Investigation of decay modes of superheavy nuclei

Investigation of decay modes of superheavy nuclei 1 corresp first-author manjunathhc@rediffmail.com manjunathhc@rediffmail.com manjunathhc@rediffmail.com 1 1 2 1 3 1 3 Department of Physics, Government College for Women, Kolar-563101, Karnataka, India Department of Physics, Government College for Women, Kolar-563101, Karnataka, India Department of Physics, Government First Grade College, Kolar-563101 Karnataka, India Department of Physics, Government First Grade College, Kolar-563101 Karnataka, India Department of Physics, St. Joseph’s college, Affiliated to Bharathidasan University, Tiruchirappalli-Tamil Nadu, India Department of Physics, St. Joseph’s college, Affiliated to Bharathidasan University, Tiruchirappalli-Tamil Nadu, India A detailed investigations of different decay modes, namely, alpha decay, beta decay, cluster decay, including heavy particle emission ( Z c & gt; 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. 292-293 Og were identified as 86 Kr emitters, and 298 122 and 300 122 were identified as 94 Zr emitters, whereas heavy-particle radioactivity from 91 Y was also observed in 299 123. Furthermore, the nuclei 300 124 and 306 126 exhibit 96 Mo 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 decay Beta decay Heavy-particle radioactivity Branching ratios Introduction 1 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 μ s, can be synthesized using cold and hot fusion reactions. Cold fusion reactions involve either lead or bismuth as targets [ 1 1 ], whereas hot fusion reactions include 48 Ca beams on various actinide targets. [ 2 2 , 3 3 ]. Many theoretical predictions, such as microscopic–macroscopic [ 4 4 ] (single-particle potential) and self-consistent approaches, including nucleus–nucleus potential [ 5 5 , 6 6 ], relativistic field models [ 7 7 , 8 8 ], and multinucleon transfer reactions [ 9 9 ], provide information regarding the nucleus structure, shell closure location, and decay modes in heavy and superheavy nuclei. The discovery of superheavy elements [ 10 10 , 11 11 ] points to the island of stability. Boilley et al. [ 12 12 ] predicted the evaporation residue cross sections in superheavy elements and the influence of shell effects [ 13 13 ]. The entrance channel dynamics were studied using 48 Ca as a projectile and 208 Pb as target [ 14 14 ]. In 1966, two groups of researchers, namely, Mayers and Swiatecki, and Viola and Seaborg [ 15 15 ], separately predicted the presence of heavy nuclei near the island of stability. Later, Sobiczewski et al. [ 16 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 17 ] proposed a shell model which includes deformation property of the nuclei. Bender et al. [ 18 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 19 ]. The investigation of isomers of the superheavy nucleus 254 No is a stepping stone toward the island of stability [ 20 20 ]. Previous researchers [ 21 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 22 - 23 23 22-23 ]. The possible decay modes in the superheavy nuclei Z =119 and 120 were predicted in Ref. [ 34 34 ]. From Ref. [ 35 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 284 Fl, α -decay is dominant in the nuclei 286-289 Fl, and β + is dominant in the nucleus 290 Fl. Furthermore, the concept of heavy-particle radioactivity [ 36 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 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 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 39 ]. Various phenomenological and microscopic models, such as the fission model [ 40 40 ], the cluster model [ 41 41 ], generalized liquid drop model [ 42 42 ], and the unified model for alpha decay and alpha capture [ 43 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 44 - 46 46 44-46 ]. Simple empirical formulas are also available for determining the decay half-lives [ 47 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 1 Alpha decay and cluster decay 2 In the effective liquid-drop model (ELDM), the α -decay half-life is computed using the relation <math display="block" id="M1"><mrow><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>ln</mi><mn>2</mn></mrow><mrow><msub><mi>ν</mi><mn>0</mn></msub><mi>P</mi><msub><mi>P</mi><mi>α</mi></msub></mrow></mfrac><mn>,</mn></mrow></math> where ν 0 is the assault frequency on the barrier, and ν 0 =1.8×10 22 s -1 [ 48 48 ]. P α is the preformation factor, which is closely related to the shell structure [ 49 49 ]. The empirical formula for P α is expressed as <math display="block" id="M2"><mtable columnalign="left"><mtr><mtd><mi>log</mi><msub><mi>P</mi><mi>α</mi></msub><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><mo>+</mo><msub><mi>p</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>Z</mi><mo>−</mo><msub><mi>Z</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>2</mn></msub><mo>−</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>p</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><msub><mi>N</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>N</mi><mn>2</mn></msub><mo>−</mo><mi>N</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi><mn>4</mn></msub><mi>A</mi><mn>,</mn></mtd></mtr></mtable></math> where N , Z , and A are the neutron, charge, and mass number of the parent nucleus, respectively, Z 1 and Z 2 are the proton magic numbers around Z ( Z 1 ≤ Z ≤ Z 2 ), and N 1 and N 2 are the neutron magic numbers around N ( N 1 ≤ N ≤ N 2 ). p 1 , p 2 , and p 3 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 50 ]. P is the Gamow penetrability factor, given by the expression <math display="block" id="M3"><mrow><mi>P</mi><mo>=</mo><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mi>ℏ</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>ζ</mi><mn>0</mn></msub></mrow><mrow><msub><mi>ζ</mi><mi>c</mi></msub></mrow></msubsup></mrow></mstyle><msqrt><mrow><mn>2</mn><mi>μ</mi><mo stretchy="false">[</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo>−</mo><mi>Q</mi><mo stretchy="false">]</mo></mrow></msqrt><mtext>d</mtext><mi>ζ</mi></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where μ is the inertial coefficient resulting from the Werner—Wheeler approximation [ 51 51 ]. The limits of integration ζ 0 and ζ c are the inner and outer turning points, expressed as ζ 0 = R p - <math id="M4"><mrow><msub><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mn>1</mn></msub></mrow></math> and <math id="M5"><mrow><msub><mi>ζ</mi><mi>c</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mi>Q</mi></mfrac></mrow></math> . R p is the radius of the parent nucleus, and <math id="M6"><mrow><msub><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mn>1</mn></msub></mrow></math> 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 52 , 53 53 ]. Hence, we can use the effective one-dimensional total potential energy as follows: <math display="block" id="M7"><mrow><mi>V</mi><mo>=</mo><msub><mi>V</mi><mtext>c</mtext></msub><mo>+</mo><msub><mi>V</mi><mtext>s</mtext></msub><mo>+</mo><msub><mi>V</mi><mi mathvariant="script">l</mi></msub><mn>.</mn></mrow></math> To evaluate the Coulomb contribution in terms of the deformation parameter, we used V c as defined in Ref. [ 54 54 ]: <math display="block" id="M8"><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mtext>c</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub></mrow><mi>R</mi></mfrac><mo>+</mo><mn>3</mn><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub><msup><mi>e</mi><mn>2</mn></msup><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>λ</mi><mn>,</mn><mi>i</mi><mo>=</mo><mn>1,2</mn></mrow></munder></mstyle></mtd></mtr><mtr><mtd><mo>×</mo><mfrac><mrow><msubsup><mi>R</mi><mi>i</mi><mi>λ</mi></msubsup><mo stretchy="false">(</mo><msub><mi>α</mi><mi>i</mi></msub><mn>,</mn><mi>T</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>λ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><msubsup><mi>Y</mi><mi>λ</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mrow><mo>[</mo><mrow><msub><mi>β</mi><mrow><mi>λ</mi><mi>i</mi></mrow></msub><mo>+</mo><mfrac><mn>4</mn><mn>7</mn></mfrac><msubsup><mi>β</mi><mrow><mi>λ</mi><mi>i</mi></mrow><mn>2</mn></msubsup><msubsup><mi>Y</mi><mi>λ</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> with <math display="block" id="M9"><mrow><msub><mi>R</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>R</mi><mrow><mi>o</mi><mi>i</mi></mrow></msub><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>λ</mi></munder></mstyle><msub><mi>β</mi><mrow><mi>λ</mi><mi>i</mi></mrow></msub><msubsup><mi>Y</mi><mi>λ</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>α</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where β λ i is the deformation parameter, Y λ (0) are the spherical harmonics, and <math id="M10"><mrow><msub><mi>R</mi><mrow><mi>o</mi><mi>i</mi></mrow></msub><mo>=</mo><mn>1.28</mn><msubsup><mi>A</mi><mi>i</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mo>−</mo><mn>0.76</mn><mo>+</mo><mn>0.8</mn><msubsup><mi>A</mi><mi>i</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></math> . The effective surface potential can be calculated by <math display="block" id="M11"><mrow><msub><mi>V</mi><mtext>s</mtext></msub><mo>=</mo><msub><mi>σ</mi><mrow><mtext>eff</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>S</mi><mn>1</mn></msub><mo>+</mo><msub><mi>S</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where S 1 and S 2 are the surface areas of the spherical fragments. σ eff is the effective surface tension, which is defined as <math display="block" id="M12"><mrow><msub><mi>σ</mi><mrow><mtext>eff</mtext></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mo stretchy="false">(</mo><msup><mi>R</mi><mn>2</mn></msup><mo>−</mo><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac><mrow><mo>(</mo><mrow><mi>Q</mi><mo>−</mo><mfrac><mn>3</mn><mrow><mn>20</mn><mi>π</mi><msub><mi>ϵ</mi><mn>0</mn></msub></mrow></mfrac><msup><mi>e</mi><mn>2</mn></msup><mrow><mo>[</mo><mrow><mfrac><mrow><msup><mi>Z</mi><mn>2</mn></msup></mrow><mi>R</mi></mfrac><mo>−</mo><mfrac><mrow><msubsup><mi>Z</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msubsup><mi>Z</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow></mfrac></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where R 2 is the final radius of the daughter fragment. The centrifugal potential energy is determined by <math display="block" id="M13"><mrow><msub><mi>V</mi><mi mathvariant="script">l</mi></msub><mo>=</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup><mi mathvariant="script">l</mi><mo stretchy="false">(</mo><mi mathvariant="script">l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mi>μ</mi><msup><mi>ζ</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> 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 55 , 56 56 ], the selection rules follow the condition <math display="block" id="M14"><mrow><mo>|</mo><msub><mi>J</mi><mtext>p</mtext></msub><mo>−</mo><msub><mi>J</mi><mtext>d</mtext></msub><mo>|</mo><mo>≤</mo><msub><mi mathvariant="script">l</mi><mi>α</mi></msub><mo>≤</mo><mo>|</mo><msub><mi>J</mi><mtext>p</mtext></msub><mo>+</mo><msub><mi>J</mi><mtext>d</mtext></msub><mo>|</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mfrac><mrow><msub><mi>π</mi><mtext>p</mtext></msub></mrow><mrow><msub><mi>π</mi><mtext>d</mtext></msub></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><msup><mrow><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><msub><mi mathvariant="script">l</mi><mi>a</mi></msub></mrow></msup><mn>,</mn></mrow></math> where J p , π p and J d , π d are the spin and parity of the parent and daughter nuclei, respectively. μ = <math id="M15"><mrow><mfrac><mrow><msub><mi>M</mi><mn>1</mn></msub><msub><mi>M</mi><mn>2</mn></msub></mrow><mrow><msub><mi>M</mi><mn>1</mn></msub><mo>+</mo><msub><mi>M</mi><mn>2</mn></msub></mrow></mfrac></mrow></math> is the reduced mass of the fragments, where M 1 and M 2 denote their atomic masses. In ELDM, a system with two intersecting spherical nuclei with different radii is considered [ 52 52 ]. A schematic diagram for the representation of four independent coordinates, namely R 1 , R 2 , ζ , and ξ , is shown in Fig. 1 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: <math display="block" id="M16"><mrow><msub><mi>R</mi><mn>1</mn></msub><msup><mrow/><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>ζ</mi><mo>−</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>R</mi><mn>2</mn></msup><mo>−</mo><msup><mi>ξ</mi><mn>2</mn></msup><mn>.</mn></mrow></math> 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 53 , 57 57 ]. Assuming that nuclear matter is incompressible, the constraint for the conservation of the total volume of the system is <math display="block" id="M17"><mtable columnalign="left"><mtr><mtd><mn>2</mn><mo stretchy="false">(</mo><msub><mi>R</mi><mn>1</mn></msub><msup><mrow/><mn>3</mn></msup><mo>+</mo><msub><mi>R</mi><mn>2</mn></msub><msup><mrow/><mn>3</mn></msup><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><mo stretchy="false">[</mo><msub><mi>R</mi><mn>1</mn></msub><msup><mrow/><mn>2</mn></msup><mo stretchy="false">(</mo><mi>ζ</mi><mo>−</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>R</mi><mn>2</mn></msub><msup><mrow/><mn>2</mn></msup><mi>ξ</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>ζ</mi><mo>−</mo><mi>ξ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mi>ξ</mi><mn>3</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>=</mo><mn>4</mn><msup><mi>R</mi><mn>3</mn></msup><mn>,</mn></mtd></mtr></mtable></math> where R = r 0 A 1/3 is the radius of the parent nucleus ( r 0 =1.34 fm is an adjustable parameter), with A being the mass number of the parent. The radius of the α particle, R 1 , is assumed to be constant in the varying mass asymmetry shape description: <math display="block" id="M18"><mrow><msub><mi>R</mi><mn>1</mn></msub><mo>−</mo><msub><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>=</mo><mn>0,</mn></mrow></math> where <math id="M19"><mrow><msub><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Z</mi><mi>i</mi></msub></mrow><mi>Z</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mi>R</mi></mrow></math> , i =1,2; <math id="M20"><mrow><msub><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mn>1</mn></msub></mrow></math> provides the final radius of the α particle. Here, Z 1 , Z 2 , and Z are the atomic numbers of the α particle, daughter nucleus, and parent nucleus, respectively. Beta decay 2 For all types of β processes, the expression for the half-life T β is given by [ 58 58 ] <math display="block" id="M21"><mrow><mfrac><mn>1</mn><mrow><msub><mi>T</mi><mi>β</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>T</mi><mrow><msup><mi>β</mi><mo>+</mo></msup></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>T</mi><mrow><msup><mi>β</mi><mo>−</mo></msup></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>E</mi><mi>C</mi></mrow></mfrac><mn>.</mn></mrow></math> Here, EC is the electron capture. For a particular type of β -decay, the half-life is expressed as follows: <math display="block" id="M22"><mrow><msub><mi>f</mi><mn>0</mn></msub><msup><mrow/><mi>b</mi></msup><msub><mi>T</mi><mi>b</mi></msub><mo>=</mo><mi>ln</mi><mn>2</mn><msup><mrow><mrow><mo>[</mo><mrow><mfrac><mrow><msup><mi>g</mi><mn>2</mn></msup><msub><mi>m</mi><mtext>e</mtext></msub><msup><mrow/><mn>5</mn></msup><msup><mi>c</mi><mn>4</mn></msup></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>3</mn></msup><msup><mi>ℏ</mi><mn>7</mn></msup></mrow></mfrac><mo>|</mo><msub><mi>M</mi><mrow><mtext>if</mtext></mrow></msub><msup><mo>|</mo><mn>2</mn></msup></mrow><mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>.</mn></mrow></math> Here, <math id="M23"><mrow><msubsup><mi>f</mi><mn>0</mn><mi>b</mi></msubsup></mrow></math> 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 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 60 ]. Equation (15) is reduced to <math display="block" id="M24"><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><mo stretchy="false">[</mo><msubsup><mi>f</mi><mn>0</mn><mi>b</mi></msubsup><msub><mi>T</mi><mi>b</mi></msub><mo stretchy="false">(</mo><mtext>sec</mtext><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mn>5.7</mn><mo>±</mo><mn>1.1,</mn></mrow></math> <math display="block" id="M25"><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><mo stretchy="false">[</mo><msubsup><mi>f</mi><mn>0</mn><mi>b</mi></msubsup><msub><mi>T</mi><mi>b</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mn>4.7.</mn></mrow></math> β ± decay 3 The Fermi function for β -decay is expressed as <math display="block" id="M26"><mrow><msubsup><mi>f</mi><mn>0</mn><mi>β</mi></msubsup><mo>±</mo><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>1</mn><mrow><msub><mi>E</mi><mn>0</mn></msub></mrow></msubsup></mrow></mstyle><mi>F</mi><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>E</mi><mn>0</mn></msub><mo>−</mo><mi>E</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mtext>d</mtext><mi>E</mi><mn>.</mn></mrow></math> 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 <math display="block" id="M27"><mrow><msub><mi>F</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>p</mi><mi>R</mi><msup><mo stretchy="false">)</mo><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>γ</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mi>exp</mi><mrow><mo>[</mo><mrow><mfrac><mrow><mi>π</mi><mi>ξ</mi></mrow><mi>p</mi></mfrac></mrow><mo>]</mo></mrow><mo>|</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>i</mi><mfrac><mi>ε</mi><mi>p</mi></mfrac><msup><mrow><mo stretchy="false">)</mo><mo>|</mo></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>Γ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mn>2</mn><mi>γ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> Here, <math id="M28"><mrow><mi>γ</mi><mo>=</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>α</mi><mn>2</mn></msup><msup><mi>Z</mi><mn>2</mn></msup></mrow></msqrt></mrow></math> , 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 <math display="block" id="M29"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo>∓</mo><msub><mi>V</mi><mn>0</mn></msub><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><msup><mtext>æ</mtext><mo>∓</mo></msup><mo stretchy="false">(</mo><mi>E</mi><mo>∓</mo><msub><mi>V</mi><mn>0</mn></msub><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>E</mi><mo>∓</mo><msub><mi>V</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo>∓</mo><msub><mi>V</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi>E</mi></mrow></mfrac><mn>.</mn></mrow></math> Here, V 0 =1.81 α 2 Z 4/3 , is the finite wavelength of the β particle, <math id="M30"><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><msup><mi>E</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></math> is the momentum of the β particle, E 0 =1+ Q β ± / m e c 2 is the total limit energy of the β decay, E =1+ ε / m e c 2 , and ε is the kinetic energy of the β particle The expression for the energy released in β + decay is <math display="block" id="M31"><mrow><msub><mi>Q</mi><mrow><msup><mi>β</mi><mo>+</mo></msup></mrow></msub><mo>=</mo><mi>M</mi><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>M</mi><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><msub><mi>m</mi><mtext>e</mtext></msub><msup><mi>c</mi><mn>2</mn></msup><mn>.</mn></mrow></math> Similarly, for β - decay, <math display="block" id="M32"><mrow><msub><mi>Q</mi><mrow><msup><mi>β</mi><mo>−</mo></msup></mrow></msub><mo>=</mo><mi>M</mi><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>M</mi><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Electron capture 3 The value of Q for electron capture is given by <math display="block" id="M33"><mtable columnalign="left"><mtr><mtd><msub><mi>Q</mi><mrow><mtext>EC</mtext></mrow></msub><mo>=</mo><mi>M</mi><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>M</mi><mo stretchy="false">(</mo><mi>A</mi><mn>,</mn><mi>Z</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>B</mi><mtext>e</mtext></msub></mtd></mtr><mtr><mtd><mo>=</mo><msub><mi>Q</mi><mrow><msup><mi>β</mi><mo>+</mo></msup></mrow></msub><mo>+</mo><mn>2</mn><msub><mi>m</mi><mtext>e</mtext></msub><msup><mi>c</mi><mn>2</mn></msup><mo>−</mo><msub><mi>B</mi><mtext>e</mtext></msub><mn>.</mn></mtd></mtr></mtable></math> 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 <math display="block" id="M34"><mrow><msubsup><mi>f</mi><mn>0</mn><mrow><mtext>EC</mtext></mrow></msubsup><mo>=</mo><msubsup><mi>f</mi><mn>0</mn><mi>K</mi></msubsup><mo>+</mo><msubsup><mi>f</mi><mn>0</mn><mrow><msub><mi>L</mi><mtext>I</mtext></msub></mrow></msubsup><mo>+</mo><msubsup><mi>f</mi><mn>0</mn><mrow><msub><mi>L</mi><mrow><mtext>II</mtext></mrow></msub></mrow></msubsup><mn>.</mn></mrow></math> In general, for any shell, <math display="block" id="M35"><mtable columnalign="left"><mtr><mtd><msubsup><mi>f</mi><mn>0</mn><mi>X</mi></msubsup><mo>=</mo><mfrac><mtext>π</mtext><mn>2</mn></mfrac><msup><mrow><mo>[</mo><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>Q</mi><mrow><mtext>EC</mtext></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>E</mi><mtext>X</mtext></msub></mrow><mo>]</mo></mrow><mn>2</mn></msup><mrow><mo>[</mo><mrow><msubsup><mi>g</mi><mi>X</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>X</mtext></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>f</mi><mi>X</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><msub><mi>Z</mi><mi>X</mi></msub><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>X</mi><mo>=</mo><mi>K</mi><mo>,</mo><msub><mi>L</mi><mtext>I</mtext></msub><mo>,</mo><msub><mi>L</mi><mrow><mtext>II</mtext></mrow></msub><mo>.</mo></mtd></mtr></mtable></math> Here, E X is the total energy of the electron: <math display="block" id="M36"><mrow><msub><mi>E</mi><mi>K</mi></msub><mo>=</mo><mi>γ</mi><mn>,</mn><msub><mi>E</mi><mi>L</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>γ</mi></mrow><mn>2</mn></mfrac><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mn>,</mn></mrow></math> where Z X is the effective charge, which considers the screening of the Coulomb field of the nucleus by other electrons [ 61 61 ]: <math display="block" id="M37"><mrow><msub><mi>Z</mi><mi>K</mi></msub><mo>=</mo><mi>Z</mi><mo>−</mo><mn>0.35</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msub><mi>Z</mi><mi>L</mi></msub><mo>=</mo><mi>Z</mi><mo>−</mo><mn>4.15.</mn></mrow></math> The non-zero components of the radial parts ( g X & f X ) of the wave function of the relativistic electron of orbit X are <math display="block" id="M38"><mrow><msubsup><mi>g</mi><mi>K</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>4</mn><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>α</mi><mi>Z</mi><mi>R</mi><msup><mo stretchy="false">)</mo><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>γ</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>α</mi><mi>Z</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>γ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M39"><mrow><msubsup><mi>g</mi><mrow><msub><mi>L</mi><mtext>I</mtext></msub></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>2</mn><mi>γ</mi><mo>+</mo><msup><mrow><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>2</mn><mi>γ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>α</mi><mi>Z</mi><mi>R</mi><msup><mo stretchy="false">)</mo><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>γ</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi>α</mi><mi>Z</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>γ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>2</mn><mi>γ</mi><mo>+</mo><msup><mrow><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>2</mn><mi>γ</mi><mo>+</mo><msup><mrow><mn>2</mn><mo stretchy="false">)</mo></mrow><mi>γ</mi></msup></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M40"><mrow><msubsup><mi>g</mi><mrow><msub><mi>L</mi><mrow><mtext>II</mtext></mrow></msub></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>3</mn><mrow><mn>16</mn></mrow></mfrac><msup><mrow><mo stretchy="false">(</mo><mi>α</mi><mi>Z</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><msubsup><mi>g</mi><mrow><msub><mi>L</mi><mtext>I</mtext></msub></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Spontaneous fission 2 Spontaneous fission decay is studied by employing the quantum tunneling effect through the potential barrier. The decay constant of spontaneous fission is expressed as <math display="block" id="M41"><mrow><mi>λ</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mn>2</mn></mrow><mrow><msub><mi>T</mi><mrow><mtext>sf</mtext></mrow></msub></mrow></mfrac><mo>=</mo><mi>ν</mi><mi>S</mi><msub><mi>P</mi><mtext>s</mtext></msub><mn>,</mn></mrow></math> 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 <math display="block" id="M42"><mrow><mi>T</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mn>2</mn></mrow><mrow><mi>ν</mi><mi>P</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>h</mi><mi>ln</mi><mn>2</mn></mrow><mn>2</mn></mfrac><mfrac><mn>1</mn><mrow><msub><mi>E</mi><mi>ν</mi></msub><mi>P</mi></mrow></mfrac><mn>,</mn></mrow></math> where h is the Planck constant, and E v = hν /2 is the zero-point vibration energy. The penetration probability is evaluated using the action integral K : <math display="block" id="M43"><mrow><mi>P</mi><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mi>K</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> and hence the decimal logarithm of T ( s ) is given by <math display="block" id="M44"><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><mi>T</mi><mo>=</mo><mn>0.43429</mn><mi>K</mi><mo>−</mo><mn>20.8436</mn><mo>−</mo><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><msub><mi>E</mi><mi>ν</mi></msub><mn>.</mn></mrow></math> If E ν =0.5 MeV, then the above equation becomes log 10 T =-log 10 P -20.5426. The action integral K is evaluated as follows: <math display="block" id="M45"><mrow><mi>K</mi><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>m</mi></mrow></msqrt></mrow><mi>ℏ</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>R</mi><mi>a</mi></msub></mrow><mrow><msub><mi>R</mi><mi>b</mi></msub></mrow></msubsup></mrow></mstyle><msup><mrow><mrow><mo>(</mo><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>−</mo><mi>Q</mi><mo stretchy="false">]</mo></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mtext>d</mtext><mi>R</mi><mn>.</mn></mrow></math> The term E ( R ) is the macroscopic energy in terms of the surface, volume, Coulomb, proximity energy, shell correction, and pairing energy term [ 62 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 62 ]. In the above equation, R is the separation distance between the center of the fission fragments, and R a and R b are the turning points, which are evaluated using the boundary conditions E ( R a ) and E ( R b )= Q . However, the term B ( r ) is the inertia with respect to r and is evaluated using the semi-empirical model for inertia [ 63 63 ]: <math display="block" id="M46"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mi>μ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>k</mi><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mrow><mn>128</mn></mrow><mrow><mn>51</mn></mrow></mfrac><mo stretchy="false">(</mo><mi>r</mi><mo>−</mo><msub><mi>R</mi><mrow><mtext>sph</mtext></mrow></msub><mo>/</mo><msub><mi>R</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> 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 / R 0 =0.75 in the symmetric case. The decay constant ( λ ) and the total fission decay constant are evaluated as described in Ref. [ 62 62 ] Results and discussion 2 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 10 1 to 10 2 [ 36 36 ]. Mass excess tables such as WS4 [ 64 64 ], EBW [ 65 65 ], HFB28 and HFB29 [ 66 66 ], DZ10 [ 67 67 ], KTUY [ 68 68 ], finite-range droplet model (FRDM) [ 69 69 ], and AME16 [ 70 70 ] are available in the literature. In the present study, we used the updated AME16 [ 70 70 ] mass excess values up to Z =118, and above Z & gt;118, the mass excess values are taken from the FRDM [ 69 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 Table 1 . Similarly, Table 2 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. 1 1 Decay 1 1 Q Exp (MeV) 1 1 log <math id="M47"><mrow><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>exp</mtext></mrow></msubsup></mrow></math> [ 71 71 ] 1 1 log <math id="M48"><mrow><msubsup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mtext>PS</mtext></mrow></msubsup></mrow></math> 1 1 221 Fr→ 14 C+ 207 Tl 1 1 31.317 1 1 14.51 1 1 14.91 1 1 221 Ra → 14 C+ 207 Pb 1 1 32.396 1 1 13.37 1 1 13.56 1 1 222 Ra → 14 C+ 208 Pb 1 1 33.05 1 1 11.05 1 1 12.70 1 1 223 Ra → 14 C+ 209 Pb 1 1 31.829 1 1 15.05 1 1 13.94 1 1 224 Ra → 14 C+ 210 Pb 1 1 30.54 1 1 15.9 1 1 15.52 1 1 226 Ra → 14 C+ 212 Pb 1 1 28.2 1 1 21.29 1 1 22.74 1 1 225 Ac → 14 C+ 211 Bi 1 1 30.477 1 1 17.16 1 1 17.06 1 1 228 Th → 20 O+ 208 Pb 1 1 44.72 1 1 20.73 1 1 22.04 1 1 230 U → 22 Ne+ 208 Pb 1 1 61.4 1 1 19.56 1 1 20.21 1 1 230 Th → 24 Ne+ 206 Hg 1 1 57.571 1 1 24.61 1 1 25.07 1 1 231 Pa → 24 Ne+ 207 Tl 1 1 60.417 1 1 22.89 1 1 23.07 1 1 232 U → 24 Ne+ 208 Pb 1 1 62.31 1 1 20.39 1 1 22.25 1 1 233 U → 24 Ne+ 209 Pb 1 1 60.486 1 1 24.84 1 1 25.05 1 1 234 U → 26 Ne+ 208 Pb 1 1 59.466 1 1 25.93 1 1 25.62 1 1 234 U → 28 Mg+ 206 Hg 1 1 74.11 1 1 25.74 1 1 26.04 1 1 236 Pu → 28 Mg+ 208 Pb 1 1 79.67 1 1 21.65 1 1 22.07 1 1 238 Pu → 28 Mg+ 210 Pb 1 1 75.912 1 1 25.66 1 1 25.98 1 1 238 Pu → 30 Mg+ 208 Pb 1 1 77 1 1 25.66 1 1 26.25 1 1 238 Pu → 32 Si+ 206 Hg 1 1 91.19 1 1 25.3 1 1 26.05 1 1 242 Cm → 34 Si+ 208 Pb 1 1 96.509 1 1 23.11 1 1 24.24 1 1 Parent nuclei 1 1 log <math id="M49"><mrow><msubsup><mrow/><mrow><mtext>SF</mtext></mrow><mrow><mtext>Expt</mtext></mrow></msubsup></mrow></math> yr [ 72 72 ] 1 1 log <math id="M50"><mrow><msubsup><mrow/><mrow><mtext>SF</mtext></mrow><mrow><mtext>Th</mtext></mrow></msubsup></mrow></math> yr 1 1 254 Rf 1 1 -12.1 1 1 -10.91 1 1 256 Rf 1 1 -9.71 1 1 -8.48 1 1 258 Rf 1 1 -9.35 1 1 -7.06 1 1 260 Rf 1 1 -9.2 1 1 -6.35 1 1 262 Rf 1 1 -7.18 1 1 -6.36 1 1 258 Sg 1 1 -10 1 1 -11.33 1 1 260 Sg 1 1 -9.65 1 1 -10.17 1 1 262 Sg 1 1 -9.32 1 1 -8.722 1 1 264 Sg 1 1 -8.93 1 1 -7.98 1 1 266 Sg 1 1 -7.86 1 1 -7.96 1 1 264 Hs 1 1 -10.2 1 1 -11.02 1 1 270 Ds 1 1 -8.6 1 1 -9.46 1 1 282 Cn 1 1 -10.6 1 1 -9.39 1 1 284 Cn 1 1 -8.5 1 1 -7.98 1 1 286 Fl 1 1 -8.08 1 1 -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 Table 3 . 1 1 Parent nuclei 1 1 Q α (MeV) 1 1 log T 1/2 (Exp) 1 1 log T 1/2 (PS) 1 1 261 Bh 1 1 8.649 1 1 1.515 1 1 1.86 1 1 260 Db 1 1 9.379 1 1 -0.295 1 1 0.11 1 1 269 Sg 1 1 8.8 1 1 2.27 1 1 2.12 1 1 265 Sg 1 1 9.078 1 1 0.869 1 1 1.15 1 1 263 Sg 1 1 9.391 1 1 -0.932 1 1 0.12 1 1 261 Sg 1 1 9.803 1 1 -1.469 1 1 -1.21 1 1 272 Bh 1 1 9.3 1 1 1.025 1 1 0.78 1 1 271 Bh 1 1 9.5 1 1 0.176 1 1 0.18 1 1 270 Bh 1 1 9.3 1 1 1.785 1 1 1.02 1 1 277 Hs 1 1 8.4 1 1 -2.523 1 1 -1.02 1 1 273 Hs 1 1 9.9 1 1 -0.119 1 1 -0.32 1 1 269 Hs 1 1 9.629 1 1 0.851 1 1 0.65 1 1 274 Hs 1 1 9.5 1 1 0.079 1 1 0.21 1 1 278 Mt 1 1 9.1 1 1 0.653 1 1 1.65 1 1 276 Mt 1 1 9.8 1 1 -0.284 1 1 0.05 1 1 274 Mt 1 1 10.5 1 1 -0.357 1 1 -0.98 1 1 281 Ds 1 1 8.958 1 1 1.104 1 1 1.45 1 1 282 Rg 1 1 9.38 1 1 2 1 1 1.85 1 1 280 Rg 1 1 9.98 1 1 0.623 1 1 0.55 1 1 279 Rg 1 1 10.45 1 1 -1.046 1 1 -1.04 1 1 285 Cn 1 1 8.793 1 1 1.447 1 1 2.85 1 1 283 Cn 1 1 9.62 1 1 0.623 1 1 0.89 1 1 281 Cn 1 1 10.28 1 1 -0.886 1 1 -0.68 1 1 284 Cn 1 1 9.301 1 1 1.013 1 1 1.78 1 1 277 Cn 1 1 11.622 1 1 -2.551 1 1 -2.65 1 1 286 Nh 1 1 9.68 1 1 0.978 1 1 1.22 1 1 285 Nh 1 1 10.02 1 1 0.623 1 1 0.76 1 1 284 Nh 1 1 10.25 1 1 -0.013 1 1 0.08 1 1 283 Nh 1 1 10.6 1 1 -1.125 1 1 -0.98 1 1 289 Fl 1 1 9.847 1 1 0.279 1 1 0.96 1 1 288 Fl 1 1 9.969 1 1 -0.18 1 1 -0.16 1 1 287 Fl 1 1 10.436 1 1 -0.319 1 1 -0.28 1 1 286 Fl 1 1 10.7 1 1 -0.921 1 1 -0.87 1 1 285 Fl 1 1 11 1 1 -0.824 1 1 -1.89 1 1 290 Mc 1 1 10.3 1 1 -0.187 1 1 0.18 1 1 289 Mc 1 1 10.6 1 1 -0.481 1 1 -0.35 1 1 293 Lv 1 1 8.886 1 1 -1.244 1 1 0.12 1 1 292 Lv 1 1 10.707 1 1 -1.886 1 1 -0.96 1 1 291 Lv 1 1 11 1 1 -1.721 1 1 -1.45 1 1 289 Lv 1 1 11.7 1 1 -2.848 1 1 -2.97 1 1 294 Ts 1 1 8.963 1 1 -1.292 1 1 0.06 1 1 294 Og 1 1 8.47 1 1 -3.161 1 1 -2.45 1 1 295 Og 1 1 9.056 1 1 -1.745 1 1 0.58 1 1 298 120 1 1 13.355 1 1 -3.051 1 1 -4.68 1 1 299 120 1 1 13.105 1 1 -3.15 1 1 -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 Figure 2 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 10 2 s 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 73 ]. The obtained α -decay half-lives of all possible superheavy nuclei are presented in the heat map in Fig. 2 Fig. 2 . The right vertical bar shows the magnitude of the log T 1/2 values. The color variation from navy blue to wine indicates values in the range 10 -10 –10 2 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 -10 2 s. The inset of Fig. 2 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 . 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 86 Kr 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 96 Mo cluster emissions than those of the other studied clusters. The decay half-lives are shorter for the 91 Y 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 97 Tc and 101 Rh 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 . The range of cluster decay half-lives for superheavy elements with 104 ≤ Z ≤ 126 is shown in Fig. 5 Fig. 5 . Shorter half-lives are observed for N/Z & gt; 1.37068, and larger half-lives are observed for N/Z & lt; 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 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 μ s, 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 10 , 11 11 , 38 38 , 74 74 - 77 77 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 . 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 10 5 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 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 log T 1/2 are observed up to Z =116. In contrast, smaller half-lives are observed for higher isotopes ( Z & gt;116), and larger log T 1/2 for lower isotopes ( Z & lt;116). A similar trend was also observed in a previous study [ 78 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 Figure 7 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 Table 6 . We also identified 35 cluster emitters, which are presented in Table 4 Table 4 . 1 1 Parent nuclei 1 1 Q (MeV) 1 1 log T 1/2 1 1 264 Db 1 1 2.24 1 1 -0.04 1 1 268 Bh 1 1 2.93 1 1 -0.83 1 1 290 Fl 1 1 0.79 1 1 1.28 1 1 286 Mc 1 1 4.53 1 1 -3.68 1 1 292 Ts 1 1 4.96 1 1 -4.12 1 1 290 119 1 1 7.20 1 1 -6.27 1 1 296 119 1 1 5.75 1 1 -5.01 1 1 292 121 1 1 8.29 1 1 -7.56 1 1 294 121 1 1 8.06 1 1 -7.15 1 1 298 121 1 1 6.83 1 1 -6.32 1 1 302 121 1 1 5.12 1 1 -5.49 1 1 298 123 1 1 8.42 1 1 -8.04 1 1 300 123 1 1 8.27 1 1 -7.64 1 1 302 123 1 1 6.72 1 1 -7.23 1 1 306 123 1 1 5.73 1 1 -6.42 1 1 304 125 1 1 7.81 1 1 -8.55 1 1 306 125 1 1 7.61 1 1 -8.15 1 1 308 125 1 1 6.99 1 1 -7.75 1 1 310 125 1 1 6.47 1 1 -7.35 1 1 312 125 1 1 5.79 1 1 -6.96 1 1 Parent nuclei 1 1 Q (MeV) 1 1 log T 1/2 1 1 Cluster 1 1 292 Og 1 1 304.08 1 1 -5.08 1 1 86 Kr 1 1 293 Og 1 1 303.63 1 1 -4.63 1 1 86 Kr 1 1 298 122 1 1 338.25 1 1 -6.02 1 1 94 Zr 1 1 300 122 1 1 337.45 1 1 -6.21 1 1 94 Zr 1 1 299 123 1 1 338.66 1 1 -7.18 1 1 91 Y 1 1 300 124 1 1 356.06 1 1 -7.35 1 1 96 Mo 1 1 306 126 1 1 364.27 1 1 -8.78 1 1 96 Mo 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 Table 5 . 1 1 Parent nuclei 1 1 Q (MeV) 1 1 log T 1/2 1 1 Parent nuclei 1 1 Q (MeV) 1 1 log T 1/2 1 1 Parent nuclei 1 1 Q (MeV) 1 1 log T 1/2 1 1 256 Rf 1 1 10.15 1 1 0.92 1 1 277 Ds 1 1 10.34 1 1 -2.51 1 1 280 Lv 1 1 13.59 1 1 -6.78 1 1 257 Rf 1 1 10.05 1 1 0.78 1 1 271 Rg 1 1 11.61 1 1 -3.72 1 1 281 Lv 1 1 13.35 1 1 -6.98 1 1 258 Rf 1 1 9.94 1 1 -0.16 1 1 273 Rg 1 1 11.44 1 1 -3.56 1 1 282 Lv 1 1 13.13 1 1 -5.14 1 1 259 Rf 1 1 9.67 1 1 0.35 1 1 275 Rg 1 1 11.37 1 1 -3.42 1 1 283 Lv 1 1 12.91 1 1 -5.78 1 1 260 Rf 1 1 9.4 1 1 -1.12 1 1 277 Rg 1 1 10.88 1 1 -2.21 1 1 284 Lv 1 1 12.7 1 1 -3.89 1 1 261 Rf 1 1 9.14 1 1 1.56 1 1 279 Rg 1 1 10.44 1 1 -1.23 1 1 285 Lv 1 1 12.51 1 1 -3.99 1 1 262 Rf 1 1 8.92 1 1 0.52 1 1 280 Rg 1 1 10.24 1 1 0.62 1 1 286 Lv 1 1 12.34 1 1 -4.25 1 1 258 Db 1 1 10.45 1 1 0.55 1 1 282 Rg 1 1 9.89 1 1 2.15 1 1 287 Lv 1 1 12.19 1 1 -3.98 1 1 259 Db 1 1 10.36 1 1 -0.35 1 1 271 Cn 1 1 12.1 1 1 -4.89 1 1 288 Lv 1 1 12.06 1 1 -3.56 1 1 260 Db 1 1 10.08 1 1 0.26 1 1 272 Cn 1 1 11.96 1 1 -4.65 1 1 290 Lv 1 1 11.83 1 1 -1.75 1 1 261 Db 1 1 9.81 1 1 0.65 1 1 273 Cn 1 1 11.87 1 1 -4.33 1 1 291 Lv 1 1 11.73 1 1 -2.36 1 1 263 Db 1 1 9.34 1 1 1.52 1 1 274 Cn 1 1 11.8 1 1 -4.16 1 1 292 Lv 1 1 11.64 1 1 -1.88 1 1 270 Db 1 1 8.45 1 1 3.25 1 1 275 Cn 1 1 11.76 1 1 -3.98 1 1 293 Lv 1 1 11.55 1 1 -1.23 1 1 259 Sg 1 1 10.84 1 1 -0.15 1 1 276 Cn 1 1 11.74 1 1 -2.98 1 1 279 Ts 1 1 14.06 1 1 -7.36 1 1 260 Sg 1 1 10.74 1 1 -2.16 1 1 277 Cn 1 1 11.49 1 1 -3.15 1 1 281 Ts 1 1 14.02 1 1 -8.25 1 1 261 Sg 1 1 10.47 1 1 -0.48 1 1 278 Cn 1 1 11.25 1 1 -2.47 1 1 283 Ts 1 1 13.56 1 1 -5.36 1 1 262 Sg 1 1 10.2 1 1 -1.86 1 1 279 Cn 1 1 11.03 1 1 -2.36 1 1 285 Ts 1 1 13.14 1 1 -5.12 1 1 263 Sg 1 1 9.95 1 1 0.25 1 1 280 Cn 1 1 10.81 1 1 -1.78 1 1 287 Ts 1 1 12.78 1 1 -4.88 1 1 269 Sg 1 1 9.16 1 1 2.56 1 1 281 Cn 1 1 10.62 1 1 -0.99 1 1 289 Ts 1 1 12.5 1 1 -4.65 1 1 260 Bh 1 1 11.21 1 1 -1.42 1 1 285 Cn 1 1 10 1 1 1.56 1 1 291 Ts 1 1 12.28 1 1 -2.98 1 1 263 Bh 1 1 10.59 1 1 -1.76 1 1 273 Nh 1 1 12.4 1 1 -5.65 1 1 294 Ts 1 1 12 1 1 -1.45 1 1 265 Bh 1 1 10.12 1 1 0.12 1 1 275 Nh 1 1 12.24 1 1 -4.79 1 1 281 Og 1 1 14.44 1 1 -7.65 1 1 266 Bh 1 1 9.94 1 1 0.22 1 1 276 Nh 1 1 12.2 1 1 -4.78 1 1 282 Og 1 1 14.43 1 1 -7.63 1 1 270 Bh 1 1 9.56 1 1 1.89 1 1 277 Nh 1 1 12.18 1 1 -4.52 1 1 283 Og 1 1 14.19 1 1 -7.45 1 1 271 Bh 1 1 9.53 1 1 0.18 1 1 279 Nh 1 1 11.7 1 1 -2.89 1 1 284 Og 1 1 13.97 1 1 -6.25 1 1 272 Bh 1 1 9.27 1 1 1.12 1 1 281 Nh 1 1 11.26 1 1 -2.12 1 1 285 Og 1 1 13.76 1 1 -6.41 1 1 274 Bh 1 1 8.78 1 1 1.23 1 1 282 Nh 1 1 11.07 1 1 -1.69 1 1 286 Og 1 1 13.56 1 1 -6.24 1 1 263 Hs 1 1 11.27 1 1 -2.56 1 1 284 Nh 1 1 10.73 1 1 -0.16 1 1 287 Og 1 1 13.37 1 1 -5.25 1 1 265 Hs 1 1 10.77 1 1 -4.56 1 1 285 Nh 1 1 10.59 1 1 0.78 1 1 288 Og 1 1 13.21 1 1 -5.98 1 1 266 Hs 1 1 10.55 1 1 -1.85 1 1 286 Nh 1 1 10.47 1 1 0.88 1 1 294 Og 1 1 12.53 1 1 -3.88 1 1 267 Hs 1 1 10.37 1 1 -1.42 1 1 287 Nh 1 1 10.35 1 1 0.76 1 1 295 Og 1 1 12.44 1 1 -1.25 1 1 268 Hs 1 1 10.23 1 1 0.69 1 1 275 Fl 1 1 12.78 1 1 -4.69 1 1 285 119 1 1 14.37 1 1 -5.69 1 1 269 Hs 1 1 10.13 1 1 1.42 1 1 276 Fl 1 1 12.72 1 1 -5.12 1 1 287 119 1 1 13.96 1 1 -4.25 1 1 270 Hs 1 1 10.05 1 1 1.78 1 1 277 Fl 1 1 12.68 1 1 -5.36 1 1 289 119 1 1 13.62 1 1 -5.97 1 1 271 Hs 1 1 10 1 1 0.45 1 1 278 Fl 1 1 12.66 1 1 -5.46 1 1 292 119 1 1 13.23 1 1 -5.28 1 1 273 Hs 1 1 9.71 1 1 -0.56 1 1 279 Fl 1 1 12.42 1 1 -4.12 1 1 297 119 1 1 12.78 1 1 -3.97 1 1 275 Hs 1 1 9.23 1 1 -0.15 1 1 280 Fl 1 1 12.19 1 1 -4.36 1 1 287 120 1 1 14.56 1 1 -6.58 1 1 266 Mt 1 1 11.27 1 1 -1.93 1 1 281 Fl 1 1 11.96 1 1 -3.78 1 1 288 120 1 1 14.37 1 1 -6.25 1 1 267 Mt 1 1 11.06 1 1 -2.52 1 1 282 Fl 1 1 11.76 1 1 -3.15 1 1 290 120 1 1 14.03 1 1 -6.46 1 1 269 Mt 1 1 10.74 1 1 -2.32 1 1 283 Fl 1 1 11.56 1 1 -2.99 1 1 292 120 1 1 13.76 1 1 -5.85 1 1 271 Mt 1 1 10.57 1 1 -1.85 1 1 288 Fl 1 1 10.86 1 1 -0.25 1 1 298 120 1 1 13.2 1 1 -3.87 1 1 273 Mt 1 1 10.49 1 1 -1.23 1 1 289 Fl 1 1 10.75 1 1 0.62 1 1 299 120 1 1 13.11 1 1 -4.12 1 1 274 Mt 1 1 10.23 1 1 -0.12 1 1 277 Mc 1 1 13.19 1 1 -6.85 1 1 300 120 1 1 13.02 1 1 -4.36 1 1 275 Mt 1 1 9.99 1 1 -1.25 1 1 278 Mc 1 1 13.16 1 1 -6.48 1 1 290 121 1 1 14.59 1 1 -6.28 1 1 276 Mt 1 1 9.75 1 1 -0.36 1 1 279 Mc 1 1 13.14 1 1 -5.96 1 1 296 121 1 1 13.87 1 1 -5.48 1 1 278 Mt 1 1 9.33 1 1 1.36 1 1 280 Mc 1 1 12.9 1 1 -5.12 1 1 300 121 1 1 13.53 1 1 -5.22 1 1 268 Ds 1 1 11.62 1 1 -3.56 1 1 281 Mc 1 1 12.67 1 1 -5.36 1 1 303 122 1 1 13.77 1 1 -4.98 1 1 269 Ds 1 1 11.45 1 1 -3.24 1 1 283 Mc 1 1 12.24 1 1 -4.25 1 1 304 122 1 1 13.67 1 1 -4.99 1 1 270 Ds 1 1 11.31 1 1 -3.68 1 1 285 Mc 1 1 11.88 1 1 -3.12 1 1 304 123 1 1 14.18 1 1 -5.12 1 1 271 Ds 1 1 11.21 1 1 -0.58 1 1 288 Mc 1 1 11.47 1 1 -0.89 1 1 306 124 1 1 14.49 1 1 -5.66 1 1 272 Ds 1 1 11.14 1 1 -3.09 1 1 289 Mc 1 1 11.36 1 1 -0.52 1 1 308 124 1 1 14.28 1 1 -5.98 1 1 273 Ds 1 1 11.1 1 1 -2.96 1 1 290 Mc 1 1 11.26 1 1 -0.25 1 1 310 124 1 1 14.05 1 1 -5.87 1 1 274 Ds 1 1 11.07 1 1 -2.45 1 1 278 Lv 1 1 13.64 1 1 -6.75 1 1 275 Ds 1 1 10.81 1 1 -2.12 1 1 279 Lv 1 1 13.61 1 1 -6.12 1 1   1 1   1 1   The identified alpha emitters have half-lives of approximately μ s to 100 s in the superheavy region 104 ≤ Z ≤ 126. Table 4 Table 4 lists the identified cluster emissions with the corresponding half-lives. The amount of energy released during cluster emission, cluster emitted, and log T 1/2 values are presented in the table. The minimum cluster decay half-lives correspond to 86 Kr, 94 Zr, 91 Y, and 96 Mo for the nuclei 292-293 Og, 298,300 122, 299 123, 300 124, and 306 126, respectively. From the available literature, it is also evident that the heavy particle radioactivity of 86 Kr is observed in the superheavy nucleus Z =118 [ 36 36 , 79 79 ]. In addition, Rb, Sr, Y, Zr, Nb, and Mo cluster emissions [ 80 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 86 Kr, 94 Zr, 91 Y, and 96 Mo 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 Table 6 . The information provided Table 7 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 log T 1/2 values corresponding to spontaneous-fission, α -decay, β + -decay, and cluster-decay half-lives. For instance, the superheavy nucleus 263 Rf exhibits shorter log T 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 Table 7 . 1 1 Parent 4 1 log T 1/2 1 1 Decay mode 1 1   1 1 Parent nuclei 4 1 log T 1/2 2 1 Decay mode 1 1 nuclei 1 1 Sf 1 1 α 1 1 β + 1 1 CD 1 1   1 1   1 1   1 1 Sf 1 1 α 1 1 β + 1 1 CD 1 1   1 1   1 1 263 Rf 1 1 1.25 1 1 1.98 1 1 1.65 1 1 36.6 1 1 Sf =57% 1 1 β + =43% 1 1 296 120 1 1 19.58 1 1 -4.87 1 1 -4.38 1 1 -5.45 1 1 CD =53% 1 1 α =47% 1 1 262 Db 1 1 2.31 1 1 -0.48 1 1 -0.51 1 1 34.76 1 1 β + =52% 1 1 α =48% 1 1 291 120 1 1 25.45 1 1 -6.21 1 1 -5.85 1 1 -4.69 1 1 α =51% 1 1 β + =49% 1 1 264 Bh 1 1 0.71 1 1 -1.65 1 1 -1.75 1 1 20.58 1 1 β + =51% 1 1 α =49% 1 1 297 120 1 1 15.36 1 1 -4.25 1 1 -4.6 1 1 -3.62 1 1 β + =52% 1 1 α =48% 1 1 262 Bh 1 1 -3.25 1 1 -3.25 1 1 -2.22 1 1 17.73 1 1 Sf =50% 1 1 Sf =50% 1 1 289 120 1 1 21.36 1 1 -6.31 1 1 -6.27 1 1 -4.3 1 1 α =50% 1 1 β + =50% 1 1 266 Bh 1 1 1.62 1 1 -1.89 1 1 -1.29 1 1 23.6 1 1 α =59% 1 1 β + =41% 1 1 293 120 1 1 25.32 1 1 -5.18 1 1 -5.43 1 1 -5.06 1 1 β + =51% 1 1 α =49% 1 1 264 Hs 1 1 -2.98 1 1 -3.15 1 1 -1.11 1 1 15.37 1 1 α =51% 1 1 Sf =49% 1 1 293 121 1 1 29.32 1 1 -6.98 1 1 -6.29 1 1 -6.47 1 1 α =52% 1 1 CD =48% 1 1 270 Mt 1 1 2.11 1 1 -2.45 1 1 -2.08 1 1 19.43 1 1 α =54% 1 1 β + =46% 1 1 299 121 1 1 18.56 1 1 -4.75 1 1 -5.05 1 1 -5.43 1 1 CD =52% 1 1 β + =48% 1 1 272 Mt 1 1 1.35 1 1 -1.65 1 1 -1.63 1 1 22.66 1 1 α =50% 1 1 β + =50% 1 1 303 121 1 1 -4.36 1 1 -3.52 1 1 -4.22 1 1 2.35 1 1 Sf =51% 1 1 β + =49% 1 1 268 Mt 1 1 0.99 1 1 -2.78 1 1 -2.54 1 1 16.53 1 1 α =52% 1 1 β + =48% 1 1 291 121 1 1 25.87 1 1 -5.94 1 1 -6.7 1 1 -6.65 1 1 β + =50% 1 1 CD =50% 1 1 276 Ds 1 1 -1.72 1 1 -1.65 1 1 -0.09 1 1 23.61 1 1 Sf =51% 1 1 α =49% 1 1 295 121 1 1 30.36 1 1 -5.78 1 1 -5.88 1 1 -6.34 1 1 CD =52% 1 1 β + =48% 1 1 272 Rg 1 1 2.22 1 1 -3.15 1 1 -3.34 1 1 12.55 1 1 β + =51% 1 1 α =49% 1 1 297 121 1 1 26.9 1 1 -4.77 1 1 -5.46 1 1 -6.33 1 1 CD =54% 1 1 β + =46% 1 1 276 Rg 1 1 2.42 1 1 -2.16 1 1 -2.44 1 1 18.44 1 1 β + =53% 1 1 α =47% 1 1 301 121 1 1 8.25 1 1 -5.36 1 1 -4.64 1 1 -1.47 1 1 α =54% 1 1 β + =46% 1 1 269 Rg 1 1 -3.98 1 1 -3.86 1 1 -2.95 1 1 9.04 1 1 Sf =51% 1 1 α =49% 1 1 294 122 1 1 30.65 1 1 -5.98 1 1 -6.51 1 1 -6.89 1 1 CD =51% 1 1 β + =49% 1 1 270 Rg 1 1 -1.56 1 1 -3.87 1 1 -3.79 1 1 10.12 1 1 α =51% 1 1 β + =49% 1 1 295 122 1 1 34.25 1 1 -5.99 1 1 -6.73 1 1 -6.78 1 1 CD =50% 1 1 β + =50% 1 1 278 Rg 1 1 -1.32 1 1 -1.54 1 1 -1.99 1 1 21.54 1 1 β + =56% 1 1 α =44% 1 1 296 122 1 1 36.89 1 1 -6.24 1 1 -6.1 1 1 -6.83 1 1 CD =52% 1 1 α =48% 1 1 274 Rg 1 1 3.56 1 1 -3.22 1 1 -2.89 1 1 15.4 1 1 α =53% 1 1 β + =47% 1 1 297 122 1 1 34.22 1 1 -6.96 1 1 -6.32 1 1 -6.77 1 1 α =51% 1 1 CD =49% 1 1 283 Nh 1 1 -1.55 1 1 -1.89 1 1 -1.55 1 1 18.55 1 1 α =55% 1 1 Sf =45% 1 1 305 122 1 1 -0.58 1 1 -4.98 1 1 -4.68 1 1 -0.68 1 1 α =52% 1 1 β + =48% 1 1 274 Nh 1 1 0.65 1 1 -4.01 1 1 -4.6 1 1 6.88 1 1 β + =53% 1 1 α =47% 1 1 302 122 1 1 18.21 1 1 -5.27 1 1 -4.87 1 1 -6.31 1 1 CD =54% 1 1 α =46% 1 1 280 Nh 1 1 4.98 1 1 -2.78 1 1 -3.27 1 1 14.3 1 1 β + =54% 1 1 α =46% 1 1 301 122 1 1 23.56 1 1 -5.96 1 1 -5.5 1 1 -6.46 1 1 CD =52% 1 1 α =48% 1 1 278 Nh 1 1 5.88 1 1 -3.87 1 1 -3.72 1 1 11.51 1 1 α =51% 1 1 β + =49% 1 1 299 122 1 1 30.89 1 1 -6.14 1 1 -5.91 1 1 -6.7 1 1 CD =52% 1 1 α =48% 1 1 274 Fl 1 1 -5.12 1 1 -4.89 1 1 -3.95 1 1 3.62 1 1 Sf =51% 1 1 α =49% 1 1 303 122 1 1 13.77 1 1 -5.11 1 1 -5.09 1 1 -4.33 1 1 α =50% 1 1 β + =50% 1 1 285 Fl 1 1 -0.45 1 1 -2.11 1 1 -1.97 1 1 16.71 1 1 α =52% 1 1 β + =48% 1 1 303 123 1 1 27.89 1 1 -6.36 1 1 -5.96 1 1 -3.66 1 1 α =52% 1 1 β + =48% 1 1 282 Mc 1 1 11.02 1 1 -3.78 1 1 -4.55 1 1 0.52 1 1 β + =55% 1 1 α =45% 1 1 297 123 1 1 38.48 1 1 -6.17 1 1 -7.18 1 1 -7.51 1 1 CD =51% 1 1 β + =49% 1 1 287 Mc 1 1 1.32 1 1 -2.98 1 1 -2.4 1 1 0.5 1 1 α =55% 1 1 β + =45% 1 1 301 123 1 1 35.02 1 1 -6.18 1 1 -6.37 1 1 -6.86 1 1 CD =52% 1 1 β + =48% 1 1 284 Mc 1 1 8.98 1 1 -3.58 1 1 -4.12 1 1 0.37 1 1 β + =54% 1 1 α =46% 1 1 307 123 1 1 2.18 1 1 -4.98 1 1 -5.15 1 1 2.81 1 1 β + =51% 1 1 α =49% 1 1 276 Mc 1 1 -3.12 1 1 -5.89 1 1 -5.86 1 1 1.45 1 1 α =50% 1 1 β + =50% 1 1 305 123 1 1 15.58 1 1 -5.48 1 1 -5.56 1 1 -0.65 1 1 β + =50% 1 1 α =50% 1 1 277 Lv 1 1 -6.85 1 1 -5.78 1 1 -5.42 1 1 0.18 1 1 Sf =54% 1 1 α =46% 1 1 308 123 1 1 -4.89 1 1 -5.26 1 1 -6.01 1 1 3.61 1 1 β + =53% 1 1 α =47% 1 1 289 Lv 1 1 3.15 1 1 -3.05 1 1 -2.83 1 1 -1.32 1 1 α =52% 1 1 β + =48% 1 1 303 124 1 1 40.25 1 1 -6.25 1 1 -6.83 1 1 -6.77 1 1 β + =50% 1 1 CD =50% 1 1 293 Ts 1 1 -2.98 1 1 -3.58 1 1 -2.84 1 1 -2.88 1 1 α =55% 1 1 Sf =45% 1 1 305 124 1 1 33.21 1 1 -5.12 1 1 -6.42 1 1 -4.29 1 1 β + =56% 1 1 α =44% 1 1 284 Ts 1 1 13.25 1 1 -5.25 1 1 -5.83 1 1 -2.5 1 1 β + =53% 1 1 α =47% 1 1 302 124 1 1 40.25 1 1 -5.96 1 1 -6.6 1 1 -7.13 1 1 CD =52% 1 1 β + =48% 1 1 286 Ts 1 1 14.55 1 1 -5.16 1 1 -5.4 1 1 -2.6 1 1 β + =51% 1 1 α =49% 1 1 304 124 1 1 35.85 1 1 -6.01 1 1 -6.19 1 1 -6.45 1 1 CD =51% 1 1 β + =49% 1 1 290 Ts 1 1 9.18 1 1 -4.36 1 1 -4.55 1 1 -3.09 1 1 β + =51% 1 1 α =49% 1 1 307 124 1 1 22.14 1 1 -5.97 1 1 -6.02 1 1 -0.09 1 1 β + =50% 1 1 α =50% 1 1 288 Ts 1 1 13.78 1 1 -4.87 1 1 -4.98 1 1 -2.91 1 1 β + =51% 1 1 α =49% 1 1 301 124 1 1 44.14 1 1 -6.04 1 1 -7.23 1 1 -7.53 1 1 CD =51% 1 1 β + =49% 1 1 280 Ts 1 1 0.68 1 1 -6.25 1 1 -6.69 1 1 -2.01 1 1 β + =52% 1 1 α =48% 1 1 309 124 1 1 6.87 1 1 -5.11 1 1 -5.62 1 1 3.17 1 1 β + =52% 1 1 α =48% 1 1 282 Ts 1 1 8.58 1 1 -5.85 1 1 -6.26 1 1 -2.19 1 1 β + =52% 1 1 α =48% 1 1 309 125 1 1 26.25 1 1 -6.51 1 1 -6.49 1 1 0.26 1 1 α =50% 1 1 β + =50% 1 1 290 Og 1 1 16.98 1 1 -3.78 1 1 -3.91 1 1 -4.72 1 1 CD =55% 1 1 β + =45% 1 1 307 125 1 1 38.21 1 1 -6.21 1 1 -6.89 1 1 -3.75 1 1 β + =53% 1 1 α =47% 1 1 291 Og 1 1 14.89 1 1 -3.78 1 1 -4.13 1 1 -4.9 1 1 CD =54% 1 1 β + =46% 1 1 311 125 1 1 11.58 1 1 -5.75 1 1 -6.09 1 1 2.43 1 1 β + =51% 1 1 α =49% 1 1 289 Og 1 1 18.74 1 1 -5.25 1 1 -4.56 1 1 -4.46 1 1 α =54% 1 1 β + =46% 1 1 305 125 1 1 45.35 1 1 -6.87 1 1 -7.29 1 1 -7.5 1 1 CD =51% 1 1 β + =49% 1 1 284 119 1 1 5.95 1 1 -6.68 1 1 -7.53 1 1 -3.17 1 1 β + =53% 1 1 α =47% 1 1 303 125 1 1 45.25 1 1 -6.96 1 1 -7.69 1 1 -8.07 1 1 CD =51% 1 1 β + =49% 1 1 293 119 1 1 18.29 1 1 -3.98 1 1 -4.57 1 1 -5.15 1 1 CD =53% 1 1 β + =47% 1 1 315 126 1 1 -1.58 1 1 -5.75 1 1 -6.17 1 1 2.82 1 1 β + =52% 1 1 α =48% 1 1 286 119 1 1 14.98 1 1 -7.25 1 1 -7.11 1 1 -3.79 1 1 α =50% 1 1 β + =50% 1 1 313 126 1 1 15.25 1 1 -6.24 1 1 -6.57 1 1 0.85 1 1 β + =51% 1 1 α =49% 1 1 294 119 1 1 15.98 1 1 -3.89 1 1 -5.43 1 1 -5.15 1 1 β + =51% 1 1 CD =49% 1 1 308 126 1 1 48.21 1 1 -6.87 1 1 -7.12 1 1 -5.37 1 1 β + =51% 1 1 α =49% 1 1 288 119 1 1 19.35 1 1 -5.85 1 1 -6.69 1 1 -4.15 1 1 β + =53% 1 1 α =47% 1 1 311 126 1 1 32.21 1 1 -5.12 1 1 -6.96 1 1 -0.97 1 1 β + =58% 1 1 α =42% 1 1 298 119 1 1 -4.85 1 1 -3.22 1 1 -4.59 1 1 0.23 1 1 Sf =51% 1 1 β + =49% 1 1 310 126 1 1 39.21 1 1 -6.32 1 1 -6.73 1 1 -1.99 1 1 β + =52% 1 1 α =48% 1 1 291 119 1 1 21.25 1 1 -4.58 1 1 -4.99 1 1 -4.74 1 1 β + =51% 1 1 CD =49% 1 1 309 126 1 1 44.58 1 1 -6.58 1 1 -7.36 1 1 -3.69 1 1 β + =53% 1 1 α =47% 1 1 295 119 1 1 11.35 1 1 -4.23 1 1 -4.15 1 1 -5.03 1 1 CD =54% 1 1 α =46% 1 1 312 126 1 1 26.12 1 1 -5.74 1 1 -6.33 1 1 0.01 1 1 β + =52% 1 1 α =48% 1 1 295 120 1 1 22.15 1 1 -4.78 1 1 -5.02 1 1 -5.46 1 1 CD =52% 1 1 β + =48% 1 1 307 126 1 1 51.32 1 1 -7.11 1 1 -7.75 1 1 -6.99 1 1 β + =52% 1 1 α =48% 1 1 294 120 1 1 23.58 1 1 -4.22 1 1 -4.79 1 1 -5.36 1 1 CD =53% 1 1 β + =47% 1 1 314 126 1 1 7.56 1 1 -6.21 1 1 -5.94 1 1 1.7 1 1 α =51% 1 1 β + =49% Finally, Fig. 8 Fig. 8 shows the lifetimes of the superheavy elements after the competition between different decay modes was studied. 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A. , Nazarewicz W. et al. , Cluster radioactivity of <math id="M51"><mrow><msubsup><mrow/><mrow><mn>118</mn></mrow><mrow><mn>294</mn></mrow></msubsup></mrow></math> Og176 . Phys. Rev. C 99 , 0 41304 ( 2019 ). doi: 10.1103/PhysRevC.99.041304 http://doi.org/10.1103/PhysRevC.99.041304 Cluster radioactivity of 118294 Og176 Poenaru D. N. , Gherghescu R. A. , Greiner W. , Heavy-particle radioactivity . J. Physics Conference Series 436 , 0 12056 ( 2013 ). doi: 10.1088/1742-6596/436/1/012056 http://doi.org/10.1088/1742-6596/436/1/012056 Heavy-particle radioactivity All authors contributed to the study conception and design. Conceived of the original idea, developed the theory formulation of work and writing by H.C. Manjunatha and N. Sowmya. Performed the computations and performed, analytic calculations and graphical representation by P.S.Damodara Gupta, K.N.Sridhar and A.M. Nagaraja, data analysis by L.Seenappa and S. Alfred Cecil Raj. All authors read and approved the final manuscript. 10.1007/s41365-021-00967-y 2021-07-13 2021-08-28 2021-08-31 2021-11-23 2021-11-01 2021-12-06 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00967-y http://dx.doi.org/10.1007/s41365-021-00967-y . 2021 Nuclear Science and Techniques 11 32