Ratio of isoscalar to isovector core polarization for magnetic moments

In calculations of isoscalar magnetic moments of odd-odd $N=Z$ nuclei, it was found that, for medium- to heavy-mass nuclei, large-scale shell-model calculations yielded results that were very close to those obtained with the much simpler single-$j$ shell model. To understand this, we compare isoscalar and isovector core-polarization configuration-mixing contributions to the magnetic moments of mirror pairs in first-order perturbation theory, using a spin-dependent $\ensuremath{\delta}$ interaction. We fit the strength of the $\ensuremath{\delta}$ interaction by looking at isovector and isoscalar mirror pairs. We then use the same interaction to calculate corrections due to first-order core polarization of the magnetic moments of odd-odd nuclei.