The Faddeev approach to high energy atomic collisions and the significance of correspondence identities

Abstract

The Faddeev equations enable one to express the scattering amplitude for a three-body process interacting via binary potentials in terms of the two-body off-shell Coulomb T matrices. In the first order iterate the equations reduce to a sum of two terms. A form of the two-body off-shell Coulomb T matrix is presented which can be expressed partially as a sum over the classical trajectories of the scattering particles. It is shown that in the limit of high energies and non-zero scattering angles an on-shell correspondence identity exists for this T matrix, so that only the classical path term contributes to the scattering amplitude. The classical path approximation is applied to elastic and inelastic collisions of electrons on hydrogen atoms. For elastic collisions the first order Faddeev approximation predicts differential cross sections considerably larger than those predicted by the first Born approximation at energies where the born approximation is expected to be good. At angles of scattering above 30o our results are identical to those calculated using a different exact form of the Coulomb T matrix (Chen and Sinfailam, 1972). In the Faddeev approximation the differential cross section diverges at small angles of scattering. The angular and energy distributions of the differential cross section for inelastic collisions are close to the predictions of the Coulomb projected Born approximation (Geltman and Hidalgo, 1971), though the latter approximate predicts much smaller cross sections. Singularities in the on-shell Coulomb T matrix associated with the long range nature of the Coulomb potential are responsible for both the zero angle divergence and the overestimation of the differential cross section in the Faddev approximation. In conclusion we comment on the importance of recent calculations of the second-order iterate of the Faddeev equations for electron-hydrogen scattering (Chen et. al., 1973) where it is shown that the differential cross section for elastic scattering approaches the Born result. This demonstrates that cancellation occurs between the singularities in the first-order and second-order terms of the Faddeev equations.