Decay Characteristics of Neutron Excess Potassium Nuclei

Bevelacqua Resources, 7531 Flint Crossing Circle SE, Owens Cross Roads, AL 35762 USA


Introduction
The nucleosynthesis of heavy elements occurs by three basic processes that add protons or neutrons to a nuclear system 1,2 .The p-process adds protons and the s-or slow process and r-or rapid process adds neutrons.Capture of protons by nuclear systems produces predominantly proton-rich nuclei that tend to decay by positron emission and electron capture 1,2 .Neutron capture creates neutron-rich nuclei, and the resulting nuclear systems depend 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 hole 1 .The matter ejected from these structures and from supernovae explosions is an important source of rapid neutron capture (r-process) nucleosynthesis 1 .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 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 to determine the structure of unstable neutron excess nuclei.Theoretical studies would complement experiments that 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 determining 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 potassium, is important for a comprehensive astrophysical interpretation of nucleosynthesis.For example, Terasawa et al. 21studied the role of light neutron-rich nuclei during r-process nucleosynthesis in supernovae.Specifically, Ref. 21 noted that light neutron excess systems can significantly affect the heavy-element abundances.
Recent studies emphasize the importance of studying potassium isotopes as well as their astrophysical significance 22- 24 .These studies include both theoretical as well as experimental efforts, and provide additional data to assist in clarifying a picture of the evolution of nuclear structure with increasing neutron number Refs.22-24 have both theoretical nuclear physics as well as astrophysical importance in predicting the production of neutron excess potassium nuclei.The continuing interest in neutron excess systems suggests the importance of evaluating potassium systems considerably heavier that those investigated in Refs.22 -24.In particular, this paper evaluates 47 K -69 K that span a much greater range than investigated in previous calculations.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.

Calculational Methodology
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 25 and Petrovich et.al. 26 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.N 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 27 , and the methodology of Ref.

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][14][15][16][17][18][19][20]26 that is similar to the approach of Ref. 29. The ingleparticle 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 30 .The central potential strength of the Rost interaction 30 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 30 : r 0 = 1.262 (1.295) fm, r so = 0.908 (1.194) fm, a = 0.70 (0.70) fm, and γ = 17.5 (28.2) for protons (neutrons) 26,30 .The spin-orbit interaction strength V so is related to γ by the relationship 30 :

[ ( ) ]
[ ] The scaling relationships of Eqs. 2 and 3 yield reasonable fits to observed single-particles levels in 120 Sn and 138 Ba.
The pairing correction term of Blomqvist and Wahlborn 31 is used in the calculations presented herein.The pairing correction improves the predicted energies of occupied levels in 120 Sn, 138 Ba, and 208 Pb 26 .
When applied to specific nuclei, this methodology requires modification.For example, Ray and Hodgson 32 note that 40 Ca and 48 Ca require different potentials to properly fit their single-particle level structure.Schwierz, Wiedenhöver, and Volya 33 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 , sodium 13 , magnesium 14 , aluminum 15 , silicon 16 , phosphorous 17 , sulfur 18 , chlorine 19 , and argon 20 nuclei.Similar issues also apply to potassium systems.
In view of the results of Refs.32 and 33, the following modification is made to obtain the potassium 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 32,33 .In previous neutron excess 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 , sodium 13 , magnesium 14 , aluminum 15 , silicon 16 , phosphorous 17 , sulfur 18 , chlorine 19 , and argon 20 was determined by the available experimental data [34][35][36] .Given the proximity to the A = 19 system, a value of λ = 1.5 is also utilized for potassium.Since the paper's primary purpose is investigation of the neutron excess nuclei, determining a consistent a(A) value for the heaviest potassium systems is desirable.
The heaviest mass A = 19 isotope [34][35][36] suggested experimentally is 54 K. Given the expected order of energy levels, 54 K would have a partially filled 1f 5/2 neutron single-particle level structure.Isotopes heavier than 54 K would require filling of the 1f 5/2 , 2p 1/2 , and 1g 9/2 neutron single-particle levels.The possibility of bound potassium isotopes with A ≥ 55 is addressed in subsequent discussion.

Calculation of Half-Lives
Using Eq. 4, single-particle levels are calculated for A ≥ 47 potassium isotopes.A ≥ 47 potassium nuclei were evaluated for stability with respect to alpha decay, beta decay, positron decay, electron capture, and spontaneous fission.
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 69 ≥ A ≥ 47 potassium isotopes are summarized in Table 1, and compared to available data [34][35][36] and calculations incorporated in the Japanese data compilation 36 .The alpha decay energies are calculated using the relationship based on Ref. 37 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 25 .Fortunately, no alpha decay modes occurred in the Table 1 summary of 69 ≥ A ≥ 47 potassium isotope decay properties.
The beta decay half-lives are determined following the log ft methodology of Wong 37 .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. 25. Spontaneous fission half-lives are addressed using the methods noted in Refs.38-52.

Results and Discussion
Using Eq. 4, the a(A) value was varied in increments of 0.0001 to assess the applicability of the proposed model to predict the decay properties of 47 ≥ A ≥69 potassium isotopes.In view of uncertainties in the model and associated interaction, a smaller increment was not deemed to be justified.
The issues associated with fitting all calcium, iron, fluorine, zinc, neon, sodium, magnesium, aluminum, silicon, phosphorous, sulfur, chlorine, and argon nuclei with a single potential [32][33] were noted in Refs.8-20.These considerations are also applicable to the potassium systems considered in this paper.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>54 potassium isotopes.The 69 ≥ A ≥ 55 potassium isotopes fill the 1f 5/2 , 2p 1/2 , and 1g 9/2 neutron single-particle levels, and are summarized in Table 1.These systems represent the heaviest possible neutron excess systems that could occur in the Z=19 system.The neutron excess systems summarized in Table 1 were based on an evaluation of alpha, beta, electron capture, and positron decay modes.Spontaneous fission half-lives were also evaluated, but tend to be larger that the aforementioned 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 1 must be viewed with this limitation.However, since the neutron decay modes tend to be much shorter than the alpha, beta, electron capture, and positron decay modes [34][35][36] , the model results provide upper bounds on the half-lives of neutron excess potassium isotopes.

47 ≥ A ≥ 54 Potassium Isotopes with Experimental Half-Life Data
The 47 K system completes filling of the 1f 7/2 neutron shell. 47K is fit with an a(A) value of -0.0382. 48K -51 K fill the 2p 3/2 neutron shell, and are best fit with a(A) values between -0.0409 and -0.0201 with an average value of about -0.0287. 52K -54 K partially fill the 1f 5/2 neutron shell, and are best fit with a(A) values between -0.0482 and 0.0447.For 52 K -54 K, the average a(A) value is about 0.0001.
54 K is the heaviest known neutron excess potassium system.There is no experimental half-life data for A > 54 potassium systems.
The a(A) values for the 55 K -69 K systems are based on the decreasing lifetime trends of neutron excess silicon, phosphorous, sulfur, chlorine, argon, and potassium systems 34 .Using the 50 K -54 K values, a linear extrapolation was utilized to obtain the a(A) values for the 55 K -69 K.The derived a(A) values are listed in Table 1.
Table 1 lists the half-life of the limiting decay transition (i.e., the transition that has the shortest decay half-life).For example, 47 K has four beta decay transitions that are possible within the scope of the aforementioned single-particle model (i.e., allowed 1d 5/2 (n) to 1d As noted in Table 1, the model predicts the proper decay mode for the known 69 ≥ A ≥ 47 potassium [34][35][36] systems.
The results for the known nuclei, summarized in Table 1, suggest that the model predictions of the neutron excess potassium systems are reasonably credible.
For nuclei filling the 1f 7/2 neutron shell, model predictions for 47 K are within about 0.6% of the experimental half-lives 34 . 47K decays via beta emission through a allowed 1d transitions.Model predictions for 52 K -54 K agree with the experimental half-lives 34 .

69 ≥ A ≥ 55 Potassium Isotopes without Experimental Half-Life Data
The a(A) values for 69 ≥ A ≥ 55 potassium isotopes were derived from a fit based on the half-lives of 50 K -54 K.This approach is consistent with the a(A) extrapolation methodology noted in Refs.8 -20.The a(A) values for 69 ≥ A ≥ 55 potassium systems are provided in Table 1.
Table 1 also summarizes calculated single-particle decay properties of potassium systems that have not yet been observed [34][35][36] .These systems are nuclei of interest in astrophysical applications  .
The existence of 69 ≥ A ≥ 55 potassium systems, as predicted by the proposed model, is dependent on the characteristics of the interaction of Eq. 4.Although the existence of some of these systems may be an artifact of the model interaction, their study is of critical importance in understanding the role of neutron excess potassium systems in nucleosynthesis.
The 55 K -57 K systems complete filling of the 1f 5/2 neutron shell. 55K -57 K have beta decay half-lives that decrease from 9.23 to 4.48 ms.These systems decay through allowed 1f 5/2 (n) to 1f 7/2 (p) beta decay transitions.Japanese Data Compilation calculations 36 for 55 K -57 K are consistent with the model results.
The 58 K and 59 K systems fill the 2p 1/2 neutron shell.These systems decay through an allowed 1f 5/2 (n) to 1f 7/2 (p) beta decay transition.The 58 K and 59 K half-lives decrease from 3.28 to 2.45 ms, respectively.Japanese Data Compilation calculations 36 for 58 K and 59 K are also consistent with the model results.
The 60 K -69 K systems fill the 1g 9/2 neutron shell.These systems decay through an allowed 1f 5/2 (n) to 1f 7/2 (p) beta decay transition.The 60 K -69 K half-lives decrease from 1.88 to 0.318 ms.Japanese Data Compilation calculations 36 for 60 K -69 K are consistent with the model results.
No potassium systems with A > 69 are predicted by the model.This occurs because the 1g 9/2 neutron single-particle level is the last bound neutron state, and only 50 neutrons are bound in potassium systems.However, in view of the model potential uncertainties, the calculated properties of the heaviest potassium systems summarized in Table 1 are not definitive.
The predicted A = 55 -69 potassium isotopes have no experimental data, but the model predicts beta decay half-lives in the range of 0.315 -9.23 ms.Based on calculations in Z = 9 -18, 20, 26, and 30 systems [8][9][10][11][12][13][14][15][16][17][18][19][20] and calculations summarized in the Japanese Data Compilation 36 , the results summarized in this paper likely overestimate the beta decay half-lives of A = 55 -69 neutron excess potassium 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 = 55 -69 potassium nuclei.
28 is utilized to obtain a converged solution.Refs.8 -20 and 26 provide a more complete description of the model, its numerical solution, and further definition of the individual terms appearing in Eq. 1.