Partial-Wave Analysis of the Inelastic Scattering of Electrons by Nuclei. I. Results for Quadrupole Excitations

The problem of nuclear Coulomb excitation by inelastic scattering is approached by using a distorted partial-wave description for the initial and final states of the electron and a multipole expansion of the interaction with the nucleus which is treated in the semiclassical approximation. The method is facilitated by use of a closed analytic form for the contribution to the scattering of partial waves of sufficiently high angular momenta to be insensitive to the shape (but not the charge) of the target nucleus. The contribution of partial waves which do penetrate the nuclear charge distribution is found by numerical evaluation of the various matrix elements using electron wave functions obtained by numerical solution of the radial part of the Dirac equation in a central field. The particular case of electric quadrupole excitation assuming a uniform charge distribution is calculated and compared with the usual Born approximation treatment of inelastic electron scattering. The results of the calculations for 200-MeV electrons differ from the Born approximation result in two ways: for scattering angles less than 90\ifmmode^\circ\else\textdegree\fi{} the Coulomb field causes an apparent, and significant, change in the radius of the charge distribution; for scattering angles greater than 90\ifmmode^\circ\else\textdegree\fi{} the cross section differs qualitatively from the Born approximation result.