Beta-decay properties of neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes from deformed quasiparticle random-phase approximation
aa r X i v : . [ nu c l - t h ] A p r Beta-decay properties of neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes fromdeformed quasiparticle random-phase approximation
P. Sarriguren ∗ Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, Spain (Dated: August 24, 2018) β -decay properties of even and odd- A neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes involvedin the astrophysical rapid neutron capture process are studied within a deformed proton-neutronquasiparticle random-phase approximation. The underlying mean field is described self-consistentlyfrom deformed Skyrme Hartree-Fock calculations with pairing correlations. Residual interactionsin the particle-hole and particle-particle channels are also included in the formalism. The isotopicevolution of the various nuclear equilibrium shapes and the corresponding charge radii are investi-gated in all the isotopic chains. The energy distributions of the Gamow-Teller strength as well asthe β -decay half-lives are discussed and compared with the available experimental information. Itis shown that nuclear deformation plays a significant role in the description of the decay propertiesin this mass region. Reliable predictions of the strength distributions are essential to evaluate decayrates in astrophysical scenarios. PACS numbers: 21.60.Jz, 23.40.Hc, 27.60.+j, 26.30.-k
I. INTRODUCTION
The rapid structural changes occurring in the groundstate and low-lying collective excited states of neutron-rich nuclei in the mass region A = 80 −
128 have beenextensively studied both theoretical and experimentally(see e.g. [1, 2] and references therein). From the theo-retical side, the equilibrium nuclear shapes in this massregion have been shown to suffer rapid changes as a func-tion of the number of nucleons with competing spherical,axially symmetric prolate and oblate, and triaxial shapesat close energies. Both relativistic [3, 4] and nonrelativis-tic [5–9] approaches agree in the general description of thenuclear structural evolution in this mass region, which issupported experimentally by spectroscopic studies [10–12], 2 + lifetime measurements [13–15] and quadrupolemoments for rotational bands [15], as well as by laserspectroscopy measurements [16].However, the nuclear structure richness is not the onlyattractive feature characterizing these nuclei. Anotherremarkable property of nuclei in this mass region is thatthey are involved in the astrophysical rapid neutron cap-ture process ( r process), which is considered to be oneof the main nucleosynthesis mechanisms leading to theproduction of heavy neutron-rich nuclei in the universe[17, 18]. The r -process nucleosynthesis involves manyneutron-rich unstable isotopes, whose neutron capturerates, masses, and β -decay half-lives ( T / ) are crucialquantities to understand the possible r -process paths, theisotopic abundances, and the time scales of the process[18–20]. Although much progress has been done measur-ing masses (see for example the Jyv¨askyl¨a mass database[21]) and half-lives [22–24], most of the nuclear propertiesof relevance for the r process are experimentally unknown ∗ Electronic address: [email protected] due to their extremely low production yields in the lab-oratory. Therefore, reliable nuclear physics models arerequired to simulate properly the r process.The quasiparticle random-phase approximation(QRPA) is considered a well suited model to de-scribe medium-mass open-shell nuclear properties andspecifically β -decay properties. QRPA calculations forneutron-rich nuclei have been carried out within differentspherical formalisms, such as Hartree-Fock-Bogoliubov(HFB) [25], continuum QRPA with density function-als [26], and relativistic mean field approaches [27].However, the mass region we are dealing with requiresnuclear deformation as a relevant degree of freedomto characterize the nuclear structure involved in thecalculation of the β -strength functions. The deformedQRPA formalism was developed in Refs. [28–31], usingphenomenological mean fields. A Tamm-Dancoff ap-proximation with Sk3 interaction was also implementedin Ref. [32]. More recently, deformed QRPA calculationsusing deformed Woods-Saxon potentials and realisticCD-Bonn residual forces have been performed in [33, 34].First-forbidden transitions were also considered in thosereferences, showing that their effect in this mass regioncan be neglected. Various self-consistent deformedQRPA calculations to describe the β -decay properties,either with Skyrme [35] or Gogny [36] interactions arealso available in the literature.In Refs. [37, 38] the decay properties of neutron-rich Zr and Mo isotopes were studied within a deformedproton-neutron QRPA based on a self-consistent Hartree-Fock (HF) mean field formalism with Skyrme interactionsand pairing correlations in BCS approximation. Resid-ual spin-isospin interactions were also included in theparticle-hole and particle-particle channels [39, 40]. Inthis work this study is extended to the neighboring re-gions including even and odd- A neutron-rich Ge, Se, Kr,Sr, Ru, and Pd isotopes. These calculations are timelybecause they address a mass region which is at the bor-derline of present experimental capabilities for measur-ing half-lives at MSU and RIKEN [22–24]. In addition,theoretical calculations can be tested with the availableexperimental information on half-lives providing simul-taneously predictions for the underlying Gamow-Tellerstrength distributions and for the half-lives of more exoticnuclei not yet measured. Finally, this more comprehen-sive study allows one to judge better the extent to whichthe method is able to describe the decay properties of nu-clei in a wider mass region that includes spherical, welldeformed, and weakly deformed transitional isotopes, aswell as isotopes exhibiting shape coexistence. Therefore,the theoretical method will be tested over a rich set of dif-ferent nuclear structures that will reveal the limitationsof the model.The paper is organized as follows. In Sec. II a reviewof the theoretical formalism used is introduced. Sec-tion III contains the results obtained for the potentialenergy curves (PEC), Gamow-Teller (GT) strength dis-tributions, and β -decay half-lives, which are comparedwith the experimental data. Section IV summarizes themain conclusions. II. THEORETICAL FORMALISM
A summary of the theoretical framework used in thispaper to describe the β -decay properties in neutron-richisotopes is shown in this section. More details of theformalism can be found elsewhere [39, 40]. The methodstarts from a self-consistent calculation of the mean fieldby means of a deformed Skyrme Hartree-Fock procedurewith pairing correlations in BCS approximation. Single-particle energies, wave functions, and occupation ampli-tudes are generated from this mean field. The Skyrmeinteraction SLy4 [41] is used as a representative of mod-ern Skyrme forces. It has been very successful describingnuclear properties all along the nuclear chart and hasbeen extensively studied [6, 7, 42].The solution of the HF equation, assuming time rever-sal and axial symmetry, is found by using the formalismdeveloped in Ref. [43]. The single-particle wave functionsare expanded in terms of the eigenstates of an axiallysymmetric harmonic oscillator in cylindrical coordinates,using twelve major shells. The pairing gap parametersfor protons and neutrons in the BCS approximation aredetermined phenomenologically from the odd-even massdifferences [44]. In a further step, constrained HF cal-culations with a quadratic constraint are performed toconstruct the PECs, analyzing the nuclear binding ener-gies in terms of the quadrupole deformation parameter β . Calculations for GT strengths are performed subse-quently for the various minima in the energy curves in-dicating the equilibrium shapes of each nucleus. Sincedecays connecting different shapes are disfavored, simi-lar shapes are assumed for the ground state of the parentnucleus and for all populated states in the daughter nu-cleus. The validity of this assumption was discussed for example in Refs. [28, 30].To describe GT transitions, a separable spin-isospinresidual interaction in the particle-hole (ph) and particle-particle (pp) channels is added to the mean field andtreated in a deformed proton-neutron QRPA [28–32, 39,40, 45]. An optimum set of coupling strengths could bechosen following a case by case fitting procedure and onewill finally get different answers depending on the nu-cleus, shape, and Skyrme force. However, since the pur-pose here is to test the ability of QRPA to account forthe GT strength distributions in this mass region withas few free parameters as possible, the same couplingstrengths are used for all the nuclei considered in thispaper, which are taken from previous works [37, 38]. Weuse χ phGT = 0 .
15 MeV and κ ppGT = 0 .
03 MeV for the resid-ual interaction in the ph and pp channels, respectively.The sensitivity of the GT strength distributions tothe various ingredients contributing to the deformedQRPA calculations, namely to the nucleon-nucleon effec-tive force, to deformation, to pairing correlations, and toresidual interactions, have been investigated in the past[39, 40, 46–48]. In this work the most reasonable choicesfound in those references are used. Summarizing the var-ious sensitivities, the conclusion is that the main featuresof the GT strength distributions are in general very ro-bust against the Skyrme force used, showing some moresensitivity in the spherical cases, where the location of thesingle-particle energies is more critical to determine theexcitation energies of the GT transitions. Deformationhas been shown to be an important issue to describe theprofiles of the GT strength distributions. First, becausethe degeneracy of the spherical shells is broken makingthe GT strength distributions more fragmented than thecorresponding spherical ones. Secondly, because the en-ergy levels of deformed orbitals cross each other in a waythat depends on the magnitude of the quadrupole defor-mation as well as on the oblate or prolate character. Thislevel crossing may lead in some instances to sizable differ-ences in the GT profiles, a fact that has been exploited tolearn about the nuclear shape from the measured β -decaypattern [49–51]. Pairing correlations are also importantto describe nuclei out of closed shells. Their influence onthe GT profiles was studied in Ref. [40], concluding thatthe main effect is to decrease slightly the strength at lowenergies and to create new transitions, mainly at highenergies, that are forbidden in the absence of such corre-lations. The effect of the ph and pp residual interactionsis also well known. The repulsive ph interaction redis-tributes the GT strength by shifting it to higher excita-tion energies causing a displacement of the GT resonance.It also reduces somewhat the total strength. The attrac-tive pp interaction moves the strength to lower energies.Its effect on the GT resonance is in general negligible,but nevertheless, the changes induced in the low-energyregion are of great relevance in the calculation of the β -decay half-lives, which are only sensitive to the strengthcontained in the energy region below the Q -energy win-dow.The GT transition amplitudes in the intrinsic frameconnecting the ground state | + i of an even-even nucleusto one phonon states in the daughter nucleus | ω K i ( K =0 ,
1) are found to be (cid:10) ω K | σ K t ± | (cid:11) = ∓ M ω K ± , (1)where M ω K − = X πν ( q πν X ω K πν + ˜ q πν Y ω K πν ) , (2) M ω K + = X πν (˜ q πν X ω K πν + q πν Y ω K πν ) , (3)with ˜ q πν = u ν v π Σ νπK , q πν = v ν u π Σ νπK , (4)in terms of the occupation amplitudes for neutrons andprotons v ν,π ( u ν,π = 1 − v ν,π ) and the matrix elements ofthe spin operator, Σ νπK = h ν | σ K | π i , connecting protonand neutron single-particle states, as they come out fromthe HF+BCS calculation. X ω K πν and Y ω K πν are the forwardand backward amplitudes of the QRPA phonon operator,respectively.Once the intrinsic amplitudes in Eq. (1) are calculated,the GT strength B ω ( GT ± ) in the laboratory system fora transition I i K i (0 + → I f K f (1 + K ) can be obtainedas B ω ( GT ± ) = X ω K h(cid:10) ω K =0 (cid:12)(cid:12) σ t ± (cid:12)(cid:12) (cid:11) δ ( ω K =0 − ω )+2 (cid:10) ω K =1 (cid:12)(cid:12) σ t ± (cid:12)(cid:12) (cid:11) δ ( ω K =1 − ω ) i , (5)in [ g A / π ] units. To obtain this expression, the initialand final states in the laboratory frame have been ex-pressed in terms of the intrinsic states using the Bohr-Mottelson factorization [52].The specific treatment of odd- A systems has been de-scribed [31, 47] by considering two types of GT contri-butions. One type is due to phonon excitations in whichthe odd nucleon acts only as a spectator. The transitionamplitudes in the intrinsic frame are in this case basicallythe same as in the even-even case, but with the blockedspectator excluded from the calculation. The other typeof transitions involves the odd nucleon and is treated per-turbatively by taking into account phonon correlations tofirst order in the quasiparticle transitions. The excitationenergies of the GT states with respect to the ground statein the daughter nuclei have been discussed in Ref. [47] forboth types of transitions in terms of the QRPA phononenergy and the quasiparticle energies. E ( M e V ) -0.40 0.4 β Ge Se Kr Sr Zr Mo Ru Pd FIG. 1: Potential energy curves for even-even neutron-richGe, Se, Kr, Sr, Zr, Mo, Ru, and Pd isotopes obtained fromconstrained HF+BCS calculations with the Skyrme forceSLy4. -0.2-0.100.10.2 β prolateoblateg.s.
80 82 84 86 88 90 92 94 A r c (f m ) Ge (a)(b) π exp ( Ge) = (9/2+)J π exp ( Ge) = (5/2+)J π exp ( Ge) = (3/2+, 5/2+)
FIG. 2: (Color online) Isotopic evolution of the quadrupoledeformation parameter β (a) and charge radius (b) corre-sponding to the energy minima obtained from the Skyrmeinteraction SLy4 for Ge isotopes. Ground state results areencircled. -0.3-0.2-0.100.10.20.30.4 β prolateoblateg.s.
86 88 90 92 94 96 98 100 A r c (f m ) Se (a)(b) π exp ( Se) = (5/2+)J π exp ( Se) = (5/2+)
FIG. 3: (Color online) Same as in Fig. 2, but for Se isotopes. -0.4-0.3-0.2-0.100.10.20.30.40.5 β
90 92 94 96 98 100 102 104 A r c (f m ) prolateoblateg.s.exp Kr (a)(b) π exp ( Kr)=5/2(+)J π exp ( Kr)=1/2+ J π exp ( Kr)=1/2(+)J π exp ( Kr)=(3/2+)
FIG. 4: (Color online) Same as in Fig. 2, but for Kr isotopes.Experimental charge radii are from [59]. -0.3-0.2-0.100.10.20.30.40.5 β
94 96 98 100 102 104 106 108 A r c (f m ) prolateoblateg.s.exp Sr (a)(b) π exp ( Sr) = 1/2+J π exp ( Sr) = 1/2+J π exp ( Sr) = 3/2+J π exp ( Sr) = (5/2-)
FIG. 5: (Color online) Same as in Fig. 2, but for Sr isotopes.Experimental charge radii are from [59]. -0.3-0.2-0.100.10.20.30.4 β prolateoblateg.s.
110 112 114 116 118 120 122 A r c (f m ) Ru (a)(b) π exp ( Ru) = 5/2+J π exp ( Ru) = (1/2+)J π exp ( Ru) = (3/2+)isomer
Ru = (7/2-)isomer
Ru = (7/2-)
FIG. 6: (Color online) Same as in Fig. 2, but for Ru isotopes. -0.2-0.100.10.20.3 β proloblateg.s.
114 116 118 120 122 124 126 128 A r c (f m ) Pd (a)(b) π exp ( Pd) = (1/2+) isomer (7/2-)J π exp ( Pd) = (3/2+) isomer (9/2-)
FIG. 7: (Color online) Same as in Fig. 2, but for Pd isotopes. B ( G T ) B ( G T ) ex [MeV]0.11100 5 10 15 20 25E ex [MeV] Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Q β FIG. 8: (Color online) QRPA-SLy4 Gamow-Teller strengthdistributions for Ge isotopes as a function of the excitationenergy in the daughter nucleus. The calculations correspondto the various equilibrium deformations found in the PECs. B ( G T ) B ( G T ) ex [MeV]0.11100 5 10 15 20 25E ex [MeV] Se Se Se Se Se Se Se Se Se Se Se Se Se Se Se Q β FIG. 9: (Color online) Same as in Fig. 8, but for Se isotopes. B ( G T ) B ( G T ) ex [MeV]0.11100 5 10 15 20 25E ex [MeV] Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Q β FIG. 10: (Color online) Same as in Fig. 8, but for Kr isotopes. B ( G T ) B ( G T ) ex [MeV]0.11100 5 10 15 20 25E ex [MeV] Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Q β FIG. 11: (Color online) Same as in Fig. 8, but for Sr isotopes. B ( G T ) B ( G T ) ex [MeV]0.11100 5 10 15 20 25E ex [MeV] Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Q β FIG. 12: (Color online) Same as in Fig. 8, but for Ru isotopes. B ( G T ) B ( G T ) ex [MeV]0.11100 5 10 15 20 25E ex [MeV] Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Q β FIG. 13: (Color online) Same as in Fig. 8, but for Pd isotopes.
The β -decay half-life is obtained by summing all theallowed transition strengths to states in the daughter nu-cleus with excitation energies lying below the correspond-ing Q -energy, Q β ≡ Q β − = M ( A, Z ) − M ( A, Z + 1) − m e ,written in terms of the nuclear masses M ( A, Z ) and theelectron mass ( m e ), and weighted with the phase spacefactors f ( Z, Q β − E ex ), T − / = ( g A /g V ) D X In this section I present first the PECs in the isotopicchains studied. Quadrupole deformation parameters aswell as charge r.m.s. radii ( r c ) are analyzed as a functionof the mass number. Then, energy distributions of theGT strength corresponding to the local minima in thePECs are calculated. Finally, half-lives are evaluated andcompared with the experiment. A. Structural isotopic evolution In Fig. 1 the PECs, i.e., the energies relative to thatof the ground state, are plotted as a function of thequadrupole deformation β for the neutron-rich Ge, Se,Kr, Sr, Zr, Mo, Ru, and Pd isotopes. The results corre-spond to the SLy4 interaction. The isotopes covered in this study include middle-shell nuclei with proton num-bers between shell closures Z = 28 and Z = 50, namely Z = 32 , , , , , , , 46 and neutron numbers be-tween shell closures N = 50 (as in Ge) and N = 82 (asin the heaviest Pd).In most of the isotopic chains one can see the appear-ance of several equilibrium nuclear shapes, whose relativeenergies change with the number of neutrons. In Ge iso-topes, prolate shapes that are ground states in the lighterisotopes are found with the only exception of Ge, wherea spherical shape is found in accordance with its N = 50semi-magic character. At N = 58 , 60 ( , Ge) oblateand prolate shapes are practically degenerate in energyand oblate shapes become ground states for heavier iso-topes. The case of Se isotopes is similar with oblate andprolate minima all along the isotopic chain. The lighter(heavier) isotopes have prolate (oblate) ground stateswith transitional isotopes around N = 58 , 60 ( , Se). Inthis case the energy barriers are more pronounced thanin the case of Ge isotopes. Kr isotopes show compet-ing shapes in the lighter isotopes that become oblate at N = 58 , 60 ( , Kr) and then turn into prolate shapesbeyond Kr. In the heavier isotopes, as in the case ofSe isotopes, shape coexistence is found with very well de-veloped oblate and prolate minima separated with highenergy barriers. Sr isotopes show a transition from oblateat N = 58 ( Sr) to prolate at N = 60 ( Sr) with a twominima structure for heavier isotopes. The cases of Zrand Mo isotopes were discussed in Refs. [37, 38]. Bothoblate and prolate minima are observed in the lighter iso-topes of Zr and Mo with prolate ground states. Whereasthe prolate shape remains ground state in most of theheavier Zr isotopes, oblate shapes are lower in energy forthe heavier Mo isotopes. Finally, Ru and Pd isotopesshow oblate and prolate minima in the lighter isotopesand a gradual transition into spherical shapes as one ap-proaches the shell closure at N = 82.In summary, a large diversity of nuclear structures arefound in this mass region, from spherical to well deformedshapes, passing through soft transitional nuclei and evenpossible shape-coexistence structures. This rich varietyof shapes represents a challenge to any theoretical modeltrying to describe them in a unified manner. In the nextsubsections the results are compared with the availableexperimental data, which are restricted at present to β -decay half-lives. Then, the theoretical approach will betested against this information and the limitations of themodel will be established.These results are in qualitative agreement with sim-ilar calculations obtained in this mass region from dif-ferent theoretical approaches, including macroscopic-microscopic methods based on liquid drop models withshell corrections [54, 55], relativistic mean fields [56], aswell as nonrelativistic calculations with Skyrme [5] andGogny [8, 9, 57] interactions. Thus, a consistent theoret-ical description emerges, which is supported by the stillscarce experimental information available [2, 10–16, 58].The isotopic evolution can be better appreciated inFigs. 2–7, where quadrupole deformations β (a) andr.m.s. charge radii r c (b) of the various energy minimaare plotted as a function of the mass number A . Thedeformation corresponding to the ground state for eachisotope is encircled. Also shown in these figures for odd- A isotopes, are the spin and parity ( J π ) of the differentshapes and the experimental assignments [44]. The ex-perimental assignments based on systematics estimatedfrom trends in neighboring nuclides have not been in-cluded.In Fig. 2 for Ge isotopes one can see clearly the shapetransition at A = 90 − 92 ( N = 58 − 60) from prolateshapes with β ≈ . β ≈ − . β in theprolate and oblate sectors are very similar. Fig. 3 showsthe analogous results for Se isotopes. In this case onecan see the transition from prolate ( β ≈ . 2) to oblate( β ≈ − . 2) at A = 92 ( N = 58). The prolate shapegrows in the heavier isotopes ( β ≈ . β ≈ . 15) to oblate( β ≈ − . 25) and a subsequent transition from oblateto prolate ( β ≈ . 35) shapes. The radii are sensitive tothis transitions, although the measured radii [59] seem tofavored prolate shapes in the lighter isotopes. Sr isotopesin Fig. 5 show a clear transition from oblate to strongprolate ( β ≈ . 4) deformations at A = 96 − 98 ( N =58 − Sr and Sr both theoreticaland experimentally [59]. In the case of Ru (Pd) isotopesshown in Fig. 6 (7), one can see a smooth transition fromdeformed oblate (prolate) solutions in the lighter isotopesto spherical shapes in the heavier ones. This change isfelt in the trends of the radii, but no experiments are yetavailable to compare with.Spins and parities in odd- A isotopes can be comparedwith their experimental assignments. In the Ge isotopesthe calculations agree reasonably well with the assign-ments taking into account that oblate and prolate shapesare very close in energy and that a 1 / + isomer is ob-served experimentally in Ge at 248 keV. In the lighterSe isotopes, 1 / + and 3 / + states are obtained, whereasexperimental assignments are (5 / + ). In both isotopes,5 / + states very close in energy to the ground statesare also obtained, although somewhat above. Similarly,in the lighter Kr isotopes the experimental assignmentsare obtained very close in energy to the ground states,although slightly above. On the other hand, a 7 / + iso-mer is experimentally observed in Kr at 355 keV thatcorresponds to the ground state here. Sr isotopes ex-hibit a nice agreement. The measured spin and paritiesof ground states in , , Sr correspond to the prolatecalculations. A (7 / + ) state is also observed experimen-tally in Sr at 56 keV. In Sr the observed 1 / + ground state appears as an excited state. It is also worth notingthat the prolate ground state (3 / − ) for this isotope isobserved experimentally at 645 keV. In the case of Ru iso-topes the measured J π are difficult to reproduce. Theyare found in the calculations, but not as ground states.On the other hand, the negative parity 7 / − states foundin the calculations are also seen experimentally at low en-ergies. In particular, an isomeric state (7 / − ) at an un-determined energy has been seen in Ru. Finally, in Pdisotopes the negative parity isomers, which are oblate inthis description, are reproduced in the calculations, butnot the ground states. Σ B ( G T ) Σ B ( G T ) ex [MeV]0480 2 4 6 8 10 12 14 16 18E ex [MeV]048 Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Ge Q β S n FIG. 14: (Color online) QRPA-SLy4 accumulated GTstrengths in Ge isotopes calculated for the various equilib-rium shapes. Q β and S n energies are shown by solid anddashed vertical arrows, respectively. B. Gamow-Teller strength distributions In the next figures, the energy distributions of the GTstrength corresponding to the various deformed equilib-rium shapes are shown for each isotopic chain. The re-sults are obtained from QRPA with the force SLy4 with Σ B ( G T ) Σ B ( G T ) ex [MeV]0480 2 4 6 8 10 12 14 16E ex [MeV]048 Se Se Se Se Se Se Se Se Se Se Se Se Se Se Se Q β S n FIG. 15: (Color online) Same as in Fig. 14, but for Se iso-topes. pairing correlations and with residual interactions withthe parameters written in Sec. II. The GT strength isplotted versus the excitation energy of the daughter nu-cleus with a quenching factor 0.77. Zr and Mo isotopeswere already studied in Refs. [37, 38] and are not re-peated here.Figures 8–13 contain the results for Ge, Se, Kr, Sr,Ru, and Pd isotopes. The energy distributions of the in-dividual GT strengths corresponding to the ground stateshapes are shown, together with continuous distributionsfor the ground state shapes as well as for the other possi-ble shapes, obtained by folding the strength with 1 MeVwidth Breit-Wigner functions. Q β values are shown withvertical arrows. In both cases, even and odd isotopes,the Q β values increase with the number of neutrons ineach isotopic chain and the values in the odd- A isotopes( Z, N + 1) are about 2-3 MeV larger than the valuesin the neighbor even-even isotopes ( Z, N ). The generalstructure of the GT distributions is characterized by theexistence of a GT resonance, which is placed at increasingexcitation energy as the number of neutrons N increasesin a given isotopic chain. The total GT strength also Σ B ( G T ) Σ B ( G T ) ex [MeV]048 prolateoblate0 2 4 6 8 10 12 14 16E ex [MeV]048 Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Kr Q β S n FIG. 16: (Color online) Same as in Fig. 14, but for Kr iso-topes. increases with N , as it is expected to fulfill the Ikedasum rule. The various shapes produce quite similar GTstrength distributions on a global scale. Nevertheless, thesmall differences among the various shapes at the low en-ergy tails (below the Q β ) of the GT strength distributionsthat can be appreciated because of the logarithmic scale,lead to sizable effects in the β -decay half-lives.Unfortunately, comparison with experiment is still notpossible for the GT strength distributions, the measuredhalf-lives will be compared to the calculations in the nextsubsection. Comparison with calculated GT distribu-tions from other theoretical approaches is also restrictedto the few cases where these results have been published[33, 35]. In Refs. [33, 60] the authors performed QRPAcalculations with deformed Woods-Saxon potentials andrealistic CD-Bonn residual forces using the G -matrix for-malism and compared these results with the results ob-tained from separable forces. While in Ref. [33] thecomparison between the results obtained from realisticor separable residual interactions is restricted to the half-lives, in Ref. [60] the authors compared those results inthe context of two-neutrino double-beta decay, conclud-0 Σ B ( G T ) Σ B ( G T ) ex [MeV]0240 2 4 6 8 10 12 14E ex [MeV]048 Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Sr Q β S n FIG. 17: (Color online) Same as in Fig. 14, but for Sr iso-topes. ing that both approaches, realistic and separable, leadto similar results. On the other hand, in Ref. [35] theSkyrme force SLy4 was used to generate the mean fieldas it is done in this work. The residual interaction inthe ph channel was self-consistently introduced and notreduced to a separable form. Finally the pp residual in-teraction was written as a contact force with a couplingstrength fitted to reproduce the half-life in Zr. TheGT strength distributions in neutron-rich Zr isotopes ob-tained from this approach were compared with the cor-responding distributions obtained with separable forcesin Figs. 5-6 in Ref. [35]. From this comparison one canconclude that in many aspects the main characteristics ofthe consistent force are maintained by a separable forcewith a much lower computational cost. The compari-son of the half-lives shows also a remarkable agreementbetween both approaches.In the next figures, Figs. 14–19, one can see in moredetail the accumulated GT strength in the energy re-gion below the corresponding Q β energy of each isotope,which is the relevant energy range for the calculation ofthe half-lives. The vertical solid (dashed) arrows show Σ B ( G T ) Σ B ( G T ) ex [MeV]0240 2 4 6 8 10 12E ex [MeV]024 Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Ru Q β S n FIG. 18: (Color online) Same as in Fig. 14, but for Ru iso-topes. the Q β ( S n ) energies, taken from experiment [44]. Inthese figures the sensitivity of these distributions to de-formation can be appreciated and one can understandthat measurements of the GT strength distribution from β -decay can be, in particular cases, an additional sourceof information about the nuclear deformation, as it wasshown in Refs. [49–51]. The GT strength distribution inodd- A isotopes is found to be displaced to higher energies(typically about 2-3 MeV) with respect to the even-evencase. The shift corresponds roughly to the breaking of aneutron pair and therefore it amounts to about twice theneutron pairing gap. Below this energy only transitionsinvolving the odd nucleon are possible.The energy distribution of the GT strength is funda-mental to constrain the underlying nuclear structure. Fora theoretical model, it represents a more demanding testthan just reproducing half-lives or total GT strengthsthat are integral quantities obtained from these strengthdistributions properly weighted with phase factors (seeEq. 6). These quantities might be reproduced even withwrong strength distributions. This is of especial impor-tance in astrophysical scenarios of high densities and tem-1 prolateoblatespherical Σ B ( G T ) Σ B ( G T ) ex [MeV]0120 2 4 6 8 10E ex [MeV]0123 Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Pd Q β S n FIG. 19: (Color online) Same as in Fig. 14, but for Pd iso-topes. peratures that cannot be reproduced in the laboratory.Given that the phase factors in the stellar medium are dif-ferent from those in the laboratory, the stellar half-livesbecome dependent on the electron distribution in thestellar plasma that eventually may block the β -particleemission [61]. Therefore, to describe properly the decayrates under extreme conditions of density and tempera-ture, it is not sufficient to reproduce the half-lives in thelaboratory. One needs, in addition, to have a reliabledescription of the GT strength distributions [62, 63]. C. Beta-decay Half-lives The calculation of the half-lives in Eq. (6) involvesknowledge of the GT strength distribution and of the β energies ( Q β − E ex ), which are evaluated by using Q β values obtained from the mass differences between par-ent and daughter nuclei obtained from SLy4 with a zero-range pairing force and Lipkin-Nogami obtained from thecode HFBTHO [64].In Figs. 20–25 the measured β -decay half-lives (solid 80 82 84 86 88 90 92 94 A -3 -2 -1 T / ( s ) expexp sysprolateoblate Ge FIG. 20: (Color online) Measured β -decay half-lives for Geisotopes compared to theoretical QRPA-SLy4 results calcu-lated from different shapes. Circles are experimental values(open circles are experimental values from systematics) [44]. dots, open dots stand for experimental values from sys-tematics) [23, 44] are compared with the theoretical re-sults obtained with the prolate, oblate, and sphericalequilibrium shapes, for the various isotopic chains. InFig. 20 one can see the half-lives for Ge isotopes. Thelighter isotopes are not well reproduced, being largelyoverestimated. This point will be discussed later. Thehalf-lives obtained from oblate shapes are larger than thecorresponding prolate ones. This feature is correlatedwith the GT strength contained below the Q β energy inFigs. 8 and 14. Prolate shapes, which are closer to exper-iment, are also the ground states in this range of massesaccording to the calculations (see Fig. 2). For heavierisotopes, the half-lives for oblate and prolate shapes arevery similar. In the case of Se isotopes in Fig. 21, thecalculations also overestimate the half-lives of the lighterisotopes, but the agreement with experiment is in thiscase much better. In the middle region the experimentalhalf-lives, which are taken from systematics, are reason-ably well reproduced. The half-lives of heavier isotopesexhibit a rather flat behavior. Half-lives of Kr isotopesare shown in Fig. 22. As in the previous figures, thehalf-lives from the oblate shapes are larger than the pro-late ones in the lighter Kr isotopes, but the situation isreversed at Kr. This is again nicely correlated with the2 86 88 90 92 94 96 98 100 A -3 -2 -1 T / ( s ) expexp sysprolateoblate Se FIG. 21: (Color online) Same as in Fig. 20, but for Se iso-topes. GT strength at low excitation energies shown in Fig. 16.In general, the half-lives in the middle region are well de-scribed. This is also true for Sr and Ru isotopes in Figs.23 and 24, respectively, where the trends observed exper-imentally are well reproduced, except for the lighter Srisotopes that are clearly underestimated and the heavierRu isotopes, where the data from systematics fall downfaster than the calculations. Finally, in the case of Pdisotopes, shown in Fig. 25, the calculations underesti-mate (overestimate) the measured half-lives in the lighter(heavier) isotopes.All in all, the agreement with experiment is reason-able, especially in the middle regions. These regions con-tain in general well deformed nuclei, where the presentapproach is more suitable. On the other hand, weaklydeformed transitional isotopes, such as light Ge and Seisotopes and heavy Ru and Pd isotopes are not so welldescribed. Furthermore, in the light isotopes of all theisotopic chains, which are closer to the valley of stability,the half-lives are larger because of the small Q β energiesinvolved. In these cases the half-lives are determined ex-clusively by the very low energy tail of the GT strengthdistribution contained in the narrow window below Q β .Therefore, tiny variations in the description of the GTstrength distribution in the low-lying energy region candrive sizable effects in the half-lives. Of course it is alsoimportant to describe the half-lives of the long-lived iso- 90 92 94 96 98 100 102 104 A -3 -2 -1 T / ( s ) expexp sysprolateoblate Kr FIG. 22: (Color online) Same as in Fig. 20, but for Kr iso-topes. Experimental half-lives are from [23, 44]. topes, but their significance to constrain the GT strengthdistribution is minor since the half-lives are insensitive tomost of this distribution.Half-lives for neutron-rich Kr, Sr, Zr, and Mo isotopescalculated from self-consistent deformed QRPA calcula-tions with the Gogny D1M interaction and experimentalvalues of Q β [36] agree with the results in this work withinthe uncertainties of the calculations. The agreement isalso very reasonable between the calculated half-lives andthose obtained from deformed QRPA calculations usingdeformed Woods-Saxon potentials to generate the meanfield and complemented with realistic CD-Bonn residualforces [33, 34]. The agreement is also good with the re-sults in Ref. [35] using the Skyrme force SLy4 with con-sistent residual interactions in the ph channel as men-tioned earlier. Fig. 7 in that reference displays this com-parison.It is also worth noticing that the worst agreement withexperiment occurs in the light Ge isotopes, as well asin heavy Pd isotopes. In these cases the calculationsoverestimate the experiment leaving room for contribu-tions coming from first forbidden (FF) transitions. Onecan understand from simple qualitative arguments thatthe role of FF transitions is expected to be more impor-tant in lighter Ge and in Pd isotopes. Thus, for Ge, Se, Kr, and Sr isotopes, the last occupied protonorbitals come basically from the 2 p / , f / and 2 p / 94 96 98 100 102 104 106 108 A -3 -2 -1 T / ( s ) expexp sysprolateoblate Sr FIG. 23: (Color online) Same as in Fig. 20, but for Sr iso-topes. Experimental half-lives are from [23, 44]. negative-parity spherical shells. On the other hand theneutrons in − Ge isotopes occupy orbitals belonging tothe 1 g / , d / and 1 g / positive-parity spherical shells.Therefore, in the β -decay one neutron in a positive-paritystate is transformed into a proton that would sit in anegative-parity state, thus suppressing GT and favoringFF transitions in the low-lying transitions. This is par-ticularly true for the lighter Ge isotopes. In the heavierones, other neutron states with negative parity (1 h / )have to be considered because of deformation effects. Thesame argument can be applied to the lighter Se, Kr, andSr isotopes, but in these cases proton states from positiveparity (1 g / ) are closer in energy and would participatein the decay favoring GT transitions. The situation isdifferent in the case of Ru and Pd isotopes. Now theavailable proton states for the decay are of positive parity(1 g / ), whereas most of the last occupied neutrons be-long to negative-parity states (1 h / ), thus favoring FFtransitions. According to calculations [26, 33] of the FFtransitions in this mass region, minor effects are expectedfrom them. Nevertheless, it would be very interesting inthe future to study systematically the FF contributionsin all the isotopes in this mass region.Another feature observed in the present calculations isthe existence of some odd-even staggering effect in thecalculated half-lives, which is not observed experimen-tally. This effect is particularly evident in Ru and Pd 110 112 114 116 118 120 122 124 A -2 -1 T / ( s ) expexp sysprolateoblatespherical Ru FIG. 24: (Color online) Same as in Fig. 20, but for Ru iso-topes. isotopes. There are not many calculations involving si-multaneously even-even and odd- A isotopes, but some ofthem exhibit some sort of staggering effect as well [33].The appearance of this effect in the half-lives suggestssome deficiency in the model that might be related tothe determination of ground-state energies in the odd- A systems [65]. Unfortunately, there are more sources ofuncertainty related to the odd- A systems that should beconsidered as well [66], such as the spin and parity as-signments, the blocking procedures or the treatment ofthe 1qp excitations involving the odd nucleon. This is-sue will be the subject of a future investigation in thisdirection.It is also interesting to look for the simultaneous ap-pearance of structural effects that eventually can appearin different observables. One example can be seen in theevolution of the experimental half-lives with the numberof neutrons in the isotopic chains. At some points oneobserves discontinuities in the general trends of behavior,such as in the mass regions , Se, , Kr, , Sr, and , Ru. These experimental findings on the half-livesare correlated with the shape transitions in Figs. 3–6 pre-dicted in the model. One cannot state firmly that thesesharp changes in the behavior of the half-lives are signa-tures of shape transitions, but certainly this correlationcannot be discarded given that a change of the deforma-tion in the nuclear system involves a structural change4 114 116 118 120 122 124 126 128 A -2 -1 T / ( s ) expexp sysprolatespherical Pd FIG. 25: (Color online) Same as in Fig. 20, but for Pd iso-topes. to whom the half-lives are also sensitive.Finally, the impact of deformation on the decay prop-erties can be better appreciated in a systematic compar-ison of the half-lives calculated with both the sphericalapproximation and the deformation that corresponds tothe minimum of the PEC for each isotope. Then, Fig.26 shows the ratios of the calculated and experimentalhalf-lives for two sets of data corresponding to a spher-ical calculation (open dots) and to a deformed calcula-tion (solid dots) at the self-consistent deformation thatgives the minimum of the PECs. These ratios are plottedas a function of the experimental half-lives (a) and as afunction of the quadrupole deformation at the minimumof the PECs (b). To increase the size of the sample,besides the isotopes considered in this work with mea-sured half-lives, I have also included the set of Zr andMo neutron-rich isotopes studied in Ref. [38] with mea-sured half-lives. In the upper panel of Fig. 26 (a) onecan see how deformation improves the description of thehalf-lives. Practically all the full black dots are containedwithin the horizontal lines defining the region of one or-der of magnitude agreement. On the other hand, theresults from the spherical calculation are more spreadout with larger discrepancy with experiment. One canalso see that the results are better in both spherical anddeformed calculations for shorter half-lives, whereas theresults for larger half-lives show sizable deviations. The -0.2 -0.1 0 0.1 0.2 0.3 0.4 β -2 -1 T / ( ca l c ) / T / ( e xp ) -2 -1 T (exp) [s]10 -2 -1 T / ( ca l c ) / T / ( e xp ) deformedspherical -2 -1 T (exp) [s]10 -2 -1 T / ( ca l c ) / T / ( e xp ) -0.2 -0.1 0 0.1 0.2 0.3 0.4 β -2 -1 T / ( ca l c ) / T / ( e xp ) deformedspherical (a)(b) FIG. 26: (Color online) Ratio of calculated to experimental β -decay half-lives for two sets of calculations, with the spheri-cal approximation (open dots) and with the deformation thatcorresponds to the minimum of the PECs (solid dots). Theratios are plotted as a function of the experimental half-lives(a) and as a function of the quadrupole deformation at theminimum of the PECs (b). latter correspond to isotopes close to the valley of sta-bility with small Q β -values, where the half-lives are onlysensitive to the small portion of the GT strength distri-bution at low excitation energies below Q β . In the lowerpanel (b) one can see the results from a different point ofview and it can be studied whether deformation improvesthe results evenly in the whole range of deformations orwhether its effect is stronger at large deformations. Threeregions of accumulation of results can be distinguished.Two of them correspond to well deformed nuclei locatedat β ≈ − . β ≈ . 35. In these regions the de-formed calculations clearly improve the results from thespherical ones that show a tendency to underestimate theexperiment. The other region corresponds to 0 < β < . r = log (cid:20) T / (calc) T / (exp) (cid:21) . (9)Then, the average position of the points, M r , the stan-dard deviation, σ r , and the total error, Σ r , are definedas M r = 1 n n X i =1 r i ; σ r = " n n X i =1 ( r i − M r ) / ;Σ r = " n n X i =1 ( r i ) / , (10)and their corresponding factors M r = 10 M r , σ r = 10 σ r ,and Σ r = 10 Σ r . The analysis of the results shown in Fig.26 involving n = 81 nuclei leads to the values M r =1 . σ r = 10 . 21, and Σ r = 10 . 24 in the spherical caseand M r = 0 . σ r = 3 . 09, and Σ r = 3 . 09 in thedeformed one, showing clearly the improvement achievedwith the deformed formalism. IV. CONCLUSIONS A microscopic approach based on a deformed QRPAcalculation on top of a self-consistent mean field ob-tained with the SLy4 Skyrme interaction has been usedto study the nuclear structure and the decay propertiesof even and odd neutron-rich isotopes in the mass region A ≈ − β -decay half-lives have been computedfor the equilibrium shapes.The isotopic evolution of the GT strength distributionsexhibits some typical features, such as GT resonances in-creasing in energy and strength as the number of neu-trons increases. Effects of deformation are hard to seeon a global scale, but they become apparent in the lowexcitation energy below Q β energies, a region that deter-mines the half-lives. Half-lives have been calculated using Q β energies calculated with the force SLy4. In general,a reasonable agreement with experiment is obtained, es-pecially in the short-lived nuclei of Ge, Se, Kr, Sr, andRu isotopes. The results are comparable to other calcula-tions using different approaches for the mean field and/orresidual interactions. Special difficulties are found to de-scribe properly the half-lives of the lighter Ge isotopesand the Pd isotopes. These are examples of transitionalnuclei where the nuclear structure is more involved andthe concept of a well defined shape might not be mean-ingful.A systematic comparison of the ratios of the calcu-lated and experimental half-lives has been done usingboth spherical and deformed calculations, showing thatthe inclusion of deformation improves significantly thedescription of the decay properties.Experimental information on the energy distribution ofthe GT strength is a valuable piece of knowledge aboutnuclear structure in this mass region. The study of thesedistributions is within the current experimental capabil-ities in the case of the lighter isotopes considered in thiswork. Here, I have presented theoretical predictions forthem based on microscopic calculations. Similarly, mea-suring the half-lives of the heavier isotopes will be highlybeneficial to model the r process and to constrain theoret-ical nuclear models. This possibility is also open withinpresent capabilities at RIKEN. 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