$\beta$-decay half-life of ${}^{50}\mathrm{V}$ calculated by the shell model

In this work we survey the detectability of the ${\beta }^{-}$ channel of ${}_{23}^{50}\mathrm{V}$ leading to the first excited $2^{+}$ state in ${}_{24}^{50}\mathrm{Cr}$. The electron-capture (EC) half-life corresponding to the transition of ${}_{23}^{50}V$ to the first excited $2^{+}$ state in ${}_{22}^{50}\mathrm{Ti}$ had been measured earlier. Both of the mentioned transitions are 4th-forbidden non-unique. We have performed calculations of all the involved wave functions by using the nuclear shell model with the GXPF1A interaction in the full f-p shell. The computed half-life of the EC branch is in good agreement with the measured one. The predicted half-life for the ${\beta }^{-}$ branch is in the range $\approx 2\times {10}^{19}$ yr whereas the present experimental lower limit is $1.5\times {10}^{18}$ yr. We discuss also the experimental lay-out needed to detect the ${\beta }^{-}$-branch decay.

Figure 1

Decay scheme of ${}^{50}\mathrm{V}$ showing the two available decay channels. Both the ${\beta }^{-}$ and the EC branch are 4th-FNU transitions from the $6^{+}$ ground state of ${}^{50}\mathrm{V}$ to the first excited $2^{+}$ states in ${}^{50}\mathrm{Cr}$ and ${}^{50}\mathrm{Ti}$.

Figure 2

Comparison of shell model results with experimental data for ${}^{50}\mathrm{Ti}$, ${}^{50}\mathrm{V}$, and ${}^{50}\mathrm{Cr}$.

Figure 3

Summary of lower limits on the ${\beta }^{-}$ decay branch as a function of time, including one positive evidence claimed and the calculation performed in this work.