Diffraction theory analysis of pion elastic scattering from the calcium isotopes

We show that the experimental results for pion elastic scattering from the calcium isotopes in the ${\Delta }_{33}$ resonance region can be well represented at small angles in terms of the variables of diffraction theory. The parameters describing the amplitude reflect physics that can be attributed to pion and ${\Delta }_{33}$ dynamics in a microscopic approach, as well as to details of the neutron and proton densities. Using a phenomenological determination of the former, we show that the expermental results require a neutron skin in excess of that obtained from scaling the proton density according to the relative number of neutrons and protons. The experimental results are consistent with microscopic Hartree-Fock densities, provided that the ratio of isovector to isoscalar optical potential is substantially enhanced over the value obtained from the free pion-nucleon scattering amplitude. The size of this enhancement agrees with pion single-charge-exchange scattering from the calcium isotopes.