Quadrupolar correlations and deformation effect on two neutrino $\varepsilon β^{+}$ and $\varepsilon \varepsilon$ modes of $^{156}$Dy isotope

The two-neutrino positron double-$β$ decay modes of $^{156}$Dy isotope are studied in the Projected Hartree-Fock-Bogoliubov framework for the $0^{+}\to 0^{+}$ transition. Theoretically calculated half-lives of electron-positron conversion and double-electron capture modes are presented. The effect of the quadrupolar deformation on nuclear transition matrix element $M_{2ν}$ is also investigated.

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