Deformed shell model results for two neutrino positron double beta decay of ${}^{84}$Sr

Half-lives $T_{1\mathrm{/}2}^{2\nu }$ for two neutrino positron double beta decay modes ${\beta }^{+}$EC/ECEC are calculated for ${}^{84}$Sr, a nucleus of current experimental interest, within the framework of the deformed shell model based on Hartree-Fock states employing a modified Kuo interaction in (${}^2{p}_{3\mathrm{/}2}$, ${}^1{f}_{5\mathrm{/}2}$, ${}^2{p}_{1\mathrm{/}2}$, ${}^1{g}_{9\mathrm{/}2}$) space. For a reasonable description of the spectra of ${}^{84}$Sr and ${}^{84}$Kr and to generate allowed GT strengths, the single particle energies of the proton and neutron ${}^1{g}_{9\mathrm{/}2}$ orbitals, relative to the ${}^2{p}_{3\mathrm{/}2}$ orbital energy, are chosen to be (3.5 MeV, 1.5 MeV) for both ${}^{84}$Sr and ${}^{84}$Rb and (1.5 MeV, 1.5 MeV) for ${}^{84}$Kr. With this, the calculated half-lives for the ${\beta }^{+}$EC and ECEC modes are $\sim {10}^{26}$yr and $\sim 4\times {10}^{24}$yr respectively.