Gamow-Teller transitions of neutron-rich N=82,81 nuclei by shell-model calculations
aa r X i v : . [ nu c l - t h ] F e b Gamow-Teller transitions of neutron-rich N = 82 , nuclei by shell-model calculations Noritaka Shimizu , ∗ , Tomoaki Togashi , and Yutaka Utsuno , Center for Nuclear Study, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan and Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan β -decay half-lives of neutron-rich nuclei around N = 82 are key data to understand the r -processnucleosynthesis. We performed large-scale shell-model calculations in this region using a newlyconstructed shell-model Hamiltonian, and successfully described the low-lying spectra and half-livesof neutron-rich N = 82 and N = 81 isotones with Z = 42 −
49 in a unified way. We found that theirGamow-Teller strength distributions have a peak in the low-excitation energies, which significantlycontributes to the half-lives. This peak, dominated by ν g / → π g / transitions, is enhanced onthe proton deficient side because the Pauli-blocking effect caused by occupying the valence proton0 g / orbit is weakened. I. INTRODUCTION
The solar system abundances and their peak structures indicate that major origin of most elements heavier thaniron is generated by the r -process nucleosynthesis [1]. A neutron-star merger was found by measuring the gravitationalwave which is followed by optical emission, called “kilonova” [2]. The properties of neutron-rich nuclei are key issuesto reveal the r -process nucleosynthesis which is expected to occur in kilonova phenomena.The r -process path is considered to go through the neutron-rich region of the nuclear chart. In the region wherethe r -process path comes across the magic number N = 82, these nuclei form the waiting points of neutron capturesin the r -process. The path comes along the N = 82 line in the chart bringing about the so-called second peak of thenatural abundance formed by the astrophysical r -process nucleosynthesis. In a typical r -process model, after reachingthe Sr ( Z = 38, N = 82), the β -decay and the neutron capture are repeated alternately to generate N = 82 and N = 81 nuclei up to Pd ( Z = 46, N = 82) [3]. This repeated process occurs if the β -decay rates of N = 81 aresmaller than their neutron-capture rates. Thus, the β -decay properties not only of the N = 82 isotones but also ofthe N = 81 ones are necessary to determine the r -process path, hence motivating the study of those very neutron-richnuclei from the viewpoint of nuclear structure physics. Note that the properties of nuclei near N = 82 are also awaitedin the context of fission recycling [4].On the experimental side, β -decay half-lives of neutrino-rich nuclei around N = 82 have recently been measuredby the EURICA campaign conducted at the RI Beam Factory in RIKEN Nishina Center [5, 6]. More detailed dataare now available for some nuclei. Many isomers have been identified near the N = 82 shell gap, and some of theirhalf-lives are obtained [7–10]. Furthermore, β -delayed neutron-emission probabilities and low-lying level structurehave been measured [11, 12]. These data provide a stringent test for nuclear-structure models. It should be notedthat similar experimental activities are extended to the N = 126 region, known as the third peak of the solar systemabundance, for instance by the KISS (KEK Isotope Separator System) project [13].Many theoretical efforts have also been paid to systematically calculate β -decay half-lives such as by FRDM [14],FRDM-QRPA [15], HFB-QRPA [16], DFT-QRPA [17, 18], and the gross theory [19]. Recently, further sophisticatedmethods were introduced into the systematic β -decay studies by introducing the FAM-QRPA [20] and by the rel-ativistic CDFT-QRPA [21]. Novel machine-learning techniques were also applied to predict β -decay half-lives [22].The nuclear shell-model calculation is also one of the most powerful theoretical schemes for this purpose. The previ-ous shell-model studies are, however, restricted to calculating the half-lives of the singly-magic N = 82 [23–25] and N = 126 isotones [23, 26] due to the exponentially increasing dimensions of the Hamiltonian matrices in open-shellnuclei. The present work aims to extend those previous shell-model efforts to N = 81 isotones within a unified de-scription of the structures of neutron-rich N = 82 and N = 81 isotones. The measured half-lives are well reproducedby the calculation, and we predict those for , Ru, , Tc and
Mo. It is also predicted that these nuclei haverather strong GT strengths in the low excitation energies due to the increasing number of proton holes in the g / orbit, accelerating GT decay. This paper is organized as follows. The shell-model model space and its interaction aredefined in Sect. II. Section III is devoted to the separation energies and low-lying spectra. The Gamow-Teller strengthdistribution and the half-lives are discussed in Sect. IV. Section V is devoted to the discussion of the enhancement ofthe GT transitions towards the proton-deficient nuclei and of its origin. This paper is summarized in Sect. VI. II. FRAMEWORK OF SHELL-MODEL CALCULATIONS
We performed large-scale shell-model calculations of N = 81 and N = 82 isotones. The model space for thecalculations is taken as 0 f / , 1 p / , 1 p / , 0 g / , 0 g / , 1 d / , 1 d / , 2 s / , and 0 h / for the proton orbits and0 g / , 1 d / , 1 d / , 2 s / , and 0 h / for the neutrons orbits with the Ni inert core. These orbits are shown inFig. 1. Although we focus on Z ≤
50 nuclei in this study, the single-particle orbits beyond the Z = 50 shell gap arerequired to be included in the model space explicitly so that the Gamow-Teller transition causes the single-particletransition of the valence neutrons beyond N = 50 to the same orbits and its spin-orbit partners. The model spaceis extended from that of the earlier shell-model study [23] by adding the proton 0 f / , 1 p / , and 0 h / orbits. Inthe preceding shell-model works [23, 24], the proton 0 h / orbit was omitted to avoid the contamination of thespurious center-of-mass excitation, although the neutron occupying 0 h / orbit can decay to the proton occupiedin 0 h / by the Gamow-Teller transition. In the present work, we explicitly include the proton 0 h / orbits intothe model space so that the proton single-particle orbits cover the whole neutron orbits. For fully satisfying theGamow-Teller sum rule the proton 0 h / orbit is required, but its single-particle energy is too high to significantlyaffect the Gamow-Teller strength of the low-lying states and it is omitted in the present work. The contaminationof spurious center-of-mass excitation is removed by the Lawson method [27] with β CM ~ ω/A = 10 MeV. We truncatethe model space by restricting up to 2 proton holes in pf shell and up to 3 protons occupying the orbitals beyond the Z = 50 gap so that the numerical calculation is feasible. Even if applying such a truncation the M -scheme dimensionof the shell-model Hamiltonian matrix reaches 3 . × and is quite large, and efficient usage of a supercomputer isessential. The shell-model calculations were mainly performed on CX400 supercomputer at Nagoya University andOakforest-PACS at The University of Tokyo and University of Tsukuba utilizing the KSHELL shell-model code [28],which has been developed for massively parallel computation. -20-15-10 SPE ( M e V ) proton neutron N=50Z=500f5/21p3/21p1/2 0g9/20g7/21d5/21d3/2 2s1/20h11/2 0g7/21d5/21d3/22s1/2 0h11/2N=82Z=82Z=28
FIG. 1: Single-particle energies for
Sn determined from the experimental energy levels of its one-particle and one-holeneighboring nuclei [29–34]. The single-particle orbits taken as the model space are shown.
The effective realistic interaction for the shell-model calculation is constructed mainly by combining the two estab-lished realistic interactions: the JUN45 interaction [35] for the f pg model space and the SNBG3 interaction [36] forthe neutron model space of 50 < N, Z <
82. The JUN45 and SNBG3 interactions were constructed from the G-matrixinteraction with phenomenological corrections using a chi-square fit to reproduce experimental energies. For the restpart of the two-body matrix elements (TBMEs), we adopt the monopole-based universal ( V MU ) interaction [37] whose T = 1 central force is scaled by the factor 0.75 in the same way as Ref. [38]. The single particle energies are determinedto reproduce the experimental energies of one-nucleon neighboring nuclei of Sn as shown in Fig. 1. In addition, thestrengths of the pairing interaction and the diagonal TBMEs of the ( π g / , π g / ) and ( π g / , ν h / ) orbits aremodified to reproduce the experimental energy levels of Cd,
Pd, and
In. The TBMEs are assumed to havethe mass dependence ( A/ − . .
43 44 45 46 47 48 49 50 5101020
N=82 S n , S p ( M e V ) SnInCdAg SbPd S p S n RhRuTc
FIG. 2: Separation energies of N = 82 isotones. The solid lines show the proton and neutron separation energies provided bythe present shell-model study. The filled circles and the open triangles with error bars denote the experimental values and theextrapolated values from the experimental systematics, respectively [39]. The dotted lines and the dashed lines are given bythe KTUY mass formula [40] and the FRDM [14].
43 44 45 46 47 48 49 5001020 S n , S p ( M e V ) Pd SnInCdAg S p S n RhRuTc
N=81
FIG. 3: Separation energies of N = 81 isotones. See the caption of Fig. 2 for details. III. SEPARATION ENERGIES AND EXCITATION ENERGIES
The binding energies and excitation energies of the N = 82 nuclei and those around them are important not onlyfor describing the β -decay properties, but also for confirming the validity of the shell-model interaction. Figures2 and 3 show the proton and neutron separation energies of the N = 82 and N = 81 isotones, respectively. Thepresent shell-model results reproduce the experimental values excellently. The neutron separation energy determinesthe threshold energy of the β -delayed neutron emission, which is important for the r -process nucleosynthesis. Sincethe Q -value of the β − decay is obtained using the proton and neutron separation energies as Q ( β − , Z, N ) = BE ( Z + 1 , N − − BE ( Z, N ) + ( m n − m p − m e ) c = S p ( Z + 1 , N ) − S n ( Z + 1 , N ) + 0 .
782 MeV , (1)where BE ( Z, N ) denotes the binding energy of the (
Z, N ) nucleus and 0.782 MeV is obtained from the mass differenceof a neutron, a proton and an electron. The Q values of β -decay given by the shell-model results are in good agreementwith the available experimental values, as shown as the difference of S n and S p in Figs. 2 and 3. For comparison,the result of the KTUY [40] and the FRDM [14] mass formulae are also plotted in the figures, showing very goodagreement with the experimental values except slight underestimation in the proton separation energy of In. On theproton deficient side where the experimental values are not available, the difference among the theoretical predictionsgradually increases as the proton number decreases, while the neutron separation energies of the N = 82 isotones arerather close to one another.Figure 4 shows low-lying energy levels in the neutron-rich N = 82 isotones from Z = 42 to Z = 50. For the nucleiwithout data, we plot a few lowest levels obtained by the calculation. The calculated ground states are 0 + for theeven- Z isotopes and 9 / + for the odd- Z isotopes. The experimental levels are reproduced excellently by the shell-model results. The levels of Ag are experimentally unknown, but two β -decaying states were found and tentatively SM Exp. SM Exp. SM Exp. SM Exp.SM Exp.7/2+5/2+1/2+3/2+11/2-133Sb 9/2+1/2-3/2-5/2-131In 130Cd 129Ag 128Pd7/2+5/2+3/2+11/2- 9/2+1/2-3/2-5/2- 0+2+4+6+8+ 9/2+1/2-0+2+4+ 6+8+ 0+2+4+6+8+ 0+2+4+6+8+ E x ( M e V ) FIG. 4: Excitation energies of N = 82 isotones: Sb,
In,
Cd,
Ag,
Pd,
Rh,
Ru,
Tc, and
Mo comparedbetween the shell model (SM) and experiment (Exp.) [34].
SM Exp. SM Exp.3/2+ 131Sn 130In 129Cd E x ( M e V ) FIG. 5: Excitation energies of N = 81 isotones: Sn,
In, and
Cd,
Ag,
Pd,
Rh,
Ru, and
Tc. See thecaption of Fig. 4 for details. assigned as 9 / + and 1 / − [34] without their excitation energies known. In the present calculation, the 1 / − state islocated very close to the 7 / + state. Considering a long E / − state predominantly decays through β emission.Figure 5 shows the excitation spectra of the N = 81 isotones. Unlike the N = 82 isotones, several candidates forthe ground state and some β -decaying isomers are predicted. This is partly because the 1 d / and the 0 h / neutronorbits are located very close in energy as known in the spectra of Sn and the difference of their spin numbers islarge. For
Cd, two β -decaying states with 11 / − and 3 / + were known and their order had been controversial [7].A recent experiment concluded that its ground-state spin is 11 / − and the excitation energy of 3 / + is 343(8) keV[10, 11], which is consistent with our shell-model prediction. For Pd, no experimental energy levels are known,and the present order of 11 / − and 3 / + agrees with another shell-model prediction [45]. With regard to β -decayproperties, the excitation energy of the 1 + state of In plays a crucial role in the β -decay half-life of Cd [24],whose 0 + ground state decays to the lowest 1 + state most strongly with the Gamow-Teller transition.Figure 6 shows the calculated energy levels of the N = 80 and N = 79 isotones for which the experimental dataare available. We confirm a reasonable agreement between them.The present calculation reproduces the experimental energies quite well, thus confirming the validity of the modelspace and the effective interaction employed in the present shell-model calculation. SM Exp. SM Exp. SM Exp. SM Exp.0+2+130Sn 9/2+1/2-129In 128Cd 129Sn9/2+1/2- 0+ 0+ E x ( M e V ) FIG. 6: Excitation energies of the N = 80 isotones ( Sn,
In, and
Cd) and the N = 79 isotones ( Sn and
In). Theexperimental values are taken from Refs. [11, 34]. See the caption of Fig. 4 for details. (a)
In (b)
Cd(c)
Ag (d)
PdEx (MeV) Ex (MeV) Q (cid:1) Q (cid:0) Q (cid:2) Q (cid:3) G T s t r eng t h G T s t r eng t h S n S n S n S n FIG. 7: Gamow-Teller strength functions of N = 82 isotones, (a) In, (b)
Cd, (c)
Ag, and (d)
Pd against theexcitation energies of the daughter nuclei. The dashed lines are the folded strength functions by a Lorentzian function withthe 1-MeV width. The values are shown without the quenching factor. The Q β values and the neutron separation energies areshown as the red dotted lines and the blue dotted lines, respectively. IV. GAMOW-TELLER STRENGTH FUNCTION AND β − -DECAY HALF-LIVES We calculated the Gamow-Teller β − -strength functions for N = 82 and N = 81 neutron-rich nuclei to estimate theirhalf-lives. We adopted the Lanczos strength function method [41–43] with 250 Lanczos iterations to obtain sufficientlyconverged results. The magnitude of quenching of axial vector coupling is still a challenging topic for nuclear physicsand has large uncertainty mainly caused by nuclear medium effect and many-body correlations. In the present work,the quenching factor is taken as q GT = 0 .
7, which has been most widely used [26, 44] and is consistent with theadopted value of the preceding work, q GT = 0 .
71 [24]. The first-forbidden transition is omitted in the present workbecause its contribution to the half-lives is small, around 13%, and rather independent of nuclides for the Z = 42 − N = 82 isotones in a previous shell-model study [23]. Furthermore, it is pointed out in [11] that a number of allowedtransitions are observed in the β − decays of − In and − Cd, suggesting the dominance of GT transitions inthe low excitation energies. This point will be discussed later.Figure 7 shows the Gamow-Teller distributions of N = 82 isotones, In ( Z = 49), Cd ( Z = 48), Ag ( Z = 47),and Pd ( Z = 46). Figure 8 shows those of more proton-deficient N = 82 isotones, Rh ( Z = 45), Ru ( Z = 44), Tc ( Z = 43), and Mo ( Z = 42). The Q -values are taken from the experiments for In and
Cd [34], while thepresent theoretical Q -values are used for the other nuclei. These figures present a very remarkable systematics of low- (a) Rh (b)
Ru(c)
Tc (d)
MoEx (MeV) Ex (MeV) Q β Q β Q β Q β G T s t r eng t h G T s t r eng t h S n S n S n S n FIG. 8: Gamow-Teller strength functions of proton-deficient N = 82 isotones, (a) Rh, (b)
Ru, (c)
Tc, and (d)
Mo.See the caption of Fig. 7 for details. T / (ms), N = 82 SM th SM exp SM13 SM07 SM99 Exp15 Exp16 In → Sn 156 154 247.53 260 177 261(3) 265(8) Cd → In 158 116 164.29 162 146 127(2) 126(4) Ag → Cd 44 69.81 70 35.1 52(4) Pd → Ag 28 47.25 46 27.3 35(3) Rh → Pd 13.9 27.98 27.65 11.8 20 +20 − Ru → Rh 9.2 20.33 19.76 9.6 Tc → Ru 5.7 9.52 9.44 4.3 Mo → Tc 4.0 6.21 6.13 3.5TABLE I: β -decay half-lives of the N = 82 isotones by the present shell-model calculations (SM), the shell-model study with theexperimental Q value (SM exp ), the earlier shell-model works (SM13 [23]), (SM07 [24]), (SM99 [25]), and the recent experiments(Exp15) [6], (Exp16) [7]. The half-lives are shown in the unit of ms. energy GT strength distributions, which play a crucial role in those β -decay half-lives. First, all the N = 82 isotonesconsidered here have strong GT strengths in the low-excitation energies. Except In, they are peaked at ∼ . ∼ Z and even- Z parents, respectively, and the GT strengths are more concentrated for theeven- Z isotopes. This odd-even effect is in accordance with what is found in the sd - pf shell region [48]. Second, thislow-energy GT peak grows with decreasing proton number. This is an interesting feature of low-energy Gamow-Tellertransitions predicted for this region, and more detailed discussions will be given in Sec. V.Table I shows the β -decay half-lives of the N = 82 isotones. The half-life is estimated by accumulating the transitionprobabilities from the parent ground state to the daughter states whose excitation energies are below the Q β value.The shell-model results show reasonable agreement with the experimental values. While the present half-lives of Ag and
Pd are closer to the experimental values than the earlier shell-model result, the half-life of
In isunderestimated. This underestimation is caused by the large GT transition to the lowest 7 / + state of the daughter Sn at E x = 2 . ν g / -hole state of Sn. In the pure π g − / → ν g − / single-particle transition,the corresponding B (GT) value is as much as 1.78 without the quenching factor introduced. On the other hand,the present calculation gives B (GT) = 0 .
58. This value is considerably reduced from the single-particle value due toconfiguration mixing, but further reduction is required to completely reproduce the data.For comparison, Table I also shows three shell-model results by the Strasbourg group: SM13 [23], SM07 [24], andSM99 [25]. The half-lives of
Ru,
Tc, and
Mo predicted by the present calculation are close to those of SM99[25]. The half-lives of SM13 [23] and SM07 [24] are quite close to each other. While the first-forbidden transition wasomitted and the quenching factor of the Gamow-Teller transition was taken as q GT = 0 .
71 in SM07, the first-forbiddentransition is included with q GT = 0 .
66 in SM13. The agreement of these results indicates that the contribution ofthe first-forbidden decay is rather independent of the nuclides and can be absorbed into the minor change of the
42 43 44 45 46 47 48 49050100
Proton number P n ( % ) FIG. 9: Neutron emission probabilities of N = 82 isotones. The blue filled circles, black open squares, and black open trianglesdenote the results by the present work, the earlier shell-model work [23], and the FRDM+QRPA [15], respectively. The reddiamond denotes the experimental value and the red line with an arrow at Z = 47 denotes the experimental upper limit [46, 47]. T / (ms), N = 81 SM th SM exp Exp15 Exp16 In → Sn 286 311 284(10) Cd → In 182 139 154.5(20) 147(3)
Cd (
32 + ) → In 266 181 157(8) Ag → Cd 49 59(5) Pd → Ag 32 38(2) Rh → Pd 17 19(3) Ru → Rh 11 Tc → Ru 7.0TABLE II: β -decay half-lives of the N = 81 isotones obtained by the present shell-model study (SM th ), the shell-model studywith the experimental Q value (SM exp ), and the experiments (Exp15) [6], (Exp16) [7]. The half-lives are shown in the unit ofms. The half-life of the 3 / + isomeric state of Cd is also shown.
Gamow-Teller quenching factor in this mass region. β -delayed neutron emission is important for understanding the freezeout of the r process [1]. Figure 9 show β -delayedneutron emission probabilities P n for N = 82 nuclei. In the present calculation, we accumulate the probabilities ofthe β -decay to the states above the neutron-emission threshold S n to obtain P n . The present shell-model results showan odd-even staggering similar to that of the earlier shell model [23], while the FRDM-QRPA results show weakerodd-even staggering. This odd-even staggering is caused by the difference of the peak position and the degree ofconcentration of the Gamow-Teller transition strengths. As discussed already using Figs. 7 and 8, the GT peaks ofthe even- Z parent nuclei are located at around E x = 2 MeV, which is lower than S n , causing their small P n values.For Mo, it is predicted that this low-energy GT strength is concentrated by a single peak that is located slightlybelow S n . Hence its P n is very sensitive to the detail of the energies concerned. For the odd- Z nuclei of Rh and
Tc, the low-energy GT peak is located higher than S n , enlarging their P n values.Figures 10 and 11 show the Gamow-Teller β − -strength distribution of N = 81 isotones, namely In,
Cd,
Ag,
Pd,
Rh,
Ru and
Tc obtained by the present shell-model calculations. Figures 11 shows also the distributionof the isomeric 3 / + state of Cd. The Q ( β − ) values are taken from experiments for In and
Cd [34], andtaken from shell-model values for the other nuclei. The low-energy GT peaks are obtained in all the cases calculated.They are located higher for the odd- Z parents due to pairing correlation in the daughter nuclei, but fragmented in asimilar manner. Like the case of the N = 82 isotones, those peaks are enhanced as the proton number decreases andthe proton 0 g / orbit becomes unoccupied.Table II shows the β -decay half-lives of the N = 81 isotones. The half-lives of the five nuclei with Z ≥ / + isomeric state of Cd is also shown in the table to demonstrate the capabilityto obtain the β -decay rates of isomeric states.In Tables I and II, SM th and SM exp show the shell-model results using the shell-model Q value and those using theexperimental Q value, respectively, to discuss the uncertainty of the present theoretical model. The deviations of thechoice of the Q values show up to 30% at most. The fitted quenching factor to reproduce the experimentally measuredhalf-lives of Cd,
Cd and the 3/2 + isomer by the SM exp result is q GT = 0 .
67, which shows a 9% increase of the (a)
In (b)
Cd(c)
Ag (d)
PdEx (MeV) Ex (MeV) Q β Q β Q β Q β G T s t r eng t h G T s t r eng t h FIG. 10: Gamow-Teller strength functions of N = 81 isotones, (a) In, (b)
Cd, (c)
Ag, and (d)
Pd. See the captionof Fig. 7 for details. (a)
Rh (b)
Ru(c)
Tc Ex (MeV) Ex (MeV) Q β Q β Q β G T s t r eng t h G T s t r eng t h Q β (d) 3/2 + Cd FIG. 11: Gamow-Teller strength functions of N = 81 isotones, (a) Rh, (b)
Ru, (c)
Tc, and (d) the isomeric 3 / + state of Cd. See the caption of Fig. 7 for details. half-life estimate. These differences are considered as the uncertainties of the present model.
V. POSSIBLE OCCURRENCE OF SUPERALLOWED GAMOW-TELLER TRANSITIONS TOWARD Z = 40 As mentioned in the last section, Figures 7 and 8 show that for the even- Z parents a low-energy Gamow-Teller peakemerges at ∼ Zr with Z = 40, leading to B (GT) = 2 . .
7. In this section, we focus on this growing Gamow-Teller peak toward Z = 40.At first, we discuss why this peak is enlarged with decreasing Z . By analyzing one-body transition densitiesobtained in the present calculations, one can see that those low-energy Gamow-Teller peaks are dominated by the ν g / → π g / transition. If the π g / orbit is completely filled, this transition does not occur due to the Pauli G T s t r eng t h Z(cid:4) E x (MeV) FIG. 12: Gamow-Teller strength functions of
Zr. See the caption of Fig. 7 for details. blocking. This blocking effect is weakened by removing protons from the π g / orbit, hence the enlargement of thelow-energy Gamow-Teller peak.The resulting B (GT) values of this peak are particularly large at Mo and
Zr compared to typical values. Itis known from the systematics [49] that the log f t values of allowed β decays are distributed around log f t ∼
6, whichcorresponds to B (GT) ∼ − -10 − for Gamow-Teller transitions. A well-known deviation from this systematicsis the superallowed (Fermi) transition. When isospin is a good quantum number, the Fermi transition occurs onlybetween isobaric analog states, giving a typical log f t of 3.5. With regard to Gamow-Teller transitions, however, thereare only a few cases where the log f t value is comparable to those of the superallowed Fermi transitions because ofthe fragmentation of Gamow-Teller strengths. Since B (GT) = 1 leads to log f t = 3 .
58, the B (GT) value of the orderof unity is a good criterion to compare the superallowed Fermi transition.It is proposed in [50] that such extraordinarily fast Gamow-Teller transitions be classified as Super Gamow-Tellertransitions. At that time, only two Gamow-Teller transitions, He → Li and Ne → F, were known to satisfy thecondition of Super Gamow-Teller transition defined in [50], i.e., B (GT) >
3. These large Gamow-Teller strengthsare caused by the constructive interference of j > → j > and j > → j < matrix elements [51]. It was also predictedin [50] that two N = Z doubly-magic nuclei Ni and
Sn were candidates for nuclei causing Super Gamow-Tellertransitions. Although the Gamow-Teller strengths from Ni were measured to be fragmented about a decade later[52],
Sn is now established to have a very large B (GT) value (9 . +3 . − . in [53] or 4 . +0 . − . in [54]) to a 1 + state locatedat around 3 MeV. This Gamow-Teller decay is called “superallowed Gamow-Teller” decay in [53] on the analogy ofthe superallowed Fermi decay.The B (GT) values predicted for Mo and
Zr in the present study are the order of unity, although not reach-ing the measured value of
Sn. Thus, they are new candidates for the superallowed Gamow-Teller transitions.Interestingly, those two regions of superallowed Gamow-Teller transition share the same underlying mechanism. Inthe extreme single-particle picture, the π g / orbit is completely filled and the ν g / orbit is completely empty in Sn. Since the former and the latter orbits are the highest occupied and the lowest unoccupied ones, respectively,its low-energy Gamow-Teller transition is caused by the π g / → ν g / transition. On the other hand, in Zr, the ν g / orbit is completely filled and the π g / orbit is completely empty. As for the order of single-particle levels,Fig. 13 shows the evolution of the effective single-particle energies of N = 82 isotones as a function of Z . For protons,the π g / orbit keeps the lowest unoccupied orbit in this range. For neutrons, although the ν g / orbit is the lowestat Z = 50 among the five orbits of interest, it goes up higher with decreasing Z to finally be the second highest at Z = 40. This is caused by a particularly strong attractive monopole interaction between π g / and ν g / due to acooperative attraction of the central and the tensor forces [37]. This sharp change of the ν g / orbit in going from Z = 40 to 50 is established from the energy levels of Zr and
Sn, as mentioned in [37]. In
Zr, the ν g / orbit isthus close to the highest occupied level, making a low-energy Gamow-Teller state by the ν g / → π g / transition.If one is restricted to the configuration most relevant to the low-energy Gamow-Teller transition, the final state of the Zr decay, ( ν g / ) − ( π g / ) +1 , is the particle-hole conjugation of that of the Sn decay, ( π g / ) − ( ν g / ) +1 .A schematic illustration of these configurations are given in Fig. 14. Accordingly, the B (GT) values from the vacuumto these single-particle configurations, i.e., those of Fig. 14(a) and (b), are identical.One of the important ingredients for making B (GT) large in those nuclei is that the B (GT) value obtained within0 Proton number N eu t r on ESPE ( M e V ) FIG. 13: Effective single-particle energies of the N = 82 isotones for neutron orbits (upper) and proton orbits (lower) as afunction of the proton number calculated with the Hamiltonian used in this study. π g ? g @ g π g (a) Sn A In(b) Zr B Nb FIG. 14: Schematic illustration of the dominant single-particle transition in (a) the β + decay of Sn and (b) the β − decayof Zr. The filled and open circles denote particles and holes, respectively. the single configuration of Fig. 14 (a) [and (b)] is also large. To be more specific, let us compare two cases as theinitial state, (i) | ( π g / ) ; J = 0 i and (ii) | ( π g / ) ; J = 0 i , where one proton can move to the ν g / orbit throughthe Gamow-Teller transition. The case (i) corresponds to Fig. 14(a) and yields B (GT)=17.78 (without the quenchingfactor), whereas the case (ii) gives B (GT)=3.56. The ratio of these two B (GT) values, 10 to 2, is just that of thenumber of protons in the initial state. This proportionality is well understood by remembering the Ikeda sum rule.Although the B (GT) value in the extreme single-particle picture is as large as 17.78 for the configurations of Figs. 14(a) and (b), it is reduced in reality by the quenching factor and fragmentation over other excited states. To minimizefragmentation, it is desirable to suppress the level density with the same J π near the state of interest. Sn and
Zr are doubly-magic (or semi-magic) nuclei, thus having a favorable condition for that. Another important factorto affect level density is excitation energy. As presented in Figs. 7, 11 and 12, the low-energy Gamow-Teller peakis located stably at around 2 MeV by changing Z . This excitation energy is low enough to isolate the peak, if oneremembers that the superallowed Gamow-Teller state from the Sn decay is located at ∼ N = 82 shell gap. Since these excitations typically1 J -2-1012 m a t r i x e l e m en t ( M e V ) hole-holeparticle-hole FIG. 15: Hamiltonian matrix elements concerning the π g / and ν g / orbits used in this study. The circles and the squaresare the hole-hole matrix elements, h ( π g / ) − ( ν g / ) − | V | ( π g / ) − ( ν g / ) − i J and the particle-hole matrix elements, h π g / ( ν g / ) − | V | π g / ( ν g / ) − i J , respectively. cost more than 4 MeV by estimating from the first excitation energy of Sn, they probably do not contribute muchto fragmentation.One may wonder why the low-energy Gamow-Teller peak is kept at E x ∼ Z = 48 to Z = 40 in spite ofthe sharp change of the ν g / energy as shown in Fig. 13. This is due to the nature of two-body Hamiltonian matrixelements. The low-energy Gamow-Teller state has always a neutron hole in 0 g / . For the nuclei close to Z = 50, thisstate has a few proton holes in 0 g / , and thus its excitation energy is dominated by the hole-hole matrix element h ( π g / ) − ( ν g / ) − | V | ( π g / ) − ( ν g / ) − i J =1 as well as the single-particle energy of ν g / . As presented inFig. 15, this matrix element is the most attractive among the possible J values. Hence the low-energy Gamow-Tellerstate is located lower than the simple estimate that the 0 g / orbit lies ∼ Z = 50(see Fig. 13).This situation changes as more protons are removed from the π g / orbit. For the nuclei close to Z = 40, thenumber of particles are smaller than the number of holes in the 0 g / orbit, and the particle-hole matrix element h π g / ( ν g / ) − | V | π g / ( ν g / ) − i J plays a dominant role. In Fig. 15, we also show the particle-hole matrixelements that are derived from the hole-hole matrix elements by using the Pandya transformation. The J = 1 coupledmatrix element has the largest positive value, thus losing the largest energy. This explains the calculated result thatthe low-energy Gamow-Teller state is not drastically lowered toward Z = 40 as expected from the evolution of the ν g / orbit, and also the observation that the corresponding state for the Sn decay is located at ∼ J dependence is an example of the parabolic rule that holds for short-range attractiveforces [55].To briefly summarize this section, the predicted superallowed Gamow-Teller transition toward Z = 40 occurs dueto (a) the full occupation of a neutron high- j orbit ( ν g / in this case) and the emptiness of its proton spin-orbitpartner ( π g / in this case) and (b) the low excitation energy of the J = 1 particle-hole state created by these twoorbits. Since the J = 1 proton-neutron particle-hole matrix elements are generally most repulsive among possible J ,it is needed to fulfill (b) that the ν g / orbit and the π g / orbit are closed to the highest occupied orbit and thelowest unoccupied orbit, respectively. The tensor-force driven shell evolution plays a crucial role in satisfying thiscondition. VI. SUMMARY
We have constructed a shell-model effective interaction and performed large-scale shell-model calculations ofneutron-rich N = 82 and N = 81 nuclei by utilizing our developed shell-model code and the state-of-the-art su-percomputers. We demonstrated that the experimental binding and excitation energies of neutron-rich N = 79 , , β -decay half-lives of N = 82 and N = 81 nuclei, which arereasonably consistent with the available experimental data, and several predictions for further proton-deficient nuclei.In these isotones, as the proton number decreases from Z = 49 to Z = 42, the proton 0 g / orbit becomes unoccupied2and the Gamow-Teller strengths of the low-lying states increases because of the Pauli-blocking effect. We predict thatthe low-energy Gamow-Teller strength is further enlarged in Zr to make its log f t value equivalent to that of thesuperallowed beta decay. This is quite an analogous case to the so-called “superallowed Gamow-Teller” transitionobserved in
Sn in terms of Gamow-Teller strength and underlying mechanism.In the present work, we assume the contribution of the first-forbidden transition is independent of the nuclides andcan be absorbed into a single quenching factor of the Gamow-Teller transition. Further investigation to estimate thefirst-forbidden decay especially for the N = 81 isotones is also expected. Acknowledgment
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