Electron and Muon Scattering from Nuclear Charge Distributions at Incident Momenta Between 50 and 183 MeV/c

The comparison of theoretical elastic-scattering cross sections of positrons and electrons from Woods-Saxon (WS) and "wine-bottle" (WB) charge distributions of the nucleus of Au, carried out at 183 MeV in a previous paper by the authors, is extended to lower energies and repeated for muons of comparable incident momenta. It is found that, for momentum transfers of less than 1.5 ${\mathrm{F}}^{-1\mathrm{}}$, the percent change of the cross section corresponding to a change from the WS to the WB charge distribution is largest, of the order of 30% for incident momenta of ∼100 MeV/c, particularly for positrons. At an electron energy of 50 MeV the cross section depends mainly on the mean-square radius of the nucleus, and an accuracy better than 5% is needed in order to determine additional nuclear charge distribution parameters. The mean-square radii of the WS and WB charge distributions differ by 6.5% while the corresponding electron cross sections at 50 MeV differ by a maximum of 15%. A comparison with experimental elastic positron and electron scattering cross sections for Pb measured by Miller and Robinson is carried out, and a systematic discrepancy with theory is found for both $e^{+}$ and $e^{-}$ cross sections for the 50-70-MeV energy range, while theory and experiment agree well at 87 MeV and higher energies. The calculation consists of a conventional numerical phase-shift analysis based on the Dirac equation, and the nuclei are assumed to be static, spherically symmetric extended charge distributions.