Nonrelativistic Compton scattering in Furry’s picture. II. Bethe surface by means of the complex-coordinate method

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Abstract
Bethe surface as a means of characterizing the inelastic scattering of photons and electrons on atomic targets is discussed, and framed within the two-potential Furry’s picture of scattering theory. In particular, the cross section for inelastic photon scattering is considered, and its first distorted Born approximation is identified to be given in terms of Bethe surface along the path conserving energy and momentum transfer. The difficulties in obtaining accurate cross sections in situations where the energy transfer is close to the ionization threshold are indicated, and related to the so-called Compton defect. The method for calculating the inelastic photon scattering cross section, introduced in Part I of this work [J. Chem. Phys. 80, 5669 (1984)] is summarized. The scattering cross section, and the entire Bethe surface, is obtained by means of the L2 discretization of the continuum and implemented in terms of the complex-coordinate method, without explicit calculation of the final scattering waves. The method is tested for the case of photon scattering off the hydrogen atom. The results are encouraging, and may be relevant for applications of the complex-coordinate method to calculations of more general transition amplitudes. The method is predicted to be most useful in cases close to ionization threshold (e.g., appearance edges in Compton scattering, Compton defect). Although applied to the one-electron problem the procedure is readily applicable to many electron atoms.