Decay Characteristics of Neutron Excess Magnesium Nuclei

In neutron star mergers, neutron excess nuclei and the r-process are important factors governing the production of heavier nuclear systems. A single-particle model evaluation of magnesium nuclei suggests that the heaviest Z = 12 nucleus will have mass 52 with filling of the 1f5/2 neutron shell. A = 38 – 52 magnesium isotopes have limited experimental half-life data, but the model predicts beta decay halflives in the range of 0.744 – 2.89 ms. Based on previous calculations for Z = 9 -11, 20, 26, and 30 systems and comparisons to the 38Mg – 40Mg calculations summarized in the Japanese Nuclear Data Compilation, the single-particle model results likely overestimate the half-lives of A = 38 – 52 neutron excess magnesium nuclei.

creating heavy nuclear systems. Unfortunately, the majority of these neutron excess systems have never been studied 2 .
Closing this knowledge gap was a motivation for funding facilities for rare-isotope beams (FRIB) constructed at research facilities located around the world. These facilities are located at RIKEN (Japan) 3,4 , GSI (Germany) 5 , and Michigan State University (US) 6 . The FRIB facilities enable a new class of experiments to determine the physical properties needed by theoretical models of 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 universe 2 . In particular, the study of neutron excess systems and their decay properties are significant 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 studying 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 the 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.
The study of light nuclear systems, including magnesium, is important for a comprehensive astrophysical interpretation of nucleosynthesis. For example, Terasawa et al. 14 studied the role of light neutron-rich nuclei during r-process nucleosynthesis in supernovae.
In the neighboring fluorine systems, Recio-Blanco et al. 15 noted the importance of these nuclei in nucleosynthesis, but observed that knowledge of excess neutron Z=9 systems and their associated properties are not well established. Mowlavi et al. 16 also investigated the nucleosynthesis of fluorine with a focus on asymptotic giant branch stars. Ref. 1 noted that most previous studies of the r-process have concentrated on the synthesis of heavy unstable nuclei. However, in extreme environments such as those was a direct consequence of the nuclear structure of 20 F, and its influence on the beta decay to the 20  An et al. 20 perform a theoretical study of Z = 8 -12 isotopes in the relativistic mean field model. Ref. 20 notes that the last bound neutron-rich nuclei with Z = 8, 9, 10, 11, and 12 varies with the theoretical models with upper limit mass values of 28, 33, 43, 45, and 46, respectively. The magnesium A = 46 mass limit is consistent with the predictions of this paper that also predicts 46 Mg exists. However, the singleparticle model utilized in this paper predicts that 52 Mg is the limiting neutron excess magnesium system.

Calculational Methodology
A variety of models could be applied to the investigation of neutron excess nuclei. These 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 single-particle approach.
Since the method for calculating single-particle energies in a spherically symmetric potential is wellestablished 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 21 and Petrovich et. al. 22 The binding energy E NLSJ of a particle in the field of a nuclear core is obtained by solving the radial Schrödinger Equation 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.
The method of searching for E NLSJ is provided by Brown, Gunn, and Gould 23 , and the methodology of

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 [8][9][10][11][12][13]22 that is similar to the approach of Ref. 25. The single-particle level spectrum is generated using a Woods-Saxon potential.
Parameters of the potential are obtained from a fit to the single-particle energy levels in 209 Pb and 209 Bi performed by Rost 26 . The central potential strength of the Rost interaction 26 has a standard form and can be explicitly defined as where the upper (lower) sign applies to protons (neutrons). The remaining parameters were held constant and are given by Rost 26  When applied to specific nuclei, this methodology requires modification. For example, Ray and Hodgson 28 note that 40 Ca and 48 Ca require different potentials to properly fit their single-particle level structure. Schwierz, Wiedenhöver, and Volya 29 also investigated 40 Ca and 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 a single-particle model and was noted in studies of neutron excess calcium 8 , iron 9 , fluorine 10 , zinc 11 , neon 12 , and sodium 13 nuclei. Similar issues also apply to magnesium systems.
In view of the results of Refs. 28 and 29, the following modification is made to obtain the magnesium potential strength (V A ): where λ is a potential strength multiplier that is selected to ensure consistency with available data, and a(A) is a constant that is introduced to account for the variations in potential strength with A 28,29 . In previous excess neutron nuclei calculations for calcium 8 , iron 9 , and zinc 11 , a value of λ = 1.0 was utilized.
A λ value of 1.5 for fluorine 10 , neon 12 , and magnesium 13 was determined by the available experimental data [30][31][32] . Given the proximity to the A = 9 -11 systems, a value of λ = 1.5 is also utilized for magnesium.
Since the paper's primary purpose is investigation of the neutron excess nuclei, determining a common The heaviest mass A = 12 isotope 30-32 suggested experimentally is 37 Mg. Given the expected order of energy levels, 37 Mg would have a 1f 7/2 neutron single-particle level structure. Isotopes heavier than 37 Mg would require filling of the 1f 7/2 and the more weakly bound 2p 3/2 , 2p 1/2 , and 1f 5/2 neutron single-particle levels. The possibility of bound magnesium isotopes with A ≥ 38 is addressed in subsequent discussion.
Calculations incorporated into the Japanese nuclear data compilation 32

Calculation of Half-Lives
Using Eq. 4, single-particle levels are calculated for A ≥ 20 magnesium isotopes. A ≥ 20 magnesium nuclei were evaluated for stability with respect to alpha decay, beta decay, positron decay, and electron capture. 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 52 ≥ A ≥ 20 magnesium isotopes are summarized in Table I, and compared to available data [30][31][32] and calculations incorporated in the Japanese data compilation 32 . The alpha decay energies are calculated using the relationship based on Ref. 33 where S n and S p are the binding energies of the last occupied neutron and proton single-particle levels, respectively. Alpha decay half-lives can be estimated from Q α using standard relationships 21 . Fortunately, no alpha decay modes occurred in the Table I summary of 52 ≥ A ≥ 20 magnesium isotope decay properties.
The beta decay half-lives are determined following the log ft methodology of Wong 33 . 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 order forbidden transitions were not determined. Positron and electron capture half-lives were determined following the approach of Ref. 21.

Results and Discussion
Using Eq. 4, the a(A) value was varied in increments of 0.001 -0.0001 to assess the applicability of the proposed model to predict the decay properties of most 52 ≥ A ≥ 20 magnesium isotopes. In view of uncertainties in the model and associated interaction, a smaller increment was not deemed to be justified for most magnesium systems. However, for nuclei that have half-lives that deviate from stability trends in neighboring systems, a smaller increment was utilized. For example, a(A) was adjusted in increments of 0.00001 for the stable 24 Mg, 25 Mg, and 26 Mg systems.
The issues associated with fitting all calcium, iron, fluorine, zinc, neon, and sodium nuclei with a single potential 28,29 were noted in Refs. 8-13. These considerations are also applicable to the magnesium systems considered in this paper.  33 Na -37 Na partially fill the 1f 7/2 neutron single-particle level. Given the extrapolation used in formulating the single-particle potential of Eq. 4, the results become more uncertain due to the paucity of data for A>37 magnesium isotopes. The heavier 52 ≥ A ≥ 20 magnesium isotopes that complete the 1f 7/2 , and fill the 2p 3/2 , 2p 1/2 , and 1f 5/2 neutron single-particle levels are also summarized in Table I. These systems represent the heaviest possible neutron excess systems that would occur in the Z=12 system.
Qeios, CC-BY 4.0 · Article, November 24, 2021  The neutron excess systems summarized in Table I were based on an evaluation of alpha, beta, electron capture, and positron decay modes. Other decay modes that could possibly occur in neutron excess systems (e.g., n and 2n) are not readily evaluated using a single particle model, and were not evaluated.
The results of Table I must be viewed with this limitation. However, since the neutron and proton decay modes tend to be much shorter than the alpha, beta, electron capture, and positron decay modes [30][31][32] , the model results provide upper bounds on the half-lives of neutron excess magnesium isotopes.

37 ≥ A ≥ 20 Magnesium Isotopes with Experimental Half-Life Data
The 20 Mg system completes the 1p 1/2 neutron shell. Its decay properties were best fit with an a(A) value of 0.029. 21 Mg to 26 10 , zinc (0.095) 11 , neon (0.119) 12 , and sodium (0.160) 13 . Since it is the heaviest experimentally observed magnesium system consistent with the decreasing beta decay half-life trends noted in Refs. 8 -13 and 30 -32, the 36 Mg value is used to extrapolate the half-lives of 38 Mg and heavier magnesium nuclei.
The a(A) value of 37 Mg was not utilized because it is inconsistent with the aforementioned beta decay halflife trends. As noted in Table I, the model predicts the proper decay mode for the known 37 ≥ A ≥ 20 magnesium nuclei [30][31] . The results for the known systems summarized in Table I  For nuclei filling the 1d 5/2 neutron shell, model predictions for 21 Mg, 22 Mg, and 23 Mg are within about 0.25% of the experimental positron decay half-lives 30 . 24 Mg, 25 Mg, and 26 Mg are correctly determined to be stable systems. 21 Mg, 22 Mg, and 23 Mg decay by allowed 1d 5/2 (p) to 1d 5/2 (n) positron decay transition.
The 2s 1/2 systems, 27 Mg and 28 Mg, are within 5% of their respective experimental beta decay halflives 30 . Both 27 Mg and 28 Mg decay by an allowed 1d 5/2 (n) to 1d 5/2 (p) beta decay transition. 29 Mg, 30   No magnesium isotopes with A > 52 are predicted by the model. This occurs because the 1f 5/2 neutron single-particle level is the last bound neutron state, and only 40 neutrons are bound in magnesium systems. However, in view of the model potential uncertainties, the calculated properties of the heaviest magnesium systems summarized in Table I are not definitive.
The predicted A = 38 -52 magnesium isotopes have no experimental half-life data, but the model predicts beta decay half-lives in the range of 0.744 -2.89 ms. Based on calculations in Z = 9 -11, 20, 26, and 30 systems 8-13 , these results likely overestimate the beta decay half-lives of these neutron excess magnesium nuclei. The model results are also likely to be an overestimate of the half-lives because the single-particle level calculations do not evaluate the short-lived neutron decay modes in the A = 38 -52 magnesium nuclei.

Conclusions
Single-particle level calculations suggest that neutron excess magnesium isotopes terminate with 52 Mg and filling of the 1f 5/2 neutron single-particle level. The 38 ≤ A ≤ 52 magnesium systems have predicted beta decay half-lives in the 0.744 -2.89 ms range, and likely overestimate the actual half-life values.