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 a 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.

Search for three-nucleon short-range correlations in light nuclei

“…The simulation uses a yscaling cross section model [19,30] with radiative corrections applied using the peaking approximation [31]. Coulomb corrections are applied within an improved effective momentum approximation [1,32], and are 2% or smaller for all data presented here. The uncertainty in the target thicknesses dominates the total scale uncertainty (5.1%) of the ratios, while density fluctuations and dummy subtraction (used to remove the contribution from the aluminum endcaps of the target) dominate the point-to-point systematic uncertainty of 1.3%.…”

Precise Measurement of the Nuclear Dependence of Structure Functions in Light Nuclei

“…At the energies in the present experiment, the Effective Momentum Approximation (EMA) is a good approximation [49,50,51] to the exact calculation (52]. We apply the correction as outlined in [50], by adding an energy boost to the incoming and outgoing electron energy and calculate the change in the cross section. The energy boost is given by the change in potential energy the electron experiences as it falls into the nucleus from very far away.…”

“…It has been observed [50] that it is a better approximation to assume the scattering happens closer to the surface of the nucleus. This can be achieved by using a sightly modified effective potential energy given by a factor of 0.75 -0.8 times Vo.…”

“…That boost is equal to 4V 0 /5. In this analysis, the boost was AE = 0.775V 0 , where we have just taken the central value of the range of scale factors given by Aste et al [50].…”

Focusing of high-energy particles in the electrostatic field of a homogeneously charged sphere and the effective momentum approximation

“…For the EMA to hold, it is mandatory that the wave lengths of the electron and the virtual photon are significantly smaller than the nuclear radius, i.e. ǫ f > 200 MeV and Q 2 > (200 MeV) 2 corresponding to a length scale of 1 fm should be required for 208 Pb [31,32]. Note also that the enhancement of the wave function amplitudes is not very large at high energies.…”