Electron Scattering from Deformed Nuclei

A general phenomenological theory is developed for the calculation of both elastic and inelastic transverse and longitudinal form factors for electron scattering from nuclei whose surfaces are deformed according to the expression $R+{\left(4\mathrm{}\pi \right)}^{\frac{1}{2}}{\Sigma }_{\mathrm{lm}}{\hat{a}}_{\mathrm{lm}}{Y}_{\mathrm{lm}}\left(\hat{\mathcal{r}}\right)$. Here, the ${\hat{a}}_{\mathrm{lm}}$ are operators in the nuclear Hilbert space, whose detailed dynamical character is left unspecified. The effects of deformation on the scattering, to any desired accuracy, are contained in reduced matrix elements of tensor products of the ${\hat{a}}_{\mathrm{lm}}$. These are then treated as phenomenological parameters whose values are to be fixed from an analysis of the scattering cross sections, particularly at high momentum transfer. All the static and transition multipole moments of the nucleus are thereby determined, at least in principle. The Helm model and its generalization to transverse scattering are obtained in the small-deformation limit. The theory is used to analyze the Coulomb elastic and inelastic (to the 1.3-MeV ${\frac{3}{2}}^{+}$ level) scattering of electrons from ${}_{}{}^{59}{}_{}{}^{}\mathrm{Co}$, as well as the magnetic elastic and inelastic scattering from ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}$. It is shown that the contribution of the deformation to elastic monopole scattering may be interpreted as due to an effective, oscillatory, spherically symmetric "modulating-charge" distribution, superimposed upon the more usual smeared uniform distribution. It is therefore suggested that the oscillations in the charge distribution which seem to be required in order to fit the high-energy elastic scattering from ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ and other nuclei may be due to deformations.