## FDCS in atomic units for SI of helium using a 3-body model (dashed), 4-body model (solid), 4-body model with the 3-body initial state Coulomb interaction *V*_{i} (dotted), and 4-body model with the 3-body initial state Coulomb interaction *V*_{i} and 3-body final state distorting potential *U*_{f} (dash-dotted)

_{i}

_{i}

_{f}

**Figure 9.** FDCS in atomic units for SI of helium using a 3-body model (dashed), 4-body model (solid), 4-body model with the 3-body initial state Coulomb interaction *V _{i}* (dotted), and 4-body model with the 3-body initial state Coulomb interaction

*V*and 3-body final state distorting potential

_{i}*U*(dash-dotted). All theoretical calculations are without PCI. Incident projectile energies

_{f}*E*

_{0}, ejected electron energies

*E*

_{2}, scattering angles θ

_{s}, and momentum transfers

*q*are shown in the figure.

**Abstract**

The frozen core approximation has been successfully used for many years to model 4-body collisions as 3-body collisions. We present a comprehensive comparison of 3-body and 4-body models for the process of single ionization of helium by electron impact using our 4-body distorted wave model. Differences between the two models are observed in both magnitude and peak locations. We identify four possible sources for the discrepancies between the models, and isolate the specific physical causes of the discrepancies.