Isospin Selection Rules for High-Energy Electron Scattering

The validity of Siegert's theorem and the isospin selection rule for electric and magnetic dipole transitions has been established for inelastic electron scattering, thus extending the well-known results obtained for real photons. Siegert's theorem is obtained with arbitrary electron wave functions for electric multipole transitions provided: ${\left(k_{0}R\right)}^2\ll 1$, where $k_0$ is the energy transfer, and finite nuclear size effects are ignored. However, the latter assumption follows provided ${\mathrm{}\left(\mathrm{kR}\right)\mathrm{}}^2\ll 1$ ($k$ is the momentum transfer) and this is valid only for small scattering angles ($\lesssim \frac{1}{\mathrm{ER}}$, $E$ the primary energy). For light elements the isospin selection rule operates for $E1\mathrm{}$ transitions in the forward cone only (≲30°). The $M1\mathrm{}$ selection rule also follows with ${\mathrm{}\left(\mathrm{kR}\right)\mathrm{}}^2\ll 1$ and, therefore, operates in the same angular range. The angular distributions should exhibit an anomalous depression in the forward cone of half-angle about $\frac{1}{\mathrm{ER}}$. Coulomb effects will then be decisive in determining the magnitude of the small-angle scattering. The same considerations are applicable to internal pair formation and internal conversion where the retardation assumption is valid in general under usual conditions.