Interpretation of second Born amplitudes in electron capture

Exact second Born amplitudes for $1s-1s$ electron capture are presented and interpreted in terms of contributions from intermediate states off the energy shell and on the energy shell. The classical model of Thomas corresponds to two-step scattering via one particular on-shell intermediate state. In the high-velocity limit for $1s-1s$ capture, this on-shell intermediate state of the Thomas model accounts for one-half of the total cross section, i.e., the second Born cross section. The other half comes from off-energy-shell intermediate states near the on-shell Thomas state. Above 5 MeV both the off-shell and on-shell amplitudes have a simple structure near the Thomas peak. Below 1 MeV, contributions from a broader range of intermediate states are evident in the amplitudes. Although the contribution from the Thomas state is not large below 1 MeV, other second Born effects are significant.