Two-neutrino $\beta \beta$ decays and low-lying Gamow-Teller ${\beta }^{-}$ strength functions in the mass range $A=70\text{–}176$

We apply the proton-neutron deformed quasiparticle random-phase approximation (pn-dQRPA) to describe the low-lying ($E\le 6$ MeV) $1^{+}$ Gamow-Teller (GT) strength functions in odd-odd deformed nuclei which participate as intermediate nuclei in two-neutrino double-$\beta$-decay ($2\nu \beta \beta$) transitions within the mass range $A=70\text{–}176$. In deriving equations of motion we use a single-particle basis with projected angular momentum, provided by the diagonalization of a spherical mean field furnished with a quadrupole-quadrupole interaction. The schematic residual Hamiltonian contains pairing and proton-neutron interaction terms in particle-hole (ph) and particle-particle (pp) channels, with constant strengths. By adopting constant particle-hole and particle-particle strengths we are able to describe the positions of the giant GT resonance and the measured half-lives of the $2\nu \beta \beta$ decays over the whole mass range $A=70\text{–}176$. At the same time we obtain a good agreement with the measured low-lying GT ${\beta }^{-}$ strength functions. By using the adopted ph and pp strengths, we predict the half-lives of a number of deformed $2\nu \beta \beta$ emitters and the low-lying GT strength functions of the corresponding odd-odd intermediate nuclei for their possible experimental tests in the future.

Figure 1

Computed cumulative GT strength (17) (solid lines) vs experimental values (dashed lines) for ${}^{76}\mathrm{Ge}$ [12, 45] (a), ${}^{82}\mathrm{Se}$ [45] (b), ${}^{96}\mathrm{Zr}$ [13] (c), and ${}^{100}\mathrm{Mo}$ [14] (d). The excitation energy is always relative to the ground state of the odd-odd daughter nucleus.

Figure 2

Same as in Fig. 1 for ${}^{128}\mathrm{Te}$ [15] (a), ${}^{130}\mathrm{Te}$ [15] (b), ${}^{136}\mathrm{Xe}$ [10] (c), and ${}^{150}\mathrm{Nd}$ [11] (d).

Figure 3

Cumulative GT strength vs the mass number $A$ summed over energy intervals $E\left(\omega \right)=n-(n+0.5)$ MeV (solid lines) and $E\left(\omega \right)=(n+0.5)-(n+1)$ MeV (dashed lines) for $n=0$ (a), $n=1$ (b), $n=2$ (c), $n=3$ (d), $n=4$ (e), and $n=5$ (f), given in Table 3.