Gamow-Teller transitions of neutron-rich $N=82,81$ nuclei by shell-model calculations

$\beta$-decay half-lives of neutron-rich nuclei around $N=82$ are key data to understand the $r$-process nucleosynthesis. We performed large-scale shell-model calculations in this region using a newly constructed shell-model Hamiltonian, and successfully described the low-lying spectra and half-lives of neutron-rich $N=82$ and $N=81$ isotones with $Z=42-49$ in a unified way. We found that their Gamow-Teller strength distributions have a peak in the low-excitation energies, which significantly contributes to the half-lives. This peak, dominated by $\nu 0g_{7/2} \to \pi 0g_{9/2}$ transitions, is enhanced on the proton deficient side because the Pauli-blocking effect caused by occupying the valence proton $0g_{9/2}$ orbit is weakened.


I. INTRODUCTION
The solar system abundances and their peak structures indicate that major origin of most elements heavier than iron is generated by the r-process nucleosynthesis [1]. A neutron-star merger was found by measuring the gravitational wave which is followed by optical emission, called "kilonova" [2]. The properties of neutron-rich nuclei are key issues to 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 where the r-process path comes across the magic number N = 82, these nuclei form the waiting points of neutron captures in the r-process. The path comes along the N = 82 line in the chart bringing about the so-called second peak of the natural abundance formed by the astrophysical r-process nucleosynthesis. In a typical r-process model, after reaching the 120 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 128 Pd (Z = 46, N = 82) [3]. This repeated process occurs if the β-decay rates of N = 81 are smaller than their neutron-capture rates. Thus, the β-decay properties not only of the N = 82 isotones but also of the N = 81 ones are necessary to determine the r-process path, hence motivating the study of those very neutron-rich nuclei from the viewpoint of nuclear structure physics. Note that the properties of nuclei near N = 82 are also awaited in the context of fission recycling [4].
On the experimental side, β-decay half-lives of neutrino-rich nuclei around N = 82 have recently been measured by the EURICA campaign conducted at the RI Beam Factory in RIKEN Nishina Center [5,6]. More detailed data are now available for some nuclei. Many isomers have been identified near the N = 82 shell gap, and some of their half-lives are obtained [7][8][9][10]. Furthermore, β-delayed neutron-emission probabilities and low-lying level structure have been measured [11,12]. These data provide a stringent test for nuclear-structure models. It should be noted that similar experimental activities are extended to the N = 126 region, known as the third peak of the solar system abundance, 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 sophisticated methods were introduced into the systematic β-decay studies by introducing the FAM-QRPA [20] and by the relativistic 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 previous shell-model studies are, however, restricted to calculating the half-lives of the singly-magic N = 82 [23][24][25] and N = 126 isotones [23,26] due to the exponentially increasing dimensions of the Hamiltonian matrices in open-shell nuclei. The present work aims to extend those previous shell-model efforts to N = 81 isotones within a unified description of the structures of neutron-rich N = 82 and N = 81 isotones. The measured half-lives are well reproduced by the calculation, and we predict those for 125,126 Ru, 124,125 Tc and 124 Mo. It is also predicted that these nuclei have rather strong GT strengths in the low excitation energies due to the increasing number of proton holes in the g 9/2 orbit, accelerating GT decay. This paper is organized as follows. The shell-model model space and its interaction are defined in Sect. II. Section III is devoted to the separation energies and low-lying spectra. The Gamow-Teller strength distribution and the half-lives are discussed in Sect. IV. Section V is devoted to the discussion of the enhancement of the 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 the calculations is taken as 0f 5/2 , 1p 3/2 , 1p 1/2 , 0g 9/2 , 0g 7/2 , 1d 5/2 , 1d 3/2 , 2s 1/2 , and 0h 11/2 for the proton orbits and 0g 7/2 , 1d 5/2 , 1d 3/2 , 2s 1/2 , and 0h 11/2 for the neutrons orbits with the 78 Ni inert core. These orbits are shown in Fig. 1. Although we focus on Z ≤ 50 nuclei in this study, the single-particle orbits beyond the Z = 50 shell gap are required to be included in the model space explicitly so that the Gamow-Teller transition causes the single-particle transition of the valence neutrons beyond N = 50 to the same orbits and its spin-orbit partners. The model space is extended from that of the earlier shell-model study [23] by adding the proton 0f 5/2 , 1p 3/2 , and 0h 11/2 orbits. In the preceding shell-model works [23,24], the proton 0h 11/2 orbit was omitted to avoid the contamination of the spurious center-of-mass excitation, although the neutron occupying 0h 11/2 orbit can decay to the proton occupied in 0h 11/2 by the Gamow-Teller transition. In the present work, we explicitly include the proton 0h 11/2 orbits into the model space so that the proton single-particle orbits cover the whole neutron orbits. For fully satisfying the Gamow-Teller sum rule the proton 0h 9/2 orbit is required, but its single-particle energy is too high to significantly affect the Gamow-Teller strength of the low-lying states and it is omitted in the present work. The contamination of spurious center-of-mass excitation is removed by the Lawson method [27] with β CM ω/A = 10 MeV. We truncate the 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 dimension of the shell-model Hamiltonian matrix reaches 3.1 × 10 9 and is quite large, and efficient usage of a supercomputer is essential. The shell-model calculations were mainly performed on CX400 supercomputer at Nagoya University and Oakforest-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.  FIG. 1: Single-particle energies for 132 Sn determined from the experimental energy levels of its one-particle and one-hole neighboring nuclei [29][30][31][32][33][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 established realistic interactions: the JUN45 interaction [35] for the f 5 pg 9 model space and the SNBG3 interaction [36] for the neutron model space of 50 < N, Z < 82. The JUN45 and SNBG3 interactions were constructed from the G-matrix interaction with phenomenological corrections using a chi-square fit to reproduce experimental energies. For the rest part 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 determined to reproduce the experimental energies of one-nucleon neighboring nuclei of 132 Sn as shown in Fig. 1. In addition, the strengths of the pairing interaction and the diagonal TBMEs of the (π0g 9/2 , π0g 9/2 ) and (π0g 9/2 , ν0h 11/2 ) orbits are modified to reproduce the experimental energy levels of 130 Cd, 128 Pd, and 130 In. The TBMEs are assumed to have the mass dependence (A/132) −0.3 .

III. SEPARATION ENERGIES AND EXCITATION ENERGIES
The binding energies and excitation energies of the N = 82 nuclei and those around them are important not only for describing the β-decay properties, but also for confirming the validity of the shell-model interaction. Figures  2 and 3 show the proton and neutron separation energies of the N = 82 and N = 81 isotones, respectively. The present shell-model results reproduce the experimental values excellently. The neutron separation energy determines the threshold energy of the β-delayed neutron emission, which is important for the r-process nucleosynthesis. Since the Q-value of the β − decay is obtained using the proton and neutron separation energies as  Figure 4 shows low-lying energy levels in the neutron-rich N = 82 isotones from Z = 42 to Z = 50. For the nuclei without data, we plot a few lowest levels obtained by the calculation. The calculated ground states are 0 + for the even-Z isotopes and 9/2 + for the odd-Z isotopes. The experimental levels are reproduced excellently by the shellmodel results. The levels of 129 Ag are experimentally unknown, but two β-decaying states were found and tentatively   assigned as 9/2 + and 1/2 − [34] without their excitation energies known. In the present calculation, the 1/2 − state is located very close to the 7/2 + state. Considering a long E3 half-life in such a case, it is reasonable to assume that the 1/2 − state predominantly decays through β emission. Figure 5 shows the excitation spectra of the N = 81 isotones. Unlike the N = 82 isotones, several candidates for the ground state and some β-decaying isomers are predicted. This is partly because the 1d 3/2 and the 0h 11/2 neutron orbits are located very close in energy as known in the spectra of 131 Sn and the difference of their spin numbers is large. For 129 Cd, two β-decaying states with 11/2 − and 3/2 + were known and their order had been controversial [7]. A recent experiment concluded that its ground-state spin is 11/2 − and the excitation energy of 3/2 + is 343(8) keV [10,11], which is consistent with our shell-model prediction. For 127 Pd, no experimental energy levels are known, and the present order of 11/2 − and 3/2 + agrees with another shell-model prediction [45]. With regard to β-decay properties, the excitation energy of the 1 + state of 130 In plays a crucial role in the β-decay half-life of 130 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 data are available. We confirm a reasonable agreement between them.
The present calculation reproduces the experimental energies quite well, thus confirming the validity of the model space and the effective interaction employed in the present shell-model calculation.

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 their half-lives. We adopted the Lanczos strength function method [41][42][43] with 250 Lanczos iterations to obtain sufficiently converged results. The magnitude of quenching of axial vector coupling is still a challenging topic for nuclear physics and 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 the adopted value of the preceding work, q GT = 0.71 [24]. The first-forbidden transition is omitted in the present work because its contribution to the half-lives is small, around 13%, and rather independent of nuclides for the Z = 42 − 48, N = 82 isotones in a previous shell-model study [23]. Furthermore, it is pointed out in [11] that a number of allowed transitions are observed in the β − decays of 121−131 In and 121−125 Cd, suggesting the dominance of GT transitions in the low excitation energies. This point will be discussed later.    energy GT strength distributions, which play a crucial role in those β-decay half-lives. First, all the N = 82 isotones considered here have strong GT strengths in the low-excitation energies. Except 131 In, they are peaked at ∼ 3.5 MeV and ∼ 2 MeV for the odd-Z and even-Z parents, respectively, and the GT strengths are more concentrated for the even-Z isotopes. This odd-even effect is in accordance with what is found in the sd-pf shell region [48]. Second, this low-energy GT peak grows with decreasing proton number. This is an interesting feature of low-energy Gamow-Teller transitions 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 transition probabilities 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 129 Ag and 128 Pd are closer to the experimental values than the earlier shell-model result, the half-life of 131 In is underestimated. This underestimation is caused by the large GT transition to the lowest 7/2 + state of the daughter 131 Sn at E x = 2.4 MeV, which might imply the need for further improvement of the theoretical model. This state is considered to be dominated by the ν0g 7/2 -hole state of 132 Sn. In the pure π0g −1 9/2 → ν0g −1 7/2 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 to configuration 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], and SM99 [25]. The half-lives of 126 Ru, 125 Tc, and 124 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 was omitted and the quenching factor of the Gamow-Teller transition was taken as q GT = 0.71 in SM07, the first-forbidden transition is included with q GT = 0.66 in SM13. The agreement of these results indicates that the contribution of the first-forbidden decay is rather independent of the nuclides and can be absorbed into the minor change of the  [23], and the FRDM+QRPA [15], respectively. The red diamond denotes the experimental value and the red line with an arrow at Z = 47 denotes the experimental upper limit [46,47].  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-delayed neutron emission probabilities P n for N = 82 nuclei. In the present calculation, we accumulate the probabilities of the β-decay to the states above the neutron-emission threshold S n to obtain P n . The present shell-model results show an odd-even staggering similar to that of the earlier shell model [23], while the FRDM-QRPA results show weaker odd-even staggering. This odd-even staggering is caused by the difference of the peak position and the degree of concentration of the Gamow-Teller transition strengths. As discussed already using Figs. 7 and 8, the GT peaks of the 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 124 Mo, it is predicted that this low-energy GT strength is concentrated by a single peak that is located slightly below S n . Hence its P n is very sensitive to the detail of the energies concerned. For the odd-Z nuclei of 127 Rh and 125 Tc, the low-energy GT peak is located higher than S n , enlarging their P n values. They are located higher for the odd-Z parents due to pairing correlation in the daughter nuclei, but fragmented in a similar manner. Like the case of the N = 82 isotones, those peaks are enhanced as the proton number decreases and the proton 0g 9/2 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 ≥ 45 show reasonable agreement with the available experimental values, indicating the validity of the present shell-model calculation. The half-life of the 3/2 + isomeric state of 129 Cd is also shown in the table to demonstrate the capability to 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 the experimental Q value, respectively, to discuss the uncertainty of the present theoretical model. The deviations of the choice of the Q values show up to 30% at most. The fitted quenching factor to reproduce the experimentally measured half-lives of 129 Cd, 130 Cd and the 3/2 + isomer by the SM exp result is q GT = 0.67, which shows a 9% increase of the 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 peak emerges at ∼ 2 MeV and that its magnitude is enhanced as the proton number decreases. As depicted in Fig. 12, this peak is finally concentrated in a single state at 122 Zr with Z = 40, leading to B(GT) = 2.7 calculated with the quenching factor 0.7. 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 densities obtained in the present calculations, one can see that those low-energy Gamow-Teller peaks are dominated by the ν0g 7/2 → π0g 9/2 transition. If the π0g 9/2 orbit is completely filled, this transition does not occur due to the Pauli blocking. This blocking effect is weakened by removing protons from the π0g 9/2 orbit, hence the enlargement of the low-energy Gamow-Teller peak. The resulting B(GT) values of this peak are particularly large at 124 Mo and 122 Zr compared to typical values. It is known from the systematics [49] that the log f t values of allowed β decays are distributed around log f t ∼ 6, which corresponds to B(GT) ∼ 10 −3 -10 −2 for Gamow-Teller transitions. A well-known deviation from this systematics is the superallowed (Fermi) transition. When isospin is a good quantum number, the Fermi transition occurs only between isobaric analog states, giving a typical log f t of 3.5. With regard to Gamow-Teller transitions, however, there are only a few cases where the log f t value is comparable to those of the superallowed Fermi transitions because of the fragmentation of Gamow-Teller strengths. Since B(GT) = 1 leads to log f t = 3.58, the B(GT) value of the order of 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-Teller transitions. At that time, only two Gamow-Teller transitions, 6 He→ 6 Li and 18 Ne→ 18 F, were known to satisfy the condition of Super Gamow-Teller transition defined in [50], i.e., B(GT) > 3. These large Gamow-Teller strengths are caused by the constructive interference of j > → j > and j > → j < matrix elements [51]. It was also predicted in [50] that two N = Z doubly-magic nuclei 56 Ni and 100 Sn were candidates for nuclei causing Super Gamow-Teller transitions. Although the Gamow-Teller strengths from 56 Ni were measured to be fragmented about a decade later [52], 100 Sn is now established to have a very large B(GT) value (9.1 +3.0 −2.6 in [53] or 4.4 +0.9 −0.7 in [54]) to a 1 + state located at around 3 MeV. This Gamow-Teller decay is called "superallowed Gamow-Teller" decay in [53] on the analogy of the superallowed Fermi decay.
The B(GT) values predicted for 124 Mo and 122 Zr in the present study are the order of unity, although not reaching the measured value of 100 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. In the extreme single-particle picture, the π0g 9/2 orbit is completely filled and the ν0g 7/2 orbit is completely empty in 100 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 π0g 9/2 → ν0g 7/2 transition. On the other hand, in 122 Zr, the ν0g 7/2 orbit is completely filled and the π0g 9/2 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 π0g 9/2 orbit keeps the lowest unoccupied orbit in this range. For neutrons, although the ν0g 7/2 orbit is the lowest at 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 π0g 9/2 and ν0g 7/2 due to a cooperative attraction of the central and the tensor forces [37]. This sharp change of the ν0g 7/2 orbit in going from Z = 40 to 50 is established from the energy levels of 91 Zr and 101 Sn, as mentioned in [37]. In 122 Zr, the ν0g 7/2 orbit is thus close to the highest occupied level, making a low-energy Gamow-Teller state by the ν0g 7/2 → π0g 9/2 transition. If one is restricted to the configuration most relevant to the low-energy Gamow-Teller transition, the final state of the 122 Zr decay, (ν0g 7/2 ) −1 (π0g 9/2 ) +1 , is the particle-hole conjugation of that of the 100 Sn decay, (π0g 9/2 ) −1 (ν0g 7/2 ) +1 . A schematic illustration of these configurations are given in Fig. 14. Accordingly, the B(GT) values from the vacuum to 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 within the single configuration of Fig. 14 (a) [and (b)] is also large. To be more specific, let us compare two cases as the initial state, (i) |(π0g 9/2 ) 10 ; J = 0 and (ii) |(π0g 9/2 ) 2 ; J = 0 , where one proton can move to the ν0g 7/2 orbit through the Gamow-Teller transition. The case (i) corresponds to Fig. 14(a) and yields B(GT)=17.78 (without the quenching factor), whereas the case (ii) gives B(GT)=3.56. The ratio of these two B(GT) values, 10 to 2, is just that of the number 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 minimize fragmentation, it is desirable to suppress the level density with the same J π near the state of interest. 100 Sn and Hamiltonian matrix elements concerning the π0g 9/2 and ν0g 7/2 orbits used in this study. The circles and the squares are the hole-hole matrix elements, (π0g 9/2 ) −1 (ν0g 7/2 ) −1 |V |(π0g 9/2 ) −1 (ν0g 7/2 ) −1 J and the particle-hole matrix elements, π0g 9/2 (ν0g 7/2 ) −1 |V |π0g 9/2 (ν0g 7/2 ) −1 J , respectively.
cost more than 4 MeV by estimating from the first excitation energy of 132 Sn, they probably do not contribute much to fragmentation. One may wonder why the low-energy Gamow-Teller peak is kept at E x ∼ 2 MeV from Z = 48 to Z = 40 in spite of the sharp change of the ν0g 7/2 energy as shown in Fig. 13. This is due to the nature of two-body Hamiltonian matrix elements. The low-energy Gamow-Teller state has always a neutron hole in 0g 7/2 . For the nuclei close to Z = 50, this state has a few proton holes in 0g 9/2 , and thus its excitation energy is dominated by the hole-hole matrix element (π0g 9/2 ) −1 (ν0g 7/2 ) −1 |V |(π0g 9/2 ) −1 (ν0g 7/2 ) −1 J=1 as well as the single-particle energy of ν0g 7/2 . As presented in Fig. 15, this matrix element is the most attractive among the possible J values. Hence the low-energy Gamow-Teller state is located lower than the simple estimate that the 0g 7/2 orbit lies ∼ 3 MeV below the Fermi surface at Z = 50 (see Fig. 13).
This situation changes as more protons are removed from the π0g 9/2 orbit. For the nuclei close to Z = 40, the number of particles are smaller than the number of holes in the 0g 9/2 orbit, and the particle-hole matrix element π0g 9/2 (ν0g 7/2 ) −1 |V |π0g 9/2 (ν0g 7/2 ) −1 J plays a dominant role. In Fig. 15, we also show the particle-hole matrix elements that are derived from the hole-hole matrix elements by using the Pandya transformation. The J = 1 coupled matrix element has the largest positive value, thus losing the largest energy. This explains the calculated result that the low-energy Gamow-Teller state is not drastically lowered toward Z = 40 as expected from the evolution of the ν0g 7/2 orbit, and also the observation that the corresponding state for the 100 Sn decay is located at ∼ 3 MeV [53]. It should be noted that this J dependence is an example of the parabolic rule that holds for short-range attractive forces [55].
To briefly summarize this section, the predicted superallowed Gamow-Teller transition toward Z = 40 occurs due to (a) the full occupation of a neutron high-j orbit (ν0g 7/2 in this case) and the emptiness of its proton spin-orbit partner (π0g 9/2 in this case) and (b) the low excitation energy of the J = 1 particle-hole state created by these two orbits. 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 ν0g 7/2 orbit and the π0g 9/2 orbit are closed to the highest occupied orbit and the lowest unoccupied orbit, respectively. The tensor-force driven shell evolution plays a crucial role in satisfying this condition.

VI. SUMMARY
We have constructed a shell-model effective interaction and performed large-scale shell-model calculations of neutron-rich N = 82 and N = 81 nuclei by utilizing our developed shell-model code and the state-of-the-art supercomputers. We demonstrated that the experimental binding and excitation energies of neutron-rich N = 79, 80, 81 nuclei are well reproduced by the available experimental data including the low-lying excited states. The present study gives the Gamow-Teller strength functions and the β-decay half-lives of N = 82 and N = 81 nuclei, which are reasonably 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 0g 9/2 orbit becomes unoccupied and the Gamow-Teller strengths of the low-lying states increases because of the Pauli-blocking effect. We predict that the low-energy Gamow-Teller strength is further enlarged in 122 Zr to make its log f t value equivalent to that of the superallowed beta decay. This is quite an analogous case to the so-called "superallowed Gamow-Teller" transition observed in 100 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 and can be absorbed into a single quenching factor of the Gamow-Teller transition. Further investigation to estimate the first-forbidden decay especially for the N = 81 isotones is also expected.