Realistic shell model calculation of $2\nu \beta \beta$ nuclear matrix elements and role of shell structure in intermediate states

Abstract

We discuss two conditions needed for correct computation of $2\nu \beta \beta$ nuclear matrix-elements within the realistic shell-model framework. An algorithm in which intermediate states are treated based on Whitehead's moment method is inspected, by taking examples of the double GT${}^{+}$ transitions \mbox{^{36}Ar}\rightarrow\mbox{^{36}S}, \mbox{^{54}Fe}\rightarrow\mbox{^{54}Cr} and \mbox{^{58}Ni} \rightarrow\mbox{^{58}Fe}. This algorithm yields rapid convergence on the $2\nu \beta \beta$ matrix-elements, even when neither relevant GT${}^{+}$ nor GT${}^{-}$ strength distribution is convergent. A significant role of the shell structure is pointed out, which makes the $2\nu \beta \beta$ matrix-elements highly dominated by the low-lying intermediate states. Experimental information of the low-lying GT${}^{\pm }$ strengths is strongly desired. Half-lives of T^{2\nu}_{1/2}({\rm EC}/{\rm EC}; \mbox{^{36}Ar}\rightarrow\mbox{^{36}S})=1.7\times 10^{29}\mbox{yr}, T^{2\nu}_{1/2}({\rm EC}/{\rm EC};\mbox{^{54}Fe}\rightarrow \mbox{^{54}Cr})=1.5\times 10^{27}\mbox{yr},T^{2\nu}_{1/2}({\rm EC} /{\rm EC};\mbox{^{58}Ni}\rightarrow\mbox{^{58}Fe})=6.1\times 10^{24}\mbox{yr}and T^{2\nu}_{1/2}(\beta^+/{\rm EC};\mbox{^{58}Ni} \rightarrow\mbox{^{58}Fe})=8.6\times 10^{25}\mbox{yr} are obtained from the present realistic shell-model calculation of the nuclear matrix-elements.

Comments: LaTeX 28 pages & 5 Postscript figures, to appear in Nucl. Phys. A