On higher Born approximations in potential scattering

Abstract

Many discussions of higher Born approximations have followed the work of Distel (1932) and Sauter (1933 b ), 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.