclei using alpha-decay data. Especially, it would be helpful in the cases where the target nucleus is stable against alpha decay or alpha-decay data are currently not available.

Unlike alpha clustering in light nuclei which could be studied by using microscopic methods, probing alpha clustering in medium-mass and heavy/superheavy nuclei is more challenging thanks to intrinsic difficulties in solving quantum systems with a large number of nucleons. In literature, alpha-cluster formation in medium-mass and heavy/superheavy nuclei is usually investigated systematically by exploiting experimental data on alpha decay.
However, neither all the nuclei with alpha-cluster structures permit alpha decays, nor all the nuclei permitting alpha decays have their decay widths be measured. Therefore, new probes of alpha-cluster formation besides those based on the alpha decay are necessary. In this note, we would like to study the possibility to adopt the cluster-daughter overlap as a new probe of alpha-cluster formation in medium-mass and heavy nuclei, motivated by the observation that alpha-cluster formation is suppressed when the cluster-daughter overlap is large and favored when the cluster-daughter overlap is small. In Section II, we analyze the relation between the cluster-daughter overlap and the alpha-cluster formation quantitatively within the framework of the quartetting wave function approach [46][47][48], which is a microscopic method proposed recently to estimate alpha-cluster formation probabilities. The inclusion of shell-model properties in the quartetting wave function approach is also discussed recently in Ref. [49]. We introduce a dimensionless parameter O to measure quantitatively the degree of the cluster-daughter overlap. In Section III, the O parameter is then adopted to explore the landscape of alpha-cluster formation probabilities in medium-mass and heavy even-even nuclei, with O being calculated directly from experimentally measured charge radii. Special attentions are paid to the closed-(sub)shell effects. In Section IV, we give the conclusions.

TION IN THE LIGHT OF QUARTETTING WAVE FUNCTION APPROACH
We first study the relation between the cluster-daughter overlap and the alpha-cluster formation probability within the framework of the quartetting wave function approach. The quartetting wave function approach is inspired by the THSR wave function for light selfconjugate nuclei, and has been applied successfully in studying alpha-cluster formation in heavy/superheavy nuclei. Within this framework, we divide the wave function of the four valence nucleons uniquely as the product of the center-of-mass (COM) part and the intrinsic part. Following Ref. [46][47][48][49], by adopting the local-density approximation and ignoring the derivative terms of the intrinsic wave function, the Schrödinger equation for the COM wave function could be given by W (r) is the effective potential felt by the COM motion of the alpha cluster. A detailed derivation of Eq. (1) could be found in Ref. [46]. The alpha-cluster formation probability P α could be then obtained as follows: Here, ρ Mott B = 0.02917 fm −3 is the Mott density higher than which the the alpha cluster is believed to dissolve and merge with the daughter nucleus. The root-mean-square (rms) . 1: A schematic representation of the cluster-daughter system. In this work, we assume that the centers of mass of the alpha particle and the daughter nucleus coincide with their centers of charge. R α and R d are the rms point radii of the alpha particle and the daughter nucleus. R i is the rms intercluster separation. O is the center of mass of the parent nucleus. R 1 (R 2 ) is the displacement vector between O and the center of mass of the daughter nucleus (alpha particle). r i (r j ) is the displacement vector between the center of mass of the daughter nucleus (alpha particle) to a representative nucleon inside the daughter nucleus (alpha particle).
intercluster separation is given by where φ(r) is the normalized radial wave function of Φ(r). To measure the degree of the cluster-daughter overlap quantitatively, we introduce a dimensionless parameter O to measure the relative size between the rms intercluster separation R i and the sum of the rms point radii of the alpha particle R α ≡ R 2 1/2 α and the daughter nucleus R d ≡ R 2 1/2 d (see Fig. 1 for a schematic representation), As a result, a large (small) overlap between the alpha cluster and the daughter nucleus corresponds to a small (large) value of the O parameter.
In the rest part of this section, we would like to adopt the quartetting wave function approach to study the relation between the O parameter and the alpha-cluster formation probability. We take 212 Po = 208 Pb + α as an example, which could be viewed as an alpha cluster moving on the top of the doubly magic nucleus 208 Pb and has been investigated intensively by previous studies [41,42,[50][51][52][53]. For the density distributions for the daughter nucleus, we adopt [47,54] ρ π (r) = 0.0628948 as the proton and neutron distributions, respectively. The critical radius that marks the dissolution of the alpha cluster is then given by r cluster = 7.43825 fm. In other words, the alpha cluster persists only for r > r cluster and dissolves once r < r cluster . When r > r cluster , the effective potential W (r) = W ext (r) + W intr (r), with W ext (r) being the external potential inside which the alpha cluster moves and W intr (r) being the intrinsic energy of the alpha cluster inside the nuclear medium. The external potential W ext (r) could be approximated For the alpha emitter 212 Po, the two free parameters c and d could be determined by fitting the emission energy Q α and the alpha-decay half-life T 1/2 . The intrinsic potential W intr (r), on the other hand, is given by [46] and E (0) α = −28.3 MeV being the energy of the alpha particle in the vacuum. When r < r cluster , the cluster state of the four valence nucleons merges with the shell-model state of the daughter nucleus. In the local density approximation, we have W (r) = µ 4 , with µ 4 being a constant related to the Fermi energy of the four valence nucleons.
In Fig. 2, we study the relation between the cluster-daughter overlap measured by the O parameter and the alpha-cluster formation probability P α by varying the values of µ 4 from 0.95 Q α to 1.05 Q α . Fig. (1a) shows the behavior of the effective potential W (r) with the critical radius r cluster = 7.43825 fm displayed by the vertical line. As µ 4 becomes larger, the horizontal platform of W (r) within the critical radius r cluster becomes higher, and the energy pocket beyond the critical radius r cluster becomes deeper. Fig. (1b) shows the normalized radial wave function φ(r). As µ 4 becomes larger, more and more parts of the radial wave function φ(r) are pushed into the outer region r > r cluster . According to Eq. (2), this indicates that the alpha-cluster formation probability becomes larger and larger. Fig. (1c) displays the relation between the dimensionless parameter O and the alpha-cluster formation probability P α , hinting that there might exist approximately a simple positively-correlated linear relation between P α and O. Here, the rms point radii of the alpha particle, the daughter nuclei 208 Pb, and the alpha emitter 212 Po are compiled from the experimental values of their rms charge radii recorded in Ref. [55] by using [56]. This linear relation allows us to probe the landscape of alpha-cluster formation probabilities by studying that of the cluster-daughter overlap.

III. THE LANDSCAPE OF ALPHA-CLUSTER FORMATION PROBABILITIES PROBED BY CLUSTER-DAUGHTER OVERLAP
We investigate the landscape of alpha-cluster formation probabilities in medium-mass and heavy even-even nuclei based on discussions in Section II. Explicitly, we will study the landscape of the dimensionless parameter O along various isotopic and isotonic chains. Then, according to the linear relation between P α and O, the same landscape could also reveal important features of the landscape of alpha-cluster formation probabilities. To extract the O parameter from experimental nuclear ground-state charge radii, we make use of the relation [38,40,[57][58][59], Here, R 2 1/2 ch,P , R 2 1/2 ch,α , and R 2 1/2 ch,d are the rms charge radii of the parent nucleus, the alpha particle, the daughter nucleus, while R 2 1/2 i is the rms separation between the centers of mass of alpha particle and the daughter nucleus. Eq. (7) assumes that the centers of mass of the alpha particle and the daughter nucleus coincide with their centers of charge, and follows directly from the definition of rms charge radius of the parent nucleus: An explanation of the variables here could be found in the caption of Fig. 1 Fig. (1a) shows the behavior of the effective potential W (r). Fig. (1b) shows the normalized radial wave function φ(r). Fig. (1c) shows the relation between the dimensionless parameter O and the alpha-cluster formation probability P α . The critical radius r cluster = 7.43825 fm is displayed explicitly by the vertical line in Figs. (1a) and (1b).
Eq. (7). The experimental values of R 2 1/2 ch,P , R 2 1/2 ch,α , and R 2 1/2 ch,d could be found in Ref. [55], from which the value of R i ≈ R 2 1/2 i is obtained by solving Eq. (7). The O parameter could then be calculated according to its definition in Eq. (4). In this work, we will study the landscape of the O parameters of not only stable but also decaying nuclei which are unbound. For the latter case, the definition of the rms radius could be nontrivial.
However, the lifetimes of these decaying nuclei are typically very long. For instance, the lifetime of 210 Po is about 140 days, long enough to allow the electron scattering experiment to determine its charge radius practically. As a result, for our current purposes we can talk about the rms radii of these decaying nuclei safely.
We start with the medium-mass nuclei Te, Xe, and Ba, whose O-parameter landscapes are plotted in Fig. 3. It is found that, in these three nuclei, the O parameter (the alphacluster formation probability P α ) decreases first and reaches its minimum at around the N = 82 shell closure. After crossing N = 82, the O parameter (the alpha-cluster formation probability P α ) gets suddenly a large increase, manifesting clearly the effect of the N = 82 shell closure on P α . Such a behavior is consistent with previous studies on heavy nuclei in Ref. [28,48], which could be viewed as a nontrivial check of the assumption that the alpha-cluster formation probability P α is approximately proportional to the dimensionless O parameter. Compared with Ref. [45], which studies the alpha-cluster formation probabilities in Te, Xe, and Ba isotopes near the N = 50 shell based on limited alpha-decay data, the analysis here allows one to get a complementary understanding on how P α changes around the N = 82 shell, thanks to the availability of rich experimental data on ground-state charge radii in this region. We also study the effects of the Z = 50 shell along the N = 64 and N = 66 isotonic chains in Fig. 4. As expected, a large increase of the O parameter (the alpha-cluster formation probability P α ) is found from Z = 50 to Z = 52.
As mentioned before, lots of works have been devoted to study the alpha-cluster formation probabilities of heavy nuclei by exploiting alpha-decay data. Here, we would like to provide a complementary calculation by using experimental data of nuclear rms radii. The Oparameter landscapes of the heavy nuclei Hg, Pb, Po, Rn, and Ra around the N = 126 major shell could be found in Fig. 5. For the Po, Rn, and Ra isotopic chains in Fig. (5c), (5d) and (5e), the O parameters decrease first and get a large increase across the N = 126 shell, consistent with previous studies based on alpha-decay data [28,48]. In Fig. (5a), we plot the O parameters for the Hg isotopes in the region N = 102 − 122. It is interesting We study further the Z = 82 shell effects along the N = 114, 116, 118, and 120 isotonic chains in Fig. 6. Similar to the medium-mass nuclei, a large increase of the O parameter (the alpha-cluster formation probability P α ) is found from Z = 82 to Z = 84, consistent with Refs. [34,48].

IV. CONCLUSIONS
In this note, we study the possibilities to use the cluster-daughter overlap as a new probe of alpha-cluster formation in medium-mass and heavy even-even nuclei. Quantitatively, the cluster-daughter overlap is measured by a dimensionless O parameter which is the ratio between the rms intercluster separation and the sum of the rms point radii of the alpha particle and the daughter nucleus. The larger (smaller) the cluster-daughter overlap is, the smaller (larger) the O parameter will be. Within the quartetting wave function approach, previous studies based on alpha-decay data, the new probe could be used in cases where the target nucleus is stable against the alpha decay or the alpha-decay data are temporarily unavailable. The study here could be helpful for future theoretical and experimental studies on alpha-cluster formation in medium-mass and heavy regions, as well as providing extra motivations to carry out new experiments to measure charge radii in the medium-mass and heavy regions precisely. In this work, we assume both the alpha cluster and the daughter nucleus to be spherical. It is also important to consider further the impact of nuclear deformations, which according to previous studies such as Ref. [60,61] would enhance the alpha-cluster formation probability. Hopefully, this would be the topic of our future studies.