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Partial wave completion technique for scattering amplitudes in Coulomb dipole excitation

When a charged particle collides with an ion, the dominant contribution to the inelastic scattering amplitude may come from partial waves having large values of the angular momentum ℓ of the colliding particle. It is important, when calculating the scattering amplitude by summing over ℓ, to ensure that this contribution is taken into account since numerical truncation of the infinite sum can lead to unphysical oscillations in the angular distributions of the amplitude. We derive, in the case of Coulomb dipole transitions, an analytic expression for the contribution to the scattering amplitudes coming from partial waves with ℓ greater than some prescribed value Λ. We make use of the Coulomb Bethe approximation to obtain this result.