Isospin Selection Rules for High-Energy Electron Scattering
Judah M. Eisenberg, Morris E. Rose · Physical Review · 1963
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.