Systematics of α-preformation factors in closed-shell regions

Systematics of α -preformation factors in closed-shell regions 1 corresp first-author dengdaming@smail.nju.edu.cn dengdaming@smail.nju.edu.cn dengdaming@smail.nju.edu.cn 1 2 3 corresp zren@nju.edu.cn zren@nju.edu.cn zren@nju.edu.cn Key Laboratory of Modern Acoustics and Department of Physics, Nanjing University, Nanjing 210093, China Key Laboratory of Modern Acoustics and Department of Physics, Nanjing University, Nanjing 210093, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou 730000, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou 730000, China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, China The α -preformation factors of medium and heavy-mass nuclei are calculated by using the cluster-formation model. The obtained preformation factors of even-even, odd- A , and odd-odd nuclei consist in both magnitude and trend with the general features predicted by standard calculations. The variation of α clustering affected by the evolution of nuclear structure is observed from different behaviors of preformation factors. We typically analyze the variation of preformation factors in the closed-shell N =126 and Z =82 regions, and discuss in detail the structural effects on α -cluster formation. This work shows the strong correlation between α -preformation factors and the shell structure, which would be a useful reference for microscopic cluster-model calculations of α -decay half-lives. α -preformation factors α clustering α decay Medium and heavy-mass nuclei 1 Introduction 1 As one of the fundamental decay modes of unstable nuclei, α decay receives constant attention due to its importance in investigations of nuclear stability and structure. Experimentally, the identifications of newly-synthesized superheavy elements are mostly achieved by detecting α -decay chains [ 1 1 ]. Meanwhile, measurement of α decay energy also provides reliable information for the study of nuclear mass and excitation level [ 2 2 - 4 4 2-4 ]. On the other hand, since α -decay half-life is closely related to the structural properties of the decaying system [ 5 5 - 7 7 5-7 ], α decay is used as a powerful tool to extract the structural information such as deformation [ 8 8 , 9 9 ], nucleon density distribution [ 10 10 , 11 11 ], α -preformation probability [ 12 12 - 16 16 12-16 ], etc. As a kind of microscopic theory of α decay, the cluster model, which considers the decaying nucleus as an α -core system, has been able to reproduce the half-lives of most known α -radioactive nuclides within permissible range of deviation [ 17 17 , 18 18 ]. Furthermore, the α -decay fine structure (branching ratios to different excitation levels) was well described in recent coupled-channel calculations [ 19 19 , 20 20 ]. Although the present cluster-model description can well reproduce most experimental data, the calculation still can be improved because the α -preformation factor employed in the theory is not consistently evaluated due to the complexity of nuclear many-body problem. More importantly, this factor is considered to carry most information of nuclear structure, which is essential to understand how α clustering happens in heavy nuclei. Within the cluster-model description, the preformation factor ( P α ), which is also named spectroscopic factor in some literatures, is introduced to describe the probability of an α cluster forming inside the parent nucleus. Usually, this factor is presumed as a constant for a certain kind of nuclei (even-even/odd- A /odd-odd) whereby the value is determined through minimizing the deviation between theoretical and experimental decay widths. This simple treatment yields satisfactory results for most open-shell nuclei, and also there are statistical analyses suggesting that the P α factor should vary smoothly in the open-shell region [ 21 21 , 22 22 ]. As for nuclei around closed magic shells, however, such an assumption may cause large discrepancy of α -decay half-life between theoretical predictions and the experimental data [ 23 23 , 24 24 ]. Progress on microscopic calculation of P α has been made on a typical nucleus 212 Po, which can be regarded as an α cluster plus a double-magic core 208 Pb. Varga et al. successfully reproduced the experimental decay width of 212 Po by using a wavefunction with combined shell-model and cluster-model components as the initial state of parent nuclei, and the P α factor was consistently obtained as 0.3 [ 25 25 , 26 26 ]. Recently, Röpke et al. attempted to describe the transition of α -cluster state in 212 Po from an unbound four-nucleon shell model state to an intrinsic bound α -like state as the baryonic density decreases at the nuclear surface [ 27 27 ]. The α formation is considered to take place at a critical density as 0.03 fm -3 . The microscopic calculations above were performed due to the simplicity of structure for 212 Po, but as more valence nucleons are involved, the complexity of depicting the α correlation in nuclei would largely increase. Therefore, it would be very difficult to pursue a fully microscopic description of α -preformation process for nuclei without an α -core structure. As a simple method to obtain the information of P α for known α emitters, one can roughly extract the P α factors through dividing the experimental decay width by the theoretical value. Numbers of works are devoted to this approach and reasonable results are obtained [ 10 10 , 28 28 - 31 31 28-31 ]. However, in some cases such a simple treatment can not ensure a smooth variation of P α for adjacent nuclei in open-shell regions, because the decay width is strongly dependent on the chosen α -daughter potential. To obtain a reasonable P α variation, the recently proposed cluster-formation model (CFM) suggests that the P α factors can be phenomenologically extracted from the experimental binding energies [ 33 33 , 34 34 ]. Based on simple quantum-mechanical considerations, the relation between P α and binding energy is derived by considering the interaction between surface nucleons as a predominant factor to the formation of α cluster. More recently, we successfully generalized the model to odd- A and odd-odd nuclei by including the effect from unpaired nucleons [ 35 35 ]. The results obtained reasonably agree with the major features predicted in previous researches. In the present study, in order to obtain a comprehensive view of the systematics of P α in the closed-shell region, the calculation of P α factors is extended by including more experimental data. Up to 505 nuclei are investigated and analyzed within the CFM, and we analyze in detail the behavior of P α in the typical N =126 and Z =82 shell regions. The paper is organized as follows. In Sect. 2, we briefly present the theoretical formulism of the CFM, specially the content of extracting the α -cluster formation energy. In Sect. 3, we show the numerical results and discuss in detail the correlations between P α variation and its associated structural effects. A summary of the results is given in Sect. 4. 2 Theoretical descriptions 1 Within the CFM, the clusterization state, Ψ i , is introduced to describe different kinds ( i ) of possible clustering of nucleons. The clusterization state is the eigenstate of its corresponding clusterization Hamiltonian, H i , which is derived from the theoretical separation of the many-body Hamiltonian for the parent nucleus into the one for a specific clusterization configuration [ 33 33 ]. Accordingly, the initial state, Ψ , of the parent nucleus can be defined as the superposition of all these possible clusterization states, and the total Hamiltonian for the system, H , can be written as the summation of corresponding clusterization Hamiltonians, <math display="block" id="M1"><mrow><mi>Ψ</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mi>i</mi><mi>N</mi></munderover></mstyle><msub><mi>a</mi><mi>i</mi></msub><msub><mi>Ψ</mi><mi>i</mi></msub><mn>,</mn></mrow></math> <math display="block" id="M2"><mrow><mi>H</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mi>i</mi><mi>N</mi></munderover></mstyle><msub><mi>H</mi><mi>i</mi></msub><mn>,</mn></mrow></math> where a i is the superposition coefficient for Ψ i . Since all these clusterization states describe the same system, they are assumed to share a same eigenenergy, which equals to the eigenenergy, E , for the total wavefunction. Consequently, considering the orthonormality of the clusterization wavefunctions, one can derive the following relation in terms of energy [ 33 33 ], <math display="block" id="M3"><mrow><mi>E</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mi>i</mi><mi>N</mi></munderover></mstyle><mo>|</mo><msub><mi>a</mi><mi>i</mi></msub><msup><mo>|</mo><mn>2</mn></msup><mi>E</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mi>i</mi><mi>N</mi></munderover></mstyle><msub><mi>E</mi><mrow><mtext>f</mtext><mi>i</mi></mrow></msub><mn>,</mn></mrow></math> where E f i is the formation energy for the cluster in clusterization state Ψ i . Specifically in the case of α clusterization, the preformation probability of α cluster can be properly defined as <math display="block" id="M4"><mrow><msub><mi>P</mi><mi>α</mi></msub><mo>=</mo><mo>|</mo><msub><mi>a</mi><mi>α</mi></msub><msup><mo>|</mo><mn>2</mn></msup><mo>=</mo><mfrac><mrow><msub><mi>E</mi><mrow><mtext>f</mtext><mi>α</mi></mrow></msub></mrow><mi>E</mi></mfrac><mn>,</mn></mrow></math> where a α denotes the superposition coefficient for α clusterization state Ψ α . In principle, to accurately determine this coefficient, one should calculate the overlap integral between α clusterization state and the initial state wavefunction by solving the time-independent Schrödinger equation. However, according to Eq. (4), the P α factor can be alternatively evaluated if the ratio of the formation energy to the total energy is determined. For medium and heavy-mass nuclei, it is suggested that the formation of α cluster happens at the nuclear surface [ 36 36 - 38 38 36-38 ]. One can reasonably consider the α correlation of surface nucleons contributes predominantly to the cluster formation. Since the α correlation is based on the pairing and n-p correlations, which are widely investigated on the basis of binding energy [ 39 39 - 41 41 39-41 ], it would be possible to determine these two energy values by extracting from experimental binding energies. Within the CFM, the α formation energy, E f α , represents the intrinsic correlation energy of the α cluster, whereas the energy, E , corresponds to the total energy of the considered four-nucleon system. For even-even nuclei, both these energies were determined by Ahmed and co-workers after they systematically analyzed the nucleon-nucleon correlations in the heavy-mass region [ 33 33 ]. In our recent research, we generalized the model to odd- A and odd-odd nuclei by expressing these two energies in a separation-energy systematics [ 35 35 ], <math display="block" id="M5"><mrow><msub><mi>E</mi><mrow><mi>f</mi><mi>α</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mn>2</mn><msub><mi>S</mi><mtext>p</mtext></msub><mo>+</mo><mn>2</mn><msub><mi>S</mi><mtext>n</mtext></msub><mo>−</mo><msub><mi>S</mi><mi>α</mi></msub></mrow></mtd><mtd><mrow><mo stretchy="false">(</mo><mtext>even</mtext><mo>−</mo><mtext>even</mtext><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mn>2</mn><msub><mi>S</mi><mtext>p</mtext></msub><mo>+</mo><msub><mi>S</mi><mrow><mtext>2n</mtext></mrow></msub><mo>−</mo><msub><mi>S</mi><mi>α</mi></msub></mrow></mtd><mtd><mrow><mo stretchy="false">(</mo><mtext>even</mtext><mo>−</mo><mtext>odd</mtext><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>S</mi><mrow><mtext>2p</mtext></mrow></msub><mo>+</mo><mn>2</mn><msub><mi>S</mi><mtext>n</mtext></msub><mo>−</mo><msub><mi>S</mi><mi>α</mi></msub></mrow></mtd><mtd><mrow><mo stretchy="false">(</mo><mtext>odd</mtext><mo>−</mo><mtext>even</mtext><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>S</mi><mrow><mtext>2p</mtext></mrow></msub><mo>+</mo><msub><mi>S</mi><mrow><mtext>2n</mtext></mrow></msub><mo>−</mo><msub><mi>S</mi><mi>α</mi></msub></mrow></mtd><mtd><mrow><mo stretchy="false">(</mo><mtext>odd</mtext><mo>−</mo><mtext>odd</mtext><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></mrow><mn>,</mn></mrow></math> <math display="block" id="M6"><mrow><mi>E</mi><mo>=</mo><msub><mi>S</mi><mi>α</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where S p ( A , Z ), S n ( A , Z ), and S α ( A , Z ) are single-proton separation energy, single-neutron separation energy, and α -cluster separation energy, respectively. The definitions of these separation energies are given by <math display="block" id="M7"><mrow><msub><mi>S</mi><mrow><mtext>2p</mtext></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>B</mi><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>2,</mn><mi>Z</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M8"><mrow><msub><mi>S</mi><mrow><mtext>2n</mtext></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>B</mi><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>2,</mn><mi>Z</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M9"><mrow><msub><mi>S</mi><mi>α</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>B</mi><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>B</mi><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>4,</mn><mi>Z</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Figure 1 Figure 1 explains schematically the detailed components of the formation energy shown in Eq. (5). In the α clusterization state, each nucleon of the α cluster contributes a S p (for proton) or S n (for neutron) to the α formation energy. The summation of four single-nucleon separation energies doubly counts the internal interactions of the α cluster when compared to the total energy, S α . As a result, the formation energy can be extracted by this summation minus the α -cluster separation energy. In the case of even-even nuclei, the single-nucleon separation energies of the two protons (neutrons) are not distinguished because all nucleons are paired in the ground state. But for odd- A and odd-odd systems, the existence of unpaired nucleon can largely suppress the cluster formation. To incorporate the influence from the unpaired nucleon(s), the term 2 S p (2 S n ) in formation energy should be replaced with a two-nucleon separation energy S 2p ( S 2n ). In this way, the hindrance caused by the unpaired nucleon can manifest itself as a reduction in formation energy, and consequently the P α would decrease if more odd nucleons are included. It should be noted that α decays of odd- A and odd-odd nuclei are split into favored and unfavored transitions. The P α factors calculated by using Eq. (5) are mainly for the favored case, because only the influence from binding energy is considered in the present model. For unfavored decays, the α formation process becomes extremely complicated due to the change of spin-parity between parent and daughter states. This makes the P α factor highly correlated to the details of the wavefunctions which are still too difficult to be acquired. Therefore, the P α given by the CFM basically correspond to favored α decays. 3 Results and discussion 1 In this work, the P α of 505 nuclides (138 even-even nuclei, 254 odd- A nuclei, and 113 odd-odd nuclei) are calculated using Eq. (4), Eq. (5), and Eq. (6). All the investigated nuclei have a positive Q α value and their corresponding binding energies are taken from the newest atomic mass evaluation (AME2012) tables [ 42 42 ]. As is known, the P α factors also can be evaluated indirectly by extracting from experimental α -decay half-lives. Calculations based on this approach have been performed by models such as the generalized density dependent cluster model (GDDCM) [ 10 10 ], generalized liquid drop model (GLDM) [ 28 28 , 29 29 ], and Hamiltonian energy density approach in terms of the SLy4 Skyrme-like effective interaction (SLy4) [ 16 16 , 31 31 ]. Before we present the details of our results, it would be interesting to compare the calculated P α factors with these models to examine the reliability of the CFM. In Fig. 2 Fig. 2 , we show the P α factors of Po isotopes calculated by different models. It can be clearly observed that the trends of P α variation predicted by these four models are very similar. The large shell effect for N =126 is reasonably reproduced by all these models. Furthermore, the P α factors by the CFM vary more smoothly than by the others. This is mainly due to the fact that the binding energy almost increases homogenously with nucleon number in this region. The major difference in this figure is the magnitude of the P α factors. As can be seen that the results by GLDM and SLy4 come close to each other and are about one order of magnitude smaller than by the CFM and GDDCM, whose magnitudes are basically of the same order. It should be noted that the P α factor of 212 Po was microscopically evaluated as 0.3 in previous study [ 25 25 ]. This value is considered reliable and comes very close to the results 0.22 and 0.25 by CFM and GDDCM, respectively. Therefore, the P α obtained by the CFM and GDDCM might be more approximate to the realistic values. In order to investigate the detailed behavior of the P α factors in the N =126 region, we plot the P α evolution along different isotopic chains in Fig. 3 Fig. 3 . Interestingly, we find the results are mostly divided into two groups. For even- N isotopic chains, P α smoothly decreases as N increases below N =126, but a sharp increase happens for all isotopic chains when N goes beyond the neutron shell closure. This behavior shows that the stabilizing effect from N =126 closed shell strongly suppresses the formation of the α cluster. The sudden enhanced P α at N =128 can be attributed to the two loosely-bounded neutrons above the shell contributing predominantly to the cluster configuration, which manifests itself as a typical signature of the shell effect. However, one can observe such an increase for odd- N isotopic chains occurs at N =129 instead of N =127. Of a similar physics, this is because the α transition would become more favored if the neutron pair above the shell closure could be utilized to form the α cluster. In fact, the N =126 shell effect in P α evolution of even-even nuclei was revealed in previous studies [ 11 11 , 16 16 , 28 28 , 29 29 ]. In our results, not only a consistent P α evolution for even-even nuclei is reproduced, we also demonstrate the N =126 shell effect for odd- A and odd-odd nuclei and point out that the indications of such effect for even- N and odd- N isotopic chains are characterized by a typical increase emerging at different locations in the P α evolution. In addition to the N =126 shell effect, the systematic behavior of P α around Z =82 is more of our interest because the indication of Z =82 shell effect for very neutron-deficient nuclei was still obscured due to the lack of precise experimental data [ 43 43 , 44 44 ]. In Fig. 4 Fig. 4 , we show the P α variation of even-even and even-odd nuclei along different isotonic chains. It can be observed clearly that the Z =82 shell emerges as a typical dip at the shell closure for all isotonic chains. Moreover, the variation of P α at Z =82 is seen less significantly when compared with the N =126 case in Fig. 3 Fig. 3 , which implies the proton shell closure exerts a smaller influence on the preformation probability. On the other hand, due to the investigated nuclei in Fig. 3 Fig. 3 located around 208 Pb, the intervention of both neutron and proton shell closures may be also responsible for a larger amplitude in P α variation at N =126. It should be noted that in a recent experiment, the researchers precisely measured the α half-lives of some very neutron deficient nuclei and then extracted the α -particle formation probability by the universal decay law (UDL) [ 45 45 ]. They successfully observed the signature of Z =82 shell closure which also shows the above characteristics. Therefore, it is encouraging that this signature is reproduced by the CFM calculation without invoking any half-life data from experiments. Another feature that should be noted in Fig. 4 Fig. 4 is that the P α factor steadily climbs with increasing Z below the proton shell closure. As can be seen from Fig. 3 Fig. 3 , the behavior of P α below N =126 is totally different, for P α decreases rapidly due to the stabilizing effect of the closed magic shell. To understand the seemingly strange behavior observed below Z =82, one should consider the excitations of protons across the Z =82 shell closure. As we know, the α -cluster formation can be explained by the configuration mixing of cross-shell excitations of correlated paired nucleons [ 39 39 ]. Below N =126, the formation process is substantially suppressed because the large gap above shell closure makes it almost impossible for neutrons to be excited to a higher major shell. But as we mentioned above, the stabilizing effect of Z =82 shell is much smaller. Once the protons below are excited to the 82 < Z ≤ 126 major shell, the formation probability would be enhanced because both correlated proton and neutron pairs now occupy the levels within the same major shell. The fewer valence neutrons exist below the proton shell closure, the more likely such excitations would happen. This explains why P α increases with larger Z below the shell closure. On the other hand, the existence of this kind of excitation can also be evidenced by the shape-coexistence phenomenon around the Z =82 region. This phenomenon is due to the nuclear deformation which results from cross-shell excitations of protons and the n-p interaction [ 46 46 ]. Therefore, the increasing trend observed in Fig. 4 Fig. 4 is probably related to the excitations of protons across the Z =82 shell. In previous studies of α -decay half-lives of exotic isotones around N =126 shell, the P α factors are suggested to follow a linear relation with Z to achieve a better agreement with experimental data [ 23 23 , 24 24 ]. This Z -dependent relation is derived from a microscopic two-level model where the shell and blocking effects can be described by considering the pairing and n-p interactions between valence nucleons [ 39 39 ]. Interestingly, this linear behavior is also observed in our resul ts. In Fig. 5 Fig. 5 , we show the evolution of P α factors with Z for different isotonic chains. The P α factors are found to be highly linear with Z according to a given linear correlation coefficient R 2 > 0.9. The success in reproducing such a structural effect further confirms reliability of the CFM. Besides, it should be noted that this linear behavior seems more valid for even- Z isotones than for odd- Z isotones as can be seen from Fig 3 Fig 3 . The vanished linear behavior for odd- Z nuclei may be attributed to the α formation being interrupted by the unpaired proton, but it still requires further investigations to reach a conclusion. 4 Summary 1 The evolution of α -preformation factors around N =126 and Z =82 shells is systematically investigated by using the cluster-formation model. The results show consistent trend and order of magnitude of P α factors with previous statistical analyses. Strong correlation between nuclear structure and the behavior of P α evolution is demonstrated and discussed. In terms of the magnitude, P α of odd- A nuclei are generally smaller than those of even-even nuclei, while for odd-odd nuclei P α became even smaller due to the existence of both unpaired proton and neutrons. 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(Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2016 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-016-0151-1 http://dx.doi.org/10.1007/s41365-016-0151-1 . 2016 Nuclear Science and Techniques 06 27