Phase-Shift Optical Model Calculations for the Elastic Scattering of Pions on Aluminum

Abstract

Theoretical optical-model exact phase-shift calculations are given for the predicted elastic scattering of pions from aluminum at the energy used (79 Mev) in the experimental studies described in the preceding paper. The predicted curves show strong diffraction minima, not present in the experimental curves. We have emphasized a comparison of the results of these calculations with those of more approximate methods. Sixteen choices of constant complex potentials were used for $r<\mathrm{}{R}_0$ for both ${\pi }^{+}$ and ${\pi }^{-}$ mesons, with a Coulomb potential for $r>\mathrm{}{R}_0=3.600\mathrm{}\lambda$. Here $\hslash k_0=\frac{\hslash }{\lambda }$ is the momentum at infinity and $\hslash (k_1+i{k}_2)$ is the momentum for $r<\mathrm{}{R}_0$. It is found that the predicted incoherent cross section ${\sigma }_a$ and positions of the diffraction minima depend strongly on $\frac{k_1}{k_0}$ for a fixed $k_2$. Values of ${\sigma }_a>\pi {R_0}^2$ are obtained for relatively long nuclear mean free paths ${\left(2\mathrm{}{k}_2\right)}^{-1\mathrm{}}$ for incoherent processes for the attractive potentials which give best fit to experiment. The results of various modified Born approximation calculations are compared with the results of the exact calculations.