H.C.Manjunatha ?N.Sowmya?P.S.Damodara Gupta?K.N.Sridhar ?A.M.Nagaraja,?L.Seenappa?S.Alfred Cecil Raj

Abstract A detailed investigation 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 heavyparticle 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.

Keywords Alpha decay.Beta decay.Heavy-particle radioactivity.Branching ratios

1 Introduction

The most important unanswered questions in Nuclear Physics are to determine the heaviest superheavy nucleithat 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 toμs,can be synthesized using cold and hot fusion reactions.Cold fusion reactions involve either lead or bismuth as targets[1],whereas hot fusion reactions includE48Ca beamson various actinidetargets[2,3].Many theoretical predictions,such as microscopic–macroscopic[4](single-particle potential)and self-consistent approaches,including nucleus–nucleuspotential[5,6],relativistic field models[7,8],and multinucleon transfer reactions[9],provide information regarding the nucleus structure,shell closurelocation,and decay modesin 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 using48Ca as a projectile and208Pb as target[14].In 1966,two groups of researchers,namely Mayers and Swiatecki,and Viola and Seaborg[15],separately predicted the presence of heavy nucleinear 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 nucleus254No 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 andα-decay are the dominant decay modes in superheavy nuclei and limit their stability.Furthermore,newly synthesized superheavy elementsare primarily identified by their decay chains from unknown nuclei to known daughter nuclei by using the parent-and-daughter correlation.

Thecompetition 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[22–33].The possible decay modes in the superheavy nuclei Z=119 and 120 are predicted in Ref.[34].From Ref.[35],it isclearly observed that theisotopes of the superheavy nuclei Z=104–112 have α-decay and spontaneous fission as dominant decay modes.However,onlyα-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 thenucleus284Fl,α-decay isdominant in the nuclei286-289Fl,andβ+is dominant in the nucleus290Fl.Furthermore,the concept of heavy-particle radioactivity[36]in thesuperheavy region hasimportant applicationsin 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,onlyα-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 betweenαdecay 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 Sect.2.The analysis of different decay modes and possible emitters in the superheavy region is explained in Sect.2.4.The paper is concluded in Sect.3.

2 Theory

2.1 Alpha decay and cluster decay

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

where ν0is the assault frequency on the barrier,and ν0=1.8×1022s-1[48].Pαis the preformation factor,which is closely related to the shell structure[49].The empirical formula for Pαis expressed as

where N,Z,and A are the neutron,charge,and mass number of the parent nucleus,respectively,Z1and Z2are the proton magic numbers around Z(Z1≤Z≤Z2),and N1and N2are the neutron magic numbers around N(N1≤N≤N2).p1,p2,and p3correspond 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

Table 1 Cluster-decay half-lives obtained from present study(PS)and available experiments(exp)

Table 2 Comparison of logarithm half-lives(years)of spontaneous fission in the superheavy region 104≤Z≤114 from present study with those from available experiments

Table 3 Comparison of alpha-decay half-lives from the present study(PS)and those from available experimental(Exp)values

Table 4 Identified cluster emitters in the superheavy nuclei region

Table 5 Identified alpha emitters in the superheavy nuclei region

Table 5 continued

Table 6 Identifiedβ+emitters in the superheavy nuclei region

Table 7 Superheavy nuclei with dual decay mode and branching ratio

Table 7 continued

where μ is the inertial coefficient resulting from the Werner–Wheeler approximation[51].The limits of integration ζ0and ζcare the inner and outer turning points,expressed as ζ0=Rp-and.Rp is the radius of theparent nucleus,andis 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:

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

with

where S1and S2are the surface areas of the spherical fragments.σeffis the effective surface tension,which is defined as

where R2is the final radius of the daughter fragment.The centrifugal potential energy is determined by

where ? 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

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 four-dimensional spherical problem to an equivalent one-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 dinuclear 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 byζandξ,respectively

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

where R=r0A1/3is 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:

2.2 Beta decay

For all types ofβprocesses,the expression for the halflife Tβis given by[58]

Here,EC isthe electron capture.For aparticular typeofβdecay,the half-life is expressed as follows:

2.2.1 β±decay

The Fermi function forβ-decay is expressed as

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

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

The expression for the energy released in β+decay is

Similarly,for β-decay,

2.2.2 Electron capture

The value of Q for electron capture is given by

Here,Be is the electron binding energy.Hence,even for the forbiddenβ+decay,electron capture is allowed.The captureof 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

In general,for any shell,

Here,EXis the total energy of the electron:

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

The nonzero components of the radial parts（gX&fX）of the wave function of the relativistic electron of orbit X are

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

whereν,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

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

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

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

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 isthe separation distance between the center of the fission fragments,and Raand Rbare 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]:

whereμand k arethe reduced massof 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 101to 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]areavailable in the literature.In the present study,we used the updated AME16[70]mass excess values up to Z=118,and above Z＞118,themassexcessvaluesaretaken 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 validatethepresent study,thecluster-decay half-livesobtained 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 theresultsarepresented 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.

As a part of this investigation,we studied theα-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.

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-livesof superheavy nucleiin theregion 104≤Z≤126 Fig.2.

Fig.2 (Color online)Map of nuclei reflecting the logarithmicαdecay 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 T1/2 values from the navy-blue region to the brown region

shows a wide range ofα-decay half-lives.For a given superheavy nucleus,the alpha decay half-lives increase as the neutron number of its isotopes increases.For instance,theα-decay half-lives are of the order of nanoseconds at N/Z=1.307692 for Rutherfordium,whereas for the same superheavy element,theα-decay half-lives are of the order of 102s at N/Z=1.504762.Similarly,all neutron-rich superheavy nuclei have comparably longerα-decay halflives,which is in agreement with the report available in Ref.[73].The obtainedα-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 log T1/2values.The color variation from navy blue to wine indicates values in the range 10-10–102s.The contrast in the blue region lies between 10-10s and 10-7s,in the green region,it lies in the range 10-6–10-4s,and the range 10-4-10-3s is presented in the yellow region.Finally,the red-to-wine region shows higher half-lives in the range 10-2-102s.The inset of Fig.2 on the top left side provides information on the magnified portion ofα-decay halflives 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 theα-decay,a search wasmade 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).Thepredicted cluster decay half-livesin 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 halflives 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 for86Kr 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 against96Mo cluster emissions than those of the other studied clusters.The decay half-lives are shorter for thE91Y emission from superheavy nuclei with Z=109,111,113,121,and 123.Similarly,the half-livesof superheavy nucleiwith Z=105 and 107 against cluster radioactivity are shorter for97Tc and101Rh than those of the other studied clusters.

Cluster radioactivity in thesuperheavy 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 nucleireflecting thelogarithmic clusterdecay 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-livesareobserved,whereasshorter cluster-decay half-livesareobserved 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 toμs 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 consistentα-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 halflives for spontaneous fission for 104＜Z＜126

The log T SF valuesvary between-50(dark blue region)and 50(dark-red region).For instance,at atomic number Z=104,for isotopes 245–275,the log T1/2（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-livesranging from nanoseconds to 105s 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 sideof Fig.6,for isotopescorresponding 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 T1/2are observed up to Z=116.In contrast,smaller half-lives are observed for higher isotopes(Z＞116),and larger log T1/2for 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 thatβ+-decay is energetically possible with Z=105,107,113,114,115,117,119,121,123,125,and 126,whereasβ--decay is not energetically possible.Furthermore,we also studiedβ-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 modefor superheavy elementsin theatomic number region 104≤Z≤126 of all possible isotopes Fig.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.

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

The identified alpha emitters have half-lives of approximatelyμs to 100 s in the superheavy region 104≤Z≤126.Table 4 liststheidentified cluster emissions with the corresponding half-lives.The amount of energy released during cluster emission,cluster emitted,and log T1/2values are presented in the table.The minimum cluster decay half-lives correspond to86Kr,94Zr,91Y,and96Mo for the nuclei292-293Og,298，300122,299123,300124,and306126,respectively.From the available literature,it is also evident that the heavy particle radioactivity of86Kr 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,with86Kr,94Zr,91Y,and96Mo cluster emissions,respectively.Similarly,approximately 20 possibleβ+emitters were identified in the superheavy region 105≤Z≤125,and they are presented in Table 6.

The information provided Table 7 regarding the halflives and branching ratios presents ambiguities in terms of determining a single decay mode.The branching ratios relativeto theminimum half-livesamong thestudied decay modes are obtained,and the second column of the table shows the log T1/2values corresponding to spontaneous-fission,α-decay,β+-decay,and cluster-decay halflives.For instance,the superheavy nucleus263Rf exhibits shorter log T1/2values for spontaneous fission andβ+-decay than for other decay modes.The branching ratio of spontaneous fission andβ+-decay was obtained,and it was found that the branching ratio corresponding to spontaneous fission andβ+was 55%and 45%,respectively.Similarly,we identified the branching ratios for the superheavy region 104≤Z≤126,which are presented in Table 7.

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,massnumber of parent and logarithmic half-livesof different decay modes(life times)for 104＜Z＜126

It can be seen that the lifetime varies from ns to min and decreases asthe 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.

3 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.Thefindingsof thisstudy werevalidated by comparison with experiments.Approximately 20β+and 7 heavy particle emitters were found in the superheavy region.Furthermore,the nuclei with almost the same halflives for the two decay modes were also reported,with the corresponding branching ratios.However,an experimental study is necessary to draw definite conclusions.

Author contributionsAll authors contributed to the study conception and design.Conceived of the original idea,developed the theory formulation of work and writing by HCM and NS.Performed the computations and performed,analytic calculations and graphical representation by PSDG,KNS and AMN,data analysis by LS and SACR.All authors read and approved the final manuscript.