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Comparative study of the three- and four-body boundary-corrected Born approximations for positronium formation

The three- and four-body boundary-corrected first Born approximations are formulated for the single-electron capture from the K shell of helium, carbon and neon atoms by impact of positrons. These approximations are applied to calculate the post and prior forms of the T-matrix elements. The differential and total cross sections obtained from these transition amplitudes are compared. In order to avoid the discrepancy between the post and prior formalisms, their average amplitude is calculated to obtain the total cross sections. For helium atoms, the total cross sections obtained from the post-, prior- and averaged-transition amplitudes calculated by the three-body approximation are compared with available experimental data as well as other theories, and also with the corresponding values obtained from a four-body formalism. In comparison with the three-body version of the specified method, the results obtained from the averaged four-body transition amplitude are in better agreement with the experiment, while the situation is the reverse for the prior calculations.