Interplay of quasiparticle-vibration coupling and pairing correlations on β-decay half-lives

Article history: Received 1 November 2017 Received in revised form 24 January 2018 Accepted 26 February 2018 Available online 28 February 2018 Editor: J.-P. Blaizot


Introduction
The nuclear β-decay is an important weak-interaction process, which not only provides information on spin and isospin properties of the nuclear effective interaction, but also sets the time scale of the rapid neutron-capture process (r-process), and hence determines the production of heavy elements in the universe [1][2][3].
Moreover, in particle physics β-decay has provided the first experimental evidence of parity violation [4] and can be employed to verify the unitarity of the CKM matrix [5,6]. Important advances in the measurement of β-decay half-lives have been achieved in recent years with the development of radioactive ion-beam facilities. For example, the β-decay half-lives of 38 neutron-rich isotopes from 36 Kr to 43 Tc with neutron number between 50 and 82 [7], of 20 neutron-rich nuclei with Z = 27 − 30 in the 78 Ni region [8], of 110 neutron-rich isotopes of elements between 37 Rb and 50 Sn (across the N = 82 shell gap) [9], and of 94 neutron-rich isotopes between 55 Cs and 67 Ho [10], were measured at RIKEN. The newly obtained experimental data provide an opti-* Corresponding author.
E-mail address: yifei .niu @eli -np .ro (Y.F. Niu). mal testground for theoretical models. The r-process nuclei at the N = 126 shell closure are, instead, still not experimentally reachable; thus, accurate theoretical models for half-life calculations are crucial for the study of the r-process nucleosynthesis.
QRPA is one of the most commonly used models for the calculation of the β-decay half-lives. Many versions of QRPA have been implemented to this aim, from models that are based at least in part on phenomenological ingredients [11][12][13][14][15] to self-consistent models without adjustable parameters based on both nonrelativistic [16][17][18][19][20][21] and relativistic Energy Density Functionals (EDFs) [22][23][24][25][26][27][28][29][30][31]. In the QRPA calculations, the half-lives are usually overestimated; the remedy often consists in tuning up the strength of an attractive isoscalar (IS) proton-neutron (pn) pairing force which is introduced in QRPA and does not affect ground-state properties [14,16,[26][27][28][29]. However, the isoscalar pn pairing has no, or little, effect on closed-shell nuclei like 78 Ni and 132 Sn as well as on the nuclei before the shell closure [16,[26][27][28][29]. Therefore, there must exist other correlations than isoscalar pn pairing, that are capable to reduce such half-lives. One possible candidate is an attractive tensor force as it has been pointed out in Refs. [32,33]. Also this requires the introduction of new terms in the Hamiltonian and the tuning of one or more parameters. Another possible solution is the inclusion of new correlations beyond QRPA without introducing new parameters. This is particularly evident in the case of magic nuclei, in which QRPA becomes simple RPA. The RPA approach is restricted to configurations of one particle-one hole (1p-1h) nature. It is well known that this is not sufficient for the description of the spreading width of giant resonances. In order to reproduce the observed widths, one must consider the damping caused by the coupling to more complicated states, like 2p-2h configurations [34][35][36][37][38]. An effective way to account for (most of) the observed spreading widths is to take into account the coupling of single-nucleon states to the collective lowlying (mainly surface) nuclear vibrations or phonons (that is, to include 1p-1h-1 phonon configurations in the model space) [39]. We call this model RPA plus PVC, and we note that self-consistent versions of such a model have been realized based on both relativistic and nonrelativistic energy functionals. Good agreement with the experiment has been obtained for the line shape of the Gamow-Teller (GT) and spin-dipole strength distribution [40][41][42][43]. In our application of the RPA+PVC model, based on Skyrme energy density functionals [42,43], it has been found that the coupling to phonons produces a downward shift of the GT strength, accompanied by the development of broadening and/or fragmentation.
Since β-decay is dominated by the low-lying GT transitions below the Q β value, it is expected that the effect of PVC on the GT excitations can help in reducing the half-lives. To demonstrate the effect of PVC on half-lives, we have applied our self-consistent Skyrme RPA+PVC model to magic nuclei, and have shown that the inclusion of PVC considerably improves the agreement of β-decay half-lives with experiment [44]. However, the above investigation was limited to magic nuclei since pairing correlations were not included. Recently, the self-consistent QRPA with quasiparticlevibration coupling (QRPA+QPVC) model, based either on Skyrme EDFs [45] or on relativistic EDFs [46], was developed and used to study the GT transitions in superfluid nuclei. This new development opens up the possibility to explore the β-decay half-lives for a whole isotopic chain. In Ref. [46], the half-lives of Ni isotopes were investigated by means of the QRPA+QPVC model based on the relativistic EDFs, and it was found that the inclusion of QPVC can reduce the half-lives. However, isoscalar pairing was not included in the calculations.
In this Letter, we are going to use the self-consistent QRPA+ QPVC model based on the Skyrme EDFs, with the inclusion of isoscalar pairing, to investigate the half-lives of Ni and Sn isotopes. Since Ni and Sn isotopes have a magic number of protons, they will be considered as spherical in our calculations. In particular, the effects of isovector pairing and isoscalar pairing, as well as their interplay with effects related to QPVC, are explored in detail. The comparison of the theoretical results with experimental data can give relevant indications on the value of the isoscalar pairing strength, which is still under active discussion. Several pieces of information, that have been reviewed in [47], point to a value of the IS pairing strength of the order of 1.5 that of the IV pairing. Further independent confirmations of this finding would be very valuable.

Formalism
We first carry out a self-consistent HFB+QRPA calculation of the GT strength, using the SkM * effective interaction. We choose the interaction SkM * because it gives a good description of GT resonances in magic and superfluid nuclei and β-decay half-lives in magic nuclei at the (Q)PVC level, consistently improving the results obtained in QRPA, as shown in previous studies [42][43][44][45]. The detailed formulas of charge-exchange QRPA on top of HFB can be found in Ref. [48]. In our formalism, only the IV T = 1 pairing correlations are taken into account to describe the ground-state. Since we do not consider nuclei close to N = Z this should be a reasonable approximation. Pairing must be included also in the residual QRPA force, and for a zero-range force only its T = 0 component plays a role for the GT excitations. We adopt a density-dependent, zero-range, surface pairing force which is parameterized as follows [48]: where r = (r 1 + r 2 )/2, ρ 0 is taken to be 0.16 fm −3 , and P σ is the spin exchange operator. The QRPA in its standard matrix form produces a set of 1 + eigenstates |n with forward-going and backward-going amplitudes ab , having energies E n and associated values of the B(GT) strength B n . We then couple these QRPA states with a set of doorway states consisting of a two-quasiparticle excitation coupled to a collective vibration. The properties of these collective vibrations, i.e. phonons |nL , are, in turn, obtained by computing the QRPA response with the same Skyrme interaction, for states of natural parity L π = 0 + , 1 − , 2 + , 3 − , 4 + , 5 − , and 6 + . The phonons are built out of proton-proton or neutron-neutron excitations, so only the IV pairing is included in the residual interaction entering the QRPA calculation. We have retained in our model space the phonons with energy less than 20 MeV and absorbing a fraction of the non-energy weighted isoscalar or isovector sum rule (NEWSR) strength larger than 5%.
The self-energy of the QRPA state |n is given by where the matrix elements W ↓ ab,a b (E M ) are spreading terms associated with the coupling of two-quasiparticle configurations with the doorway states, and defined in Ref. [45]. They are complex and energy-dependent, and they must be calculated by using an averaging parameter η that avoids divergences and represents, in an approximate way, the coupling of the doorway states to even more complicated configurations. We have found that the lifetimes converge for small values of η, which are expected to be appropriate for low-lying, discrete states. In our calculations we have set η = 200 keV. One can then calculate the GT strength distribution from the Gaussian averaging [49], where σ n is defined as σ n = ( Once the strength function has been obtained, the β-decay halflife of an even-even nucleus is calculated in the allowed GT approximation by using the expression [16,29,44,50] where D = 6163.4 s, and we set g A equal to 1 rather than to its free value of 1.26, to account for the quenching of isovector spin matrix elements [51,16], as is commonly done in QRPA calculations [16,15,[18][19][20]28,29]. Notice that here the energy E refers to the ground state of the daughter nucleus, which is related to the energy E M referred to the ground state of the mother nucleus by where p e , E e , and F 0 (Z + 1, E e ) denote the momentum, energy and Fermi function of the emitted electron, respectively. ω is the energy difference between the initial and final nuclear state, con- MeV is the mass difference between neutron and proton). respectively. In order to study the effect of isovector pairing on the β-decay half-lives, we vary these pairing strength parameters by ±10%, i.e., we change f iv from 0.9 to 1.1 (see Eq. (1)). The resulting pairing gaps vary by about ±35% and are shown in Fig. 1.

Isovector pairing
In Fig. 2, the β-decay half-lives of Ni and Sn isotopes, calculated either without pairing correlations ( f iv = 0.0) or with different isovector pairing strengths, are compared with the experimental data. In panels (a) and (c), the half-lives are calculated at the QRPA level, while in panel (b) and (d) they are calculated within the QRPA+QPVC model. So far, isoscalar pairing correlations are not included ( f is = 0.0). The first general observation is that the influence of IV pairing correlations is small, compared to the effects of QPVC. Looking in more detail at the results, one notices that the effect of f iv is negligible for the isotopes close to, and especially beyond, the magic nuclei 78 Ni and 132 Sn (notice that contrary to experiment, the isotopes 130-134 Sn are predicted to be stable in QRPA and then do not appear in Fig. 2(a)). In the case of Ni isotopes, the effect of f iv increases progressively going from 78 Ni towards 68 Ni, that is, towards the middle of the shell where n is maximum (see Fig. 1). Larger values of f iv lead to longer lifetimes. In the case of Sn isotopes, the lifetimes of 140-146 Sn instead first decrease when pairing IV correlations are switched on going from f iv = 0 to f iv = 0.9, and then increase going to f iv = 1.1.
This behaviour can be understood considering that pairing correlations have two competing effects on lifetime. On the one hand, a larger value of f iv leads to a larger pairing gap and then to larger GT excitation energies, which in turn reduces the β-decay phase space and increases the lifetime. On the other hand, pairing correlations modify the occupation of single-particle levels and in some cases induce new low-energy transitions that tend instead to reduce the lifetime. In order to have a more microscopic understanding of the dependence of the lifetime on f iv for the nuclei shown in Fig. 2, we plot the single-particle spectra of 80 Ni and 134 Sn in Fig. 3. Let us consider Ni isotopes first. Going from 68 Ni to 78 Ni the 1ν g 9/2 level is being filled; pairing correlations slightly modify the occupation factors and do not lead to new transitions. The most important transition determining the lifetime is ν2p 1/2 → π2p 3/2 . In the presence of pairing correlations, the energy of the corresponding 2qp excitation is increased and the lifetime becomes longer for increasing values of f iv . This transition remains the dominant one also in the case of 80-84 Ni, but is less influenced by pairing correlations due to the shift of the chemical potential to the shell above. As a consequence, the lifetime is insensitive to the value of f iv . Turning now to Sn isotopes, the situation for 136-140 Sn is qualitatively similar to the case of ing pairing on tends to reduce the lifetime; further increasing the value f iv the competing effect on quasiparticle energies prevails and the lifetime increases, as was already noticed in Fig. 2(c).
The QRPA+QPVC results are shown in Fig. 2(b) and (d), again for different values of the isovector pairing strength. With the inclusion of QPVC, the half-lives are reduced systematically. The experimental half-lives of Ni isotopes are well reproduced, while the half-lives of Sn isotopes are still overestimated, although much less than before. We remind that no isoscalar pairing has been introduced so far. At the QPVC level, we find that the effect of isovector pairing is greatly reduced; we will discuss the underlying reason making reference to Fig. 4, in which 70 Ni has been picked up as an example.
In Fig. 4, the GT strength distributions (with respect to the daughter nucleus), the cumulative sums of the GT strengths, and  the arrows indicate the experimental energies of the measured 1 + states. At the QRPA level, one observes a single peak accompanied by a small strength at lower energy, which is only visible in panel (d) when multiplied by a factor of 10, in each case for f iv = 0, 1.0 and 1.1. The QRPA excitation energy is increased as the pairing strength increases, which leads to an increase of the halflife, as discussed above. The QRPA excitation energy is much higher than the experimental 1 + state energies, resulting in a strong overestimation of the experimental half-life (see Fig. 2(a)).
With the inclusion of QPVC, the QRPA peak is shifted downwards and some GT strength appears in the region of the experimental 1 + states. The value of the shift is about 2 MeV for f iv = 0 and about 2.8 MeV for f iv = 1.0 and f iv = 1.1 (cf. Figs. 4(a) and (d)). This difference is related to the fact that the collectivity of the 2 + phonon and the associated QPVC vertex are much reduced in the absence of pairing correlations. The difference in the values of the energy shifts counterbalances the effect found at the QRPA level. As a consequence, the excitation energies and the lifetimes calculated with and without pairing turn out also to be similar (see Fig. 4(c) and Fig. 2(b)). Obviously, however, the experimental properties of low-lying collective phonons are much better described when pairing is included.

The role of isoscalar pairing
We investigate the role of isoscalar pairing in Fig. 5, where the β-decay half-lives of Ni and Sn isotopes, calculated by QRPA and QRPA+QPVC model without isoscalar pairing ( f is = 0.0) or with different isoscalar pairing strengths, are shown and compared with the experimental data. The strength of isovector pairing is fixed at f iv = 1.0. IS pairing has almost no effect for nuclei with neutron number just before the shell closure (small effects are visible in Fig. 5 in the case of 68,70 Ni), but starts to play a role when neutron number goes across the shell closure, at both QRPA and  [52]. In panel (d), the strengths indicated by arrows are multiplied by a factor of 10 to be visible. In these panels, the excitation energies calculated with respect to the mother nucleus are transformed to E, namely the excitation energies referred to the ground state of daughter nucleus, by using the experimental binding energy difference; accordingly, the vertical dashed lines show the experimental value of Q β . QRPA+QPVC level. The effects are particularly pronounced for Sn isotopes, where the influence of isoscalar pairing is very significant right after the magic nucleus 132 Sn, at variance with the case of IV pairing. The case of 134 Sn is analysed in detail in Fig. 6.
One can refer to the single-particle spectra shown in Fig. 3 in the following discussion. Before the N = 50 or N = 82 shell is filled, the transitions that contribute to the lowest GT energy state are mainly the back spin flip transitions, ν2p 1/2 → π2p 3/2 (for Ni) and ν2d 3/2 → π2d 5/2 (for Sn). For these configurations involving low angular momentum states, the IS pairing matrix elements are small. Furthermore, in the QRPA calculation these matrix elements are multiplied by the factor u π u π u ν u ν + v π v π v ν v ν = u 2 ν factors (the initial nucleus is closed shell in protons, so that v π 2p 3/2 , v π 2d 5/2 = 0) which is also small. Thus, IS pairing does not play a significant role for nuclei with neutron number smaller than N = 50 and N = 82 shell, as already found in refs. [16,26]. On the other hand, after crossing the N = 50 and N = 82 shell closures, the transitions ν1g 7/2 → π1g 9/2 (for Ni) and ν1h 9/2 → π1h 11/2 (for Sn) start to contribute in an important way, because these configurations are associated with large attractive IS pairing matrix elements and large u 2 ν factors. For nuclei just above the shell closure like 80-84 Ni and 134-138 Sn, where the ν1g 7/2 (ν1h 11/2 ) orbital is barely occupied, at f is = 0, the configuration of ν1g 7/2 → π1g 9/2 (for Ni) or ν1h 9/2 → π1h 11/2 (for Sn) has higher unperturbed energy than that of ν2p 1/2 → π2p 3/2 (for Ni) and ν2d 3/2 → π2d 5/2 (for Sn), and hence the lowest GT state is dominated by ν2p 1/2 → π2p 3/2 (for Ni) and ν2d 3/2 → π2d 5/2 (for Sn). With the inclusion of IS pairing, the configuration of ν1g 7/2 → π1g 9/2 (for Ni) or ν1h 9/2 → π1h 11/2 (for Sn) is pushed to lower energy, and attracts strength from other transitions. This produces a low-lying, collective state in the QRPA calculation, which is shown in Fig. 6(d) in the case of 134 Sn. The energy of this state is very sensitive to the value of f is . For nuclei that are further away from the N = 50 and N = 82 shell closures, like 140-146 Sn, the ν1g 7/2 → π1g 9/2 (for Ni) or ν1h 9/2 → π1h 11/2 (for Sn) transition already dominates the lowest GT state in the f is = 0 case; nonetheless, the inclusion of IS pairing further lowers the GT energy due to the attractive character of the matrix elements involving these configurations. At the QPVC level, the effect of IS pairing is similar to that found in QRPA, because the isoscalar pairing does not enter the calculation of the self-energies.
For the Ni isotopes we have considered, since the effect of isoscalar pairing is small, the inclusion of QPVC is crucial to reproduce the experimental half-lives. In the case of Sn isotopes, isoscalar pairing and QPVC cooperate and improve the agreement with experimental data, as compared to the QRPA result with f is = 0 (see Fig. 5(d)). The experimental half-lives are well reproduced in the QRPA+QPVC model with isoscalar pairing strength f = 1.4. This value is in good agreement with that reported in Ref. [47].
In Fig. 6, the GT strength distributions (with respect to the daughter nucleus), the cumulative sums of the strengths, and the cumulative sums of 1/T 1/2 are shown in the case of 134 Sn, in a similar way as in Fig. 4 but for various values of the isoscalar pairing strength. In QRPA, the excitation energy of the main low-lying peak is almost unchanged by increasing f is , whereas at the QPVC level it is not changed going from f is = 0 to f is = 1.2 but it moves downwards going from f is = 1.2 to f is = 1.4. Both in QRPA and QPVC, starting from f is = 1.

Conclusions
In summary, we have calculated the half-lives of a chain of Ni and Sn isotopes around the closed shell nuclei 78 Ni and 132 Sn, in the framework of the self-consistent QRPA+QPVC model based on Skyrme density functional SkM * , which has been shown to lead to a good agreement with experiment for half-lives of magic nuclei [44] and GT transitions of both magic [42] and superfluid nuclei [45]. We have investigated in detail the effects of IV pairing and IS pairing, both at QRPA and QRPA+QPVC level. It is found that IV pairing mainly plays a role for nuclei before closed shell, and it increases the half-lives. However, this effect is greatly reduced when QPVC is taken into account, as compared to QRPA. On the contrary, IS pairing mainly plays a role for nuclei after closed shell, and it decreases the half-lives. The effect of IS pairing is similar at QRPA and QRPA+QPVC level. In our calculations, IS pairing plays an important role in the case of Sn isotopes beyond 132 Sn, where it reinforces the action of QPVC in reducing the half-lives; good agreement with experiment is found assuming a value of the isoscalar pairing strength f is = 1.4. Most of the Ni isotopes we have considered lie before 78 Ni, so that the effect of isoscalar pairing is small; also in this case good agreement with experiment is found once the QPVC is included. In the present work, only GT transition is considered in the half-life calculation. The study on the role of forbidden transitions in the framework of QRPA+QPVC is envisaged.