Comprehensive Study of 3-Body and 4-Body Models of Single Ionization of Helium

For decades the frozen core approximation has been successfully used to model 4-Body collisions as 3-Body processes. In recent years, full 4-Body models have been used to calculate fully differential cross sections (FDCS) for single ionization of helium, and these 4-Body models show discrepancies with the 3-Body models. We have identified four possible sources of the discrepancies, which are: the initial state helium wave function, the final state He$^{+}$ wave function, the final state potential for the outgoing electrons, and the perturbation. To identify which of these four sources causes in the differences in FDCS, we have performed a comprehensive study of 3-body and 4-body models for a wide range of incident projectile energies, ionized electron energies, and scattering angles.