Decay Characteristics of Neutron Excess Zinc Nuclei

In neutron star mergers, neutron excess nuclei and the r-process are important factors governing the production of heavy nuclear systems. An evaluation of zinc nuclei suggests that the heaviest Z = 30 nucleus will have mass 88 with filling of the 3s1/2 neutron shell. A = 84 – 88 zinc isotopes have limited experimental half-life data, but the model predicts beta decay half-lives in the range of 60 – 100 ms. Based on comparisons to Z = 20 and Z = 26 systems, these results likely overestimate the experimental half-lives of these A = 84 – 88 neutron excess zinc nuclei. 1.0 Introduction The nucleosynthesis of heavy elements occurs by three basic processes that add protons or neutrons to a nuclear system1,2. The p-process adds protons and the sor slow process and ror rapid process adds neutrons. Capture of protons by nuclear systems produces predominantly proton-rich nuclei that tend to decay by positron emission and electron capture1,2. Neutron capture creates neutron-rich nuclei, and the resulting nuclear system depends upon the rate of neutron addition and the beta decay rates of the residual nuclei. In the s-process neutron capture chain, the time between successive neutron captures is sufficiently long for the product nucleus to beta decay to a stable system. Within the r-process, the time between neutron captures is too short to permit decays except for very rapid beta transitions. Therefore, the r-process must occur in an environment that has a high density of neutrons. The s-process typically occurs in red giant stars. The r-process occurs in a variety of astronomical events, including supernovae explosions and stellar mergers. Binary neutron star or neutron star and stellar-mass black hole mergers can form a massive rotating torus around a spinning black hole1. The matter ejected from these structures and from supernovae explosions is an important source of rapid neutron capture (r-process) nucleosynthesis1. Fully understanding the r-process requires knowledge of the properties of neutron excess nuclei involved in creating heavy nuclear systems. Unfortunately, the majority of these neutron excess systems have never been studied2. Closing this knowledge gap was a motivation for funding facilities for rare-isotope beams constructed at research facilities located around the world3-8. These facilities enable a new class of experiments to determine the physical properties needed by theoretical models to determine the structure of unstable neutron excess nuclei. Theoretical studies would complement the forthcoming experiments that will provide critical information on the unstable nuclei that must be understood in order to explain nuclear abundances observed in the universe2. In particular, the study of neutron excess Qeios, CC-BY 4.0 · Article, December 20, 2020 Qeios ID: JZI1LG · https://doi.org/10.32388/JZI1LG 1/12 systems and their decay properties are important considerations in understanding the r-process, and its importance in producing the observed elements in the universe. The study of neutron excess systems is also important for evaluating nuclear decay properties, nuclear structure under extreme conditions, and nuclear reaction mechanisms. Existing theoretical models have not been extensively applied to many of these neutron excess nuclei. This paper attempts to partially fill this void by calculating the decay properties of neutron excess systems that are important in nucleosynthesis. These theoretical studies should also assist in planning future experiments associated with neutron excess systems that are far removed from the line of stability. Neutron excess nuclei that merit study occur throughout the periodic table2-8 including nuclei in the 11 ≤ Z ≤ 32 range8. Previous studies provided half-life and structure calculations for neutron excess calcium9 and iron10 systems. This paper extends the approach of Refs. 9 and 10 to zinc systems as an additional investigation of neutron excess nuclei that are of potential astrophysical significance. An additional study of neutron excess fluorine systems11 was performed using a similar methodology. 2.0 Calculational Methodology A variety of models could be applied to the investigation of neutron excess nuclei. These models vary in sophistication, but the proposed model utilizes a basic single-particle approach. This is a reasonable first step because there are uncertainties in the nuclear potential that likely are more significant than the limitations introduced by a singleparticle approach. Since the method for calculating single-particle energies in a spherically symmetric potential is well-established, only salient features are provided. The model used to describe the particle plus core system represents an application of the standard method of Lukasiak and Sobiczewski12 and Petrovich et al.13 The binding energy ENLSJ of a particle in the field of a nuclear core is obtained by solving the radial Schrödinger Equation h2 2(2π)2μ d2 dr2 − L(L + 1) r2 − ENLSJ − VLSJ(r) UNLSJ(r) = 0(1) where r is the radial coordinate defining the relative motion of the nuclear core and the particle; VLSJ(r) is the model interaction; ENLSJ is the core plus particle binding energy; UNLSJ(r) is the radial wave function; and L, S, and J are the orbital, spin, and total angular momentum quantum numbers, respectively. The N quantum number is the radial quantum number, and μ is the reduced mass. Additional details of the model, as applied to neutron excess nuclei, are provided in Ref. 9. 3.0 Nuclear Interaction Nuclear stability with respect to alpha decay, beta decay, positron decay, and electron capture is addressed using the method previously published by the author and coworkers13 that is similar to the approach of Ref. 14. The singleparticle level spectrum is generated using a Woods-Saxon potential based on the Rost interaction15. The Rost interaction yields reasonable fits to observed single-particle levels in 120Sn and 138Ba. The pairing [ ( ) ] Qeios, CC-BY 4.0 · Article, December 20, 2020 Qeios ID: JZI1LG · https://doi.org/10.32388/JZI1LG 2/12 correction term of Blomqvist and Wahlborn16 is used in the calculations presented herein. The pairing correction improves the predicted energies of occupied levels in 120Sn, 138Ba, and 208Pb13. When applied to zinc nuclei, this methodology requires modification. Ray and Hodgson17 note that 40Ca and 48Ca require different potentials to properly fit their structure. Schwierz, Wiedenhöver, and Volya18 also investigated 40Ca and 48Ca and noted that a proper fit to the single-particle levels required a different potential for each energy level. Difficulties in the selection of an appropriate potential is an additional motivation for the utilization of single-particle levels in this study of neutron excess zinc nuclei. Similar issues in calculating the nuclear structure are noted for 70Fe, Z = 56 – 80 systems19,20, and in the zinc system for mean field and dispersive optical potential models21,22. The importance of nuclear correlations in describing the structure of 71Zn was noted in Ref. 23. The results in zinc and neighboring systems suggest that collective effects19 and nuclear correlations23 will also become important in zinc systems as the neutron number increases. These effects require the alteration of the nuclear potential as noted in Refs. 9 11, 17, 18, and 21 23. In view of the results of Refs. 9 – 11 and 17 23, the following modification is made to the Rost interaction: V0 = 51.6 1 ± 0.73 N − Z A [1 ± a(A)]MeV(2) where a(A) is a constant that was introduced in Ref. 9 to account for the variations in potential strength with A17-20. It is preferable that a(A) be constant for as many zinc isotopes as possible. Since the paper’s primary purpose is investigation of the neutron excess nuclei, determining a common a(A) value for the heaviest zinc systems is desirable. 4.0 Calculation of Half-Lives Using Eq. 1, single-particle levels are calculated for A ≥ 54 zinc isotopes. A ≥ 54 zinc nuclei were evaluated for stability with respect to alpha decay, beta decay, positron decay, electron capture, and two-proton (2p) decay. These calculations were performed to ensure that the nuclear structure contained no interloping states or structural defects, and that any decay modes in conflict with data were identified. The decay modes and half-lives of 88 ≥ A ≥ 54 zinc isotopes are summarized in Table I and compared to available data24,25. The alpha decay energies are calculated using the relationship based on Ref. 26. Qα = 28.3MeV − 2Sn − 2Sp(3) where Sn and Sp are the binding energies of the last occupied neutron and proton single-particle levels, respectively. Alpha half-lives can be estimated from Qα using standard relationships12. No alpha decay modes occur in the Table I summary of 88 ≥ A ≥ 54 zinc isotope decay properties. [ ] Qeios, CC-BY 4.0 · Article, December 20, 2020 Qeios ID: JZI1LG · https://doi.org/10.32388/JZI1LG 3/12

systems and their decay properties are important considerations in understanding the r-process, and its importance in producing the observed elements in the universe.
The study of neutron excess systems is also important for evaluating nuclear decay properties, nuclear structure under extreme conditions, and nuclear reaction mechanisms. Existing theoretical models have not been extensively applied to many of these neutron excess nuclei.
This paper attempts to partially fill this void by calculating the decay properties of neutron excess systems that are important in nucleosynthesis. These theoretical studies should also assist in planning future experiments associated with neutron excess systems that are far removed from the line of stability.
Neutron excess nuclei that merit study occur throughout the periodic table [2][3][4][5][6][7][8] including nuclei in the 11 ≤ Z ≤ 32 range 8 . Previous studies provided half-life and structure calculations for neutron excess calcium 9 and iron 10 systems. This paper extends the approach of Refs. 9 and 10 to zinc systems as an additional investigation of neutron excess nuclei that are of potential astrophysical significance. An additional study of neutron excess fluorine systems 11 was performed using a similar methodology.

Calculational Methodology
A variety of models could be applied to the investigation of neutron excess nuclei. These models vary in sophistication, but the proposed model utilizes a basic single-particle approach. This is a reasonable first step because there are uncertainties in the nuclear potential that likely are more significant than the limitations introduced by a singleparticle approach.
Since the method for calculating single-particle energies in a spherically symmetric potential is well-established, only salient features are provided. The model used to describe the particle plus core system represents an application of the standard method of Lukasiak and Sobiczewski 12 and Petrovich et al. 13 The binding energy E NLSJ of a particle in the field of a nuclear core is obtained by solving the radial Schrödinger where r is the radial coordinate defining the relative motion of the nuclear core and the particle; V LSJ (r) is the model interaction; E NLSJ is the core plus particle binding energy; U NLSJ (r) is the radial wave function; and L, S, and J are the orbital, spin, and total angular momentum quantum numbers, respectively. The N quantum number is the radial quantum number, and µ is the reduced mass. Additional details of the model, as applied to neutron excess nuclei, are provided in

Nuclear Interaction
Nuclear stability with respect to alpha decay, beta decay, positron decay, and electron capture is addressed using the method previously published by the author and coworkers 13 that is similar to the approach of Ref. 14. The singleparticle level spectrum is generated using a Woods-Saxon potential based on the Rost interaction 15 .
The Rost interaction yields reasonable fits to observed single-particle levels in 120 48 Ca and noted that a proper fit to the single-particle levels required a different potential for each energy level. Difficulties in the selection of an appropriate potential is an additional motivation for the utilization of single-particle levels in this study of neutron excess zinc nuclei. Similar issues in calculating the nuclear structure are noted for 70 Fe, Z = 56 -80 systems 19,20 , and in the zinc system for mean field and dispersive optical potential models 21,22 . The importance of nuclear correlations in describing the structure of 71 Zn was noted in Ref. 23. The results in zinc and neighboring systems suggest that collective effects 19 and nuclear correlations 23 will also become important in zinc systems as the neutron number increases. These effects require the alteration of the nuclear potential as noted in Refs. 9 -11, 17, 18, and 21 -23.
In view of the results of Refs. 9 -11 and 17 -23, the following modification is made to the Rost interaction: where a(A) is a constant that was introduced in Ref. 9 to account for the variations in potential strength with A 17-20 . It is preferable that a(A) be constant for as many zinc isotopes as possible. Since the paper's primary purpose is investigation of the neutron excess nuclei, determining a common a(A) value for the heaviest zinc systems is desirable.

Calculation of Half-Lives
Using Eq. 1, single-particle levels are calculated for A ≥ 54 zinc isotopes. A ≥ 54 zinc nuclei were evaluated for stability with respect to alpha decay, beta decay, positron decay, electron capture, and two-proton (2p) decay. These calculations were performed to ensure that the nuclear structure contained no interloping states or structural defects, and that any decay modes in conflict with data were identified.
The decay modes and half-lives of 88 ≥ A ≥ 54 zinc isotopes are summarized in Table I and compared to available data 24,25 . The alpha decay energies are calculated using the relationship based on Ref. 26.
where S n and S p are the binding energies of the last occupied neutron and proton single-particle levels, respectively.
Alpha half-lives can be estimated from Q α using standard relationships 12 . No alpha decay modes occur in the Table I summary of 88 ≥ A ≥ 54 zinc isotope decay properties.
[ ] Qeios, CC-BY 4.0 · Article, December 20, 2020   The beta decay half-lives are determined following the log ft methodology of Wong 26 . Allowed (first forbidden) transition half-lives were derived using the values of log ft = 5 (8). Given the uncertainties in the calculated level energies, second and higher forbidden transitions were not determined. Positron and electron capture half-lives were determined following the approach of Ref. 12.
The single-particle model is used to calculate the alpha, beta, positron, and electron capture decay half-lives. The 2p decay mode is evaluated using the methodology of Ref. 27. Since the 2p decay mode involves two protons, it is not easily evaluated using a single-particle model. The methodology of Ref. 27, as applied to the 54 Zn nucleus, is addressed in more detail in subsequent discussion.

Selection of Experimental Half-Lives
The

Results and Discussion
Using Eq. 2, the a(A) value was varied in increments of 0.005 to assess the applicability of the proposed model to predict the decay properties of A ≥ 54 zinc isotopes. In view of uncertainties in the model and associated interaction, a smaller increment was not deemed to be justified unless noted in subsequent discussion. The issues associated with fitting all nuclei in this mass region with a single potential 15-18 were noted previously.
Within the single particle model, 54 Zn -58 Zn nuclei fill the 1f 7/2 neutron shell. 54 Zn is a 2p emitter and was evaluated shells fill. This trend holds for the 2p 3/2 , 1f 5/2 , and 2p 1/2 neutron shells. However, as the neutron number increases to fill the 1g 9/2 , 2d 5/2 , and 3s 1/2 shells, single-particle model effects are supplemented by other contributions. This is characterized by an increasing a(A) value, which represents a greater contribution from other degrees of freedom including collective effects. A similar phenomenon was noted in Ref. 19  Although, there is no decay data for the 86 Zn nucleus, it was also modeled using an a(A) value of 0.095.
The heaviest zinc neutron excess systems (i.e., 87 Zn and 88 Zn) fill the 3s 1/2 neutron shell. There is no decay data for the 87 Zn and 88 Zn nuclei 24,25 . Following our previous discussion, an a(A) value based on the heaviest zinc isotopes, with measured half-lives, is used to determine the half-lives of the 87 Zn and 88 Zn systems.
Spherical single-particle energy calculations produce reasonable results for the observed beta, positron, and electron capture decay modes. Using the methodology of Ref. 27, a credible result is obtained for the 54 Zn 2p decay mode.
No alpha decay transitions were predicted by the model calculations for the nuclei summarized in Table I. Table I lists the half-life of the limiting decay transition (i.e., the transition that has the shortest decay half-life). For example, 60 Zn has two positron decay transitions that are possible within the scope of the aforementioned single-particle model (i.e., allowed 2p 3/2 (p) to 2p 3/2 (n) [2.42 min] and allowed 2p 3/2 (p) to 2p 1/2 (n) [17.4 h]). For 60 Zn, the limiting positron decay mode is the allowed 2p 3/2 (p) to 2p 3/2 (n) [2.42 min] transition.
The model generally predicts the proper decay mode for 54 ≤ A ≤ 88 zinc nuclei 24,25 . The results for known zinc systems summarized in Table I suggest that the model predictions of the neutron excess zinc systems tend to improve as the number of neutrons increases.
The 54 Zn -58 Zn systems are the least massive zinc isotopes and fill the 1f 7/2 neutron shell. 54 Zn is a 2p emitter and its decay half-life was evaluated following the effective liquid drop model (ELDM) approach of Gonçalves et al. 27 . The ELDM model was utilized for 54 Zn because a single-particle model is not directly applicable to a 2p decay process. Using the ELDM approach 27 , the 54 Zn 2p decay mode half-life is calculated to be 4.35 ms, which is in reasonable agreement with the experimental value of 3 ms 24 .
The single-particle model predicts a positron decay half-life of 1370 ms for 54 Zn using an a(A) value of 0.115.
However, this is not the limiting decay mode since the positron decay half-life is over 450 times larger than the 2p decay half-life. The single-particle model correctly predicts the 55  shell and are correctly predicted to be stable by the single-particle model. 69 Zn and 70 Zn fill the 2p 1/2 neutron shell. The 69 Zn βdecay mode is correctly predicted by the model, and its halflive is overestimated by 18%. 70 Zn is correctly predicted to be a stable nuclear system by the single-particle model.   The Table II results reemphasize the previous discussion that asserted 88 Zn was the heaviest bound zinc system.