Analytical approximation to pion-nucleus scattering near the (3,3) resonance

Starting from the eikonal approximation to the scattering amplitude an analytical approximation is obtained, including Coulomb effects. The amplitude is characterized by three numbers, one of which is a radius parameter $b_1$. These numbers are related in a simple way to the optical potential $U\left(r\right)\mathrm{}$ for $r\simeq b_1$. The quality of the analytical result is assessed by comparing it to the exact eikonal amplitude for a model of $U$ which is linear in density. It reproduces accurately the position and depth of the first minimum, the magnitude of the first secondary maximum, and the differences of the ${\pi }^{+}$ and ${\pi }^{-}$ cross sections. It is furthermore shown that at 180 MeV the analytical approximation reproduces semiquantitatively the magnitude and shape of the "model exact" solution of the Klein-Gordon equation at forward angles. An application to scattering by the calcium isotopes is taken as an example to show the sensitivity to the neutron distribution as the $f_{\frac{7}{2}}$ shell is filled.