Coulomb distortion of relativistic electrons in the nuclear electrostatic field

Abstract

Continuum states of the Dirac equation are calculated numerically for the electrostatic field generated by the charge distribution of an atomic nucleus. The behavior of the wave functions of an incoming electron with given asymptotic momentum in the nuclear region is discussed in detail and the results are compared to different approximations used in the data analysis for quasielastic electron scattering off medium and highly charged nuclei. It is found that most of the approximations provide an accurate description of the electron wave functions in the range of electron energies above 100 MeV typically used in experiments for quasielastic electron scattering off nuclei only near the center of the nucleus. It is therefore necessary that the properties of exact wave functions are investigated in detail in order to obtain reliable results in the data analysis of quasielastic (e, e'p) knockout reactions or inclusive quasielastic (e, e') scattering. Detailed arguments are given that the effective momentum approximation with a fitted potential parameter is a viable method for a simplified treatment of Coulomb corrections for certain kinematical regions used in experiments. Numerical calculations performed within the framework of the single-particle shell model for nucleons lead to the conclusion that our results are incompatible with calculations performed about a decade ago, where exact electron wave functions were used in order to calculate Coulomb corrections in distorted-wave Born approximation. A discussion of the exact solutions of the Dirac equation for free electrons in a Coulomb field generated by a point-like charge and some details relevant for the numerical calculations are given in the appendix.