Superheavy Nuclei XI: 1500 ≤ A < 1600 Systems

Decay properties of nuclei are calculated in the mass region 1500 ≤ A < 1600. The calculations are performed using the adjusted Rost interaction. Model calculations suggest that two new islands of stability could exist in the vicinity of the Z = 414 N = 1088 (X(414, 1502)) and Z = 420 N = 1120 (X(420, 1540))systems.

The use of single-particle energy levels to evaluate nuclear stability is appropriate since extrapolations to the superheavy mass regions are speculative. Using a more sophisticated method is not warranted in view of the uncertainties encountered in these calculations. Methods that are more sophisticated are appropriate when data are available to examine fine model details and interaction characteristics. As was demonstrated in Refs. 3 and 5, singleparticle energy level calculations are entirely appropriate for initial calculations into a superheavy mass region where there is no experimental data to guide the calculations. Moreover, theoretical calculations are currently the only way to investigate the 1500 ≤ A < 1600 mass region because an experimental investigation is not currently feasible.

Calculational Methodology
Since the method for calculating single-particle energies in a spherically symmetric potential is well established 3,5,21-32 , only salient features are provided. Details of the methodology were provided in Ref. 21, which extended the approach of Petrovich et al. 5 Specific details of the numerical method, model, and convergence criteria are provided in Refs. 2, 5, 21-35.

Theoretical Model
The model describing the nucleon plus nuclear core system represents an application of the standard method of Lukasiak and Sobiczewski 3 and Petrovich et al. 5 The calculational method used to generate a single-particle level spectrum determines the binding energy E NLSJ of a particle in the field of a spherical nuclear core 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. Additional details of the model and associated interactions are provided in Refs. 2, 5 and 21-35.

Determination of Q Values and Half-Lives
The reader is strongly cautioned not to interpret the calculated half-lives as representing a definitive value. As noted in where S n and S p are the binding energies of the last occupied neutron and proton single-particle energy levels, respectively. Alpha half-lives (T 1/2 α ) were estimated from Q α using standard relationships provided in Ref. 3.
The beta decay half-lives (T 1/2 β ) are determined following the log ft methodology of Wong 1 . Allowed (first-forbidden) transition half-lives were derived using the values of log ft = 5 (8). Given the uncertainties in the calculated single-particle level energies, second and higher forbidden transitions were not determined. The beta half-life values in Table 1 listed as stable are either beta particle stable or decay by these higher order forbidden transitions.

Nuclear Interaction
Nuclear stability with respect to alpha decay, beta decay, positron decay, electron capture, and spontaneous fission is addressed using the method previously published by the author 21-32 and coworkers 5 that is similar to the approach of Ref.
3. The single-particle level spectrum is generated using a Woods-Saxon potential with parameters optimized to permit extrapolation into the superheavy region 2,24,29 .
Uncertainties in the nuclear interaction for A ≥ 1200 superheavy nuclei preclude absolute theoretical predictions of nuclear properties including single-particle energies, half-lives, and Q-values. However, a model potential can be developed to predict trends in these properties and suggest islands of stability in A ≥ 1200 nuclei 29 .
A specific interaction for investigating A ≥ 1200 systems was developed in Ref. 29. Any potential applicable to A ≥ 1200 systems must be constructed in a manner that is consistent with the general uncertainties in the nuclear interaction. Ref.
29 reviewed a representative sample of these uncertainties in order to guide the determination of the strength of an interaction applicable for use in A ≥ 1200 systems. The adjusted Rost interaction for use in A ≥ 1200 systems is based on calculations and associated uncertainties that span a wide range of nuclear systems including structure and single-particle level calculations in (1)  To account for the 10% potential strength uncertainty in calculating the properties of A ≥ 1200 systems, this paper uses the adjusted Rost interaction 29 : V 0 = 51.6λ 1 ± 0.73

Results and Discussion
The calculations presented in this paper are based on the adjusted Rost interaction 29 which has a potential strength that is 10% stronger that the Rost interaction 2 used in Refs. 5 and 21-23. Accordingly, direct comparison of half-lives with the A = 298 -472, 5 570 ≤ A ≤ 620 21 , 620 < A <700 22 , and 700 ≤ A<800 23 mass regions is not appropriate because they were based on the unmodified Rost interaction 2 . Similarly, comparison to calculations based on the modified Rost interaction 24 used in 800 ≤ A<1200 25-28 systems is also not appropriate since the interactions are not the same. However, comparisons to existing nuclear and 1200 ≤ A < 1500 systems [30][31][32] are outlined in subsequent discussion.
The effective half-life (Eq. 4) for nuclei with 1500 ≤ A < 1600 is plotted in Fig. 1. The alpha decay Q value (Q α ), and beta (T 1/2 β ) and alpha (T 1/2 α ) decay half-lives for the most stable 1500 ≤ A < 1600 systems are provided in Table 1  All 1500 ≤ A < 1600 systems decay through alpha emission. Beta decays occur in most bound 1500 ≤ A < 1600 systems through the transitions addressed in subsequent discussion. The most stable 1500 ≤ A < 1600 systems (i.e., X(414, 1502) and X(420, 1540)) are beta stable.
In general, it is expected that any bound superheavy nucleus will be strongly influenced by its shell structure. Based on previous calculations 3, 5, 21-32 , a bound superheavy nucleus is formed from the extra binding energy from closed-shell effects. The importance of these shell effects are noted in subsequent discussion.
The 1500 ≤ A < 1600 calculations suggest that two new islands of stability could exist in the vicinity of the X(414, 1502) and X(420, 1540) systems. Maximum stability occurs in the doubly-closed X(414, 1502) nucleus, which has closed 5g 7/2 neutron and 2j 15/2 proton shells.
As noted in Table 1, effective half-lives ≥0.1 s occur in a subset of the 25 bound nuclei within 1500 ≤ A < 1600 systems. These enhanced stability regions occur at Z = 414 (N = 1088) and Z = 420 (N = 1120). As used in this paper, an effective half-life includes the combined effect of the alpha and beta decay modes: Most of the 1500 ≤ A < 1600 systems summarized in Fig. 1 have effective half-lives less than one second. The calculated half-lives of most 1500 ≤ A < 1600 nuclei are shorter than the observed half-lives in Z = 114 -118 systems 36 .
The longest-lived systems summarized in Table 1  The results of the calculations suggest that for a given A value, S p tends to decrease and S n tends to increase as Z increases. This usually results in increasing Q α values as Z increases for a fixed A value. The beta decay systematics are more complex, and depend on the occupancy of specific single-particle levels, single-particle level quantum numbers, and single-particle energy level values that permit an allowed or forbidden transition to occur. The specific trends in alpha and beta stability are addressed in the subsequent discussion of nuclear stability.
A few general items are noted and are consistent with the trends noted in Refs. 21-32. For a given A value, alpha decay half-lives tend to decrease and beta decay half-lives tend to increase as Z increases. For a fixed Z, alpha decay half-lives tend to increase and beta decay half-lives tend to decrease as A increases.
In general, most decays in the 1500 ≤ A < 1600 systems occur through both alpha and beta pathways. The specific beta decay mode varies in the bound 1500 ≤ A < 1600 systems and is noted in subsequent discussion.
The discussion of specific nuclear systems focuses on the nuclides summarized in Table 1. These nuclei have the longest half-lives of the 25 even-even systems found to theoretically exist within the 1500 ≤ A < 1600 mass region.
Most of the calculated 1500 ≤ A < 1600 half-lives are shorter than the longest-lived Z = 114 -118 nuclei 36

1510 ≤ A < 1520 Systems
The 1510 ≤ A < 1520 mass region has a lower level of stability than the 1500 ≤ A < 1510 systems. No 1510 ≤ A < 1520 systems have an effective half-life ≥ 1 s.
The 1510 ≤ A < 1520 systems have effective half-lives in the range of 0.04 -0.2 s. In the 1510 ≤ A < 1520 mass region, beta decays predominantly occur through first forbidden 6d 3/2 (n) to 4p 3/2 (p) beta decay transitions.

1520 ≤ A < 1530 Systems
No bound 1520 ≤ A < 1530 systems were identified by the calculational model.
The X(414, 1502) and X(420, 1540) systems are the most stable nuclei in the 1500 ≤ A < 1600 mass region. Both of these systems have doubly-closed neutron and proton shells and are beta stable. As noted in Table 1, their half-lives are on the order of 10 10 yr.

Shell Closure
The most stable system in the 1500 ≤ A < 1600 mass region is X(414, 1502) that is stable with respect to beta decay, and has an alpha decay half-life of 2.9x10 10 yr. X(414, 1502) is doubly-closed, and has closed 5g 7/2 neutron and 2j 15 the closed shell proton (neutron) level energy and the 4p 3/2 (6p 3/2 ) level that lies above it 30 .
A comparison to other systems should only be made for calculations using the same model interaction. The adjusted Rost interaction 29 was developed for A ≥ 1200 systems. Therefore, only a comparison of the 1500 ≤ A < 1600 systems summarized in this paper to 1200 ≤ A < 1500 system 30-32 is appropriate.
More specific comments regarding the relative stability to previously investigated A = 298 -472 doubly-closed shell nuclei 5 and nuclear systems in the 570 ≤ A ≤ 620 21 , 620 < A <700 22 , 700≤A<800 23 , 800 ≤ A < 900 25 , 900 ≤ A < 1000 26 , 1000 ≤ A < 1100 27 and 1100 ≤ A < 1200 28 mass regions are not appropriate since these calculations used either the unmodified Rost interaction 2 or the modified Rost interaction 24 . As noted previously, the 1500 ≤ A < 1600 calculations only provide regions of possible stability and should only be compared with calculations that utilize the adjusted Rost interaction 29 . Table 1 summarizes the current list of most stable 1500 ≤ A < 1600 systems that utilize the adjusted Rost interaction 29 .
As a matter of comparison, X(354, 1226) Z = 354 N = 872 with an effective half-life of 4.8x10 12 yr 30 is the most stable system determined to date using the adjusted Rost interaction.

Model Weaknesses
The adjusted Rost interaction 29 is extrapolated from Z ≤ 82 data without the benefit of experimental benchmarks in the 1400 ≤ A < 1500 mass region. Although this is a necessity due to the lack of experimental data, it must be acknowledged as a weakness in the present approach. This weakness will be applicable for any current theoretical investigation in the 1500 ≤ A < 1600 mass region.
In Ref. 30, there were eight 1200 ≤ A < 1300 systems with effective half-lives >10 10 y which is on the order of the current estimate for the age of the Universe (≈1.4 x 10 10 y). In the 1500 ≤ A < 1600 mass region, the X(414, 1502) and X(420, 1540) systems have half-lives of 2.9x10 10 and 5.6x10 9 yr, respectively. exist. These decay modes would then be more likely to dominate all decay processes of these superheavy systems. This is a significant weakness of the proposed extension of the theory beyond its origin via connection to known isotopes.
Another weakness of the approach outlined in this paper is treating all evaluated nuclei as spherically symmetric systems. Many of these systems are likely deformed, and these deformations should be included in subsequent investigations. These calculations have been initiated. However, it seems unlikely that any given A(N, Z) nuclear system will have a deformed structure that is more stable than the spherically symmetric configuration utilized in the model  reaches the 1500 ≤ A < 1600 mass region, but this approach is not yet viable. In the interim, the author hopes that other theoretical work will challenge and refine the conclusions of this paper, and experimentalists will develop accelerator techniques to collide multiple beams or establish other approaches to reach the 1500 ≤ A < 1600 mass region.
A possible experimental approach is offered by the high alpha particle energies emitted by the postulated 1500 ≤ A < 1600 systems. The alpha particle energies of these theoretical superheavy nuclei are more than 100% larger than the measured Z = 114-118 values 36 . This substantial increase in alpha particle energies offers a possible avenue for the experimental verification of 1500 ≤ A < 1600 nuclei.
Compared to Z = 114 -118 nuclei, the higher alpha particle energies from the 1500 ≤ A < 1600 nuclei have a longer range in a material medium. This range manifests itself as a longer track length as the alpha particle is attenuated by the medium. Measuring alpha track lengths is a well-established approach in applied physics including the measurement of the 222 Rn air concentration 38,39 . Since the track length is related to the alpha particle energy, it provides a possible method to verify the existence of a 1500 ≤ A < 1600 superheavy system.
A final possible verification approach is based on the fact that various lead isotopes are the endpoint of known heavy element decay chains. If lead targets were vaporized, and then accelerated in a charged particle accelerator, they could then be separated by mass. Within this mass spectrum could be the remnants of the long-lived parent superheavy nuclei summarized in Table I. Extreme precision would be required to detect these primordial superheavy trace isotopes. At the

Conclusions
Calculations in the 1500 ≤ A < 1600 mass region suggest that two new islands of stability could exist in the vicinity of the Z = 414 N = 1088 (X(414, 1502)) and Z = 420 N = 1120 (X(420, 1540))systems. Using the adjusted Rost interaction 29 , 25 even-even nuclear systems are predicted in the 1500 ≤ A < 1600 mass region. The most stable system in the 1500 ≤ A < 1600 mass region is X(414, 1502) that is stable with respect to beta decay, and has an alpha decay half-life of 2.9x10 10 yr. X(414, 1502) is doubly-closed, and has closed 5g 7/2 neutron and 2j 15/2 proton shells.
There is considerable uncertainty in extrapolating nuclear potentials to the 1500 ≤ A < 1600 mass region. Therefore, many of the quantitative details regarding half-lives presented in this paper may be incorrect. However, the qualitative results, including the general predictions of the range of N and Z combinations associated with stability are expected to be more reliable. It is hoped that this paper will foster more sophisticated investigations of the 1500 ≤ A < 1600 mass region.

Acknowledgments
The author acknowledges the assistance of Dr. John X. Wang from Poly Software International in using the PSI-Plot program to produce the graphics used in Figs. 1 and 2.