Unique first-forbidden β-decay transitions in odd–odd and even–even heavy nuclei

Abstract

The allowed Gamow–Teller (GT) transitions are the most common weak nuclear processes of spin–isospin $(\sigma \tau )$ type. These transitions play a key role in numerous processes in the domain of nuclear physics. Equally important is their contribution in astrophysics, particularly in nuclear synthesis and supernova-explosions. In situations where allowed GT transitions are not favored, first-forbidden transitions become significant, specifically in medium heavy and heavy nuclei. For neutron-rich nuclei, first-forbidden transitions are favored mainly due to the phase-space amplification for these transitions. In this work we calculate the allowed GT as well as unique first-forbidden (U1F) $|\mathrm{\Delta }J|=2$ transitions strength in odd–odd and even–even nuclei in mass range $70\le A\le 214$. Two different pn-QRPA models were used with a schematic separable interaction to calculate GT and U1F transitions. The inclusion of U1F strength improved the overall comparison of calculated terrestrial β-decay half-lives in both models. The ft values and reduced transition probabilities for the $2^{-}⟷0^{+}$ transitions were also calculated. We compared our calculations with the previously reported correlated RPA calculation and experimental results. Our calculations are in better agreement with measured data. For stellar applications we further calculated the allowed GT and U1F weak rates. These include ${\beta }^{\pm }$-decay rates and electron/positron capture rates of heavy nuclei in stellar matter. Our study shows that positron and electron capture rates command the total weak rates of these heavy nuclei at high stellar temperatures.

Introduction

The connection between the generation of energy in stars and weak interaction leads to large-scale stellar events. Supernova explosions and related physics are one of the most studied phenomena to be known in astrophysics. Supernovae are intimately connected with the nucleosyntheis problem (e.g. [1]). The weak decay processes are the crucial constituents in all major astrophysical events. The core of the massive star collapses due to weak interaction reactions activating a supernova explosion. Another key phenomenon where weak interactions play a significant part includes neutronization of stellar core via capturing of electrons by nuclei and by free protons. This process effects the creation of heavier elements beyond iron through r-process during late phases of evolution of massive stars. The weak rates determine the mass of the core and provide a fair estimate of the fate and strength of the shock wave produced by the supernova outburst [2], [3]. The β-decay properties of neutron rich nuclei are essential in order to understand the r-process. Though in astrophysical context the site of the r-process is not known with certainty, it is normally accepted that it takes place in an explosive environment possessing extremely high neutron densities ($\ge {10}^{20}{\text{ cm}}^{-3}$) and high temperatures ($T\ge {10}^9\text{ K}$). Under these conditions, neutrons are captured more rapidly than the competing β-decays and in the nuclear chart the r-process path goes through the neutron-rich domain with comparatively small (and having round-about constant) neutron separation energies of ≲ 3 MeV [4].

The theoretical explanation of weak processes, especially the double β-decay process, is an open question for the nuclear-structure theories and it has significance to explore physics beyond the standard model [5]. The compiled theoretical and experimental results can be seen in [6]. The analysis of some of the recent developments, both in experiments and theory were discussed in [7]. The estimation of β-decay half-lives, in agreement with the experimental values, is one of the challenging difficulties for nuclear theorists. The β-decay rates and half-lives of nuclei are determined widely by using various nuclear models. Takahashi et al. [8] calculated these rates using the gross theory which is statistical in nature. In this model the shell structure of nucleons is not entirely accounted for and the theory takes into account the average values of β-strength functions. Very soon it was realized to use a microscopic model for a more reliable calculation of β-decay half-lives. In studies of nuclear β-decay properties the proton–neutron quasi particle random phase approximation (pn-QRPA) theory has been widely used in literature. In pn-QRPA a quasiparticle basis via pairing interaction is constructed first, and then the equation of RPA having schematic Gamow–Teller (GT) residual interaction is solved. Sorensen and Halbleib [9] developed this model by simplifying the usual RPA to calculate the relevant transitions. The pn-QRPA calculations were then extended to deformed nuclei by many authors [10], [11], [12], [13], [14], [15]. Microscopic calculations of allowed weak rates, from atomic number 6 to 114, were first performed by [16]. Later the pn-QRPA model was used to calculate β-decay properties of nuclei far from line of stability both in the β-decay [17] and electron capture direction [18]. These calculations [17], [18] highlighted the strong and reliable predictive power of the pn-QRPA model specially as one moves far from the line of stability. The pn-QRPA model was later used for calculation of unique first-forbidden (U1F) transitions ($|\mathrm{\Delta }J|=2$) by [19], [20] under terrestrial conditions. This calculation demonstrated that for near-magic and near-stable nuclei, greater contribution to the total transition strength came from U1F transitions (see Fig. 9 and Table IX of [19]). It was Nabi and Klapdor-Kleingrothaus who used the pn-QRPA model, for the first time, to calculate weak interaction rates in stellar matter [21], [22], [23]. The same model was later modified to calculate U1F rates in stellar matter by Nabi and Stoica [24]. The QRPA studies based on the Fayans energy functional was extended by Borzov for a consistent treatment of allowed and first-forbidden (FF) contributions to r-process half-lives [2]. In [2] it was shown that the first forbidden transitions contribute dominantly to the total weak decay half-life, mostly for the nuclei having closed Z and N shells. More recently the authors of Ref. [4] calculated the half-lives for r-process waiting-point nuclei using the large scale shell-model including FF contributions.

There are several experimental and theoretical results available about the allowed transitions but literature is rather scarce when it comes to discussion of FF transitions. Suhonen [25] studied the electron capture/${\beta }^{+}$ transitions for ¹³⁶La(1⁺) → ¹³⁶Ba($J^{\pi }$) and β-decay properties for ¹³⁶Cs(5⁺) → ¹³⁶Ba($J^{\pi }$) and ¹³⁶I(2⁻) → ¹³⁶Xe($J^{\pi }$). The β-decay strength were calculated in allowed and FF approximations including only the ground state transitions. The β-decay properties for odd–odd nuclei to the excited levels of neighboring even–even nuclei were discussed in detail within the QRPA theory. A common vacuum was assumed and harmonic oscillator basis were employed in this calculation [25]. The effects of spin–isospin dependent interactions on even–even and odd–odd nuclei and FF beta decay transitions for $|\mathrm{\Delta }J|=0$, 2 were studied by Civitarese et al. [26] using RPA technique for two different quasiparticle excitations. Here the authors assumed that the relativistic β-moment $M^{\pm }$ $({\rho }_A,\lambda =0)$ is proportional to the matrix element of non-relativistic one. For $0^{+}⟷0^{-}$ the experimental and calculated ft values were in better agreement. For U1F transitions the renormalization effects improved the theoretical values (see Fig. 3 of [26]). The $0^{+}⟷0^{-}$ FF beta transitions for some spherical nuclei in the mass region $90\le A\le 214$ were later studied by Çakmak et al. [27] using the pn-QRPA method. In this work the relativistic β-moment matrix elements were calculated directly and without any assumption. The results obtained by [27] were in better agreement with the experimental data and previous calculations. Using the shell-model, the β-decay half-lives of $N=126$ isotones have been calculated by Suzuki et al. [28], considering both the GT and FF transitions. It was found that the FF transitions reduce the half-lives, by nearly twice to several times, from those by the GT contributions only. Recently the β-decay half-lives and β-delayed neutron emission probabilities (${\mathrm{P}}_n$) for 5409 nuclei have been studied by [29] by using the pn-RQRPA model. It was concluded that FF transitions contribute significantly to the total decay rate and must be taken into account regularly in modern evaluations of half-lives.

In this paper we calculate U1F β-decay transitions for both spherical and deformed odd–odd and even–even nuclei using two different pn-QRPA models, having separable residual GT interactions. Earlier rank 0 ($0^{+}⟷0^{-}$) FF transitions for these heavy nuclei were studied by Çakmak et al. [27] using the pn-QRPA(WS) method. We are currently working on the calculation of rank 0 FF transitions using the pn-QRPA(N) model and hope to report this in future. Accordingly we restrict ourselves to calculations of U1F β-decay transitions in this paper. For the first pn-QRPA(WS) calculation, we employ the Woods–Saxon potential with Chepurnov parametrization [30]. Only the ph term of β-decay effective interaction was considered for calculation of U1F transitions. The pairing correlation constants were taken as $C_n=C_p=12/\sqrt{A}$ for open shell nuclei. The strength parameter of the effective interaction was taken as ${\chi }_{U1F}=350A^{-5/3}\text{MeV}{\text{fm}}^{-2}$. The second pn-QRPA(N) model used a deformed Nilsson potential. Further GT interaction was explored both in the particle–hole (ph) and particle–particle (pp) channels in the pn-QRPA(N) model. Using the Nilsson basis, the strength parameters of the ph and pp forces were chosen so as to best reproduce the values of experimental β-decay half-lives. In this project the effective ratio of axial and vector coupling constants $(g_A/g_V)^2{}_{eff}$ was taken as: $(g_A/g_V)^2{}_{eff}=0.7(g_A/g_V)^2{}_{bare}$, with $(g_A/g_V)^2{}_{bare}=-1.254$. The same quenching factor of 0.7 was also used in [26].

The theoretical description of allowed and U1F transitions using the pn-QRPA(WS) and pn-QRPA(N) models are discussed in next section. The pn-QRPA(N) model was later used to calculate allowed GT and U1F rates in stellar matter. We are currently working on calculation of stellar rates using the pn-QRPA(WS) model and this would be treated as a future assignment. In Sec. 3 we discuss the results of our calculations and also compare them with experimental data and other theoretical models. We finally conclude our findings in Sec. 4.