High-energy higher-order Born approximations: Theoretical development

The second and third terms of the generalized Born series are analyzed in an attempt to gain a series approximation to the differential cross section, valid through $O\left(\frac{1}{k^2}\right)$, which treats all Born terms analogously. The resulting expressions for the second and third Born terms, derived from assumptions similar to those of Glauber theory, are compared with other analyses for the case of small-angle elastic scattering of electrons by hydrogen atoms. A most notable result is that, in addition to the Glauber-like term, there is a second term of $O\left(\frac{1}{k^2}\right)$ which contributes to the real part of the third Born term. In addition, the angular range of maximum validity of the Glauber assumptions is established for inelastic collisions.