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

Abstract

The impact of the strongly attractive electromagnetic field of heavy nuclei on electrons in quasielastic (e, e ′ ) scattering is often accounted for by the effective momentum approximation. This method is a plane wave Born approximation which takes the twofold effect of the attractive nucleus on initial and final state electrons into account, namely the modification of the electron momentum in the vicinity of the nucleus, and the focusing of electrons towards the nuclear region leading to an enhancement of the corresponding wave function amplitudes. The focusing effect due to the attractive Coulomb field of a homogeneously charged sphere on a classical ensemble of charged particles incident on the field is calculated in the highly relativistic limit and compared to results obtained from exact solutions of the Dirac equation. The result is relevant for the theoretical foundation of the effective momentum approximation and describes the high energy behavior of the amplitude of continuum Dirac waves in the potential of a homogeneously charged sphere. Our findings indicate that the effective momentum approximation is a useful approximation for the calculation of Coulomb corrections in (e, e ′ ) scattering off heavy nuclei for sufficiently high electron energies and momentum transfer.

Coulomb distortion effects in quasi-elastic scattering on heavy nuclei

“…The reason is twofold. Firstly, the energy of the final state electrons is relatively small, and the focusing effect becomes more important [51], such that the DWBA cross section becomes significantly larger than the EMA prediction. Secondly, the quasi-elastic peak is shifted towards smaller energy transfer, where the optical potential increasingly distorts the wave functions of the final state nucleons.…”

“…which interpolates between the non-relativistic and relativistic regime [51]. We additionally solved the Dirac equation for Dirac nucleons in the energydependent volume-central part of the optical potential used in this work for different energies, such that the amplitude of the eikonal phase corrected nucleon spinors could also be corrected by the corresponding energy-dependent amplitude modification.…”

“…, where j µ,EM A e is calculated with plane electron waves with effective momenta. A detailed discussion of the focusing effect can be found in [43].…”

Cross Sections of Charged Current Neutrino Scattering off132Xe for the Supernova Detection

“…where eff is the effective Coulomb potential. In a recent study using exact Dirac wave functions, it has been shown that an accurate approximation for the effective electron momenta is obtained by using the mean value of the Coulomb potential, eff = 4 (0)/5, where (0) = −3 /(2 ) corresponds to the electrostatic potential evaluated at the center of the nucleus [66,67].…”

QRAP: A numerical code for projected (Q)uasiparticle (RA)ndom (P)hase approximation

“…with V ef f = 4V C (0)/5 = −6Z f α/5R A [38,67]. These two approximations for the Coulomb correction were tested in the calculation of the inclusive cross section for neutrino scattering on 208 Pb [38].…”