Coulomb Effect of Positron for (e,e') Reactions in Quasielastic Region

Abstract

The static Coulomb field of medium and heavy nuclei has been one of persistent problems in medium energy electron scattering. In particular, in the early 1990’s, Ohio group for the ðe; e0Þ reactions treated exactly the Coulomb distortion using partial wave expansion of the electron wave function, referred to the full distorted wave Born approximation (DWBA) in quasielastic region. This calculation is able to compare various nuclear models and furnish an invaluable check for several approximate methods of the Coulomb distortion. However, it is numerically challenging and computation time increases rapidly with higher incident electron energy. In particular, it is not possible to separate the cross section into the longitudinal and the transverse structure functions. In order to avoid the difficult problems related to the DWBA calculations, there are two approaches to treat approximately the electron Coulomb distortion. The first approach is known as the effective momentum approximation (EMA). This is that the electron momenta are modified by the value of the Coulomb potential at the center of target nucleus. This approximation describes the Coulomb distortion for light nuclei and for high incident electron energies but does not for heavy nuclei and for intermediate incident electron energies. Recently, there were experimental data measured from Saclay for C and Pb with positron beam. They improved the EMA calculation with the effective momentum p0 1⁄4 p ð2=3ÞVð0Þ, where Vð0Þ is the Coulomb potential at the origin. With this method, they claimed that if one uses this new EMA the total structure functions induced by positrons have roughly the same magnitude as those by electrons as a function of the energy transfer. As the other approach, from the middle of 1990’s, Kim and Wright have developed an approximate treatment of the electron Coulomb distortion. This approximation leads to an r-dependent momentum which includes the Coulomb distortion arising from the four potential A . The advantage of this approximation is that the separation of the cross section into a longitudinal term and a transverse term is possible. This approximation of the Coulomb distortion agreed with the full DWBA results (about 1–2%) near the peaks of the cross sections for heavy nucleus like Pb on ðe; e0Þ and ðe; e0pÞ reactions with the momentum transfer greater than approximately 300MeV/c. More recently, Kim and Wright improved the above method of the r-dependent momentum for the electron (or positron) current by averaging the phase shift of the leptons, referred to ad hoc approximation (see ref. 8 in detail). They compared the ad hoc approximation, the full DWBA, and the EMA calculations for the Coulomb distortion with a simple harmonic oscillation model. They showed that the total structure functions did not have the same magnitude because of the phase shift of the leptons even though they used the same kinematics as the Saclay experiments. In this report, we examine the Coulomb distortion effect of the positron for the cross section with the ad hoc approximation. In addition, we also investigate the positron Coulomb distortion effect at high incident positron energy of 2.02 GeV for several nuclei, C, Fe, and Au, which is the same kinematics as the SLAC data for the electron scattering. We use a relativistic single particle model for the bound state based on the –! model and apply the same potential for the outgoing nucleon as the bound state coupled with a free relativistic current operator. Since only electrons are observed in the ðe; e0Þ reaction, we do not take into account whether or not the ejected nucleons are ‘‘lost’’, which is no nucleon flux loss. This assumption guarantees current conservation and contains the final state interaction. In the plane wave Born approximation (PWBA) in which the electrons or the positrons are described as plane waves, the cross section for the inclusive ðe; e0Þ scattering can be written as d d d! 1⁄4 M q q4 RLðq; !Þ þ tan 2 q 2q2 ! RTðq; !Þ # ; ð1Þ