Electron-capture decay in isotopic transfermium chains from self-consistent calculations
aa r X i v : . [ nu c l - t h ] S e p Electron-capture decay in isotopic transfermium chains from self-consistentcalculations
P. Sarriguren
Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, Spain ∗ Weak decays in heavy nuclei with charge numbers Z = 101 −
109 are studied within a micro-scopic 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 obtained from phenomenological formulas. Transfermium isotopes of Md, No, Lr,Rf, Db, Sg, Bh, Hs, and Mt that can be produced in the frontier of cold and hot fusion-evaporationchannels are considered. Several isotopes are identified whose β + /EC - and α -decay half-lives arecomparable. The competition between these decay modes opens the possibility of new pathwaystowards the islands of stability. I. INTRODUCTION
Understanding superheavy nuclei (SHN) is a topicalissue that attracts a great deal of research activity. Thesynthesis of new SHN is a long story that has already ledin the last decades to the discovery of a large number ofnew elements (see [1–7] for a review).Cold-fusion reactions were first used to synthesize SHNwith Z = 107 −
113 by using target nuclei
Pb and
Bi and medium-mass stable isotopes of Ti, Cr, Fe, Ni,and Zn as projectiles [1, 3, 8]. In these cold reactions thecompound nucleus produced has relatively low excitationenergy and typically evaporates one or two neutrons. At-tempts to produce heavier nuclei with these type of reac-tions failed because of the fast decreasing of the produc-tion cross sections for increasing charge of the projectiles.This difficulty was solved by using more asymmetric re-actions with both target and projectile having a largeneutron excess and thus, decreasing the Coulomb repul-sion. In practice, these hot-fusion reactions were carriedout with long-lived actinide nuclei from
U to
Cf astargets and the double magic nucleus Ca as projectiles.The method was successfully accomplished to produceSHN with Z =112–118 in the neutron-evaporation chan-nels ( xn -channels) [2, 4, 9]. The compound nuclei pro-duced in these hot reactions are created in highly excitedstates that evaporate typically between 2 and 5 neutronsbefore starting a chain of α decays ending with a sponta-neous fission (SF). Tracing back these paths allows oneto arrive to the SHN originally produced.However, the SHN synthesized so far are still far fromthe theoretically predicted ”islands of stability” for SHN.Calculations of binding energies within macroscopic-microscopic models [10–16] predict the existence of par-ticularly stable structures for spherical SHN with Z =114 and N = 184, as well as for deformed configurationswith Z = 108 and N = 162. These results agree withmore recent microscopic calculations performed withinself-consistent relativistic and non-relativistic mean-field ∗ Electronic address: [email protected] models [17–22] that predict stabilized regions with shellclosures at Z = 114 , N = 184, Z = 120 , N = 172, and Z = 126 , N = 184, depending on the interactions andtheir parametrizations.Further experimental investigation into the moreneutron-rich SHN region following the direction of thepredicted island of stability is a difficult task in the xn channels because of the limited number of available sta-ble projectiles and targets to reach those nuclei, as wellas because of the small production cross sections. Studiesof the optimal combinations of target and beam partnersin a search for more efficient reactions to synthesize SHNcan be found in [23–25].Alternative ways to produce more neutron-rich iso-topes through fusion-evaporation reactions that includenot only xn channels, but also the emission of chargedparticles from the compound nucleus ( pxn and αxn chan-nels), are presently being explored. The products ofthese reactions might fill the gap of unknown nuclidesbetween those produced in cold and hot reactions inthe xn fusion-evaporation channels. A number of suchreactions, including isotopes of Md, No, Lr, Rf, Db,Sg, Bh, Hs, and Mt have been studied in asymmet-ric hot fusion-evaporation reactions [26, 27], predictingcross sections that are about one or two orders of mag-nitude smaller than those of the xn channels. Never-theless, although the cross sections are smaller, the pxn and αxn channels will allow the production of new iso-topes that are unreachable in the xn channels due to thelack of proper projectile-target combinations. Produc-tion of superheavy isotopes with Z = 111 −
117 has alsobeen studied in the charged particle-evaporation chan-nels of Ca-induced actinide-based fusion reactions [28].Recent measurements [29] or reanalysis [30] of proton-evaporation rates in the Ti(
Bi, xn ) − x Db reactionshave demonstrated the viability of these fusion reactionsto approach the island of stability. The average crosssections for proton evaporation are found experimentallyto be between 10 and 100 times smaller than the crosssections in the neutron-evaporation channels. Althoughsmall, the former may represent an alternative way toproduce more neutron-rich SHN.Similarly, β + /EC -decay may open new pathways to-wards the predicted region of stability and may help tofill the gap between the nuclei produced in cold and hotfusion reactions [31, 32]. Experimental evidence of thisdecay has been already found in Db [33]. However, the β + /EC -decay half-lives in SHN have not been sufficientlystudied yet. There are phenomenological parametriza-tions [34] that can be used to extrapolate to regionswhere the half-lives are unknown. There are also cal-culations that neglect nuclear structure effects, such asthose in [31, 32], where only allowed transitions connect-ing parent and daughter ground states are considered.The nuclear matrix elements of the transitions were as-sumed to be a constant phenomenological value givenby log( f t ) = 4 . Q EC energies were taken from the masses of the finite rangedroplet model (FRDM) [36]. In a different approach,half-lives for β + /EC -decay were also evaluated within aproton-neutron quasiparticle random-phase approxima-tion (pnQRPA), which is based on a phenomenologicalfolded-Yukawa single-particle Hamiltonian, using massesfrom FRDM and standard phase factors. However, only β + /EC half-lives smaller than 100 s were published [37].In this work, the β + /EC -decay half-lives of someselected even-even and odd- A isotopes in the region Z = 101 −
109 and N = 151 −
168 are studied.Namely,
Md, , No, , , , , , Lr, − Rf, , , , , , , Db, − Sg, , , , , , , Bh, − Hs, and , , , , , , Mt. We use a microscopicapproach based on a deformed self-consistent Hartree-Fock calculation with Skyrme interactions and pairingcorrelations that describes the nuclear structure of par-ent and daughter nuclei involved in the β + /EC -decay.The method has been already used to calculate thosehalf-lives in different SHN [38] and therefore, the presentwork is an extension of the previous study to nuclei inthe transfermium region.The paper is organized as follows. Section II containsthe basis of the theoretical method used to calculateGamow-Teller (GT) strength distributions and β + /EC half-lives. The phenomenological models used to calcu-late α -decay half-lives are also presented. Section III con-tains the results for the SHN mentioned above and theyare discussed in terms of their relevance in terms of thecompetition between β + /EC - and α -decay modes. Fi-nally, Section IV contains the summary and conclusions. II. THEORETICAL FORMALISM
A brief summary of the theoretical framework used inthis work to account for the β + /EC -decay half-lives inSHN is presented in this section. The procedure usedhere follows closely the approach used in [38]. Furtherdetails of the formalism can be found elsewhere [39–44] The β + /EC -decay half-life, T β + /EC , is calculated bysumming all the allowed GT transition strengths connect-ing the parent ground state with states in the daughternucleus with excitation energies, E ex , lying below the Q i energy ( i = β + , EC ), T − i = ( g A /g V ) D X 1) + m e , (2)In this work these energies are taken from experiment[45]. They are shown in table I together with their un-certainties.To get the half-lives, the GT strengths are weightedwith phase-space factors f i ( Z, Q i − E ex ). The two com-ponents of these factors, positron emission f β + and elec-tron capture f EC , are computed numerically for eachvalue of the energy, as explained in [46].Concerning the nuclear structure involved in the cal-culation of the GT strength B ( GT, E ex ), a self-consistentcalculation of the mean field is first performed by meansof a deformed Hartree-Fock procedure with Skyrme inter-actions and pairing correlations in the BCS approxima-tion with fixed gap parameters. This calculation providesus single-particle energies, wave functions, and occupa-tion probabilities. The Skyrme interaction SLy4 [47] ischosen for this study because of its proven ability to de-scribe successfully nuclear properties throughout the en-tire nuclear chart [48–50]. The solution of the HF equa-tions is found by using the formalism developed in [51],under the assumption of time reversal and axial sym-metry. The single-particle wave functions are expandedin terms of the eigenstates of an axially symmetric har-monic oscillator in cylindrical coordinates, using 16 ma-jor shells. This basis size is sufficiently large to achieveconvergence of the HF energy. One should also take intoaccount that the use of an axially deformed harmonic os-cillator basis, which is tuned in terms of two parameters(oscillator length and axis ratio), accelerates the conver-gence as compared to the spherical basis. Deformation-energy curves (DECs) are constructed by constrained HFcalculations that allow to analyze the nuclear bindingenergies as a function of the quadrupole deformation pa-rameter β .Calculations of the GT strengths are performed subse-quently for the deformed ground states that correspondto the absolute minima in the DEC. Nuclear deformationhas been revealed as a key element to describe β -decayproperties in many different mass regions [39–42] and itis also expected to play a significant role in SHN [38].A deformed pnQRPA with residual spin-isospin interac-tions is used to obtain the energy distribution of the GTstrength needed to calculate the half-lives. In the case ofSHN the coupling strengths of the residual interactionsthat scale with the inverse of the mass number are ex-pected to be very small and their effect is neglected. TheGT strength distributions in the following figures are re-ferred to the excitation energy in the daughter nucleus.In the case of odd- A nuclei the procedure is based onthe blocking of the state that corresponds to a given spinand parity ( J π ), using the equal filling approximation(EFA) to calculate its nuclear structure [42]. The EFAprescription is commonly used in self-consistent mean-field calculations because it preserves time-reversal in-variance with the corresponding numerical advantagesassociated to this symmetry. In this approximation halfof the unpaired nucleon sits in a given orbital and theother half in the time-reversed partner. The reliabil-ity of this approximation has been demonstrated [52] bycomparing the results from EFA with those from moresophisticated approaches, including the exact blockingprocedure with time-odd mean fields fully taken into ac-count. It was shown [52] that both procedures are strictlyequivalent when time-odd terms are neglected and thatthe impact of the time-odd terms is quite small. The fi-nal conclusion was that the EFA is sufficiently precise formost practical applications. The blocked state is chosenamong the states in the vicinity of the Fermi level as thestate that minimizes the energy.This model has been successfully used in the past tostudy different mass regions including neutron-deficientmedium-mass [53–56] and heavy nuclei [57–59], neutron-rich nuclei [60–64], and f p -shell nuclei [65–67]. The ef-fect of various ingredients of the model like deformationand residual interactions on the GT strength distribu-tions, which finally determine the decay half-lives, wasalso studied in the above references. In particular, thesensitivity of the GT distributions to deformation hasbeen used to learn about the nuclear shapes when com-paring with experiment [68].In this work, only allowed β + /EC decays are consid-ered. Forbidden transitions are in general much smallerand therefore, they can be safely neglected, especiallyin nuclei with small Q EC -energies, such as those stud-ied here. Allowed transitions correspond to ∆ π = 0 and∆ J = 0 , ± Q EC en-ergies involved, only the low-lying excitations connectingproton with neutron states in the vicinity of the Fermilevel obeying the above selection rules will contribute. Inthe case of the decay of even-even nuclei one has 0 + → + transitions. In the case of odd- A nuclei one needs the J π of the parent nucleus that will determine the allowed J ′ π reached in the daughter nucleus.As mentioned earlier, the spin and parity of the de-caying nucleus are chosen by selecting the state occupiedby the odd nucleon that minimizes the energy. How- ever, other possibilities of states in the neighborhood ofthe Fermi level different from the ground states are alsoconsidered to study the β + /EC -decay. This is becausethe calculations including deformation produce in manycases a high density of states around the Fermi level witha given ordering, which can be altered by small changesin the theoretical treatment. In addition, fusion reactionsproduce compound nuclei and subproducts in excitedstates that could decay directly. Thus, it is interesting toknow the β + /EC -decay of the predicted ground states,as well as the decay of excited states close to it. Since the β + /EC -decay half-lives are sensitive to the spin-parity ofthe odd nucleon, calculations are performed for severalstates with opposite parity close to the Fermi energy.Comparison between α - and β + /EC -decay modes iscrucial to understand the possible branching and path-ways of the original compound nucleus leading to sta-bility. Unfortunately, not all the α -decay half-lives ofnuclei in this mass region have been measured yet. Inthose cases where experimental information on the totalhalf-life and the percentage of the α -decay mode inten-sity are available, T α values are extracted and plottedin the following figures. To complete this information inother cases, the α -decay half-lives have been estimatedfrom phenomenological formulas that depend on the Q α energies, using values taken from experiment [45] thatcan be seen in table I with their uncertainties. Followingthe same approach as in [38], four different parametriza-tions are used, which have been fitted to account forthe properties of SHN. Namely, they are the formula byParkhomenko and Sobiczewski [69] (label 1 in the x-axisof the next figures), the Royer formula [70] (label 2), andthe Viola-Seaborg formula [71] with two different sets ofparameters from [69] (label 3) and [31, 72] (label 4). III. RESULTS AND DISCUSSION DECs are first shown in figure 1 for a selected iso-tope, Sg, which is representative of the nuclei in thismass region. The energies in figure 1 are relative tothe ground state energy and are plotted as a function ofthe quadrupole deformation β . The results correspond-ing to the interaction of reference in this work (SLy4)show a ground state corresponding to a prolate shapeat β ≈ . 25 and two excited configurations with oblate( β ≈ − . 25) and superdeformed prolate ( β ≈ . β ≈ . 25. Deformationsof parent and daughter decay partners are assumed tobe the same in this work. This is a widely used ap-proximation based on the spin-isospin character of theGamow-Teller operator that does not contain any radialdependency. Thus, the spatial functions of parent anddaughter wave functions are expected to be as close aspossible to overlap maximally. As a result, transitionsconnecting different radial structures in the parent anddaughter nuclei are suppressed. This is justified by thefact that core polarization effects in the daughter nu-clei are negligible, as it can be seen, for example, fromGogny calculations [74], where the DECs of parent andcorresponding daughter isotopes considered in this workare practically the same with ground states at β ≈ . T β + /EC on the left-hand panels and T α onthe right-hand ones, grouped by isotopes. In the case ofodd- A nuclei, T β + /EC are calculated for several J π val-ues. For T α there are calculations using the four modelsdescribed above. The errors in the half-lives correspondto calculations using the extreme values of the Q EC and Q α given by their experimental uncertainties in table I.Experimental data for T β + /EC ( T α ) are extracted andshown under the label ’exp’ in those cases where boththe total experimental half-lives and the percentage of β ( α ) decay intensities are measured.Figure 2 contains the results for Md ( Z = 101) andNo ( Z = 102) isotopes. In this case the β + /EC -decayhalf-lives are orders of magnitude larger than the corre-sponding α -decay half-lives for a given isotope and haveno special interest.Figure 3 shows the results for the odd- A Lr isotopes( Z = 103). In this case J π is determined by the odd-proton state. The experimental J π assignments [45] aremostly uncertain (values within parenthesis) or estimatedfrom systematic trends in neighboring nuclei (denotedwith J π = (1 / − ) g . s . and (7 / − ) in Lr, J π = (1 / − ) in Lr, and J π = 1 / − Lr. In thepresent calculations, states 1 / − , 7 / − and 7 / + veryclose in energy around the Fermi surface are obtained,and then, calculations for these three possibilities areperformed. The β + /EC -decay half-lives of the positive-parity states turn out to be much shorter than the cor-responding half-lives of the negative-parity states. Thelatter are always orders of magnitude larger than the α -decay half-lives, but the former can compete in some in-stances with α -decay. Namely, in the case of the positive-parity states we obtain comparable β + /EC - and α -decayhalf-lives for , Lr isotopes. In the case of Lr thedifference is within a factor of 10, whereas for Lr thedifference is already about two orders of magnitude. Figure 4 corresponds to Rf isotopes ( Z = 104). Even-even and odd- N isotopes are considered. In the caseof odd- N nuclei, J π is determined by the odd neutronstate. The systematics of the J π in the various isotopesis not clear since each isotope has a different number ofneutrons and the last neutron occupies a different or-bital. Experimentally, the J π assignments are as follows, J π = (9 / − ) g . s . and 5 / + Rf, J π = (1 / + ) g . s . and(11 / − ) in Rf, J π = 7 / + g . s . , (3 / + ) and (9 / + ) in Rf, J π = 3 / + g . s . and 9 / + Rf, J π = 3 / + Rf, and J π = 3 / + Rf. In the calculations,various possibilities for the odd-neutron states are ob-tained including those mentioned above. β + /EC -decayhalf-lives are performed for representative J π of bothpositive and negative parities. One can observe againthat the β + /EC -decay half-lives of the positive-paritystates are shorter than the corresponding half-lives of thenegative-parity states. For the positive-parity states, thehalf-lives of β - and α -decays are similar in , Rf. Inthe case of Rf ( Rf) the T β + /EC are within a factorof 10 (100) larger than T α .In the case of the odd- Z Db isotopes ( Z = 105) in fig-ure 5, the experimental J π is found to be J π = (9 / + ) g . s . and (1 / − ) in Db and J π = (9 / + ) in Db. Thesestates are found in the calculations close to the Fermi sur-face, as well as 5 / − states and half-lives are calculatedfor these options. Referring again to the positive-paritystates that exhibit smaller half-lives than the negativeones, one can see that the half-lives of β - and α -decaysare comparable in , , Db, while there is one orderof magnitude of difference in Db and two orders in , Db.Next we consider Sg ( Z = 106) isotopes in figure6. The spin-parity experimental assignments of theodd isotopes are as follows: J π = 1 / + Sg, J π = (3 / + ) g . s . and (11 / − ) in Sg, J π = 7 / + g . s . and 3 / + Sg, and J π = 9 / + g . s . and 3 / + Sg. In this case the half-lives T β + /EC are quite simi-lar for both parities in the lighter isotopes and they startto diverge from Sg. The β - and α -decay half-livesare comparable in Sg, whereas they differ by aboutone order of magnitude in Sg and by two orders in , , Sg.The half-lives of Bh isotopes ( Z = 107) are shown infigure 7. The only experimental value of J π assigned is(5 / − ) in Bh. The calculations give 5 / − and 9 / + states at the Fermi level and β -decay calculations aremade for both of them. In the lighter isotopes consid-ered there is not big difference between the β + /EC -decayhalf-lives calculated with the different spin-parities. Forisotopes heavier than N = 162 the difference is muchlarger. The β - and α -decay half-lives are comparable in Bh, they differ by one order of magnitude in Bhand by two orders of magnitude in Bh.Even-even isotopes of Hs ( Z = 108) are shown in figure8. The spin-parity experimental assignments are 7 / + Hs, 3 / + g . s . and 9 / + Hs, 5 / + Hs, 9 / + Hs, and 3 / + Hs. Similarto the case of Sg isotopes, the β + /EC -decay half-livescalculated with positive-parity states are very close tothose from negative-parity states in the lighter isotopesup to N = 159. Heavier isotopes show a more drasticdependence on the parity of the states. The β - and α -decay half-lives differ by less than a factor of 10 in Hsand about two orders of magnitude in , Hs.Finally, in Mt isotopes ( Z = 109), the half-lives plot-ted in figure 9 show that the difference between T α and T β + /EC is more than two orders of magnitude in all iso-topes except Mt, where it is about two orders.From the above figures one can also learn about theuncertainties associated with different aspects of the cal-culations, in particular with the Q EC energies that areplotted as error bars and with the J π assignments in thecase of odd- A isotopes.Tables II-IV summarize the main results obtained inthis work regarding the comparison between the half-livesof the β + /EC - and α -decay modes. Table II containsthe calculated half-lives for the isotopes with comparable T β + /EC and T α . Experimental values extracted from [45]are also shown when available. In the case of T α thevalues shown correspond to the average value obtainedfrom the four formulas considered in this work. SF in thismass region is another possible decay mode that mightalso compete with α and β decay in some isotopes. Forcomparison, the available experimental half-lives for SF, T SF , obtained from the total half-lives and percentage ofthe SF decay mode intensity from [45] are quoted in thelast column of the table.Table III shows the same information as in table II, butfor the isotopes whose T β + /EC and T α values are within afactor of ten. Similarly, table IV contains the informationon the isotopes with T β + /EC and T α differing by abouttwo orders of magnitude.To understand why in the case of odd- A isotopes the T β + /EC for positive-parity states are always significantlylower than the same half-lives for negative-parity states,one has to analyze the different scenarios for the decaywhen the odd nucleon in the parent nucleus has a posi-tive or a negative parity. This parity will determine theparity of all the states reached in the daughter nucleus(allowed transitions). For that purpose figure 10 showsa Nilsson-like diagram, where the single-particle energiesare plotted as a function of the quadrupole deformation β for protons (left) and neutrons (right) in the case of Sg with Z = 106 and N = 160. The calculations cor-respond to the Skyrme interaction SLy4. Fermi levelsfor protons ( ε π ) and neutrons ( ε ν ) are plotted as thickdotted black lines. Positive-parity states are shown withsolid lines, whereas negative-parity states are shown withdashed lines. The spherical shells are shown at β = 0with their spherical quantum numbers. The boxes cen-tered at β = 0 . 25 correspond to the regions of interestdetermined by the quadrupole deformation of the equi-librium ground state in this mass region. The color codeused to plot the different components of the angular mo-menta is also shown. The spherical shells involved in the β + /EC -decay in increasing order of energy are the follow-ing: h / , f / , i / , f / , p / for protons and i / , g / , j / , g / , d / , d / , s / , h / , j / forneutrons. In this analysis it is sufficient to focus on thestates in the vicinity of the Fermi energy because onlylow-lying transitions involving the odd nucleon are rele-vant to calculate half-lives in nuclei with low Q -energies,which is the case of the isotopes studied here. In the β + /EC -decay one proton is transformed into one neu-tron. In the case of odd-proton isotopes, the odd-protonis directly involved in the low-lying transitions below the Q -window that determine the β + /EC -decay half-lives.In the case of odd-neutron isotopes, the low-lying GTexcitations corresponding to β + /EC -decay involve pro-ton states in the vicinity of the Fermi level that matchthe allowed quantum-numbers given by the odd neutron.Focusing on the proton single-particle energies withinthe box around β = 0 . 25, one can see that among theodd-proton isotopes considered, Lr ( Z = 103) wouldhave the odd proton placed in one of the orbitals 7 / − ,7 / + , and 1 / − close to the Fermi energy. They origi-nate in the spherical shells h / , i / , f / , respectively.In the case of Db ( Z = 105) and Bh ( Z = 107), thestates involved would be 5 / − ( f / ) and 9 / + ( i / ),whereas in the case of Mt( Z = 109) one finds the states9 / + ( i / ), 9 / − ( h / ), 3 / − ( f / ), and 11 / + ( i / )close to the Fermi level. All of above states have beenconsidered in the decay of these odd- A isotopes. There-fore, in the odd-proton isotopes, the states mentionedabove would decay into neutron states in the vicinityof the neutron Fermi energy, which is shown in theright panel of figure 10 within the black box centeredat β = 0 . 25. For neutron numbers between N =150-162 the states involved are 9 / − ( j / ), 7 / + ( g / ),9 / + ( i / ), 1 / + ( d / ), 3 / + ( g / ), and 11 / − ( j / ).Beyond N =162, new states appear with 13 / − ( j / ),9 / + ( g / ), 5 / + ( g / ), 3 / + ( d / ), 3 / − ( h / ), and1 / − ( h / , j / ).Then, it is easy to understand that the transitions in-volving the positive-parity proton states, namely 7 / + and 9 / + , would match the neutron states 5 / + , 7 / + ,and 9 / + , while the negative-parity proton states 5 / − cannot match the 9 / − , 11 / − or 13 / − and only in theheavier isotopes the 9 / − proton states states can matchthe 9 / − and 11 / − neutron states. This explains quali-tatively why the decays from even-parity states are muchfaster than the decays from odd-parity states. In thecase of odd-neutron isotopes, the argument is similar, butnow, the odd neutron in the parent nucleus determinesthe proton states involved in the transitions.The comparison of the calculations with the availableexperimental half-lives in both cases T β + /EC and T α isin general quite satisfactory, which helps to be confidentin the reliability of the calculations. More specifically,in the case of β + /EC -decay, the experimental T β + /EC in Lr lies within the calculated half-lives with positive- ornegative-parity states, whereas in the case of Lr and Rf the experiment is very close to the calculation withthe 7 / + state. On the other hand, in Db, the ex-periment is closer to the calculation with negative parityand in Db the experiment lies between the predictionswith positive or negative parities. Finally, in Sg theexperiment is close to the calculations with both parities.In the case of α -decay the experimental information onhalf-lives is more abundant. In general, the predictions ofthe different formulas considered agree within one orderof magnitude with the measurements. IV. CONCLUSIONS In this work, the β + /EC -decay half-lives of someselected even-even and odd- A isotopes in the region Z = 101 − 109 and N = 151 − 168 are studied.Namely, Md, , No, odd − Lr, − Rf, odd − Db, − Sg, odd − Bh, − Hs, and odd − Mt. The microscopic formalism used to describethe nuclear structure of the decay partners is based on adeformed Skyrme HF+BCS approach.Uncertainties in the experimental Q EC energies aretranslated into uncertainties of the half-lives calculatedwith them. In the case of odd- A nuclei, different J π as-signments are considered to learn about their influenceon the final half-lives. It is found that in odd- A iso-topes the T β + /EC for the positive-parity states are alwaysshorter than the T β + /EC for the negative-parity states.The results for T β + /EC are compared with the α -decay half-lives T α obtained from phenomenological formulasusing experimental Q α energies and their uncertainties.The agreement between the calculated and the availableexperimental half-lives in both cases T β + /EC and T α isfound to be always within a factor of 10, granting thecategory of trustable predictions. T α are in most cases lower than the corresponding T β + /EC for a given isotope. This difference is abouttwo orders of magnitude in Lr, Rf, , Db, , , Sg, Bh, , Hs, and Mt. In the casesof Lr, Rf, Db, Sg and Bh the difference isonly about one order of magnitude. Finally, the isotopes , Lr, , Rf, , , Db, Sg, Bh, and Hshave comparable values of the half-lives for the β + /EC -and α -decay modes. Therefore, these different modeswill compete in the latter cases favoring new branchesof decay in the β + /EC direction that have not yet beensufficiently studied. This opens new possibilities to reachunexplored roads towards the predicted islands of stabil-ity. Acknowledgments I would like to thank G. G. Adamian for the sugges-tion of this problem and for useful discussions and valu-able advice. This work was supported by Ministerio deCiencia e Innovaci´on MCI/AEI/FEDER,UE (Spain) un-der Contract No. PGC2018-093636-B-I00. [1] Hofmann S and M¨unzenberg G 2000 Rev. Mod. Phys. J. Phys. G R165[3] Hamilton J H, Hofmann D and Oganessian Y T 2013 Annu. Rev. Nucl. Part. Sci. Rep. Prog. Phys. Nucl. Phys. A et al. Eur. Phys. J. A Rev. Mod. Phys. J.Phys. Soc. Japan (10) 2593[9] Oganessian Y T et al. Phys. Rev. C Nucl. Phys. Phys.Lett. Nucl. Phys. A Nucl. Phys. A J. Phys. G Phys. Rev. C Phys. Rev. C Phys. Rev. C Phys. Rev. C Phys.Lett. B Rev.Mod. Phys. Prog. Part. Nucl. Phys. Nucl. Phys. 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A TABLE I: Experimental Q EC and Q α energies (MeV) from AME2012 [45] Nucleus Q EC Q α Nucleus Q EC Q α Nucleus Q EC Q α Md 0.0 ± ± No -0.9 ± ± No 0.0 ± ± Lr 3.140 ± ± Lr 2.36 ± ± Lr 1.74 ± ± Lr 1.11 ± ± Lr 0.60 ± ± Lr - 7.23 ± Rf 4.38 ± ± Rf 2.48 ± ± Rf 3.26 ± ± Rf 1.56 ± ± Rf 2.51 ± ± Rf 0.87 ± ± Rf 1.76 ± ± Rf 0.29 ± ± Rf 1.06 ± ± Rf -0.20 ± ± Rf 0.46 ± ± Rf -1.5 ± ± Rf - 7.89 ± Db 4.34 ± ± Db 3.63 ± ± Db 2.93 ± ± Db 2.32 ± ± Db 1.80 ± ± Db 0.63 ± ± Db - 8.49 ± Sg 3.45 ± ± Sg 4.57 ± ± Sg 2.88 ± ± Sg 3.76 ± ± Sg 2.11 ± ± Sg 3.08 ± ± Sg 1.42 ± ± Sg 2.31 ± ± Sg 0.88 ± ± Sg 1.76 ± ± Sg -0.2 ± ± Sg 0.67 ± ± Sg -0.7 ± ± Sg - 8.89 ± Bh 5.13 ± ± Bh 4.31 ± ± Bh 3.56 ± ± Bh 2.93 ± ± Bh 1.67 ± ± Bh 1.23 ± ± Bh 0.62 ± ± Hs 5.22 ± ± Hs 3.51 ± ± Hs 4.55 ± ± Hs 3.03 ± ± Hs 3.89 ± ± Hs 2.02 ± ± Hs 3.11 ± ± Hs 0.86 ± ± Hs 1.78 ± ± Hs 0.22 ± ± Hs 1.34 ± ± Hs -0.1 ± ± Hs 0.93 ± ± Mt 5.78 ± ± Mt 5.14 ± ± Mt 4.72 ± ± Mt 3.33 ± ± Mt 2.53 ± ± Mt 2.01 ± ± Mt 1.28 ± ± TABLE II: Experimental [45] and calculated T β + /EC [s] and T α [s] for isotopes with comparable half-lives. Experimentalhalf-lives for spontaneous fission [45], T SF [s], are also shown for comparison.Nucleus T β + /EC T α T SF exp. calc. exp. calc. exp. Lr 120 42.6 42 28.6 Lr 50231 30925 Rf 7.2 3.5 1.9 3.2 Rf 28211 25260 396 Db 230 4.1 2.5 5.4 46 Db 165 233 Db 6192 24700 16560 Sg 254 636 206 130 Bh 192 770 Hs 26.1 27 5.6TABLE III: Same as in Table II, but for isotopes with T β + /EC and T α half-lives differing by about one order of magnitude.Nucleus T β + /EC T α T SF exp. calc. exp. calc. exp. Lr 5414 670 Rf 25.4 14.3 6.0 1.6 371 Db 420 127 78.4 20.4 51.8 Sg 128 18.4 9.9 184 Bh 44.8 22 6.4 TABLE IV: Same as in Table II, but for isotopes with T β + /EC and T α half-lives differing by about two orders of magnitude.Nucleus T β + /EC T α T SF exp. calc. exp. calc. exp. Lr 1033 2609 8.0 23.1 28.2 Rf 79.1 1.6 0.007 Db 10.7 0.5 0.10 Db 40.1 16.7 1.4 6.2 Sg 15.6 0.14 0.08 0.034 Sg 58.8 1.1 0.9 7.2 Sg 1483 19.1 0.46 Bh 16.9 1.2 0.33 Hs 15.5 0.07 0.08 0.28 Hs 86.9 1.4 0.36 Mt 28.6 0.4 0.36 -0.4 -0.2 0 0.2 0.4 0.6 0.8 β E ( M e V ) SLy4SG IID1S Gogny Sg FIG. 1: Deformation-energy curves for the Sg isotope obtained from constrained HF+BCS calculations with the Skyrmeforces SLy4 [47] and SGII [73], as well as with the D1S Gogny [74] interactions. 158 159 N T / ( s ) T α model Md No No T α T EC/ β + FIG. 2: Calculated half-lives for Md, No, and No isotopes. (left) Microscopic calculations of T β + /EC for the groundstate prolate configurations. J π is indicated for odd- A isotopes. (right) T α from four different phenomenological formulas,labeled from 1 up to 4 in the x axis (see text). T / in the y-axis is a short notation for both T β + /EC and T α . 152 154 156 158 160 162 N -1 T / ( s ) T α model Lr Lr Lr Lr Lr Lr T α T EC/ β + expexp exp FIG. 3: Same as in figure 2, but for the odd- A Lr isotopes from Lr up to Lr. 152 154 156 158 160 162 N -1 T / ( s ) T α model Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf Rf T α T EC/ β + exp exp FIG. 4: Same as in figure 2, but for − Rf isotopes. 152 154 156 158 160 162 164 N -2 -1 T / ( s ) T α model Db Db Db Db Db Db Db T α T EC/ β + exp exp exp FIG. 5: Same as in figure 2, but for the odd- A Db isotopes from Db up to Db. 152 154 156 158 160 162 164 N -3 -2 -1 T / ( s ) T α model Sg Sg Sg Sg Sg Sg Sg Sg Sg Sg Sg Sg Sg Sg T α T EC/ β + exp exp FIG. 6: Same as in figure 2, but for − Sg isotopes. 154 156 158 160 162 164 166 N -3 -2 -1 T / ( s ) T α model Bh Bh Bh Bh Bh Bh Bh T α T EC/ β + exp FIG. 7: Same as in figure 2, but for the odd- A Bh isotopes from Bh up to Bh. 156 158 160 162 164 166 N -4 -3 -2 -1 T / ( s ) T α model Hs Hs Hs Hs Hs Hs Hs Hs Hs Hs Hs Hs Hs T α T EC/ β + exp FIG. 8: Same as in figure 2, but for − Hs isotopes. 156 158 160 162 164 166 168 N -5 -4 -3 -2 -1 T / ( s ) T α model Mt Mt Mt Mt Mt Mt Mt T α T EC/ β + exp FIG. 9: Same as in figure 2, but for the odd- A Mt isotopes from Mt up to Mt. β -12-10-8-6-4-202 ε s . p . ( M e V ) β -16-14-12-10-8-6-4-202 ε s . p . ( M e V ) Sg (Z=106) Sg (N=160) NeutronsProtons (a) (b) ε π ε ν FIG. 10: Single-particle energies obtained with SLy4 for (a) protons and (b) neutrons in Sg as a function of the quadrupoledeformation β . The Fermi levels for protons ( ε π ) and neutrons ( ε νν