Differential cross sections for ionization of the hydrogen atom by fast charged particles in the binary-encounter theory and bethe theory

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Abstract
Differential cross sections for ionization of the hydrogen atom from the n = 1 and 2 states by fast charged particles are calculated in the binary-encounter and Bethe (first Born) theory. Comparison shows good agreement between the two theories for large momentum and energy transfer. It is shown that use of an unrealistic velocity distribution of the target electron in the binary-encounter theory can lead to very poor results. If in the Bethe theory the Coulomb final-state wave functions are approximated by plane final-state wave functions, the same correct cross sections are obtained at and near the Bethe ridge for large momentum and energy transfer as in the Bethe and binary-encounter theories, but very poor results are obtained for small momentum and energy transfer. In the Bethe theory, the (numerically calculated) differential cross sections per unit energy transfer (in rydberg units) are found to be proportional to -2 + (4/3n2)-3 for large , in agreement with the corresponding binary-encounter cross-section formula. For ionization from the n = 2 states we have calculated (Bethe theory) the proportionality factors in the asymptotic total cross-section formula, and the continuum contributions to the optical oscillator strength and the polarizability.