Exploring the ground state bulk and decay properties of the nuclei in superheavy island ¹¹footnotemark: 1

In our previous works and others, various relativistic parametrizations have been successfully applied to study the ground-state properties such as binding energy, separation energy, quadrupole deformation parameter, $root-mean-square$ rms charge radii, $\alpha$-decay energies, and the alpha decay half-lives of actinides and superheavy nuclei [38, 40, 43, 54, 56, 60, 78, 82, 90, 91, 92, 93, 94, 95, 96]. From these analyses, it is found that the results obtained from the NL3^∗ parameter set provide a reasonable fit to the experimental data. Since experimental data is only available for nuclei up to Z = 118, so, we have further shown the $Q-$values for these nuclei being calculated by using the relativistic mean-field formalism with NL3^∗ parameter set [54] and references therein. The $\alpha$-decay half-lives for these nuclei are calculated by using the four different semi-empirical formulae namely UDL, VSS, MUDL, and MBrown (will be discussed in the subsequent subsection). In our previous study also [54], we have investigated the $Q-$values and $\alpha$-decay half-lives using NL3^∗ parameter set with four different formulae, namely, Viola-Seaborg, Alex-Brown, Parkhomenko-Sobiczewski, and Royer. The results for Q-values and $\alpha-$decay half-lives are shown in Figs. 4, 5. It can be analyzed from the graphs and also from the previous study [54] that the results obtained from NL3^∗ parameter set gives the reliable fit with the experimental data for known superheavy nuclei. Hence, it will be interesting and also crucial to extend the work for the unknown region from Z = 122 to 126 along with two different semi-empirical formulae namely; UDL and MUDL for the calculation of decay half-lives.

The $\alpha$-decay Energy ($Q_{\alpha}$): In this subsection, we analyze the $\alpha$-decay energy ($Q_{\alpha}$) and the alpha decay half-lives ($T_{1/2}^{\alpha}$) of the considered 204 superheavy nuclei associated with various $\alpha$-decay chains. The $\alpha$-decay energy ($Q_{\alpha}$) plays a significant role in understanding the decay properties and for calculating the $\alpha$-decay half-lives ($T_{1/2}^{\alpha}$) of a nucleus. For example, the $\alpha$-decay half-life of the nuclei having large comparative values in contrast to neighbouring nuclei shows the stability and/or shell closure. The consideration of the $\alpha$-decay of the nuclei determines the island of stability in a superheavy valley. The $Q_{\alpha}$ energy is calculated using the following relation, $Q_{\alpha}(N,Z)$ = BE $(N-2,Z-2)$ + BE $(2,2)$ - BE $(N,Z)$. Here, BE $(N,Z)$, BE $(N-2,Z-2)$, and BE $(2,2)$ are the binding energies of the parent, daughter, and $\alpha$ particle (BE = 28.296 MeV), respectively. In the case of FRDM predictions, the $Q_{\alpha}$ value is calculated by using the mass excess values given in Ref. [8]. The calculated $Q_{\alpha}$ energy values by RMF formalism are compared with the available FRDM data, WS4 predictions [97], KUTY mass estimates [9] and the experimental data [98] as shown in Fig. 6.

To examine the consistency of the present calculation, we estimate the mean deviation (MD) in the $Q_{\alpha}$-values for RMF (NL3^∗), FRDM, WS4, KUTY mass estimates to the available experimental data. It is worth mentioning that there are only five experimental data available among the 204 considered nuclei. Hence we also estimated the possible mean deviation for RMF (NL3^∗) with other theoretical models, viz., FRDM, WS4, and KUTY mass estimates. All calculated mean deviations are listed in Table 1. One notices that all the estimated mean deviations (MDs) are in the acceptable range from the table. Further, the comparison MD for RMF (NL3^∗) with-respect-to experimental data is good compared to FRDM and KUTY mass estimates, whereas WS4 has a lower value. For example, the MD for NL3^∗ is $\sim-0.23$, and other theoretical models, namely, FRDM, WS4, KUTY mass estimates, have $\sim 0.25,0.03,$ and $0.55$, respectively. This implies that the RMF (NL3^∗) calculation for $Q$-values is slightly underestimated to the experimental data, wherever available. But other theoretical models considered here for comparison are slightly overestimated to the experimental data. The relative MDs for NL3^∗ to experimental data and other theoretical models show that the results are reasonably good agreements with each other. In other words, the estimated $Q$-values from RMF (NL3^∗) and other theoretical models apply to the estimation of decay half-life of the unknown region of the superheavy island. Furthermore, the present analysis gives qualitative and quantitative theoretical information on the superheavy island within the microscopic relativistic mean-field model.

The $\alpha$-decay Half-lives: Geiger and Nuttall predicted the linear relationship between $Q$-value and half-lives of $\alpha$-decay. However, the measured experimental data of parent nuclei emitting $\alpha$-particle in both heavy and superheavy regions with $Z$ = 118 produces several linear segments with varying slopes and intercepts, unlike the Geiger and Nuttall law. To overcome this difficulty, a logarithmic form of experimental $\alpha$-decay half-lives is plotted as a function of different semi-empirical parameters. We have used four different recently developed semi-empirical formulae to calculate the alpha decay half-lives of the superheavy nuclei in this present work. The formulae will be briefly described below:
Formula-1: Qi et al. [99] presented a linear universal decay law (UDL) for $\alpha$-decay and cluster decay modes. Two parameters $\chi^{\prime}$ and $\rho^{\prime}$ presented in this formula [given in Eq. (6)] depend on the Q-value. The UDL formula relate the half-life of monopole radioactive decay with the Q-value of the emitting particles and the masses and charges of the nuclei involved in the decay. Simply, it is given as,

	$\displaystyle\log_{10}T_{1/2}^{UDL}$	$\displaystyle=$	$\displaystyle aZ_{\alpha}Z_{d}\sqrt{\frac{A}{Q_{\alpha}}}+b\sqrt{AZ_{\alpha}Z_{d}(A_{d}^{1/3}+A_{\alpha}^{1/3}})+c$		(6)
			$\displaystyle=a\chi^{\prime}+b\rho^{\prime}+c.$

Here A is the reduced mass $A=A_{d}A_{\alpha}/(A_{d}+A_{\alpha}$) and $A_{d}$, $A_{\alpha}$ are the mass number of daughter and alpha particle. The adjustable parameter $a$, $b$, and $c$ are 0.3949, -0.3693, and -23.7615 respectively are taken from Ref. [57].

Formula-2: The Viola and Seaborg (VSS) in 1966 predicted a simple Gamov model-based simple formula for calculating the $\alpha$-decay half-lives. The Viola-Seaborg (VSS) [57] relation is given as :

$\displaystyle\log_{10}T_{1/2}^{VSS}=(aZ+b)Q^{-1/2}_{\alpha}+(cZ+d)+h_{log},$

(7)

where $a$, $b$, $c$, $d$, and $h_{log}$ are the fitting parameter taken from Sobiczewski [100] having values $a$ = 1.66175, $b$ = -8.5166, $c$ = -0.20228, $d$ = -33.9069 and the hindrance factor $h_{log}$ for the nuclei with unpaired nucleons as,

	$\displaystyle Z=even,N=even,h_{log}=0,$
	$\displaystyle Z=odd,N=even,h_{log}=0.772,$
	$\displaystyle Z=even,N=odd,h_{log}=1.066,$
	$\displaystyle Z=odd,N=odd,h_{log}=1.114.$

Formula-3: The universal decay law formula was modified by introducing the nuclear asymmetry term ($I$) to the general procedure. A relation gives the modified universal decay law (MUDL):

	$\displaystyle\log_{10}T_{1/2}^{MUDL}$	$\displaystyle=$	$\displaystyle aZ_{\alpha}Z_{d}\sqrt{\frac{A}{Q_{\alpha}}}+b\sqrt{AZ_{\alpha}Z_{d}(A_{d}^{1/3}+A_{\alpha}^{1/3}})$		(8)
			$\displaystyle+c+dI+eI^{2}.$

Here, $I$ represents the nuclear asymmetry term, $I=(N-Z)/A$. And the fitting parameters $a$, $b$, $c$, $d$, and $e$ having values 0.41149, -0.42047, -22.89310, 12.13020, and -44.60575 respectively are taken from ref. [58].

Formula-4: In 1992, Brown proposed a universal decay law by the linear dependence of half-lives on $Z_{d}^{0.6}Q_{\alpha}^{-1/2}$ quantity for even-even parents, where $Z_{d}$ is the charge number of daughter nuclei. This formula was modified by introducing an additional hindrance term depending on parity. The modified Brown (MBrown) formula [59] used here is given as:

$\displaystyle\log_{10}T_{1/2}^{MBrown}=a(Z-2)^{b}Q^{-1/2}+c+h^{mB1}.$

(9)

The constant parameters $a$, $b$, and $c$ are 13.0705, 0.5182, and -47.8867 respectively and the hindrance term ($h^{mB1}$) depends upon parity is,

$$h^{mB1}=\begin{cases}0\hskip 17.07182ptforZ,Neven,\\ 0.6001\ for\ Z=odd,N=even,\\ 0.4666\ for\ Z=even,N=odd,\\ 0.8200\ for\ Z,N\ odd.\end{cases}$$

In this work, we have systematically analyzed the $\alpha$-decay half-life ($T_{\alpha}^{1/2}$) of $even-even$ 204 isotopes of superheavy nuclei with $Z$ = 114 to 126 by using four semi-empirical formulae, namely, UDL, VSS, MUDL and MBrown as shown in Fig. 7 to 10 respectively. To examine the consistency of the alpha decay half-lives obtained from four different formulae, we have calculated the half-lives of the known superheavy nuclei whose experimental data are known. The observed half-lives for the known nuclei are available in Ref. [101]. It is clear from the figures that there is a fair agreement between the UDL results and the experimental data while the other three empirical formulae (VSS, MUDL, and MBrown) show slightly lower values compared to the experimental data.

From the Figs. 7 to 10, one can notice that the isotopes associated with the neutron numbers $N$ = 162, 170, 174, 178, 182 and 184 have longer half-lives as compared to the other isotopes in the series, which further strengthen the predictions in Refs. [40, 42, 102]. Moreover, in Ref. [43], the authors demonstrated that alpha decay half-lives increase with the increase in neutron number and found a neutron shell closure at Z = 120, N = 184, 258. However, their study includes the structure of single-particle states and the nuclear matter properties for superheavy nuclei with Z = 90 - 120. On the other hand, the present study includes the unknown superheavy isotopes covering from neutron deficient to the neutron-rich region of the nuclear chart for Z = 122, 124, and 126. The calculation mainly involves finding the shell/sub-shell closures from their bulk and decay properties. Four semi-empirical formulae, namely, UDL, VSS, MUDL, and MBrown are used for the calculation of alpha decay half-lives. Our predictions are compared with FRDM, WS3, WS4 predictions. To find the consistency of these formulae, the standard deviation is also calculated with respect to the other theoretical results. Furthermore, we find a few shell/sub-shell closure with longer half-lives for a few new isotopes. For example, in ^292-350126, the isotopes ^348,350126 ($N$ = 222, 224) have highest half-lives over isotopic chain of Z = 126. Similarly, in ^290-346124, ^290-346122, ^288-344120, ^286-342118, ^284-340116, ^282-338114, the isotopes ³⁴⁶124 ($N$ = 222), ³⁴⁶122 ($N$ = 224), ³¹⁰120 ($N$ = 190), ³⁰⁸118 ($N$ = 190), ³⁰⁶116 ($N$ = 190), and ³³⁴114 ($N$ = 220) have relatively longer half-life as compared to other superheavy isotopes excluding the isotopes associated with the above defined neutron magic. From the above analysis, it is found that the $\alpha$-decay half-lives are dependent on the choice of the $\alpha$-decay formula. The bulk properties of these nuclei and the prediction for longer half-lives for a few isotopes are informative for the near future experimental synthesis.