Isospin dependence of second-order pion-nucleus optical potential

A universal form for the dependence of the second-order pion-nuclear optical potential on the nuclear and pionic isospin and on the densities $\rho$ and $\Delta \rho$ is derived, where $\rho$ is the total density and $\Delta \rho$ is the valence neutron density. The result is characterized by five complex parameters for each partial wave channel contributing to the optical potential. Our result applies to scattering in the vicinity of the (3,3) resonance, and the parameters may be most cleanly determined phenomenologically by applying the theory to elastic, and single- and double-charge-exchange data for medium to heavy weight $J=0\mathrm{}$, spherical nuclei. The relationship between the parameters and a microscopic cluster decomposition of the optical potential is explored. The parameters are calculated theoretically for selected second-order processes to obtian a first orientation to the magnitude of the terms. The results show, in particular, that large contributions to double charge exchange arise from nonanalog intermediate nuclear states and that these contributions have a characteristic isospin dependence which is different from that found in the simple models previously studied.