Inelastic Electron-Deuteron Scattering Cross Sections at High Energies. II. Final-State Interactions and Relativistic Corrections

Abstract

Measurements of the cross-section $\frac{{d}^{2}\ensuremath{\sigma}}{(d{\ensuremath{\Omega}}_{e}d{{E}_{e}}^{\ensuremath{'}})}$ for the inelastic electron-deuteron scattering process $e+d\ensuremath{\rightarrow}e+n+p$ have been used to determine the electromagnetic structure of the neutron. The effects on the theoretical cross section of interactions between the outgoing nucleons are examined in detail using the methods of a previous paper. The transition matrix elements connecting the initial state of the two-nucleon system (the deuteron) to a final state with specified total, orbital, and spin angular momenta are calculated using approximate wave functions which are matched to the experimentally determined neutron-proton scattering phase shifts. While individual matrix elements may be drastically changed by the distortion of the final-state wave functions by the neutron-proton interaction, the over-all corrections to the peak value of the cross section are found to be small (-1 to -2%) for electron momentum transfers in the range $q=3.4\ensuremath{-}2.6$ ${\mathrm{f}}^{\ensuremath{-}1}$. The precise magnitude of the corrections is somewhat uncertain because of the approximate nature of the wave functions, but it unlikely either that they are large, or that the corrections could become positive. The effects of final-state interactions on the cross-section $\frac{{d}^{2}\ensuremath{\sigma}}{(d{\ensuremath{\Omega}}_{e}d{{E}_{e}}^{\ensuremath{'}})}$ are also examined for final electron energies near the upper limit of the inelastic continuum. In this region, the nucleons emerge with low relative momenta, and, in agreement of the predictions of Jankus, the cross section is found to be drastically changed by the strong interactions in the final $S$ states. However, it is shown that the presence in the neutron-proton interaction of a strongly repulsive core results in a considerable diminution of the cross section relative to the predictions of Jankus for large values of $q$. This lowering of the cross section has been observed by Kendall et al. Results obtained with approximate repulsive core wave functions provide a reasonable fit both to the inelastic cross section near the end point, and to the deuteron electromagnetic form factor obtained from elastic electron-deuteron scattering. Finally, the relativistic theory of inelastic electron-deuteron scattering is examined using the methods of dispersion relations. It is found that in the region of the large peak, the cross-section $\frac{{d}^{2}\ensuremath{\sigma}}{(d{\ensuremath{\Omega}}_{e}d{{E}_{e}}^{\ensuremath{'}})}$ is given essentially correctly by a nonrelativistic calculation using a modified Hamiltonian, provided the results are interpreted correctly with respect to the kinematics. The approximations inherent in the calculation are examined in detail. The resulting cross section differs significantly from the modified Jankus cross section which has been used in the analysis of the high-energy electron-deuteron scattering data obtained by the Stanford group. It is found that the apparent values of the neutron charge form factor ${F}_{1n}$ are reduced essentially to zero for ${q}^{2}$ in the range $5 {\mathrm{f}}^{\ensuremath{-}2}\ensuremath{\le}{q}^{2}\ensuremath{\le}20 {\mathrm{f}}^{\ensuremath{-}2}$ when relativistic corrections, the effects of the deuteron $D$-states cattering, and the effects of final-state interactions are taken into account. Corresponding reductions in the value of the neutron anomalous magnetic moment form factor ${F}_{2n}$ range up to about 30%, and bring ${F}_{2n}$ into closer agreement with ${F}_{2p}$. A complete re-analysis of the experimental data will be necessary.