Correlations between magnetic moments and $\beta$ decays of mirror nuclei

We examine the magnetic moments and $\beta$-decay lifetimes of light $T=\frac{1}{2}$ mirror nuclei and obtain very tight correlations between these quantities by making use of shell-model estimates for small quantities. Using the information thus obtained, we predict values for some unknown magnetic moments of heavier $T=\frac{1}{2}$ mirror nuclei. Correlations for the magnetic moments of some $T=\frac{3}{2}$ mirror nuclei are also included. The effective operators required to reproduce the data are discussed and compared to those obtained with other methods.

Figure 1

(Top) (${\gamma }_p+\Delta {\gamma }_p$ ) vs ${\gamma }_{\beta }$ (solid circles) and $-({\gamma }_n+\Delta {\gamma }_n)$ vs ${\gamma }_{\beta }$ (open circles). The lines are the result of fit (B). (Bottom) ${\gamma }_p$ vs ${\gamma }_{\beta }$ (solid circles) and $-{\gamma }_n$ vs ${\gamma }_{\beta }$ (open circles). The lines are the result of fit (A). The single-particle model using free-nucleon values for the coupling constants is shown by the dashed line.

Figure 2

(Top) ${\gamma }_p+\Delta {\gamma }_p$ versus ${\gamma }_n+\Delta {\gamma }_n$. The line is the result of fit (B). (Bottom) ${\gamma }_p$ versus ${\gamma }_n$. The line is the result of fit (A). The single-particle model using free-nucleon values for the coupling constants is shown by the dashed line.

Figure 3

${\gamma }_p$ versus ${\gamma }_n$. Points are shown for the $A=11\text{−}43,T=1/2$ mirror pairs (solid circles); the $A=57,T=1/2$ mirror pair [open circle near (-1,1)]; the $A=1$ and $A=3,T=1/2$ mirror pairs (open circles on the upper left-hand side); and the $A=9$, 13, 17, and 35, $T=3/2$ mirror pairs (crosses). The line is the result of fit (A).