Isobar-doorway model and multiple scattering techniques in pion-nucleus scattering

An isobar-doorway model is presented for $\pi$-nucleus scattering in the (3, 3) energy region. The connection between this model and multiple-scattering and optical-model approaches is demonstrated through various approximations used in evaluating the doorway amplitude. Effective scattering amplitudes, differing from the free $\pi -N$ amplitudes as the result of nuclear effects, are suggested for use in multiple-scattering expansions. The importance of a factor ${\Gamma }^{\downarrow }$ in these amplitudes is marked. (This factor seems associated with a phenomenon analogous to collision broadening for atomic collisions.) An extension of the formalism to (${\pi }^{+}, {\pi }^0$) charge exchange between analog states is given.