Calculations of two-neutrino double-$\beta$ decay for $fpg$ shell nuclei

In this paper, we calculate the properties of two-neutrino double-$\beta$ ($2\nu \beta \beta$) decays for nuclei within $fpg$ ($1p_{3/2},0f_{5/2},1p_{1/2},0g_{9/2}$) shell model space systematically under the framework of the nuclear shell model. Both of the decays on the ${\beta }^{-}$ side ($2\nu 2{\beta }^{-}$) and the mixed ${\beta }^{+}$ and electron capture types ($2\nu 2{\beta }^{+},2\nu {\beta }^{+}\mathrm{EC}$, and $2\nu \mathrm{ECEC}$) are taken into consideration. Results of the calculated half lives, nuclear matrix elements (NMEs), phase space factors, and branching ratios are shown. Good agreements between theory and experiments have been reached. Accumulations of the NMEs are also analyzed which do not support the single-state dominance hypothesis. The half lives of the nuclei ${}^{86}\mathrm{Kr},{}^{70}\mathrm{Zn}$, and ${}^{84}\mathrm{Sr}$ are predicted to be $4.2\times {10}^{22},1.4\times {10}^{23}$, and $1.9\times {10}^{23}$ years respectively which appears attractive for future experimental probing.

Figure 1

Comparison of the calculated $T\left(GT\right)$ values with the experimental data. Each point represent a nucleus where the vertical and horizontal axis are the associated experimental and calculated $T\left(GT\right)$ values, respectively.

Figure 2

Half lives for the $2\nu \beta \beta$ decays of the $fpg$ shell nuclei. The vertical axis is for the half lives in ${log}_{10}$ frame in the units of yr while the horizontal axis for the $Q$ values is in units of MeV. Decays for the $2\nu 2{\beta }^{-}$ and $2\nu {\beta }^{+}/\mathrm{EC}$ types are represented by triangles and diamonds, respectively. Hollow squares are for the experimental data.

Figure 3

The NME as a function of $Q$ value for concerned nuclei. Triangles and diamonds represent the $2\nu 2{\beta }^{-}$ and $2\nu \mathrm{ECEC}$ types, respectively.

Figure 4

The accumulations of the NMEs of ${}^{64}\mathrm{Zn}$ decays to ground state. Decays for the $2\nu 2{\beta }^{+},2\nu {\beta }^{+}\mathrm{EC}$, and $2\nu \mathrm{ECEC}$ types are represented by the squares, circles, and triangles respectively. The SSDH, in this case, is not realized.

Figure 5

The accumulations of the NMEs of ${}^{78}\mathrm{Kr}$ decays to ground state. Decays for the $2\nu 2{\beta }^{+},2\nu {\beta }^{+}\mathrm{EC}$, and $2\nu \mathrm{ECEC}$ types are represented by the squares, circles, and triangles respectively. The SSDH is not satisfied.

Figure 6

The accumulations of the NMEs of ${}^{78}\mathrm{Kr}$ decays to the $0_1^{+}$. Decays for the $2\nu 2{\beta }^{+},2\nu {\beta }^{+}\mathrm{EC}$, and $2\nu \mathrm{ECEC}$ types are represented by the squares, circles, and triangles respectively. The SSDH, in this case, is not realized.

Figure 7

The accumulations of NMEs of ${}^{78}\mathrm{Kr}$ decays to the $2_1^{+}$. Decays for the $2\nu \beta \mathrm{EC}$ and $2\nu \mathrm{ECEC}$ types are represented by the circles and triangles respectively. The SSDH, in this case, is not satisfied.

Figure 8

The accumulation of the NME of ${}^{80}\mathrm{Se}2\nu 2{\beta }^{-}$ decay to ground state. The main contribution to the NME comes from the second intermediate $1^{+}$ state which the SSDH is not realized.

Figure 9

The accumulation of the NME for the $2\nu 2{\beta }^{-}$ decay of ${}^{70}\mathrm{Zn}$ to the ground state. The main contribution to the NME comes from the intermediate $1^{+}$ ground state (about $42\%$).