A fast particle of charge ze, mass M, and velocity v, collides with a hydrogen-like atom with one electron of charge −e, mass m, bound to a nuclear center of charge Ze. The collisions can be divided into two kinds: close collisions where the particle passes through the atom (b<d), and distant collisions where the particle passes by outside the atom (b>d). The atomic ``radius'' d can be taken as a0/Z. For the close collisions the interaction of the incident particle and the electron can be treated as a two-body collision and the energy transfer calculated from the Rutherford cross section. For the distant collisions the excitation and ionization of the atom can be considered the result of the photoelectric effect by the virtual quanta of the incident particle's fields.
For simplicity assume that for photon energies Q greater than the ionization potential I the photoelectric cross section is
σγ(Q)=1378π2(Za0)2(QI)3
(This obeys the empirical Z4λ3 law for x-ray absorption and has a coefficient adjusted to satisfy the dipole sum rule, ∫σγ(Q)dQ=2π2e2ℏ/mc.)