Exchange Moment Contributions to a Modified Shell Model

Abstract

The longitudinal exchange contribution to the nuclear magnetic moment that follows from a rather general modified shell model is studied. It is shown that for a heavy nucleus whose state can be expanded as a linear combination of certain very reasonable independent particle configurations, the longitudinal exchange contribution to the magnetic moment can be expressed, independent of the amplitudes of these states, in terms of a correction to the orbital Land\'e $g$-factor that appears in the single particle ordinary moment operator. The magnitude of this correction is simply given in terms of the fraction of the neutron-proton two-body potential that is charge exchange and spin-independent. Being independent of the configuration amplitudes, this result is expected to hold even if the core behaves as a liquid drop. Calculations assuming roughly half of the nuclear force to be charge exchange in nature reasonably account for the deviations from the values predicted by the shell model of the magnetic moments of those nuclei which consist of a single odd nucleon outside a doubly magic core. The exchange contributions generally improve the moment agreement of other nuclei in the $j=l\ensuremath{-}\frac{1}{2}$ cases, but make the agreement in the ${f}_{\frac{7}{2}}$ and ${g}_{\frac{9}{2}}$ cases poorer.