Systematic approach to β and 2 νββ decays of mass A = 100 – 136 nuclei

All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Systematic approach to β and 2vββ decays of mass A=100-136 nuclei Pirinen, Pekka; Suhonen, Jouni


I. INTRODUCTION
The research on double-β decay is both timely and of great interest for particle and nuclear physics [1][2][3]. Both the neutrinoless double-β (0νββ) and two-neutrino double-β (2νββ) decays depend sensitively on the involved nuclear matrix elements (NMEs). A large number of different nuclear models have been used to describe the associated nuclear wave functions. Among these models are the quasiparticle random-phase approximation (QRPA), its proton-neutron extension (pnQRPA) (see Ref. [4] and references therein), and its renormalized extensions [5,6]. A different category of nuclear models are formed by the interacting shell model (ISM) [7], the (proton-neutron) interacting boson model (IBA-2) [8], the Gogny-based energy-density functional approach (EDF) [9], and the projected Hartree-Fock-Bogoliubov meanfield scheme (PHFB) [10]. An extensive comparison of the double-β properties of the aforementioned models is performed in Ref. [11].
The 2νββ decays can occur in two different categories: The decays on the β − side, i.e., 2νβ − decays, have favorable decay Q values and abundances, and they have been under intensive experimental and theoretical investigation over the years. Many decay transitions have already been detected for these decays [12]. The positron-emitting modes of decays, 2νβ + β + , 2νβ + EC, and 2νECEC (generically called 2νβ + /EC hereafter, EC referring to electron capture) are much less studied due to their relatively low Q values and abundances. No such decay transitions have yet been observed in the laboratory. On the theory side, these decays have been investigated for their lepton aspects in Ref. [13] and nuclear-structure aspects, e.g., in Refs. [14][15][16][17][18][19][20][21][22]. Some of these studies refer to decays to the ground states and some refer to decays to both ground and excited states.
The phase space of the 2νβ − and 2νβ + /EC decays depends on the fourth power of the value of the weak axial-vector coupling constant g A . Hence, it is of paramount importance to know the (effective) value of g A in medium-mass and heavy-mass nuclear systems. An effective value of g A ∼ 1 has been advocated in several ISM calculations in the past (see, e.g., Refs. [23,24]). Even stronger quenchings have been proposed in Refs. [25,26]. At the same time the pnQRPA, used in the calculations of the involved NMEs, suffers from the "g pp problem," i.e., from the unsettled value of the particle-particle interaction parameter g pp describing the strength of the protonneutron interaction in the 1 + channel. Since the introduction of this parameter [27,28], several researchers have tried to fix its values by the inspection of the measured single-β-decay rates [29,30] or 2νββ decay rates [31][32][33][34].
The effective value of g A can also be studied together with the value of g pp within the pnQRPA framework. The first work on this line of study is that of Ref. [35] where both the β-decay and 2νβ − -decay data were analyzed for the A = 100,116 systems in the framework of the pnQRPA using least squares to achieve best-fit values for g A . In Ref. [35] the best-fit values g A = 0.74 (A = 100) and g A = 0.84 (A = 116) were obtained. A monotonic behavior of g A as a function of the mass number A was parametrized in Ref. [36] by analyzing the magnitudes of the NMEs of several 2νβ − decays computed by the IBA-2 model. In this study the obtained g A -versus-A slope was very flat. Simultaneous use of the 2νβ − and β − data for the A = 100,116,128 systems lead to a much quenched value of g A ∼ 0.6 in Refs. [37,38]. In Ref. [39] the interacting boson-fermion-fermion model (IBFFM-2) was used to derive effective value of g A ∼ 0.3 in the A = 128 case. In Ref. [40] an extension of the g A analysis to first-forbidden 2 − → 0 + gs β decays was achieved. In the present article we adopt a more systematic approach to attack the problem of the effective value of g A . We are going to allow the values of both g A and g pp to vary in an analysis of 24 isobaric triplets within the mass range A = 100-136. In these triplets an intermediate odd-odd isobar connects via β transitions to its two adjacent even-even isobars and the rates of the connecting transitions are experimentally known. We try to reproduce the measured transition rates in a pnQRPA calculation by varying g A and g pp in each isobaric triplet separately. In this way the present analysis is an extension of that of Ref. [41] where nine such pairs were analyzed within a very rudimentary pnQRPA approach. At the same time the present study extends the idea of Ref. [40], i.e., of using the geometric mean of the decay rates, to the Gamow-Teller β decays. The geometric means of the two decay branches are more stable against variations in g pp and g A than the individual branches of decay and are thus preferable in the analysis. By this analysis we derive a piecewise linear relationship between the mass number A and the parameter g A . By using this relationship we then predict 2νβ − and 2νβ + /EC decay half-lives for a total of 21 nuclei in the mass range A = 100-136. We also compare these predictions with those derived from a constant mean value of g A .
This article is organized as follows: In Sec. II we give a brief introduction to the underlying formalism of the Gamow-Teller β transitions, and the 2νβ − and 2νβ + /EC decays. In Sec. III we display and discuss the obtained single-β-decay and double-β-decay results. The final conclusions are drawn in Sec. IV.

II. REVIEW OF THE FORMALISM OF COMPUTATION
In this section only the basic theoretical ingredients of the computations are given. The single-and double-β decay halflives are defined in terms of the involved NMEs and the phasespace factors. The NMEs are given in terms of the singleparticle matrix elements and the one-body transition densities. For the calculation of the phase-space factors and the one-body transition densities we refer to the available literature at the appropriate places.

A. Gamow-Teller β decays
The Gamow-Teller β-decay transitions are mediated by the Pauli spin operator σ . In this work we extract the needed experimental Gamow-Teller NMEs by using the experimental comparative half-lives (log f t values). They are defined as [42] where the reduced transition probabilities B GT are given by The single-particle matrix elements are defined as where the matrix elements of the Pauli operator are given in Ref. [42]. In Eqs. (2) and (3) the final states appearing in the one-body transition densities are considered to be I + f = 0 + gs ,2 + 1 ,0 + 1 ,2 + 2 , where 0 + gs is assumed to be the vacuum state of the quasiparticle random-phase approximation (QRPA), discussed briefly in Sec. II C. The first 2 + state in the final even-even nucleus, 2 + 1 , is considered to be a one-phonon QRPA state and the first excited 0 + state, 0 + 1 , and the second 2 + state, 2 + 2 , are considered to be two-phonon QRPA states; see Sec. II C. Exact expressions for the one-body transition densities are given in Ref. [20].

B. Two-neutrino double-β decays
The two-neutrino double-β decays are of either the 2νβ − or the 2νβ + /EC type. The associated half-lives can be expressed as [13,20,43] where the expressions for the lepton phase-space integrals G α 2ν (I + ), α = β − β − are given in Ref. [1] and the integrals G α 2ν (I + ), α = β + β + , β + EC, ECEC are given in Ref. [13]. The NMEs M α 2ν (I + ) include energy denominators and a summation over all the 1 + states of the intermediate nucleus (see below). Above, the symbols EC(K) and EC(L) denote electron captures from the atomic K and L 1 shells, respectively.
The NMEs involved in the above half-life expressions can be cast in the form where in the QRPA framework one writes Here the operator c † p (c † n ) creates a proton (neutron) particle in the orbital p = n p ,l p ,j p (n = n n ,l n ,j n ), where n is the radial, l is the orbital angular-momentum, and j is the total singleparticle angular-momentum quantum number. The same holds for the particle annihilation operators, written without the dagger. The single-particle parts in Eqs. (10) and (11) are written as m(pp ,nn ) = σ p n σ pn , m(nn ,pp ) = σ n p σ np , (12) where the matrix elements of the Pauli operator were defined in Eq. (4).
The one-body transition densities involved in Eqs. (10) and (11) are given separately for the different types of I + two sets of pnQRPA states used in the calculations is given by (13) and it takes care of the matching of the corresponding states in the two sets of states based on the initial and final eveneven reference nuclei. The amplitudes X and Y (X andȲ ) come from the pnQRPA calculation starting from the initial (final) nucleus of the double-β decay (see Sec. II C). Here it is appropriate to note that the expression (13) is only an approximation if the initial and final ground states are not identical. However, the approximation is reasonable if these ground states are not very different in terms of orbital occupancies. The quantities F α k 1 (I + ) in Eq. (9) are the following energy denominators: where the normalized (by the electron rest-mass energy) intermediate energy is given by and the decay energies are Here the quantity E f is the final-state (ground-state or excitedstate) energy and where B i is the binding energy of electron i in an atomic K or L 1 orbital [13]. In k [see Ref. (20)] E k is the energy of the kth 1 + state in the intermediate nucleus.
In the pnQRPA calculations the energy E k is taken to be the average of the kth energy eigenvalues based on the two pnQRPA calculations, one for the initial and one for the final nucleus of the ββ decay. Furthermore, the energy difference E 1 − M i c 2 is taken from experiment whenever the excitation energy of the first 1 + state of the intermediate nucleus is known experimentally.

C. Nuclear models
The initial and final 1 + states of β decays and intermediate 1 + states of double-β decays (both always in odd-odd nuclei) are handled within the framework of the pnQRPA. The pnQRPA states, in an odd-odd nucleus, can be written as where |QRPA is the QRPA vacuum. The operator a † p (a † n ) creates a proton (neutron) quasiparticle in the orbital p (n). The sum runs over all proton-neutron configurations in the chosen valence space. In the pnQRPA formalism states (23) connect directly to the neighboring even-even ground states, both in the initial nucleus, 0 + i = 0 + gs , and the final one, I + f = 0 + gs . The wave functions of the final states I + f = 2 + 1 ,0 + 1 ,2 + 2 of the β and double-β decays are computed by the use of the multiple-commutator model (MCM) [44,45]. It is designed to connect excited states of an even-even reference nucleus to states of the neighboring odd-odd nucleus. The states of the odd-odd nucleus are given by the pnQRPA in the form (23). The excited states of the even-even nucleus are generated by the (charge conserving) quasiparticle randomphase approximation (QRPA) described in detail in Ref. [42].
Here the symmetrized form of the phonon amplitudes is adopted contrary to Ref. [42] so that the first excited 2 + state in the final nuclei of the β and double-β decays can be written as a QRPA phonon of the form where the amplitudes Z and W are obtained by solving the QRPA equations of motion [42]. From the above 2 + phonons (24) one can build an ideal two-phonon state of the form An ideal two-phonon state consists of partner states I π = 0 + ,2 + ,4 + that are degenerate in energy, and exactly at an energy twice the excitation energy of the 2 + 1 state. In practice this degeneracy is always lifted by the residual interaction between the one-and two-phonon states [46]. In this work we assume the I + f = 0 + 1 and I + f = 2 + 2 final states of β and double-β decays to be reasonably well described by the ansatz wave function (25).

III. RESULTS AND DISCUSSION
In this section we present and discuss the results of the calculations. For the sake of completeness we introduce each of the elements entering the calculations, referring to the already published material for details.

A. Determination of model parameters
Up to A = 108 a valence space consisting of 11 states, the entire 1p − 0f − 0g and 2s − 1d − 0h shells, was used. For A = 110 and onwards also the 2p and 1f shells were included and the valence space was expanded to 15 states. The valence spaces are visualized in Fig. 1. The single-particle bases are built by using a Coulomb-corrected Woods-Saxon potential and solving the radial Schrödinger equation. The Woods-Saxon parameters used were the ones given by Bohr and Mottelson in Ref. [47].
For the two-body part of the interactions, the renormalized Bonn-A G matrix [44,48] has been used and the neutron and proton pairing strength parameters A p pair and A n pair were fitted such that the lowest quasiparticle energies from the BCS calculation match the pairing gaps: E p qp (lowest) = p , and E n qp (lowest) = n . The pairing gaps can be calculated by the three-point formula [49] where S i is the proton or neutron separation energy. The separation energies were taken from Ref. [50]. The particle-hole and particle-particle interaction strengths are important interaction parameters in both the QRPA and pnQRPA. As a convention for the present work, in the QRPA these parameters are written as uppercase G ph and G pp and in the pnQRPA as lowercase g ph and g pp .
In QRPA, G pp has little effect on the first excited states [51] and the common value of G pp = 1.00 has been adapted for the examined nuclei. The first excited state of the presently studied even-even nuclei is a 2 + state, which is of a particle-hole nature. Therefore, the value of G ph has a significant effect on the energy of this state. The value of G ph was fixed for each nucleus separately by fitting the energy of the first 2 + state to experimental data. In pnQRPA, the particle-hole parameter g ph has a large effect on the energy location of the Gamow-Teller giant resonance (GTGR). The particle-particle parameter g pp has more to do with the Gamow-Teller β-decay transition amplitudes [51]. The value of g ph was fitted for each nucleus separately to approximately match the GTGR location to the empirical formula [42]: where E c is the Coulomb energy and E Z+1,N−1 is the energy difference between the GTGR state and the 0 + isobaric analog state of the odd-odd nucleus. The particle-particle interaction parameter g pp is often used to fit the log f t value of the transition from the first 1 + state of the odd-odd nucleus to the ground state of the even-even nucleus [48,51]. In the present work, to better examine the systematics of Gamow-Teller β decay, we assign an array of constant values of g pp = 0.6, 0.7, 0.8, 0.9. By fitting the geometric means of the β-decay matrix elements to experimental values, the behavior of the axial-vector coupling constant g A is expressed as a function of the mass number A for each value of g pp . This enables us to find a systematic behavior of g A for a given value of g pp .

B. Single-β decays and determination of the effective value of g A
In this work a selection of medium heavy nuclei in the mass region A = 100-134 are taken under investigation. This A schematic of the single-β decay triplets studied in this work. The nucleus in the middle is odd-odd, and its neighbors on the left and right are even-even. selection spans practically all the nuclei where relevant log f t data have been measured. The nuclei are grouped into triplets where an odd-odd nucleus with a 1 + ground state is in the "center" and two neighboring even-even isobars with 0 + ground states are to the "left" and "right" of the odd-odd nucleus as is represented in Fig. 2. Ground-state-to-groundstate β decays as well as decays to excited states of the even-even nuclei are predicted using the QRPA framework.
In the present analysis it is preferable to use the geometric mean of the two transition matrix elements since the geometric mean is practically independent of the value of g pp , as visible in Fig. 3. This owes to the particular behavior of the β − and β + /EC decay amplitudes as a function of g pp in a pnQRPA calculation: The β − (β + /EC) decay amplitude for a transition from an odd-odd nucleus is a decreasing (increasing) function of g pp , as shown explicitly in Ref. [29]. In this way the g pp dependencies of the two branches conspire to produce an almost flat g pp dependence for the geometric mean.
One can calculate the experimental geometric means of the left and right NMEs (multiplied by g A ) from This quantity is actually independent of the value of g A taken for theoretical calculations, which permits g A to be left as a free parameter to fit calculations to experimental data. The calculated experimental geometric means for the investigated mass region are presented in Table I. The needed experimental log f t values were taken from Ref. [52].
We first examined the ground-state-to-ground-state decays in the investigated mass region. Four rounds of pnQRPA calculations were performed using typical values of g pp = 0.6, 0.7, 0.8, 0.9 in order to analyze the left and right branches of Gamow-Teller β decay from each odd-odd nucleus. Theoretical geometric means of these matrix elements were then calculated for each value of g pp and fitted to the experimental values by altering the value of g A . The calculated matrix elements and geometric means for g pp = 0.7 are given in Table II and visualized in Fig. 4. The resulting values of g A for each value of g pp are presented in Fig. 3 as a function of the mass number A.
In   [52], except the entry marked with superscript 1 is extrapolated from systematics of similar neighboring decays and the entry marked with superscript 2 is from Ref. [53].   generally smaller than the log f t values in the right branch, so this is in good agreement with the experimentally observed behavior. In Fig. 3 one can immediately see that, with respect to increase in g pp , g A becomes more unstable with increasing A. The value of g pp = 0.9 is thus discarded as only a small variation in g A is desired. It is also evident, given a reasonable interval of g pp values, that the geometric mean does not depend very much on the value of g pp as was expected. This can be seen in Fig. 3 as the data points representing different values of g pp for a given nucleus are very close to each other. In other words, it does not take a significant change in g A to compensate for a relatively large change in g pp .
Interestingly enough, the calculations replicate the experimental geometric means of the NMEs for g pp = 0.6, 0.7, 0.8 only for values of g A < 1. This is solid evidence that an effective g A is needed when working with this mass region. Not only are the values in general smaller than the bare value of g A = 1.25 but in some cases an effective value as low as g A = 0.3 is required.
In Fig. 3 one can see an interesting rising behavior in g A as a function of A. There seem to be two mass regions in which g A behaves, on average, linearly with a positive slope. The average linear behavior is depicted in the figure as two dashed lines. These lines are represented by By using this function to generate values of g A and adopting a reasonably average value of g pp = 0.7 across the entire mass region, the Gamow-Teller matrix elements were calculated for the ground-state-to-ground-state decays. The resulting left and right log f t values are presented in Figs. 5 and 6 respectively along with results of the same calculation made with an average constant value of g A = 0.6 for comparison. The linear model appears to fare better overall in predicting the log f t values of single-β decays, especially in the right branch decays. The agreement with experiment is decent at the very least. The decays which are not quite along the line of Eq. (28) in Fig. 3 expectedly produce somewhat less accurate predictions, for example, the A = 124 triplet.
To evaluate the success of the linear model, the mean deviation of the theoretical log f t values from the experimental ones was used as a quality value. The mean deviation reads:  The examination of the linear g A model was then extended to decays from the 1 + ground state of the odd-odd nuclei to the first excited 2 + state of the even-even nuclei and to the 0 + and 2 + two-phonon states constructed from the wave function of the first 2 + state.
All processes in the investigated mass region with experimental data available for decays to these excited states have been included and the resulting log assumption of the simple structure (25) of the two-phonon states is one more limitation of the present model. Still, the accuracy of the prediction is quite passable, especially for decays to the first 2 + state, the energy of which was fitted to the experimental value. The calculated log f t values for decays to the two-phonon 2 + state are systematically too large compared to the experimental values. For these states the assumed simple two-phonon structure seems to be inadequate.
A curious case arises in the decay of rhodium to palladium via β − decay, in which the transition to the first 2 + state is predicted very slow for each A = 104, 106, 108. This can be seen in Fig. 7 as the largest deviations from the experimental values. The deviation arises from subtle cancellations among the pnQRPA X and Y amplitudes in the major components of the wave function of the ground state of the Rh nucleus and the QRPA amplitudes of the first 2 + state of the even-even reference nucleus. There are thus large cancellations in the transition amplitudes and the calculated log f t values are very high, making the comparative half-lives too long by several orders of magnitude.
The mean deviations in decays to excited states using the linear g A model are m = 0.75, 0.70, 0.81 for decays to the 2 + 1 , 0 + 2−ph , and 2 + 2−ph states respectively. For the constant value, g A = 0.6, the corresponding numbers are The constant g A seems actually preferable for the decay to the first 2 + state, but for the two-phonon states the linear model is again more accurate. This is quite peculiar, as the two-phonon states are formed from the one-phonon 2 + state. It appears that the decays to excited states follow some other g A systematic than the ground-state-to-ground-state decays. The standard QRPA framework is clearly not an accurate framework for depicting these transitions.
A disappearance of a clear two-phonon 0 + state happens at A 126. The experimental log f t values used in Fig. 8 are for the lowest excited 0 + state, but this state is always too  Fig. 7 for decays to the 0 + quadrupole two-phonon states of the even-even nuclei. high in energy to be the two-phonon state the theory predicts. Either the breaking of the degeneracy of the collective twophonon triplet increases with increasing mass number A or the formation of this particular multipolarity is prevented by some mechanism in nuclei investigated here with A 126.
The theoretical log f t values for these transitions often seem to deviate more than one unit from the experimental log f t value for the 0 + 2 state. This makes it more probable for the lowest excited experimental 0 + state to result from some other origin than two quadrupole phonons.
The β-decay rates in the A = 104 and A = 110 systems were also calculated in Ref. [30]. The computed log f t values of the present work are compared to these results in Table III. In Ref. [30] the transition rates were also computed using adjusted single-particle bases where the Woods-Saxon energies of some orbitals had been altered to better replicate experimental values. The comparison is made between results computed using the plain Woods-Saxon bases as was used in the present work. The results appear quite similar. Our present analysis predicts the ground-state-to-ground-state decays slightly more accurately. The computations of Ref. [30] systematically yield too fast transition rates while the present work gives more balanced results. The use of adjusted single-particle orbitals might enhance the accuracy of our present predictions as it did in Ref. [30].

C. Half-lives of double β decays
The half-life of every possible two-neutrino double-β decay in the examined mass region was calculated using the g A scheme of Eq. (28). The calculation was also done with a constant value of g A = 0.6 for comparison. For the particle-particle interaction parameter we adopted the value of g pp = 0.7, which functioned quite well in predicting the single-β decay comparative half-lives. One should note here TABLE V. Theoretical NMEs (columns 2-5) and half-lives (columns 7-10) of double-β − -type decays with the linear g A model and g A = 0.6 (column 6), using g pp = 0.7. Contributions from the first intermediate state only are in parentheses below the NMEs calculated using all contributions. The computation for 136 Xe was made for (a) E 1 + 1 = 0.0 MeV and (b) E 1 + 1 = 1.0 MeV. The experimental half-lives (column 11) are from Ref. [54] and the lower limits from Ref. [55].  that although this value of g pp works well for the geometric means of the left-and right-side NMEs, it does not remove the ambiguity of g pp for individual NMEs. Eleven β − -type and 10 β + /EC-type processes were examined. The phase-space integrals used to calculate the double-β decay matrix elements are tabulated in Table IV. The calculated matrix elements and half-lives for the β − processes are presented in Table V and for the β + /EC mode decays are in Table VII for systems with  A 120 and Table VIII for A 124. It should be noted that half-lives are given only for processes that are allowed by positive Q value while the values of the NMEs are calculated nevertheless.
The success of our predictions can only be evaluated by the experimental half-lives known for five β − β − processes. The  computed and experimental half-lives of the ground-state-toground-state decays of these nuclei are tabulated separately in Table VI. In only one of these cases the experimental half-life is better reproduced by our linear g A model than the constant value as seen in Table VI, namely, the decay of 116 Cd. The largest deviations from the experimental half-lives with the linear model appear in the decays of 100 Mo and 136 Xe where the discrepancy is of a factor of 3-6. One must bear in mind, though, that the A = 136 process is an extrapolation of our linear g A model and the results must be considered with caution.
To compare the linear model and constant g A we introduce a quality factor, which reads These calculated quality factors are presented in Table VI for each decay process with experimental data available. The mean of the individual quality factors yields = 0.86 for the linear g A model and = 0.34 for the constant g A = 0.6.
It seems that the 2νβ − β − decay favors the constant g A = 0.6 instead of one fitted via the systematics of single-β decay. Even with a function which more closely follows the zigzag behavior of Fig. 3 one would not find much improvement, as some of the predictions would essentially be corrected in the wrong direction. For example, the half-life of the 130 Te decay is already predicted as too short by the linear model and taking the exact value from Fig. 3 would lead to a larger g A which in turn would lead to an even shorter half-life. In the basic QRPA framework, the single-and double-β decays seem to follow somewhat different g A schemes as pointed out in Refs. [37,38] already. For the processes with experimental data available, we find that the calculated half-lives are not very far off the experimental values for either the linear or the constant g A . One might thus assume that a within-order-of-magnitude level of accuracy holds also for the processes with yet unmeasured half-lives. Of the processes with no present experimental data available, the ground-state-to-ground-state decay of 110 Pd, with a predicted half-life of 2.2 × 10 20 years appears the most attractive for future experimental probing. 054309-12 The calculated half-lives of 2νβ + /EC decays are presented in Tables VII and VIII. The matrix elements of the processes involving electron capture can be calculated separately for an electron captured from the K shell and an electron from the L shell. The values of these matrix elements are often very close to each other and, to a reasonable approximation, the mean value of these matrix elements can be used to represent the β + EC-and ECEC-type transitions. The matrix elements given in Tables VII and VIII are the mean values of the K-capture and L-capture matrix elements for the β + EC mode and KK and KL matrix elements for the ECEC mode. The half-lives are again calculated by using the linear g A model and a constant g A = 0.6.
The positron-emitting double-β + and EC decays have so far eluded experimental detection, thus making it difficult to evaluate the success of our predictions. However, the study of β − -type decays gave a general insight of the quality of the predictions therein, and a similar level of accuracy is to be expected for the positron-emitting transitions.
One should also note that the NMEs of the different modes of positron decay are quite in the same order of magnitude, but the half-lives differ largely. It is thus not the matrix element but the phase-space integral which determines the order of the half-lives of the different modes. Of the positron-decay modes, the ECEC channel to the ground state appears to be the fastest in every investigated process. The β + EC mode is not far behind in some cases. These modes in the ground-state decay of 106 Cd, 112 Sn, and 130 Ba, with predicted half-lives of the order of 10 21 years, seem favorable for experimental investigation in the future. The 2νβ − decays of the nuclei 104 Ru, 110 Pd, and 124 Sn were calculated recently in Ref. [30]. One should notice that in Ref. [30] only the lower limits of the half-lives were given due to difficulties in determining the model parameters accurately. No systematic analysis of the β-decay transitions was performed there such that it could help in determining the 2νβ − half-lives more accurately. The lower limits calculated in Ref. [30] are in good agreement with our present results. Our calculated values for both the linear and constant g A appear to systematically lie roughly one order of magnitude above the lower limits. The approach to fitting the model parameters in Ref. [30] was based on a choice of values for g A that were less quenched than the values found in the present work by varying both g A and g pp .  The double-β decays of 106 Cd were previously treated theoretically in Refs. [18,19]. In Ref. [18] the 2νβ + /EC and in Ref. [19] the 0νβ + /EC decays were computed. A comparison of our present work with the 2001 results of Ref. [18] is made in Table IX. Our linear g A model as well as the constant g A = 0.6 predict the transitions of 106 Cd systematically slightly faster than the method used in Ref. [18]. The orders of magnitude are generally the same. As there is no experimental data available for double-positron decay modes, it is difficult to evaluate the results further.
The case of 124 Xe 2νβ + /EC and 0νβ + /EC decays was treated in Ref. [22]. The decay half-lives of the 2νβ + /EC mode have a wide range of variation in Ref. [22] since the fixing of the model parameters in that work was rather loose due to the lack in experimental input and in systematic study of the model parameters. Contrariwise, in the present work the obtained systematics help pin down the calculated halflives more accurately. Comparing our calculated 2νβ + /EC half-lives with the corresponding ones of Fig. 2 of Ref. [22] we notice that our results with both the linear and constant g A predict the half-lives to be longer than those in Ref. [22], especially for the decay to the ground state of 124 Te where the present predictions do not quite fit the range of variation in Ref. [22]. In decays to higher excited states the calculated half-life intervals of Ref. [22] are so wide that they contain our present results, which still always lie toward the longer end of the half-life interval. In Ref. [22] the values of the axial-vector coupling constant were varied within g A = 1.00-1.25, which is a range that is higher than the effective value of g A given by our present analysis. The values of g pp in Ref. [22] were taken to be g pp = 0.80-0.90, corresponding to the range of g A . This range of g pp fits our analysis of Fig. 3 but here we find that the effective value of g A for these isobars is required to be quenched below 1.00 by the systematics of single-β decay. Considering the quite different adopted values of g A , the differences in the present work and Ref. [22] seem reasonable. The single-state dominance hypothesis (SSDH) was discussed systematically in Refs. [56,57]. In the present study the following nuclear systems seem to satisfy the criteria of SSDH: 100 Mo, 104 Ru, 110 Pd, 114 Cd, and 116 Cd of the β − -type processes and 102 Pd, 106 Cd, 108 Cd, and 112 Sn of the β + /EC-type processes. The realization of the SSDH appears to be dependent on the mass number A. The contributions from higher intermediate states start to dominate after A = 116, giving the transition more strength than with the first contribution only. Typical accumulation of the NMEs is depicted in Figs. 10-12 for the β − β − decay of 110 Pd and 124 Sn and the β + β + decay of 124 Xe respectively. One can see in Fig. 10

IV. CONCLUSIONS
In this work we have scanned experimental data on Gamow-Teller β − and β + /EC decay rates (comparative half-lives, i.e., log f t values) in triplets of nuclear isobars in the mass region A = 100-136. We found 24 triplets where an odd-odd nucleus appears between two even-even nuclei and the connecting β-decay transitions have measured log f t values. The decay amplitudes associated to the mentioned decays can also be calculated, in this case by the proton-neutron QRPA. In the present analysis we have used the geometric mean of the two β-decay rates since, owing to the characteristic behavior of the pnQRPA-computed β − and β + /EC decay amplitudes, that quantity is almost independent of the value of the g pp parameter of the pnQRPA. The experimental geometric means can be reproduced for each triplet separately by varying the value of the axial-vector coupling constant g A . Our study predicts a rough piecewise linear increase in the value of g A as a function of the mass number A. We have used this A dependence of g A to predict two-neutrino double-β decay half-lives, both for the 2νβ − and 2νβ + /EC transitions, in altogether 21 isobaric triplets. Decays to both the ground state and lowest few excited states have been considered and a comparison has been made between results for the linear g A and an average constant value of g A = 0.6.
The linear model for g A was found to be quite accurate in describing the ground-state-to-ground-state single-β decays with g pp = 0.7. This value of the particle-particle parameter was then adopted for the examination of the double-β decays. The calculated half-lives for β − β − decays were in decent agreement with experimental data in cases where such data was available although better accuracy was obtained by using the constant value of g A = 0.6. This seems to support the conclusions of Refs. [37,38] that the single-and double-β decays follow a different trend in terms of variation of g A .
For the half-lives of the remaining β − β − processes as well as the positron-emitting double-β + /EC decays with no present experimental data, our study produced predictions that can be expected to be not more than an order of magnitude off the true values. The most attractive processes to probe experimentally in the future seem to be ground-state-to-ground-state decays of 110 Pd (β − β − ), 106 Cd (ECEC), and 112 Sn (ECEC) with predicted half-lives of 2.