Nuclear Science and Techniques

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

Investigation of decay modes of superheavy nuclei

H.C. Manjunatha，

N. Sowmya，

P.S. Damodara Gupta，

K.N. Sridhar，

A.M. Nagaraja，

L. Seenappa，

S. Alfred Cecil Raj

Nuclear Science and Techniques

Vol.32, No.11

Article number 130

Published in print 01 Nov 2021

Available online 23 Nov 2021

DOI：10.1007/s41365-021-00967-y

6800

A detailed investigations of different decay modes, namely, alpha decay, beta decay, cluster decay, including heavy particle emission (Zc > 28), and spontaneous fission, was carried out, leading to the identification of new cluster and beta-plus emitters in superheavy nuclei with 104 ≤ Z ≤ 126. For the first time, we identified around 20 beta-plus emitters in superheavy nuclei. Heavy-particle radioactivity was observed in superheavy elements of atomic number in the range 116 ≤ Z ≤ 126. ^292-293Og were identified as ⁸⁶Kr emitters, and ²⁹⁸122 and ³⁰⁰122 were identified as ⁹⁴Zr emitters, whereas heavy-particle radioactivity from ⁹¹Y was also observed in ²⁹⁹123. Furthermore, the nuclei ³⁰⁰124 and ³⁰⁶126 exhibit ⁹⁶Mo radioactivity. The reported regions of beta-plus and heavy-particle radioactivity for superheavy nuclei are stronger than those for alpha decay. The identified decay modes for superheavy nuclei are presented in a chart. This study is intended to serve as a reference for identifying possible decay modes in the superheavy region.

Alpha decayBeta decayHeavy-particle radioactivityBranching ratios

Introduction

The most important unanswered questions in Nuclear Physics are to determine the heaviest superheavy nuclei that can exist, and to investigate whether very-long-lived superheavy nuclei exist in nature. The past ten years have been marked by remarkable progress in the science of superheavy elements and nuclei. The existence of superheavy nuclei above Z=103 can be studied in terms of whether they can occur naturally or can be synthesized in the laboratory. There are no definitive conclusions regarding the existence of superheavy nuclei in nature. In contrast, such superheavy nuclei, with half-lives ranging between days to μs, can be synthesized using cold and hot fusion reactions. Cold fusion reactions involve either lead or bismuth as targets [1], whereas hot fusion reactions include ⁴⁸Ca beams on various actinide targets. [2,3]. Many theoretical predictions, such as microscopic–macroscopic [4] (single-particle potential) and self-consistent approaches, including nucleus–nucleus potential [5, 6], relativistic field models [7, 8], and multinucleon transfer reactions [9], provide information regarding the nucleus structure, shell closure location, and decay modes in heavy and superheavy nuclei.

The discovery of superheavy elements [10, 11] points to the island of stability. Boilley et al. [12] predicted the evaporation residue cross sections in superheavy elements and the influence of shell effects [13]. The entrance channel dynamics were studied using ⁴⁸Ca as a projectile and ²⁰⁸Pb as target [14]. In 1966, two groups of researchers, namely, Mayers and Swiatecki, and Viola and Seaborg [15], separately predicted the presence of heavy nuclei near the island of stability. Later, Sobiczewski et al. [16] predicted that the nucleus Z=114 will be the center of the island of stability, with neutron number N=184. In 1955 Nilsson [17] proposed a shell model which includes deformation property of the nuclei. Bender et al. [18] used a Skyrme energy density functional model and studied the deformation properties of closed proton and neutron shells. The nuclear mass, radius, and spectroscopy far away from the valley of stability were experimentally analyzed earlier [19]. The investigation of isomers of the superheavy nucleus ²⁵⁴No is a stepping stone toward the island of stability [20]. Previous researchers [21] analyzed the nuclear shell structure and discovered additional stability near magic nuclei. The present scenario is almost near the center of the presumed island of stability, but the final landing is yet to be completed, and the intriguing question is how these superheavy nuclei are still accessible.

The identification of superheavy nuclei is based on observations of decay chains. Superheavy element region 114 ≤ Z ≤ 118 were observed owing to their consistent decay chains, which end in the isotopes of rutherfordium(Rf) and dubnium(Db). Spontaneous fission and α-decay are the dominant decay modes in superheavy nuclei, and limit their stability. Furthermore, newly synthesized superheavy elements are primarily identified by their decay chains from unknown nuclei to known daughter nuclei by using the parent-and-daughter correlation.

The competition between different decay modes, such as ternary fission, spontaneous fission, cluster decay, proton decay, β-decay, and α-decay, in the heavy and superheavy region, has been extensively studied using various theoretical models, such as Coulomb and proximity potential models, modified generalized liquid drop models, effective liquid drop models, and temperature-dependent proximity potential models [22-23]. The possible decay modes in the superheavy nuclei Z=119 and 120 were predicted in Ref. [34]. From Ref. [35], it is clearly observed that the isotopes of the superheavy nuclei Z=104–112 have α-decay and spontaneous fission as dominant decay modes. However, only α-decay is dominant in the isotopes of superheavy nuclei Z=113, 115–118. The isotopes of the superheavy nucleus Z=114 have spontaneous fission as the dominant decay mode in the nucleus ²⁸⁴Fl, α-decay is dominant in the nuclei ^286-289Fl, and β⁺ is dominant in the nucleus ²⁹⁰Fl. Furthermore, the concept of heavy-particle radioactivity [36] in the superheavy region has important applications in the synthesis of superheavy nuclei. Despite the significant experimental and theoretical progress, there are many unanswered questions related to the decay modes of superheavy nuclei. Until now, only α-decay and spontaneous fission have been successfully observed in experiments.

Experimental results suggest a considerable increase in the lifetime of nuclei as they approach closed proton and neutron shells [37]. The lifetimes of most known superheavy nuclei are governed by the competition between α-decay and spontaneous fission. The existence of the island of stability has been confirmed experimentally in the previous decade [38]. Some theoretical studies reveal that superheavy elements with 114 and 164 protons are stable against fission as well as alpha and beta decay [39]. Various phenomenological and microscopic models, such as the fission model [40], the cluster model [41], generalized liquid drop model [42], and the unified model for alpha decay and alpha capture [43], are available to study the different decay modes of superheavy nuclei. In addition, many studies have been concerned with the alpha decay and spontaneous fission of superheavy nuclei [44-46]. Simple empirical formulas are also available for determining the decay half-lives [47]. The possible isotopes of new superheavy elements are identified by studying the competition between different probable decay modes, such as α-decay, β-decay, cluster decay, and spontaneous fission. This study focuses on the different decay modes of superheavy nuclei in the atomic number range 104 ≤ Z ≤ 126. After a detailed investigation of the competition between different decay modes, the possible isotopes and their decay modes with branching ratios are identified in the superheavy nuclei region. Hence, the contribution of this study is in the prediction of the most possible decay mode in superheavy nuclei, and in the identification of possible emitters in this superheavy region. The formalism is explained in Section 2. The analysis of different decay modes and possible emitters in the superheavy region is explained in Section 2.4. The paper is concluded in Section 3.

Theory

2.1

Alpha decay and cluster decay

In the effective liquid-drop model (ELDM), the α-decay half-life is computed using the relation

T_{1 / 2} (s) = \frac{\ln 2}{ν_{0} P P_{α}},

(1)

where ν₀ is the assault frequency on the barrier, and ν₀=1.8×10²²s^-1 [48]. Pα is the preformation factor, which is closely related to the shell structure [49]. The empirical formula for Pα is expressed as

\begin{array}{l} \log P_{α} = p_{1} + p_{2} (Z - Z_{1}) (Z_{2} - Z) \\ + p_{3} (N - N_{1}) (N_{2} - N) + p_{4} A, \end{array}

(2)

where N, Z, and A are the neutron, charge, and mass number of the parent nucleus, respectively, Z₁ and Z₂ are the proton magic numbers around Z (Z₁ ≤ Z ≤ Z₂), and N₁ and N₂ are the neutron magic numbers around N (N₁ ≤ N ≤ N₂). p₁, p₂, and p₃ correspond to parameters in the region even(Z)-even(N), even(Z)-odd(N), odd(Z)-even(N), and odd(Z)-odd(N). They are presented in Table I of Ref. [50]. P is the Gamow penetrability factor, given by the expression

P = \exp [- \frac{2}{ℏ} \int_{ζ_{0}}^{ζ_{c}} \sqrt{2 μ [V (ζ) - Q]} d ζ],

(3)

where μ is the inertial coefficient resulting from the Werner—Wheeler approximation [51]. The limits of integration ζ₀ and ζc are the inner and outer turning points, expressed as ζ₀=R_p- ${\bar{R}}_{1}$ and $ζ_{c} = \frac{Z_{1} Z_{2} e^{2}}{Q}$ . R_p is the radius of the parent nucleus, and ${\bar{R}}_{1}$ is the final radius of the emitted cluster. In the ELDM, the total potential has been demonstrated to be the sum of Coulomb, proximity, and centrifugal potential [52, 53]. Hence, we can use the effective one-dimensional total potential energy as follows:

V = V_{c} + V_{s} + V_{l} .

(4)

To evaluate the Coulomb contribution in terms of the deformation parameter, we used V_c as defined in Ref. [54]:

\begin{array}{l} V_{c} (R) = \frac{e^{2} Z_{1} Z_{2}}{R} + 3 Z_{1} Z_{2} e^{2} \sum_{λ, i = 1,2} \\ \times \frac{R_{i}^{λ} (α_{i}, T)}{(2 λ + 1)} Y_{λ}^{(0)} (θ_{i}) [β_{λ i} + \frac{4}{7} β_{λ i}^{2} Y_{λ}^{(0)} (θ_{i})], \end{array}

(5)

with

R_{i} (α_{i}) = R_{o i} [1 + \sum_{λ} β_{λ i} Y_{λ}^{(0)} (α_{i})],

(6)

where βλi is the deformation parameter, Yλ(0) are the spherical harmonics, and $R_{o i} = 1.28 A_{i}^{1 / 3} - 0.76 + 0.8 A_{i}^{- 1 / 3}$ . The effective surface potential can be calculated by

V_{s} = σ_{eff} (S_{1} + S_{2}),

(7)

where S₁ and S₂ are the surface areas of the spherical fragments. σ_eff is the effective surface tension, which is defined as

σ_{eff} = \frac{1}{4 (R^{2} - R_{1}^{2} - R_{2}^{2})} (Q - \frac{3}{20 π ϵ_{0}} e^{2} [\frac{Z^{2}}{R} - \frac{Z_{1}^{2}}{R_{1}} - \frac{Z_{2}^{2}}{R_{2}}]),

(8)

where R₂ is the final radius of the daughter fragment. The centrifugal potential energy is determined by

V_{l} = \frac{ℏ^{2} l (l + 1)}{2 μ ζ^{2}},

(9)

where 𝓁 is the angular momentum of the emitted alpha/cluster and is calculated using the selection rules. In the case of alpha/cluster decay [55, 56], the selection rules follow the condition

| J_{p} - J_{d} | \leq l_{α} \leq | J_{p} + J_{d} | and \frac{π_{p}}{π_{d}} = (- {1)}^{l_{a}},

(10)

where J_p, π_p and J_d, π_d are the spin and parity of the parent and daughter nuclei, respectively. μ= $\frac{M_{1} M_{2}}{M_{1} + M_{2}}$ is the reduced mass of the fragments, where M₁ and M₂ denote their atomic masses.

In ELDM, a system with two intersecting spherical nuclei with different radii is considered [52]. A schematic diagram for the representation of four independent coordinates, namely R₁, R₂, ζ, and ξ, is shown in Fig. 1. Three constraints are used to reduce the 4-dimensional spherical problem to an equivalent 1-dimensional problem. The geometric constraint given below is introduced so that the spherical segments remain in contact:

Fig. 1.

Schematic presentation of molecular phase of the di-nuclear system (the daughter nucleus and the emitted (smaller) fragments). The distance between their geometrical centers and the distance between the center of the heavier fragment and the circular sharp neck of radius a are denoted by ζ and ξ, respectively.

R_{1}^{2} - {(ζ - ξ)}^{2} = R^{2} - ξ^{2} .

(11)

The variables ζ and ξ represent the distance between the geometrical centers and the distance between the center of the heavier fragment and the circular sharp neck of the radius, respectively [53, 57]. Assuming that nuclear matter is incompressible, the constraint for the conservation of the total volume of the system is

\begin{array}{l} 2 (R_{1}^{3} + R_{2}^{3}) + 3 [R_{1}^{2} (ζ - ξ) + R_{2}^{2} ξ] - [(ζ - ξ)^{2} + ξ^{3}] \\ = 4 R^{3}, \end{array}

(12)

where R=r₀A^1/3 is the radius of the parent nucleus (r₀=1.34 fm is an adjustable parameter), with A being the mass number of the parent.

The radius of the α particle, R₁, is assumed to be constant in the varying mass asymmetry shape description:

R_{1} - {\bar{R}}_{1} = 0,

(13)

where ${\bar{R}}_{1} = {(\frac{Z_{i}}{Z})}^{1 / 3} R$ , i=1,2; ${\bar{R}}_{1}$ provides the final radius of the α particle. Here, Z₁, Z₂, and Z are the atomic numbers of the α particle, daughter nucleus, and parent nucleus, respectively.

2.2

Beta decay

For all types of β processes, the expression for the half-life Tβ is given by [58]

\frac{1}{T_{β}} = \frac{1}{T_{β^{+}}} + \frac{1}{T_{β^{-}}} + \frac{1}{E C} .

(14)

Here, EC is the electron capture. For a particular type of β-decay, the half-life is expressed as follows:

f_{0}^{b} T_{b} = \ln 2 {[\frac{g^{2} m_{e}^{5} c^{4}}{2 π^{3} ℏ^{7}} | M_{if} |^{2}]}^{- 1} .

(15)

Here, $f_{0}^{b}$ is the Fermi function, b=β^± or EC, m_e is the mass of the electron, and M_if is the transition matrix element between the initial and final states. The right-hand side of the above may be approximated by a constant for each type of β-decay [59]. This constant is different for allowed and forbidden cases of beta decay. For allowed β-decay, this constant has been determined as 5.7±1.1 [60]. Equation (15) is reduced to

\log_{10} [f_{0}^{b} T_{b} (sec)] = 5.7 \pm 1.1,

(16)

\log_{10} [f_{0}^{b} T_{b}] = 4.7.

(17)

2.2.1

β^± decay

The Fermi function for β-decay is expressed as

f_{0}^{β} \pm = \int_{1}^{E_{0}} F (E, Z) P (E) (E_{0} - E)^{2} d E .

(18)

Here, P(E) is the momentum of the particle, and F(E,Z) can be computed at the nuclear surface using the magnitude of the radial electron/positron wave function. The first approximation of F(E,Z) is

F_{0} (E, Z) = \frac{2 (γ + 1) (2 p R)^{2 (γ - 1)} \exp [\frac{π ξ}{p}] | Γ (γ + i \frac{ε}{p} {) |}^{2}}{Γ^{2} (2 γ + 1)} .

(19)

Here, $γ = \sqrt{1 - α^{2} Z^{2}}$ , ξ=± α ZE (+ for β^- decay and - for β⁺ decay), α=1/137 is the fine structure constant, R is the radius of the nucleus, and Γ is the gamma function.

At the surface of the nucleus (for β⁺ decay), the orbital electron screening effect has a significant impact on the β electron/positron wave function. Thus, F(E,Z) becomes

F (E, Z) = F_{0} (E \mp V_{0}, Z) æ^{\mp} (E \mp V_{0}, Z) \frac{p (E \mp V_{0}) (E \mp V_{0})}{p (E) E} .

(20)

Here, V₀=1.81α²Z^4/3, is the finite wavelength of the β particle, $p (E) = \sqrt{E^{2} - 1}$ is the momentum of the β particle, E₀=1+Q_β±/m_ec² is the total limit energy of the β decay, E=1+ε/m_ec², and ε is the kinetic energy of the β particle

The expression for the energy released in β⁺ decay is

Q_{β^{+}} = M (A, Z) - M (A, Z - 1) - 2 m_{e} c^{2} .

(21)

Similarly, for β^- decay,

Q_{β^{-}} = M (A, Z) - M (A, Z - 1) .

(22)

2.2.2

Electron capture

The value of Q for electron capture is given by

\begin{array}{l} Q_{EC} = M (A, Z) - M (A, Z - 1) - B_{e} \\ = Q_{β^{+}} + 2 m_{e} c^{2} - B_{e} . \end{array}

(23)

Here, B_e is the electron binding energy. Hence, even for the forbidden β⁺ decay, electron capture is allowed. The capture of electrons of the K-shell for lower Z, and of the L-shell for higher Z is the major contributor to electron capture. The contributions of the electrons of higher shells are negligible. Thus, the Fermi function becomes

f_{0}^{EC} = f_{0}^{K} + f_{0}^{L_{I}} + f_{0}^{L_{II}} .

(24)

In general, for any shell,

\begin{array}{l} f_{0}^{X} = \frac{π}{2} {[E (Q_{EC}) + E_{X}]}^{2} [g_{X}^{2} (Z_{X}) + f_{X}^{2} (Z_{X})] \\ X = K, L_{I}, L_{II} . \end{array}

(25)

Here, EX is the total energy of the electron:

E_{K} = γ, E_{L} = (\frac{1 + γ}{2})^{1 / 2},

(26)

where ZX is the effective charge, which considers the screening of the Coulomb field of the nucleus by other electrons [61]:

Z_{K} = Z - 0.35 and Z_{L} = Z - 4.15.

(27)

The non-zero components of the radial parts (g_X & f_X) of the wave function of the relativistic electron of orbit X are

g_{K}^{2} (Z) = \frac{4 (1 + γ) (2 α Z R)^{2 (γ - 1)} {(α Z)}^{3}}{Γ (2 γ + 1)},

(28)

g_{L_{I}}^{2} (Z) = \frac{[(2 γ + {2)}^{1 / 2} + 2] (2 γ + 1) (2 α Z R)^{2 (γ - 1)} {(α Z)}^{3}}{Γ (2 γ + 1) [(2 γ + {2)}^{1 / 2} + 1] (2 γ + {2)}^{γ}},

(29)

g_{L_{II}}^{2} (Z) = \frac{3}{16} {(α Z)}^{2} g_{L_{I}}^{2} (Z) .

(30)

2.3

Spontaneous fission

Spontaneous fission decay is studied by employing the quantum tunneling effect through the potential barrier. The decay constant of spontaneous fission is expressed as

λ = \frac{\ln 2}{T_{sf}} = ν S P_{s},

(31)

where ν, S, and P_s are model-dependent quantities, namely, assault frequency, preformation probability, and barrier penetrability, respectively. In the above equation, P = SP_s and the spontaneous-fission half-lives are calculated as

T = \frac{\ln 2}{ν P} = \frac{h \ln 2}{2} \frac{1}{E_{ν} P},

(32)

where h is the Planck constant, and Ev=hν/2 is the zero-point vibration energy. The penetration probability is evaluated using the action integral K:

P = \exp (- K),

(33)

and hence the decimal logarithm of T(s) is given by

\log_{10} T = 0.43429 K - 20.8436 - \log_{10} E_{ν} .

(34)

If Eν=0.5 MeV, then the above equation becomes log₁₀T=-log₁₀P-20.5426. The action integral K is evaluated as follows:

K = \frac{2 \sqrt{2 m}}{ℏ} \int_{R_{a}}^{R_{b}} {(B (r) [E (R) - Q])}^{1 / 2} d R .

(35)

The term E(R) is the macroscopic energy in terms of the surface, volume, Coulomb, proximity energy, shell correction, and pairing energy term [62], and m is the rest mass of the neutron. A few superheavies are spherical, the rest are deformed, primarily prolate or oblate. To include this effect, deformations are also involved in the calculation of E(r), which is adopted from Ref. [62]. In the above equation, R is the separation distance between the center of the fission fragments, and Ra and Rb are the turning points, which are evaluated using the boundary conditions E(Ra) and E(Rb)=Q. However, the term B(r) is the inertia with respect to r and is evaluated using the semi-empirical model for inertia [63]:

B (r) = μ (1 + k \exp [- \frac{128}{51} (r - R_{sph} / R_{0})]),

(36)

where μ and k are the reduced mass of the fission fragments, and a semi-empirical constant (k=14.8), respectively. R_sph is the distance between the center of mass of the fission fragments, set as R_sph/R₀=0.75 in the symmetric case. The decay constant (λ) and the total fission decay constant are evaluated as described in Ref. [62]

2.4

Results and discussion

The mass excess values play a major role in the prediction of the decay mode and the corresponding half-lives. The predicted half-lives are sensitive to the Q-values, and small changes in the Q-values result in a notable change in the half-lives, with a magnitude of order 10¹ to 10² [36]. Mass excess tables such as WS4 [64], EBW [65], HFB28 and HFB29 [66], DZ10 [67], KTUY [68], finite-range droplet model (FRDM) [69], and AME16 [70] are available in the literature. In the present study, we used the updated AME16 [70] mass excess values up to Z=118, and above Z>118, the mass excess values are taken from the FRDM [69]. The dominant decay mode is identified by studying the competition between different decay modes: α-decay, β-decay, cluster decay, and spontaneous fission in the superheavy nuclei region 104 ≤ Z ≤ 126 .

A detailed literature review indicates that there is no experimental evidence for cluster radioactivity in the superheavy region. Furthermore, experimental studies of cluster decay in the actinide region are available. To validate the present study, the cluster-decay half-lives obtained in the present study in the actinide region were compared with the experiments, and good agreement was observed. With this confidence, we studied cluster decay in the superheavy region, and the results are presented in Table 1. Similarly, Table 2 shows a comparison of the studied logarithmic half-lives (in years) of spontaneous fission from the present study with those from available experiments. It can be seen that the cluster-decay and spontaneous-fission half-lives obtained in the present study are close to those of the experiments.

Cluster-decay half-lives obtained from present study (PS) and available experiments (exp)

Decay	Q_Exp(MeV)	log $T_{1 / 2}^{exp}$ [71]	log $T_{1 / 2}^{PS}$
²²¹Fr→¹⁴C+²⁰⁷Tl	31.317	14.51	14.91
²²¹Ra→¹⁴C+²⁰⁷Pb	32.396	13.37	13.56
²²²Ra→¹⁴C+²⁰⁸Pb	33.05	11.05	12.70
²²³Ra→¹⁴C+²⁰⁹Pb	31.829	15.05	13.94
²²⁴Ra→¹⁴C+²¹⁰Pb	30.54	15.9	15.52
²²⁶Ra→¹⁴C+²¹²Pb	28.2	21.29	22.74
²²⁵Ac→¹⁴C+²¹¹Bi	30.477	17.16	17.06
²²⁸Th→²⁰O+²⁰⁸Pb	44.72	20.73	22.04
²³⁰U→²²Ne+²⁰⁸Pb	61.4	19.56	20.21
²³⁰Th→²⁴Ne+²⁰⁶Hg	57.571	24.61	25.07
²³¹Pa→²⁴Ne+²⁰⁷Tl	60.417	22.89	23.07
²³²U→²⁴Ne+²⁰⁸Pb	62.31	20.39	22.25
²³³U→²⁴Ne+²⁰⁹Pb	60.486	24.84	25.05
²³⁴U→²⁶Ne+²⁰⁸Pb	59.466	25.93	25.62
²³⁴U→²⁸Mg+²⁰⁶Hg	74.11	25.74	26.04
²³⁶Pu→²⁸Mg+²⁰⁸Pb	79.67	21.65	22.07
²³⁸Pu→²⁸Mg+²¹⁰Pb	75.912	25.66	25.98
²³⁸Pu→³⁰Mg+²⁰⁸Pb	77	25.66	26.25
²³⁸Pu→³²Si+²⁰⁶Hg	91.19	25.3	26.05
²⁴²Cm→³⁴Si+²⁰⁸Pb	96.509	23.11	24.24

Comparison of logarithm half-lives (years) of spontaneous fission in the superheavy region 104 ≤ Z ≤ 114 from present study with those from available experiments.

Parent nuclei	log $_{SF}^{Expt}$ yr [72]	log $_{SF}^{Th}$ yr
²⁵⁴Rf	-12.1	-10.91
²⁵⁶Rf	-9.71	-8.48
²⁵⁸Rf	-9.35	-7.06
²⁶⁰Rf	-9.2	-6.35
²⁶²Rf	-7.18	-6.36
²⁵⁸Sg	-10	-11.33
²⁶⁰Sg	-9.65	-10.17
²⁶²Sg	-9.32	-8.722
²⁶⁴Sg	-8.93	-7.98
²⁶⁶Sg	-7.86	-7.96
²⁶⁴Hs	-10.2	-11.02
²⁷⁰Ds	-8.6	-9.46
²⁸²Cn	-10.6	-9.39
²⁸⁴Cn	-8.5	-7.98
²⁸⁶Fl	-8.08	-7.58

As a part of this investigation, we studied the α-decay properties of superheavy nuclei using the formalism explained in the theory section. The predicted alpha-decay half-lives were validated by comparison with those from available experiments in the superheavy region. The results are given in Table 3.

Comparison of alpha-decay half-lives from the present study (PS) and those from available experimental (Exp) values

Parent nuclei	Qα (MeV)	logT_1/2 (Exp)	logT_1/2 (PS)
²⁶¹Bh	8.649	1.515	1.86
²⁶⁰Db	9.379	-0.295	0.11
²⁶⁹Sg	8.8	2.27	2.12
²⁶⁵Sg	9.078	0.869	1.15
²⁶³Sg	9.391	-0.932	0.12
²⁶¹Sg	9.803	-1.469	-1.21
²⁷²Bh	9.3	1.025	0.78
²⁷¹Bh	9.5	0.176	0.18
²⁷⁰Bh	9.3	1.785	1.02
²⁷⁷Hs	8.4	-2.523	-1.02
²⁷³Hs	9.9	-0.119	-0.32
²⁶⁹Hs	9.629	0.851	0.65
²⁷⁴Hs	9.5	0.079	0.21
²⁷⁸Mt	9.1	0.653	1.65
²⁷⁶Mt	9.8	-0.284	0.05
²⁷⁴Mt	10.5	-0.357	-0.98
²⁸¹Ds	8.958	1.104	1.45
²⁸²Rg	9.38	2	1.85
²⁸⁰Rg	9.98	0.623	0.55
²⁷⁹Rg	10.45	-1.046	-1.04
²⁸⁵Cn	8.793	1.447	2.85
²⁸³Cn	9.62	0.623	0.89
²⁸¹Cn	10.28	-0.886	-0.68
²⁸⁴Cn	9.301	1.013	1.78
²⁷⁷Cn	11.622	-2.551	-2.65
²⁸⁶Nh	9.68	0.978	1.22
²⁸⁵Nh	10.02	0.623	0.76
²⁸⁴Nh	10.25	-0.013	0.08
²⁸³Nh	10.6	-1.125	-0.98
²⁸⁹Fl	9.847	0.279	0.96
²⁸⁸Fl	9.969	-0.18	-0.16
²⁸⁷Fl	10.436	-0.319	-0.28
²⁸⁶Fl	10.7	-0.921	-0.87
²⁸⁵Fl	11	-0.824	-1.89
²⁹⁰Mc	10.3	-0.187	0.18
²⁸⁹Mc	10.6	-0.481	-0.35
²⁹³Lv	8.886	-1.244	0.12
²⁹²Lv	10.707	-1.886	-0.96
²⁹¹Lv	11	-1.721	-1.45
²⁸⁹Lv	11.7	-2.848	-2.97
²⁹⁴Ts	8.963	-1.292	0.06
²⁹⁴Og	8.47	-3.161	-2.45
²⁹⁵Og	9.056	-1.745	0.58
²⁹⁸120	13.355	-3.051	-4.68
²⁹⁹120	13.105	-3.15	-1.58

From the comparison, it is observed that the predicted half-lives are in good agreement with those of the experiments. With this confidence, we obtained the alpha-decay half-lives of superheavy nuclei in the region 104 ≤ Z ≤ 126. Figure 2

Fig. 2.

(Color online) Map of nuclei reflecting the logarithmic α-decay half-lives for the isotopes of elements from Z=104 to 126. The Q-values were estimated using AME16 and FRDM95. The vertical line on the right side of the figure shows an increase in the log T_1/2 values from the navy-blue region to the brown region.

shows a wide range of α-decay half-lives. For a given superheavy nucleus, the alpha decay half-lives increase as the neutron number of its isotopes increases. For instance, the α-decay half-lives are of the order of nanoseconds at N/Z=1.307692 for Rutherfordium, whereas for the same superheavy element, the α-decay half-lives are of the order of 10²s at N/Z=1.504762. Similarly, all neutron-rich superheavy nuclei have comparably longer α-decay half-lives, which is in agreement with the report available in Ref. [73]. The obtained α-decay half-lives of all possible superheavy nuclei are presented in the heat map in Fig. 2. The right vertical bar shows the magnitude of the logT_1/2 values. The color variation from navy blue to wine indicates values in the range 10^-10–10² s. The contrast in the blue region lies between 10^-10 s and 10^-7 s, in the green region, it lies in the range 10^-6–10^-4 s, and the range 10^-4-10^-3 s is presented in the yellow region. Finally, the red-to-wine region shows higher half-lives in the range 10^-2-10² s. The inset of Fig. 2 on the top left side provides information on the magnified portion of α-decay half-lives in the superheavy region Z=104-114, whereas the bottom-right inset provides information on the magnified portion of the superheavy region Z=115-126. After the detailed investigation of the α-decay, a search was made to identify the cluster emitters in the superheavy region. Cluster radioactivity is energetically favorable if the Q-values are positive. We studied the possibility of cluster decay with 3 ≤ Z_c ≤ 45 in the superheavy region 104 ≤ Z ≤ 126. For a given parent nucleus, the half-lives corresponding to various cluster emission were evaluated, and the cluster corresponding to shorter half-lives was identified. Furthermore, the cluster emitters corresponding to shorter half-lives for different isotopes of a given superheavy element were also identified. Eventually, cluster emissions corresponding to the shortest half-lives T_c were identified; these are referred to as cluster-decay half-lives (T_c). The predicted cluster decay half-lives in the atomic number region 104 ≤ Z ≤ 126 correspond to all the studied cluster emissions, as shown in Fig. 3.

Fig. 3.

(Color online) Predicted cluster-decay logarithmic half-lives in the atomic number range Z=104–126 using AME16 and FRDM95 mass excess values. The hallow bin with different color in each panel shows the cluster emission corresponding to minimum half-lives.

This figure enables us to identify the cluster emission corresponding to the shorter half-lives of a given superheavy element. The half-lives of superheavy nuclei with Z=115–120 against cluster radioactivity are shorter for ⁸⁶Kr than those of the other studied clusters. The superheavy nuclei with Z=104, 106, 108, 110, 112, 114, 124, and 126 have shorter half-lives against ⁹⁶Mo cluster emissions than those of the other studied clusters. The decay half-lives are shorter for the ⁹¹Y emission from superheavy nuclei with Z=109, 111, 113, 121, and 123. Similarly, the half-lives of superheavy nuclei with Z=105 and 107 against cluster radioactivity are shorter for ⁹⁷Tc and ¹⁰¹Rh than those of the other studied clusters.

Cluster radioactivity in the superheavy nuclei region has shorter half-lives for cluster neutron numbers 44–48 from parent nuclei with neutron numbers 130—200, as shown in Fig. 4.

Fig. 4.

(Color online) Map of nuclei reflecting the logarithmic cluster-decay half-lives for neutron number of parent and cluster isotopes of elements with Z=104–126.

The range of cluster decay half-lives for superheavy elements with 104 ≤ Z ≤ 126 is shown in Fig. 5.

Fig. 5.

(Color online) Heat map showing the variations of lowest logarithmic half lives of clusters with 104 < Z < 126

Shorter half-lives are observed for N/Z > 1.37068, and larger half-lives are observed for N/Z < 1.37068. From the figure, it is clear that up to superheavy nuclei 104 ≤ Z ≤ 115, larger cluster-decay half-lives are observed, whereas shorter cluster-decay half-lives are observed in the superheavy region 116 ≤ Z ≤ 126. The inset of Fig. 5 on the top-left side shows a magnified portion of the logarithmic half-lives (T_c) in the superheavy region 104 ≤ Z ≤ 115, whereas the inset at the right bottom shows a magnified portion of the shorter logarithmic half-lives (T_c) in the superheavy region 116 ≤ Z ≤ 126. This figure also shows that some of the superheavy nuclei have lifetimes of the order of ns to μs, and exhibit cluster decay.

The other prominent decay mode that was studied is spontaneous fission, which is also energetically feasible in heavy and superheavy nuclei. It may occur in such nuclei owing to an increase in the Coulomb interactions. References [10, 11, 38, 74-77] report consistent α-decay chains from superheavy nuclei followed by spontaneous fission. The spontaneous fission half-lives are studied using the theory explained in Sect. 2.3. The variations of spontaneous fission half-lives in the superheavy region Z=104–126 are shown in Fig. 6.

Fig. 6.

(Color online) Heat map of the variations of logarithmic half-lives for spontaneous fission for 104 < Z < 126

The log T_SF values vary between -50(dark blue region) and 50 (dark-red region). For instance, at atomic number Z=104, for isotopes 245—275, the log T_1/2(SF) values ranging from -50 to 5 are shown, whereas the half-lives with smaller values are indicated by the color range from navy blue to blue. The half-lives ranging from nanoseconds to 10⁵ s are indicated by the color range from yellow to light orange. Similarly, in the atomic number range Z=119 and above, larger values of spontaneous-fission logarithmic half-lives are indicated by the red color range. Thus, on either side of Fig. 6, for isotopes corresponding to the atomic number range Z=104–126, smaller half-lives are observed, whereas in the middle region of the figure, larger values of logT_1/2 are observed up to Z=116. In contrast, smaller half-lives are observed for higher isotopes (Z>116), and larger logT_1/2 for lower isotopes (Z<116). A similar trend was also observed in a previous study [78], in which the half-lives of nuclei Z=92–104 were compared with experimentally available values.

A detailed investigation of the Q-values corresponding to β-decay in the superheavy region demonstrates that β⁺-decay is energetically possible with Z=105, 107, 113, 114, 115, 117, 119, 121, 123, 125, and 126, whereas β^--decay is not energetically possible. Furthermore, we also studied β-decay half-lives using the formalism explained in Sect. 2.2.1.

The competition between different possible decay modes, namely, α-decay, cluster-decay, β-decay and spontaneous fission, enables us to identify the dominant decay mode for superheavy elements in the atomic number region 104 ≤ Z ≤ 126 of all possible isotopes. Figure 7

Fig. 7.

(Color online) Chart of spontaneous fission (purple), alpha decay (brown), β⁺-decay (cyan), and cluster emitters (yellow) with atomic numbers Z=104–126. The Q-values were calculated using the FRDM95 mass tables.

shows the decay modes of the superheavy nuclei. In the studied superheavy region, we identified around 20 β⁺emitters, which are presented in Table 6. We also identified 35 cluster emitters, which are presented in Table 4.

Identified β⁺ emitters in the superheavy nuclei region

Parent nuclei	Q (MeV)	logT_1/2
²⁶⁴Db	2.24	-0.04
²⁶⁸Bh	2.93	-0.83
²⁹⁰Fl	0.79	1.28
²⁸⁶Mc	4.53	-3.68
²⁹²Ts	4.96	-4.12
²⁹⁰119	7.20	-6.27
²⁹⁶119	5.75	-5.01
²⁹²121	8.29	-7.56
²⁹⁴121	8.06	-7.15
²⁹⁸121	6.83	-6.32
³⁰²121	5.12	-5.49
²⁹⁸123	8.42	-8.04
³⁰⁰123	8.27	-7.64
³⁰²123	6.72	-7.23
³⁰⁶123	5.73	-6.42
³⁰⁴125	7.81	-8.55
³⁰⁶125	7.61	-8.15
³⁰⁸125	6.99	-7.75
³¹⁰125	6.47	-7.35
³¹²125	5.79	-6.96

Identified cluster emitters in the superheavy nuclei region

Parent nuclei	Q (MeV)	logT_1/2	Cluster
²⁹²Og	304.08	-5.08	⁸⁶Kr
²⁹³Og	303.63	-4.63	⁸⁶Kr
²⁹⁸122	338.25	-6.02	⁹⁴Zr
³⁰⁰122	337.45	-6.21	⁹⁴Zr
²⁹⁹123	338.66	-7.18	⁹¹Y
³⁰⁰124	356.06	-7.35	⁹⁶Mo
³⁰⁶126	364.27	-8.78	⁹⁶Mo

It was demonstrated that the majority of superheavy nuclei undergo α-decay and spontaneous fission. The α-emitting superheavy nuclei are listed in Table 5.

Identified alpha emitters in the superheavy nuclei region

Parent nuclei	Q (MeV)	logT_1/2	Parent nuclei	Q (MeV)	logT_1/2	Parent nuclei	Q (MeV)	logT_1/2
²⁵⁶Rf	10.15	0.92	²⁷⁷Ds	10.34	-2.51	²⁸⁰Lv	13.59	-6.78
²⁵⁷Rf	10.05	0.78	²⁷¹Rg	11.61	-3.72	²⁸¹Lv	13.35	-6.98
²⁵⁸Rf	9.94	-0.16	²⁷³Rg	11.44	-3.56	²⁸²Lv	13.13	-5.14
²⁵⁹Rf	9.67	0.35	²⁷⁵Rg	11.37	-3.42	²⁸³Lv	12.91	-5.78
²⁶⁰Rf	9.4	-1.12	²⁷⁷Rg	10.88	-2.21	²⁸⁴Lv	12.7	-3.89
²⁶¹Rf	9.14	1.56	²⁷⁹Rg	10.44	-1.23	²⁸⁵Lv	12.51	-3.99
²⁶²Rf	8.92	0.52	²⁸⁰Rg	10.24	0.62	²⁸⁶Lv	12.34	-4.25
²⁵⁸Db	10.45	0.55	²⁸²Rg	9.89	2.15	²⁸⁷Lv	12.19	-3.98
²⁵⁹Db	10.36	-0.35	²⁷¹Cn	12.1	-4.89	²⁸⁸Lv	12.06	-3.56
²⁶⁰Db	10.08	0.26	²⁷²Cn	11.96	-4.65	²⁹⁰Lv	11.83	-1.75
²⁶¹Db	9.81	0.65	²⁷³Cn	11.87	-4.33	²⁹¹Lv	11.73	-2.36
²⁶³Db	9.34	1.52	²⁷⁴Cn	11.8	-4.16	²⁹²Lv	11.64	-1.88
²⁷⁰Db	8.45	3.25	²⁷⁵Cn	11.76	-3.98	²⁹³Lv	11.55	-1.23
²⁵⁹Sg	10.84	-0.15	²⁷⁶Cn	11.74	-2.98	²⁷⁹Ts	14.06	-7.36
²⁶⁰Sg	10.74	-2.16	²⁷⁷Cn	11.49	-3.15	²⁸¹Ts	14.02	-8.25
²⁶¹Sg	10.47	-0.48	²⁷⁸Cn	11.25	-2.47	²⁸³Ts	13.56	-5.36
²⁶²Sg	10.2	-1.86	²⁷⁹Cn	11.03	-2.36	²⁸⁵Ts	13.14	-5.12
²⁶³Sg	9.95	0.25	²⁸⁰Cn	10.81	-1.78	²⁸⁷Ts	12.78	-4.88
²⁶⁹Sg	9.16	2.56	²⁸¹Cn	10.62	-0.99	²⁸⁹Ts	12.5	-4.65
²⁶⁰Bh	11.21	-1.42	²⁸⁵Cn	10	1.56	²⁹¹Ts	12.28	-2.98
²⁶³Bh	10.59	-1.76	²⁷³Nh	12.4	-5.65	²⁹⁴Ts	12	-1.45
²⁶⁵Bh	10.12	0.12	²⁷⁵Nh	12.24	-4.79	²⁸¹Og	14.44	-7.65
²⁶⁶Bh	9.94	0.22	²⁷⁶Nh	12.2	-4.78	²⁸²Og	14.43	-7.63
²⁷⁰Bh	9.56	1.89	²⁷⁷Nh	12.18	-4.52	²⁸³Og	14.19	-7.45
²⁷¹Bh	9.53	0.18	²⁷⁹Nh	11.7	-2.89	²⁸⁴Og	13.97	-6.25
²⁷²Bh	9.27	1.12	²⁸¹Nh	11.26	-2.12	²⁸⁵Og	13.76	-6.41
²⁷⁴Bh	8.78	1.23	²⁸²Nh	11.07	-1.69	²⁸⁶Og	13.56	-6.24
²⁶³Hs	11.27	-2.56	²⁸⁴Nh	10.73	-0.16	²⁸⁷Og	13.37	-5.25
²⁶⁵Hs	10.77	-4.56	²⁸⁵Nh	10.59	0.78	²⁸⁸Og	13.21	-5.98
²⁶⁶Hs	10.55	-1.85	²⁸⁶Nh	10.47	0.88	²⁹⁴Og	12.53	-3.88
²⁶⁷Hs	10.37	-1.42	²⁸⁷Nh	10.35	0.76	²⁹⁵Og	12.44	-1.25
²⁶⁸Hs	10.23	0.69	²⁷⁵Fl	12.78	-4.69	²⁸⁵119	14.37	-5.69
²⁶⁹Hs	10.13	1.42	²⁷⁶Fl	12.72	-5.12	²⁸⁷119	13.96	-4.25
²⁷⁰Hs	10.05	1.78	²⁷⁷Fl	12.68	-5.36	²⁸⁹119	13.62	-5.97
²⁷¹Hs	10	0.45	²⁷⁸Fl	12.66	-5.46	²⁹²119	13.23	-5.28
²⁷³Hs	9.71	-0.56	²⁷⁹Fl	12.42	-4.12	²⁹⁷119	12.78	-3.97
²⁷⁵Hs	9.23	-0.15	²⁸⁰Fl	12.19	-4.36	²⁸⁷120	14.56	-6.58
²⁶⁶Mt	11.27	-1.93	²⁸¹Fl	11.96	-3.78	²⁸⁸120	14.37	-6.25
²⁶⁷Mt	11.06	-2.52	²⁸²Fl	11.76	-3.15	²⁹⁰120	14.03	-6.46
²⁶⁹Mt	10.74	-2.32	²⁸³Fl	11.56	-2.99	²⁹²120	13.76	-5.85
²⁷¹Mt	10.57	-1.85	²⁸⁸Fl	10.86	-0.25	²⁹⁸120	13.2	-3.87
²⁷³Mt	10.49	-1.23	²⁸⁹Fl	10.75	0.62	²⁹⁹120	13.11	-4.12
²⁷⁴Mt	10.23	-0.12	²⁷⁷Mc	13.19	-6.85	³⁰⁰120	13.02	-4.36
²⁷⁵Mt	9.99	-1.25	²⁷⁸Mc	13.16	-6.48	²⁹⁰121	14.59	-6.28
²⁷⁶Mt	9.75	-0.36	²⁷⁹Mc	13.14	-5.96	²⁹⁶121	13.87	-5.48
²⁷⁸Mt	9.33	1.36	²⁸⁰Mc	12.9	-5.12	³⁰⁰121	13.53	-5.22
²⁶⁸Ds	11.62	-3.56	²⁸¹Mc	12.67	-5.36	³⁰³122	13.77	-4.98
²⁶⁹Ds	11.45	-3.24	²⁸³Mc	12.24	-4.25	³⁰⁴122	13.67	-4.99
²⁷⁰Ds	11.31	-3.68	²⁸⁵Mc	11.88	-3.12	³⁰⁴123	14.18	-5.12
²⁷¹Ds	11.21	-0.58	²⁸⁸Mc	11.47	-0.89	³⁰⁶124	14.49	-5.66
²⁷²Ds	11.14	-3.09	²⁸⁹Mc	11.36	-0.52	³⁰⁸124	14.28	-5.98
²⁷³Ds	11.1	-2.96	²⁹⁰Mc	11.26	-0.25	³¹⁰124	14.05	-5.87
²⁷⁴Ds	11.07	-2.45	²⁷⁸Lv	13.64	-6.75
²⁷⁵Ds	10.81	-2.12	²⁷⁹Lv	13.61	-6.12

The identified alpha emitters have half-lives of approximately μs to 100 s in the superheavy region 104 ≤ Z ≤ 126. Table 4 lists the identified cluster emissions with the corresponding half-lives. The amount of energy released during cluster emission, cluster emitted, and logT_1/2 values are presented in the table. The minimum cluster decay half-lives correspond to ⁸⁶Kr, ⁹⁴Zr, ⁹¹Y, and ⁹⁶Mo for the nuclei ^292-293Og, ^298,300122, ²⁹⁹123, ³⁰⁰124, and ³⁰⁶126, respectively. From the available literature, it is also evident that the heavy particle radioactivity of ⁸⁶Kr is observed in the superheavy nucleus Z=118 [36, 79]. In addition, Rb, Sr, Y, Zr, Nb, and Mo cluster emissions [80] were observed for Z=119–124 respectively. As in previous studies, in the present study, shorter half-lives in the superheavy region Z=118, 122-124, and 126 were observed, with ⁸⁶Kr, ⁹⁴Zr, ⁹¹Y, and ⁹⁶Mo cluster emissions, respectively. Similarly, approximately 20 possible β⁺ emitters were identified in the superheavy region 105 ≤ Z ≤ 125, and they are presented in Table 6.

The information provided Table 7 regarding the half-lives and branching ratios presents ambiguities in terms of determining a single decay mode. The branching ratios relative to the minimum half-lives among the studied decay modes are obtained, and the second column of the table shows the logT_1/2 values corresponding to spontaneous-fission, α-decay, β⁺-decay, and cluster-decay half-lives. For instance, the superheavy nucleus ²⁶³Rf exhibits shorter logT_1/2 values for spontaneous fission and β⁺-decay than for other decay modes. The branching ratio of spontaneous fission and β⁺-decay was obtained, and it was found that the branching ratio corresponding to spontaneous fission and β⁺ was 55% and 45%, respectively. Similarly, we identified the branching ratios for the superheavy region 104 ≤ Z ≤ 126, which are presented in Table 7.

Superheavy nuclei with dual decay mode and branching ratios

Parent	logT_1/2				Decay mode		Parent nuclei	logT_1/2				Decay mode
nuclei	Sf	α	β⁺	CD				Sf	α	β⁺	CD
²⁶³Rf	1.25	1.98	1.65	36.6	Sf=57%	β⁺=43%	²⁹⁶120	19.58	-4.87	-4.38	-5.45	CD=53%	α=47%
²⁶²Db	2.31	-0.48	-0.51	34.76	β⁺=52%	α=48%	²⁹¹120	25.45	-6.21	-5.85	-4.69	α=51%	β⁺=49%
²⁶⁴Bh	0.71	-1.65	-1.75	20.58	β⁺=51%	α=49%	²⁹⁷120	15.36	-4.25	-4.6	-3.62	β⁺=52%	α=48%
²⁶²Bh	-3.25	-3.25	-2.22	17.73	Sf=50%	Sf=50%	²⁸⁹120	21.36	-6.31	-6.27	-4.3	α=50%	β⁺=50%
²⁶⁶Bh	1.62	-1.89	-1.29	23.6	α=59%	β⁺=41%	²⁹³120	25.32	-5.18	-5.43	-5.06	β⁺=51%	α=49%
²⁶⁴Hs	-2.98	-3.15	-1.11	15.37	α=51%	Sf=49%	²⁹³121	29.32	-6.98	-6.29	-6.47	α=52%	CD=48%
²⁷⁰Mt	2.11	-2.45	-2.08	19.43	α=54%	β⁺=46%	²⁹⁹121	18.56	-4.75	-5.05	-5.43	CD=52%	β⁺=48%
²⁷²Mt	1.35	-1.65	-1.63	22.66	α=50%	β⁺=50%	³⁰³121	-4.36	-3.52	-4.22	2.35	Sf=51%	β⁺=49%
²⁶⁸Mt	0.99	-2.78	-2.54	16.53	α=52%	β⁺=48%	²⁹¹121	25.87	-5.94	-6.7	-6.65	β⁺=50%	CD=50%
²⁷⁶Ds	-1.72	-1.65	-0.09	23.61	Sf=51%	α=49%	²⁹⁵121	30.36	-5.78	-5.88	-6.34	CD=52%	β⁺=48%
²⁷²Rg	2.22	-3.15	-3.34	12.55	β⁺=51%	α=49%	²⁹⁷121	26.9	-4.77	-5.46	-6.33	CD=54%	β⁺=46%
²⁷⁶Rg	2.42	-2.16	-2.44	18.44	β⁺=53%	α=47%	³⁰¹121	8.25	-5.36	-4.64	-1.47	α=54%	β⁺=46%
²⁶⁹Rg	-3.98	-3.86	-2.95	9.04	Sf=51%	α=49%	²⁹⁴122	30.65	-5.98	-6.51	-6.89	CD=51%	β⁺=49%
²⁷⁰Rg	-1.56	-3.87	-3.79	10.12	α=51%	β⁺=49%	²⁹⁵122	34.25	-5.99	-6.73	-6.78	CD=50%	β⁺=50%
²⁷⁸Rg	-1.32	-1.54	-1.99	21.54	β⁺=56%	α=44%	²⁹⁶122	36.89	-6.24	-6.1	-6.83	CD=52%	α=48%
²⁷⁴Rg	3.56	-3.22	-2.89	15.4	α=53%	β⁺=47%	²⁹⁷122	34.22	-6.96	-6.32	-6.77	α=51%	CD=49%
²⁸³Nh	-1.55	-1.89	-1.55	18.55	α=55%	Sf=45%	³⁰⁵122	-0.58	-4.98	-4.68	-0.68	α=52%	β⁺=48%
²⁷⁴Nh	0.65	-4.01	-4.6	6.88	β⁺=53%	α=47%	³⁰²122	18.21	-5.27	-4.87	-6.31	CD=54%	α=46%
²⁸⁰Nh	4.98	-2.78	-3.27	14.3	β⁺=54%	α=46%	³⁰¹122	23.56	-5.96	-5.5	-6.46	CD=52%	α=48%
²⁷⁸Nh	5.88	-3.87	-3.72	11.51	α=51%	β⁺=49%	²⁹⁹122	30.89	-6.14	-5.91	-6.7	CD=52%	α=48%
²⁷⁴Fl	-5.12	-4.89	-3.95	3.62	Sf=51%	α=49%	³⁰³122	13.77	-5.11	-5.09	-4.33	α=50%	β⁺=50%
²⁸⁵Fl	-0.45	-2.11	-1.97	16.71	α=52%	β⁺=48%	³⁰³123	27.89	-6.36	-5.96	-3.66	α=52%	β⁺=48%
²⁸²Mc	11.02	-3.78	-4.55	0.52	β⁺=55%	α=45%	²⁹⁷123	38.48	-6.17	-7.18	-7.51	CD=51%	β⁺=49%
²⁸⁷Mc	1.32	-2.98	-2.4	0.5	α=55%	β⁺=45%	³⁰¹123	35.02	-6.18	-6.37	-6.86	CD=52%	β⁺=48%
²⁸⁴Mc	8.98	-3.58	-4.12	0.37	β⁺=54%	α=46%	³⁰⁷123	2.18	-4.98	-5.15	2.81	β⁺=51%	α=49%
²⁷⁶Mc	-3.12	-5.89	-5.86	1.45	α=50%	β⁺=50%	³⁰⁵123	15.58	-5.48	-5.56	-0.65	β⁺=50%	α=50%
²⁷⁷Lv	-6.85	-5.78	-5.42	0.18	Sf=54%	α=46%	³⁰⁸123	-4.89	-5.26	-6.01	3.61	β⁺=53%	α=47%
²⁸⁹Lv	3.15	-3.05	-2.83	-1.32	α=52%	β⁺=48%	³⁰³124	40.25	-6.25	-6.83	-6.77	β⁺=50%	CD=50%
²⁹³Ts	-2.98	-3.58	-2.84	-2.88	α=55%	Sf=45%	³⁰⁵124	33.21	-5.12	-6.42	-4.29	β⁺=56%	α=44%
²⁸⁴Ts	13.25	-5.25	-5.83	-2.5	β⁺=53%	α=47%	³⁰²124	40.25	-5.96	-6.6	-7.13	CD=52%	β⁺=48%
²⁸⁶Ts	14.55	-5.16	-5.4	-2.6	β⁺=51%	α=49%	³⁰⁴124	35.85	-6.01	-6.19	-6.45	CD=51%	β⁺=49%
²⁹⁰Ts	9.18	-4.36	-4.55	-3.09	β⁺=51%	α=49%	³⁰⁷124	22.14	-5.97	-6.02	-0.09	β⁺=50%	α=50%
²⁸⁸Ts	13.78	-4.87	-4.98	-2.91	β⁺=51%	α=49%	³⁰¹124	44.14	-6.04	-7.23	-7.53	CD=51%	β⁺=49%
²⁸⁰Ts	0.68	-6.25	-6.69	-2.01	β⁺=52%	α=48%	³⁰⁹124	6.87	-5.11	-5.62	3.17	β⁺=52%	α=48%
²⁸²Ts	8.58	-5.85	-6.26	-2.19	β⁺=52%	α=48%	³⁰⁹125	26.25	-6.51	-6.49	0.26	α=50%	β⁺=50%
²⁹⁰Og	16.98	-3.78	-3.91	-4.72	CD=55%	β⁺=45%	³⁰⁷125	38.21	-6.21	-6.89	-3.75	β⁺=53%	α=47%
²⁹¹Og	14.89	-3.78	-4.13	-4.9	CD=54%	β⁺=46%	³¹¹125	11.58	-5.75	-6.09	2.43	β⁺=51%	α=49%
²⁸⁹Og	18.74	-5.25	-4.56	-4.46	α=54%	β⁺=46%	³⁰⁵125	45.35	-6.87	-7.29	-7.5	CD=51%	β⁺=49%
²⁸⁴119	5.95	-6.68	-7.53	-3.17	β⁺=53%	α=47%	³⁰³125	45.25	-6.96	-7.69	-8.07	CD=51%	β⁺=49%
²⁹³119	18.29	-3.98	-4.57	-5.15	CD=53%	β⁺=47%	³¹⁵126	-1.58	-5.75	-6.17	2.82	β⁺=52%	α=48%
²⁸⁶119	14.98	-7.25	-7.11	-3.79	α=50%	β⁺=50%	³¹³126	15.25	-6.24	-6.57	0.85	β⁺=51%	α=49%
²⁹⁴119	15.98	-3.89	-5.43	-5.15	β⁺=51%	CD=49%	³⁰⁸126	48.21	-6.87	-7.12	-5.37	β⁺=51%	α=49%
²⁸⁸119	19.35	-5.85	-6.69	-4.15	β⁺=53%	α=47%	³¹¹126	32.21	-5.12	-6.96	-0.97	β⁺=58%	α=42%
²⁹⁸119	-4.85	-3.22	-4.59	0.23	Sf=51%	β⁺=49%	³¹⁰126	39.21	-6.32	-6.73	-1.99	β⁺=52%	α=48%
²⁹¹119	21.25	-4.58	-4.99	-4.74	β⁺=51%	CD=49%	³⁰⁹126	44.58	-6.58	-7.36	-3.69	β⁺=53%	α=47%
²⁹⁵119	11.35	-4.23	-4.15	-5.03	CD=54%	α=46%	³¹²126	26.12	-5.74	-6.33	0.01	β⁺=52%	α=48%
²⁹⁵120	22.15	-4.78	-5.02	-5.46	CD=52%	β⁺=48%	³⁰⁷126	51.32	-7.11	-7.75	-6.99	β⁺=52%	α=48%
²⁹⁴120	23.58	-4.22	-4.79	-5.36	CD=53%	β⁺=47%	³¹⁴126	7.56	-6.21	-5.94	1.7	α=51%	β⁺=49%

Finally, Fig. 8 shows the lifetimes of the superheavy elements after the competition between different decay modes was studied.

Fig. 8.

(Color online) Heat map showing the variations of atomic number, mass number of parent and logarithmic half-lives of different decay modes (life times) for 104 < Z < 126

It can be seen that the lifetime varies from ns to min and decreases as the atomic number increases. For instance, the average lifetime of a superheavy element with Z=104 is approximately 10 min, whereas that of a hypothetical superheavy element with Z=126 is of the order of ms.

Conclusion

We systematically investigated all possible decay modes, namely, α-decay, β-decay, cluster decay, and spontaneous fission, in the superheavy region 104 ≤ Z ≤ 126. The findings of this study were validated by comparison with experiments. Approximately 20 β⁺ and 7 heavy particle emitters were found in the superheavy region. Furthermore, the nuclei with almost the same half-lives for the two decay modes were also reported, with the corresponding branching ratios. However, an experimental study is necessary to draw definite conclusions.

References

[1]

Hebberger F. P., Hofmann S., Ackermann D. et al.,

Decay properties of neutron-deficient isotopes 256,257Db, 255Rf, 252,253Lr

. Europ. Phys. J. A 12, 57-67 (2001). doi: 10.1007/s100500170039