Exact solutions of the Schrödinger equation for the Coulomb potential are used in the scope of both stationary and time-dependent scattering theories in order to find the parameters which define regularization of the Rutherford cross-section when the scattering angle tends to zero but the distance r from the center remains fixed. Angular distribution of the particles scattered in the Coulomb field is investigated on the rather large but finite distance r from the center. It is shown that the standard asymptotic representation of the wave functions is not available in the case when small scattering angles are considered. Unitary property of the scattering matrix is analyzed and the "optical" theorem for this case is discussed. The total and transport cross-sections for scattering of the particle by the Coulomb center proved to be finite values and are calculated in the analytical form. It is shown that the considered effects can be essential for the observed characteristics of the transport processes in semiconductors which are defined by the electron and hole scattering in the fields of the charged impurity centers.
The results of numerical calculating the total Mott−Bloch correction to the Bethe stopping formula and the Lindhard−Sørensen correction in the point nucleus approximation, as well as the Mott correction and the difference between the Lindhard−Sørensen and Bloch corrections, which were obtained by some rigorous and approximate methods, are compared for the ranges of a gamma factor 1 ≲ 𝛾 ≲ 10 and the ion nuclear charge number 6 ≤ Z ≤ 114. It is shown that the accurate calculation of the Mott−Bloch correction based on the Mott exact cross section using a method previously proposed by one of the authors gives excellent agreement between its values and the values of the Lindhard−Sørensen correction in the 𝛾 and Z ranges under consideration. In addition, it is demonstrated that the results of stopping power calculations obtained by the two above-mentioned rigorous methods coincide with each other up to the seventh significant digit and provide the best agreement with experimental data in contrast with the results of some approximate methods, such as the methods of Ahlen, Jackson−McCarthy, etc.
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