$\beta \mathrm{-decay}$ rates of neutron-rich Zr and Mo isotopes in the deformed quasiparticle random-phase approximation with realistic interactions

The ${\beta }^{-}\mathrm{-decay}$ rates of neutron-rich Zr and Mo isotopes are investigated within the deformed quasiparticle random-phase approximation with realistic nucleon-nucleon interactions. Axially symmetric deformations are considered in the calculations from a mean-field description of a quasiparticle picture and of both particle-hole and particle-particle excitations. The Brückner $G$ matrix obtained with charge-dependent Bonn nucleon-nucleon forces is employed for residual particle-particle and particle-hole interactions. Contributions from both allowed Gammow-Teller and first-forbidden transitions are calculated and the sensitivity of the calculated results to the particle-particle strength is discussed. The calculated $\beta$-decay half-lives show a good agreement with the available experimental data. Moreover, predictions of $\beta$-decay half-lives are made for some extremely neutron-rich isotopes, which could be useful for future experiments.

Figure 1

Dependence of the calculated $\beta$-decay rates on the $g_{pp}$ value for the ${\beta }^{-}$ decay of ${}^{112}$Mo: (a) ratio of the calculated half-life to the experimental data versus particle-particle strength $g_{pp}$' (b) branching ratio for FF transitions versus particle-particle strength $g_{pp}$.

Figure 2

Low-lying average shape factors for GT and FF transitions as a function of the excitation energy $E_{\mathrm{ex}}$ for the ${\beta }^{-}$ decay of ${}^{106}$Zr. (a) Average shape factors for GT and FF transitions calculated in the prolate shape with ${\beta }_2=0.373$, which is suggested by the FRDM in Ref. [3]; (b) results calculated in the spherical shape with ${\beta }_2=0.0$; (c) results calculated in the oblate shape with ${\beta }_2=-0.373$. The shape factor for GT transitions is denoted by black lines and the average shape factor for FF transitions is denoted by lighter (red) lines.

Figure 3

Low-lying average shape factors for different components 0${}^{-}$, 1${}^{-}$, and 2${}^{-}$ of FF transitions as a function of the excitation energy $E_{\mathrm{ex}}$ for the ${\beta }^{-}$ decay of ${}^{106}$Zr. Calculations are performed in the prolate shape with ${\beta }_2=0.373$, which is suggested by the FRDM in Ref. [3].

Figure 4

Comparison of the calculated half-lives [denoted GT + FF] with the available experimental data (labeled Expt) for neutron-rich Zr and Mo isotopes. The theoretical results of Möller et al. (denoted FRDM + QRPA) are also shown for comparison [10], including the calculations for both GT and FF transitions. For ${}^{106-112}$Mo, calculations are also performed with the deformation parameters predicted by the RMF [5], where negative values for the quadrupole deformation parameter ${\beta }_2$ are suggested, in contrast to the FRDM predictions, and the corresponding results [denoted GT + FF (RMF)] are displayed in (b). Three regions (I–III) are distinguished, according to the source of the $Q_{\beta }$ values used in the calculations: (I) experimental $Q_{\beta }$ values [44], (II) $Q_{\beta }$ values obtained from systematics [44], and (III) theoretical $Q_{\beta }$ values obtained from the KUTY mass formula [45].