Ratio of Isoscalar to Isovector Core Polarization Contributions to Magnetic Moments

We found that large scale calculations of isoscalar magnetic moments of odd-odd N=Z nuclei yielded results remarkably similar to those of simple single j calculations. To understand why we use a delta interaction times (1+xPs) where Ps is the spin exchange operator, to calculate the ratio IS/IV of the core polarization contributions to the magnetic moments. The spin exchange contributes a factor (1-2x) to this ratio. A popular choice is x=1/3 for which (1-2x) is also 1/3. Another contribution comes from the fact that the coupling of j=l+1/2 to j'=l-1/2 via the magnetic moment operator is proportional to (gs-gl). The IS values are gl=0.5 gs=0.88; the IV values are 0.5 and 4.71. This yields a (gs-gl) ratio of 0.09 which together with the 1/3 from spin exchange tells us that the isoscalar core polarization is a mere 3\% of isovector. If we further divide by single j values to get effective charge corrections then the ration IS/IV ends up being 0.06 (or 6\%). We thus gain understanding of the resuts in ref [1] of the near equality of large scale and single j results for IS moments.\\[4pt] [1] S.Yeager, S.J.Q. Robinson, L.Zamick and Y.Y.Sharon, EPL 88, 52001 (2009)