Isospin Selection Rules for High-Energy Electron Scattering

Abstract

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: ${({k}_{0}R)}^{2}\ensuremath{\ll}1$, where ${k}_{0}$ is the energy transfer, and finite nuclear size effects are ignored. However, the latter assumption follows provided ${(\mathrm{kR})}^{2}\ensuremath{\ll}1$ ($k$ is the momentum transfer) and this is valid only for small scattering angles ($\ensuremath{\lesssim}\frac{1}{\mathrm{ER}}$, $E$ the primary energy). For light elements the isospin selection rule operates for $E1$ transitions in the forward cone only (\ensuremath{\lesssim}30\ifmmode^\circ\else\textdegree\fi{}). The $M1$ selection rule also follows with ${(\mathrm{kR})}^{2}\ensuremath{\ll}1$ 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.