On higher Born approximations in potential scattering

Source abstract

Abstract Many discussions of higher Born approximations have followed the work of Distel (1932) and Sauter (1933b), which contains an error in its development, causing many conclusions presented in the literature to be incorrect. In this paper, the second Born approximation for scattering of a Dirac electron by a Yukawa potential is calculated by a correct method. For the limiting case of the Coulomb potential, the cross-section thus obtained is just the expression obtained by McKinley & Feshbach (1948) by expansion of Mott’s complete solution. A corresponding calculation is sketched for the meson, using β-matrix formalism, and its physical interpretation considered. For the non-relativistic case, the behaviour of the Born series (up to third order) is discussed for the transition from Yukawa to Coulomb potential, disproving the conclusions of Distel (1932), M∅ller (1930) and Urban (1943) that contributions to the cross-section from higher Born approximations become infinite in this limit.