Systematic approach to $\beta$ and $2\nu \beta \beta$ decays of mass $A=100–136$ nuclei

In this work we perform a systematic study of pairs of single-$\beta$-decaying nuclei in the mass region $A=100–136$ to extract information on the effective value of the axial-vector coupling constant $g_{\mathrm{A}}$. As the many-body framework we use the quasiparticle random-phase approximation (QRPA) and its proton-neutron variant (pnQRPA) in single-particle valence bases with Woods-Saxon-calculated single-particle energies. It is found that, to a reasonable approximation, $g_{\mathrm{A}}$ is a linear function of the mass number $A$, with a slightly different parametrization below and above the mass $A=121$. Using the values of $g_{\mathrm{A}}$ extracted from the linear fit, as well as an average constant value of $g_{\mathrm{A}}$, we calculate the two-neutrino double-$\beta$ ($2\nu \beta \beta$) decay half-lives of 11 ${\beta }^{-}{\beta }^{-}$ decays and 10 mixed ${\beta }^{+}$ and electron-capture (EC) decays, ${\beta }^{+}{\beta }^{+}/{\beta }^{+}\mathrm{EC}/\mathrm{ECEC}$, to both the ground state and the lowest three excited states.

Figure 1

A schematic figure of the orbitals used to form the bases for nuclei with $A<110$ and $A\ge 110$.

Figure 2

A schematic of the single-$\beta$ 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.

Figure 3

Values of $g_{\mathrm{A}}$ as a function of the mass number $A$. The data points were produced by fitting the theoretical geometric means of the NMEs to the corresponding experimental values.

Figure 4

Theoretical $\beta$-decay matrix elements as a function of the mass number $A$. The computations were done with $g_{\mathrm{pp}}=0.7$. The effect of $g_{\mathrm{A}}$ has not yet been taken into account.

Figure 5

Experimental and theoretical $logft$ values of the ground-state-to-ground-state left branch single-$\beta$ decays across the examined mass region. The computations were made with $g_{\mathrm{pp}}=0.7$ using the linear $g_{\mathrm{A}}$ model (upper panel) and a constant value of $g_{\mathrm{A}}=0.6$ (lower panel). Solid black and hollow red (gray) symbols are used to represent the experimental and theoretical $logft$ values, respectively. Different symbols are used to distinguish different decay processes within the same mass number.

Figure 6

Same as Fig. 5 for the right branch decays.

Figure 7

Experimental and theoretical $logft$ values of decays to the first excited $2^{+}$ states of the even-even nuclei. The computations were made with $g_{\mathrm{pp}}=0.7$ using the linear $g_{\mathrm{A}}$ model (upper panel) and a constant value of $g_{\mathrm{A}}=0.6$ (lower panel). Solid black and hollow red (gray) symbols are used to represent the experimental and theoretical $logft$ values, respectively. Different symbols are used to distinguish different decay processes within the same mass number.

Figure 8

Same as Fig. 7 for decays to the $0^{+}$ quadrupole two-phonon states of the even-even nuclei.

Figure 9

Same as Fig. 7 for decays to the $2^{+}$ quadrupole two-phonon states of the even-even nuclei.

Figure 10

The accumulation of the $2\nu {\beta }^{-}{\beta }^{-}$ matrix element of ${}^{110}\mathrm{Pd}$ decay to ground state. The SSDH works to a reasonable approximation.

Figure 11

The accumulation of the $2\nu {\beta }^{-}{\beta }^{-}$ matrix element of ${}^{124}\mathrm{Sn}$ decay to ground state. The SSDH, in this case, is not realized.

Figure 12

The accumulation of the $2\nu {\beta }^{+}{\beta }^{+}$ matrix element of ${}^{124}\mathrm{Xe}$ decay to ground state. The SSDH, in this case, is not realized.