How useful is the fixed-scatterer approximation in pion physics?

We compare detailed dynamical calculations of pion-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ elastic scattering with calculations carried out in the fixed-scatterer approximation. We consider pion (laboratory) energies of 51 to 260 MeV. The dynamical calculations are (essentially) parameter free and provide an excellent fit to the data. It is shown that by introducing an energy shift parameter into the fixed-scatterer approximation calculations it is possible to obtain good agreement with the data at low energies. At the higher energies (180-260 MeV) the quality of the (energy shifted) fixed-scatterer approximation fits to the data become systematically worse, with good agreement only at small angles. These results indicate that the use of a single parameter (an energy shift) in the fixed-scatterer approximation calculations is unable to simulate the effects of nuclear binding and Fermi motion. These effects are treated in detail in the dynamical calculations.