Self-consistent calculations of electron-capture decays in Z=118, 119, and 120 superheavy isotopes
aa r X i v : . [ nu c l - t h ] F e b Self-consistent calculations of electron-capture decays in Z=118, 119, and 120superheavy isotopes
P. Sarriguren
Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, Spain
Abstract
Weak decays in superheavy nuclei with proton numbers Z = 118 −
120 and neutron numbers N = 175 −
184 arestudied within a microscopic formalism based on deformed self-consistent Skyrme Hartree-Fock mean-field calculationswith pairing correlations. The half-lives of β + decay and electron capture are compared with α -decay half-lives obtainedfrom phenomenological formulas. The sensitivity of the half-lives to the unknown Q -energies is studied by comparingthe results obtained from different approaches for the masses. It is shown that α -decay is always dominant in this massregion. The competition between α and β + /EC decay modes is studied in seven α -decay chains starting at differentisotopes of Z =118, 119, and 120. Keywords:
Weak-decay half-lives; superheavy nuclei; nuclear density-energy functional
1. Introduction
The last decades have witnessed a lot of progress in thesearch and discovery of increasingly heavy elements and itis nowadays a very fruitful line of research [1, 2, 3, 4, 5].Superheavy nuclei (SHN) with Z = 107 −
113 were syn-thesized from cold-fusion reactions by using target nu-clei
Pb and
Bi and medium-mass stable isotopes ofTi, Cr, Fe, Ni, and Zn as projectiles [1, 2, 6]. Produc-tion of heavier elements from these reactions were diffi-cult because of the strong Coulomb repulsion for increas-ing charge of the projectiles. Then, hot-fusion reactionsinvolving long-lived actinide nuclei from
U to
Cf astargets and the double magic nucleus Ca as projectileswere carried out to produce SHN with Z =112–118 in theneutron-evaporation ( xn ) channels [2, 7, 8, 9]. As a conse-quence of these experimental campaigns all the elementswith Z ≤
118 have been discovered.However, theoretical macroscopic-microscopic models[10, 11, 12] that include self-consistent treatments of theshell corrections [13, 14, 15, 16, 17], predict new regionsof particularly stable nuclear systems with proton shellsclosures at Z = 114, 120, 124 or 126 and neutron shell clo-sures at N = 172, 184, depending on the interactions andparametrizations used. Since no clear indications of closedshell at Z = 114 or N = 172 have been observed, there is astrong motivation for the search of more neutron-rich iso-topes, as well as of heavier elements in an attempt to getcloser to the predicted regions of stability. Concerning su-perheavy neutron-rich isotopes, alternative ways for theirproduction are being explored [18, 19, 20] through fusion-evaporation reactions that include not only xn channels, Email address: [email protected] (P. Sarriguren) but also the emission of charged particles from the com-pound nucleus in the pxn and αxn channels, as well asthrough multinucleon transfer reactions or fusion reactionswith radioactive ion beams [21]. The production of el-ements beyond oganesson requires complete fusion reac-tions with projectiles with
Z >
20 because of the insuf-ficient amounts of actinide targets with
Z >
98 available[4, 22]. Different possibilities of projectiles ( Ti, Cr, Fe, Ni) and targets (
Cf,
Cm,
Pu,
U) havebeen recently studied, both experimentally and theoret-ically, in a search for the most suitable combination toproduce elements with
Z > Ti +
Cf has thehighest fusion probability among the reactions studied andthus, is the best candidate for the formation of Z = 120.Similarly, Adamian et al. [24] found within a microscopic-macroscopic approach that, among the reactions studied, Ti +
Bk and Ti +
Cf have the largest crosssection for the production of evaporation residues with Z = 119 and Z = 120, respectively.The stability of the compound nuclei in the super-heavy region is generally determined by spontaneous fis-sion. However, near the predicted islands of stability, fis-sion barriers increase because of associated effects of shellclosures and the half-lives of spontaneous fission may in-crease dramatically as shown in Ref. [25]. This enablesother radioactive decay modes, such as α -decay or weak de-cays, that may come into play. In particular, the β + /EC -decay in SHN may open new pathways towards the pre-dicted region of stability [26, 27]. This possibility is alsobeing studied experimentally [28, 29]. Theoretical predic-tions of weak decays are based on different approaches. Preprint submitted to Elsevier February 23, 2021 henomenological parametrizations [30] have been devel-oped that can be used to extrapolate to regions where the β + /EC half-lives are unknown. There are also calcula-tions that neglect nuclear structure effects, such as thosein Refs. [26, 27, 31, 32], where only transitions connectingparent and daughter ground states are considered. The nu-clear matrix elements of these transitions were assumed tobe a constant value phenomenologically determined andvalid for all nuclei. However, this value can vary by al-most two orders of magnitude (from log( f t ) = 4 . f t ) = 6 . β + /EC -decay were also evalu-ated within a proton-neutron quasiparticle random-phaseapproximation (pnQRPA) based on a phenomenologicalfolded-Yukawa single-particle Hamiltonian [33].Following the work started in Refs. [34, 35], we studyhere the β + /EC -decay half-lives of some selected even-even and odd- A isotopes with Z = 118 −
120 and N =175 −
184 and the competition with α -decay. The pro-duction of new elements with Z = 119 and 120 is oneof the main objectives at worldwide leading laborato-ries such as SHI-GSI and FLEROV-JINR-DUBNA. There-fore, the study in this work addresses a highly topical is-sue. Furthermore, a comparison between α and β + /EC -decay modes is made for seven α -decay chains that fol-low the production of isotopes with Z =118, 119, and120. The method of calculation of the weak decays isbased on the pnQRPA approach with a microscopic nu-clear structure calculation consisting on a deformed self-consistent Hartree-Fock calculation with Skyrme interac-tions and pairing correlations in the BCS approximation(HF+BCS).
2. Theoretical formalism
The microscopic approach used in this work to calculate β + /EC -decay half-lives is presented here. The methodfollows closely the theoretical formalism used in Ref. [34,35] for SHN. Further details of the formalism can be foundelsewhere [36, 37].The β + /EC -decay half-life, T β + /EC , is calculated bysumming all the allowed Gamow-Teller (GT) transitionstrengths connecting the parent ground state with statesin the daughter nucleus with excitation energies, E ex , ly-ing below the Q i energy ( i = β + , EC ) and weighted withphase-space factors f i ( Z, Q i − E ex ), T − / ,i = ( g A /g V ) D X 1) + m e . (2)Usually, one takes experimental masses to evaluate Q EC ,but in the case of the SHN studied here the masses havenot been determined yet. Therefore, one has to rely on the-oretical predictions for them. There is a large number ofmass formulas available, which have been obtained fromdifferent approaches. We have considered in this worka selection of these mass formulas to evaluate the sensi-tivity of the half-lives to the unknown Q -energies. Thisbreaks the self-consistency of the microscopic calculation,but provides a measure of the uncertainties involved in thecalculated half-lives.Among the pure phenomenological approaches for themasses, the Weizsacker-Bethe (WB) nuclear mass formula[38] is used. Several macroscopic-microscopic models arealso considered. Among them we use the finite-rangedroplet model (FRDM) [39], which is corrected with mi-croscopic effects obtained from a deformed single-particlemodel based on folded-Yukawa potentials including pair-ing in the Lipkin-Nogami approach and the nuclear massformula of Ref. [40] (KTUY) that combines a gross termdescribing the general trend of the masses, an even-oddterm, and a shell correction term describing the deviationsof the masses from the general trend. The Duflo and Zuker(DZ-10) mass model [41], which is written as an effectiveHamiltonian that contains monopole and multipole terms,is used as well. Another macroscopic-microscopic massformula inspired by the Skyrme energy-density functionalis also considered. In particular, we use the Weizsacker-Skyrme formula WS4 that includes a surface diffusenesscorrection for unstable nuclei and radial basis function cor-rections (WS4+RBF) [42]. This mass formula has beenshown to be very reliable describing SHN [43]. Finally,we also compare with fully microscopic calculations basedon effective two-body Skyrme nucleon-nucleon interactionsby using the masses from the Skyrme forces SkM* andSLy4 with a zero-range pure volume pairing force [44] andLipkin-Nogami method obtained from the code HFBTHO[45]. Tables for these masses can be found on websites [46].Figure 1 shows the Q EC energies for the isotopes with Z = 118 , , 120 obtained from the mass formulas men-tioned above. It also contains (black solid circles) the av-erage values for each isotope. The results for each isotopeare typically distributed around 2 MeV. Similarly, Fig. 2shows the Q α energies, Q α = M ( A, Z ) − M ( A − , Z − − M (4 , f β + and electron capture f EC . They2 Q E C ( M e V ) WS4+RBFaverage 175 176 177 178 179 180 181 182 183 184 N01234567 Q E C ( M e V ) Q E C ( M e V ) WBFRDMKTUYDZ-10SLy4SkM* Z=118Z=119Z=120 Figure 1: Q EC energies (MeV) corresponding to different calculatedmasses for Z = 118, 119, and 120 isotopes. are computed numerically for each value of the energy us-ing the code LOGFT, as explained in Ref. [47]. f β + ( Z, W ) = Z W pW ( W − W ) λ + ( Z, W )d W , (3)with λ + ( Z, W ) = 2(1 + γ )(2 pR ) − − γ ) e − πy | Γ( γ + iy ) | [Γ(2 γ + 1)] , (4)where γ = p − ( αZ ) ; y = αZW/p ; α is the fine struc-ture constant and R the nuclear radius. W is the totalenergy of the β particle, W is the total energy availableand p = √ W − f EC , are given by f EC = π X x q x g x B x , (5)where x denotes the atomic sub-shell from which the elec-tron is captured that includes K - and L - orbits. q is theneutrino energy, g is the radial component of the bound-state electron wave function at the nuclear surface, and B stands for other exchange and overlap corrections [47] thatcome from the indistinguishability of the electrons andfrom the decrease of the nuclear charge by one unit during Q α ( M e V ) WS4+RBFaverage 175 176 177 178 179 180 181 182 183 184 N91011121314 Q α ( M e V ) DZ-10SLy4SkM* Q α ( M e V ) WBFRDMKTUY Z=118Z=119Z=120 Figure 2: Same as in Fig. 1, but for Q α energies. the decay, respectively. The bound-state radial wave func-tions and the correction factors are obtained from a rela-tivistic self-consistent mean-field calculation. They are so-lutions of the Dirac equation with a Hartree self-consistentpotential and exchange terms included in the Slater ap-proximation [48]. The nuclear potential corresponds to afinite-size nucleus with a Fermi distribution for the nuclearcharge density.Various improvements have been implemented recentlyto calculate more accurately the phase space factors. Theyhave led to a more precise evaluation of the theoreticalhalf-lives for β -decay and electron captures [49, 50]. Thefinal result found in these works is a small correction ofa few percent with respect to standard calculations in theframework of ref. [47]. This may be quite important insome specific cases when comparing with very precise ex-perimental data, but it is irrelevant in this case, wherethe purpose is to compare half-lives of different decay pro-cesses that differ by several orders of magnitude, as weshall see later. This change is also irrelevant when com-pared to the change induced by other uncertainties studiedin this work, such as the unknown Q EC energies. There-fore, the use of more accurate electron wave functions willnot change the main conclusion of this work regarding thecompetition between α and β decay modes.The nuclear structure involved in the β + /EC -decay iscontained in the energy distribution of the GT strength B ( GT, E ex ). At variance with other approaches mentioned3arlier to calculate T β + /EC in SHN, we use in this work amicroscopic approach. We start with a self-consistent cal-culation of the mean field by means of a deformed Hartree-Fock procedure with Skyrme interactions and pairing cor-relations in the BCS approximation. This calculation pro-vides us single-particle energies, wave functions, and oc-cupation probabilities. The Skyrme interaction SLy4 [51]is chosen for this study because of its proven ability to de-scribe successfully nuclear properties throughout the en-tire nuclear chart [44]. The solution of the HF equationsis found by using the formalism developed in Ref. [52], un-der the assumption of time reversal and axial symmetry.The single-particle wave functions are expanded in termsof the eigenstates of an axially symmetric harmonic oscil-lator in cylindrical coordinates using 16 major shells, afterverifying that this size is large enough to get convergenceof the HF energies. Deformation-energy curves (DECs)are constructed by constrained HF calculations that allowto analyze the nuclear binding energies as a function ofthe quadrupole deformation parameter β . The profiles ofthese curves are found to converge with the basis used.Furthermore, eventual truncation errors become largelycancelled out when subtracting energies between the twodecay partners to calculate Q-values. -0.4 -0.2 0 0.2 0.4 0.6 0.8 β E ( M e V ) Z=120, N=180 Figure 3: Deformation-energy curve for the 120 isotope obtainedfrom constrained HF+BCS calculations with the Skyrme force SLy4. Figure 3 shows the DEC of the isotope 120 as a rep-resentative example of the nuclei in this mass region. Theenergy in Fig. 3 is relative to the ground state energy. Theresults show a ground state corresponding to an almostspherical shape, as well as an excited prolate minimum at β ≈ . 5. The profile of the DEC turns out to be very sim-ilar to the DECs obtained for the other isotopes discussedin this work and agree also quite well with calculations per-formed with the finite-range Gogny D1S interaction [53].In this work we calculate energy distributions of the GTstrength and their corresponding half-lives for the groundstate configurations at β ≈ − . 05, as well as for the pro-late configuration at β ≈ . -6 -4 -2 T / α ( s ) 175 176 177 178 179 180 181 182 183 184 N10 -7 -5 -3 -1 T / α ( s ) average -7 -5 -3 -1 T / α ( s ) PSRoyerVS1VS2DZR Z=118Z=119Z=120 Figure 4: Calculated α -decay half-lives for Z = 118, 119, and 120isotopes for five different phenomenological formulas of T α , namely,PS [68], Royer [69], VS1 [70], VS2 [26, 71], and DZR [72] and theiraverage value. β -decay properties in many different mass regions [36, 37]and it is also expected to play a significant role in SHN[34, 35]. A deformed pnQRPA with residual spin-isospininteractions is used to obtain the energy distribution ofthe GT strength needed to calculate the half-lives. In thecase of SHN the coupling strengths of the residual inter-actions that scale with the inverse of the mass number areexpected to be very small and their effect is neglected.In the case of odd- A nuclei the procedure followed isbased on the blocking of a given state with a given spin andparity, using the equal filling approximation to calculate itsnuclear structure [37]. This approximation has been shownto be sufficiently precise for most practical applications[54]. The blocked state is chosen among the states in thevicinity of the Fermi level as the state that minimizes theenergy.This model of nuclear structure has been successfullyused in the past to calculate weak-decay properties in dif-ferent mass regions including neutron-deficient medium-mass [55, 56] and heavy nuclei [57, 58, 59], neutron-richnuclei [60, 61, 62, 63, 64], and f p -shell nuclei [65, 66]. Theeffect of various ingredients of the model like deformationand residual interactions on the GT strength distributions,which finally determine the decay half-lives, was also stud-ied in the above references. In particular, the sensitivityof the GT distributions to deformation has been used tolearn about the nuclear shapes when comparing with ex-4 T / β+ / E C ( s ) 175 176 177 178 179 180 181 182 183 184 N10 T / β + / E C ( s ) T / β + / E C ( s ) sphericaldeformed Z=118Z=119Z=120 Figure 5: Calculated β + /EC -decay half-lives for Z = 118, 119, and120 isotopes, using the average, maximum. and minimum Q EC val-ues in Fig. 1. Results for the ground-state spherical and excitedprolate deformed shapes are shown. periment [67]. 3. Half-lives: Results and discussion Comparison between α - and β + /EC -decay modes is cru-cial to understand the possible branching and pathways ofthe original compound nucleus leading to stability.Since no experimental information is still available onthe α -decay half-lives ( T α ), one has to rely on phenomeno-logical formulas, which in turn depend on the unknown Q α values. Thus, to get an idea of the spread of the results on T α expected from uncertainties in both Q α energies andphenomenological formulas of T α , we have calculated thehalf-lives from five parametrizations and seven mass for-mulas, as well as the average, maximum, and minimumvalues.Following the same approach as in Ref. [34], severalparametrizations are used, which were fitted to accountfor the properties of SHN. Namely, they are the formulaby Parkhomenko and Sobiczewski [68] (PS), the Royer for-mula [69] (Royer), and the Viola-Seaborg formula [70] withparameters from [68] (VS1) and [26, 71] (VS2). In additionto these formulas we also consider here a recent formula[72] (DZR) that takes into account both the blocking ef-fect of the unpaired nucleon and the contribution of thecentrifugal potential. This is an improvement of the Royerformula, which is simpler and more accurate. Figure 4 contains the results for T α . The values shownwith different symbols and colors correspond to calcula-tions from the five different parametrizations using theaverage values of Q α (see Fig. 2). The error bars foreach calculation correspond to the use of the maximumand minimum Q α values predicted by the different massevaluations. Solid black circles correspond to the averagevalues of the five formulas. Their vertical lines join themaximum and minimum values obtained from the differ-ent T α ( Q α ) alternatives. While the predictions of differentparametrizations are within one order of magnitude, theuncertainties originated from the Q α energies may vary upto seven orders of magnitude.Figure 5 contains the results for T β + /EC calculated withthe formalism described in the previous section. The re-sults plotted with a symbol (circle or triangle) correspondto the use of the average values of Q EC in Fig. 1. Themaximum and minimum values of T β + /EC for each iso-tope correspond to the minimum and maximum values of Q EC , respectively, according to the different mass formu-las. They are plotted as error bars that join these extremevalues, giving a measure of the uncertainties associatedwith the unknown Q EC energies.The results with black circles correspond to the slightlyoblate, almost spherical, ground states ( β ≈ β ≈ . β + /EC -decay one proton is transformedinto one neutron. The low-lying excitations below Q EC in the daughter nucleus that finally determine the half-lives come from transitions connecting protons around theFermi level for Z = 118 − 120 with neutrons around theFermi level in the mass region N = 175 − f / of negative parity withneutrons from the 2 g / , d / , d / , s / shells, whichare positive parity states. Thus, these transitions will bevery suppressed in nuclei with small deformations becauseof parity arguments. On the other hand, in the deformedcase, many different orbitals from different spherical shellswith positive and negative parity cross each other leadingto a much more mixed scenario where states with bothparities are found in the vicinity of the Fermi levels ofprotons and neutrons. The final result is an enhancementof the GT strength at low energy that leads to a shorterhalf-life in the case of deformed nuclei.Another interesting observation is the existence of anodd-even staggering effect in the spherical case, whichdoes not appear in the deformed case. This peculiar be-havior is related to the characteristics of the excitationsin the odd nuclei. The low-lying transitions in the oddsystem correspond basically to one-quasiparticle (1qp) ex-citations where the odd nucleon is involved in the pro-5ess. At higher excitation energies, typically beyond theenergy needed to break a pair of nucleons, the transitionsare mainly three-quasiparticle (3qp) excitations similar tothose in the even-even system but with the odd nucleonacting as a spectator. For nuclei in this mass region, whichhave rather small Q EC energies, the 3qp excitations areshifted in energy beyond Q EC , while the low-lying 1qpexcitations connecting protons from the 2 f / shell withneutrons in the 2 g / , d / , d / , s / shells are verysuppressed because of parity. The final result for sphericalnuclei is that in the odd nuclei very little strength remainswithin the Q EC window giving rise to quite large half-livesas compared with the even-even nuclei. In the deformedcase this effect is not manifest because of the higher leveldensity around the Fermi levels that involve states withboth positive and negative parities, as well as many angu-lar momentum components.Comparing the half-lives in Figs. 4 and 5, one can seethat the β + /EC -decay half-lives are systematically severalorders of magnitude larger than the corresponding average α -decay half-lives for a given isotope. The range of this dif-ference is between three and five orders of magnitude in thedeformed case and even larger in the spherical one. Onlywhen one considers the maximum values of T α allowed bythe uncertain Q α energies are then comparable to T β + /EC in the deformed case. As a consequence, α -decay in thismass region will be always dominant and much faster than β + /EC -decay.Finally, Tables 1–3 show the Q α and Q EC energies, aswell as T α and T β + /EC half-lives for nuclei involved invarious α -decay chains starting at Og and Og (Table1), 119 (Table 2), and R = T β + /EC /T α .Since the SHN produced after neutron evaporation ofthe corresponding compound nuclei are identified by their α -decay chains, the competition between α and β + /EC decay modes in the members of a given chain of α -decaysis important to analyze possible branching points in futureexperiments. Q -energies in the tables are taken from experiment [73]when available, together with their errors within parenthe-ses. Otherwise, they are calculated from the mass formulaWS4-RBF [42] and quoted without errors. Similarly, thehalf-lives are either experimental with errors or calculatedwith the Q -values given in the tables. In the latter case, T α corresponds to the average value of the phenomeno-logical parametrizations used earlier, whereas T β + /EC iscalculated with the HF+BCS+pnQRPA formalism for theground states of the nuclei. As it can be seen from the ta-bles, the α -decay mode of a given isotope is generally or-ders of magnitude faster than the corresponding β + /EC decay with ratios R of several orders of magnitude. Nev-ertheless, R decreases as we progress in odd- A chains andcan reach values close to one at the end of some of thesechains. This is the case of nuclei such as Sg, Rf, Db, and Lr. However, spontaneous fission becomes Table 1: Q α and Q EC energies [MeV], as well as T α and T β + /EC half-lives [s] for nuclei in the α -decay chains starting at Og and Og. Available experimental values appear with the errors withinparentheses. Otherwise, they are calculated values (see text). Thelast column contains the ratio R = T β + /EC /T α . Q α Q EC T α T β + /EC R Og 11.7(9) 4.38 7.1 × − × × Lv 10.89(5) 3.4(10) 2.8 × − (15) 8.1 × × Fl 10.16(5) 2.83(95) 5.2 × − (13) 1.1 × × Cn 9.94(11) 2.21(93) 4.1 × (10) 2.2 × × Ds 10.08(11) 1.63(90) 2.6 × − × × Hs 9.44(5) 0.93(84) 2.9 × − (15) 4.4 × × Sg 8.89(11) 0.16 2.9 × - - Rf 7.89(30) -0.48 1.2 × - - Og 11.73 2.58 2.3 × − × × Lv 10.774(15) 2.33 2.4 × − (12) 2.7 × × Fl 10.072(13) 1.14 7.5 × − (14) 8.0 × × Cn 9.60(20) 0.84 8.1 × × × Table 2: Same as in Table 1, but for the α -chain starting at Q α Q EC T α T β + /EC R 119 12.73 5.79 9.1 × − × × Ts 11.5(4) 4.41(85) 1.2 × − × × Mc 10.76(5) 3.82(75) 9.5 × − (60) 3.1 × × Nh 10.51(11) 3.22(75) 1.6 × − (10) 5.3 × × Rg 10.52(5) 2.65(73) 1.8 × − (11) 1.1 × × Mt 10.48(5) 2.21(72) 1.17 × − (74) 5.1 × × Bh 9.42(5) 1.16(72) 1.7 × × × Db 7.92(30) 0.63(71) 2.8 × × × − Lr 7.68(20) 0.60(57) 3.1 × × × the dominant decay mode for them. It is also worth notingthat in cases where Q EC is very small, the large T β + /EC obtained are very sensitive to fine details of the calcula-tions because they are determined from the few low-lyingtransitions located just below Q EC .As we go forward in a given chain, there is a generaltrend of decreasing values of the Q α and Q EC energies,which is translated into increasing half-lives and decreas-ing ratios. This general trend is somewhat altered in thevicinity of Z = 112 and N = 168 nuclei, where sub-shelleffects are expected at slightly prolate deformations, whichare typical of nuclei in this mass region [35]. 4. Conclusions β + /EC -decay half-lives of some selected even-even andodd- A isotopes in the region of SHN with proton numbers Z = 118 − 120 and neutron numbers N = 175 − 184 havebeen studied. Z =119 and 120 are the next elements tobe discovered and their production is object of very activeexperimental campaigns.6 able 3: Same as in Table 1, but for the α -chains starting at Q α Q EC T α T β + /EC R 120 13.25 6.15 1.7 × − × × Og 12.39 5.63 2.5 × − × × Lv 11.25 4.93 2.5 × − × × Fl 10.84 4.15 5.8 × − × × Cn 11.04(20) 3.26(62) 4.5 × − × × Ds 11.40(30) 2.74(59) 1.0 × − × × Hs 9.51(11) 1.82(50) 2.1 × × × Sg 8.63(21) 1.73(49) 2.1 × × × Rf 8.25(15) 1.03(32) 6.7 × × × 120 13.32 4.66 4.6 × − × × Og 12.21 3.79 5.0 × − × × Lv 11.26 3.30 7.4 × − × × Fl 10.83(3) 2.33(85) 3.3 × − (14) 2.0 × × 120 13.12 5.67 2.8 × − × × Og 11.9(9) 4.49(107) 2.9 × − × × Lv 11.1(3) 3.86(95) 5.6 × − × × Fl 10.56(5) 3.27(90) 2.1 × − (10) 6.7 × × Cn 10.45(5) 2.72(89) 1.8 × − (8) 5.7 × × Ds 10.83(11) 2.17(80) 6 × − (3) 5.0 × × Hs 9,70(5) 1.26(78) 1.06 × (50) 1.4 × × Sg 8.65(5) 0.61(72) 3.0 × (18) 1.6 × × Rf 7.81(30) 0.46(66) 2.5 × × × 120 12.98 3.76 2.0 × − × × Og 11.8(7) 2.94(94) 1.2 × − (5) 1.3 × × Lv 11.00(7) 2.30(93) 8 × − (3) 2.9 × × Fl 10.37(3) 1.76(93) 2.8 × − × × The nuclear structure of the decay partners is de-scribed microscopically within a pnQRPA based on a self-consistent deformed Skyrme HF+BCS approach. Uncer-tainties in the Q -energies originated from the unknownmasses are translated into uncertainties in the half-livescalculated with them. We have used different mass formu-las to evaluate the expected spread on the half-lives. Theresults for T β + /EC are compared with those of T α obtainedfrom several phenomenological formulas using Q α energiesobtained from the same mass formulas. The T α half-livesare systematically lower than the corresponding T β + /EC ones for a given isotope. The difference is always largerthan three orders of magnitude in the most favored case ofdeformed nuclei and becomes much larger for the spheri-cal ground states. Therefore, the β + /EC -decay mode willhardly compete with α -decay in the SHN studied with Z =118, 119, and 120.The competition between α and β + /EC decay modes isalso studied in seven α -decay chains starting at differentisotopes of Z =118, 119, and 120. The ratio of half-livesfor both modes indicates that α -decay is generally sev-eral orders of magnitude (up to seven orders in the chainheads) faster than β + /EC -decay. However, the half-livesbecome comparable at the end of some of the chains stud-ied. Hence, the cases in which different decay branches aremore likely to appear have been identified. This could be useful as a theoretical guide for future experimental stud-ies of these decays in new elements not yet discovered. Acknowledgments I would like to thank G. G. Adamian for useful discus-sions and valuable advice. This work was supported byMinisterio de Ciencia e Innovaci´on MCI/AEI/FEDER,UE(Spain) under Contract No. PGC2018-093636-B-I00. ReferencesReferences [1] S. Hofmann, G. M¨unzenberg, Rev. Mod. Phys. 72 (2000) 733.[2] J.H. Hamilton, D. Hofmann, Y.T. Oganessian, Annu. Rev. Nucl.Part. Sci. 63 (2013) 383.[3] Yu. Ts. Oganessian, V.K. Utyonkov, Rep. Prog. Phys. 78 (2015)036301.[4] S. Hofmann, et al., Eur. Phys. J. A 52 (2016) 180.[5] S.A. Giuliani, et al., Rev. Mod. Phys. 91 (2019) 011001.[6] K. Morita, et al., J. Phys. Soc. Japan 73(10) (2004) 2593.[7] Yu. Ts. Oganessian, et al., Phys. Rev. C 69 (2004) 021601(R).[8] Yuri Oganessian, J. Phys. G: Nucl. Part. Phys. 34 (2007) R165.[9] Yu. Ts. Oganessian, V.K. Utyonkov, Nucl. Phys. A 944 (2015)62.[10] S.G. Nilsson, et al., Nucl. Phys. A 115 (1968) 545.[11] P. M¨oller, J.R. Nix, J. Phys. G: Nucl. Part. Phys. 20 (1994)1681.[12] A.N. Kuzmina, G.G. Adamian, N.V. Antonenko, W. Scheid,Phys. Rev. C 85 (2012) 014319.[13] K. Rutz, et al., Phys. Rev. C 56 (1997) 238.[14] A.T. Kruppa, M. Bender, W. Nazarewicz, P.-G. Reinhard, T.Vertse, S. ´Cwiok, Phys. Rev. C 61 (2000) 034313.[15] M. Bender, W. Nazarewicz, P.-G. Reinhard, Phys. Lett. B 515(2001) 42.[16] J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long, L.S.Geng, Prog. Part. Nucl. Phys. 57 (2006) 470.[17] S.E. Agbemava, A.V. Afnasjev, T. Nakatsukasa, P. Ring, Phys.Rev. C 92 (2015) 054310.[18] A. Lopez-Martens, et al., Phys. Lett. B 795 (2019) 271.[19] F.P. He β berger , Eur. Phys. J. A 55 (2019) 208.[20] Juhee Hong, G.G. Adamian, N.V. Antonenko, Phys. Rev. C 94(2016) 044606; Phys. Lett. B 764 (2017) 42.[21] G.G. Adamian, N.V. Antonenko, A. Diaz-Torres, S. Heinz. Eur.Phys. J. A 56 (2020) 47.[22] G.G. Adamian, N.V. Antonenko, W. Scheid, Phys. Rev. C 69(2004) 044601.[23] H.M. Albers, et al., Phys. Lett. B 808 (2020) 135626.[24] G.G. Adamian, N.V. Antonenko, H. Lenske, L.A. Malov, Phys.Rev. C 101 (2020) 034301.[25] A. Baran, M. Kowal, P.-G. Reinhard, L.M. Robledo, A.Staszczak, and M. Warda, Nucl. Phys. A 944 (2015) 442.[26] A.V. Karpov, V.I. Zagrebaev, Y. Martinez Palazuela, L. FelipeRuiz, Walter Greiner, Int. J. Mod. Phys. E 21 (2012) 1250013.[27] V.I. Zagrebaev, A.V. Karpov, Walter Greiner, Phys. Rev. C 85(2012) 014608.[28] F.P. He β berger, et al., Eur. Phys. J. A 52 (2016) 328.[29] J. Khuyagbaatar, et al., Phys. Rev. Lett. 125 (2020) 142504.[30] X. Zhang, Z. Ren, Phys. Rev. C 73 (2006) 014305.[31] E.O. Fiset, J.R. Nix, Nucl. Phys. A 193 (1972) 647.[32] U.K. Singh, P.K. Sharma, M. Kaushik, S.K. Jain, Dashty T.Akraway, G. Saxena, Nucl. Pˆhys. A 1004 (2020) 122035.[33] P. M¨oller, M.R. Mumpower, T. Kawano, W.D. Myers, At. DataNucl. Data Tables 125 (2019) 1.[34] P. Sarriguren, Phys. Rev. C 100 (2019) 014309.[35] P. Sarriguren, J. Phys. G: Nucl. Part. Phys. 47 (2020) 125107. ∼