Accuracy, asymptotes, and applications of the Born approximations to the calculation of the Mott scattering cross section, the primary atomic displacement cross section, and the energy-loss straggling

TL;DR

This paper evaluates the accuracy of the first three Born approximations in calculating Mott scattering cross sections, atomic displacement, and energy-loss straggling across elements, providing insights into their applicability and precision.

Contribution

It offers a comprehensive analysis of the Born approximations' accuracy for various atomic calculations, including new expressions for displacement cross sections and comparisons with established formulas.

Findings

Second and third Born approximations are highly accurate for many elements.

Displacement cross sections calculated with third Born approximation are within 1% of asymptotic formulas.

Born approximations effectively estimate energy-loss straggling across a wide range of energies.

Abstract

The first, second and third Born approximations of the Mott scattering cross section are considered. The relative error of all three Born approximations averaged by angles and energies is calculated for the first 30 elements of the Mendeleev periodic table of elements and also of the second and third-Born approximation for the first hundred elements of the Mendeleev table. The accuracy of the second and third Born approximations for calculating the normalized Mott scattering cross section are compared on the wide range of nuclei of elments. The accuracy of the Born approximations for calculating the Mott correction in the Bethe-Bloch formula for for the second Born approximation and the third approximation is analyzed. An expression is obtained for the cross section of the primary displacement of the atom in the third Born approximation. For iron, silver and lead, the cross section of the primary displacement of the atom for a number of electron energies is calculated. For a number of examples, it is calculated starting from the electron energy. The difference of the cross section is obtained by the asymptotic formula from by the McKinley-Feshbach formula, it will be less than one percent. Accuracy of the Born approximations for calculating energy-loss straggling is analyzed.