In my notes, the following is mentioned:
We consider the scattering of a beam particle with energy $E$, momentum $p$ and charge $ze$ off a charge distribution $\rho (x)$ of total charge $Ze$. We will consider the target to be much heavier than the probe, so that we can neglect the recoil and the outgoing energy of the scattered particle is the same as its incoming energy. We want to calculate the cross section in perturbation theory, so we need the interaction to be small which is the case if $$zZ \alpha \lt 1$$ where $\alpha$ is the fine structure constant.
Now where does that follow from? Why the inequality above implies that the Hamiltonian interaction will be "small"?