Total Cross Sections for Inelastic Scattering of Charged Particles by Atoms and Molecules. I. A Sum Rule for the Bethe Cross Sections and Its Application to the Helium Atom

The Bethe cross section for inelastic scattering of fast charged particles by an atom or molecule is shown to obey a sum rule which permits the calculation of the total inelastic scattering cross section from the optical oscillator strength distribution and certain ground-state expectation values. The sum rule can also be utilized as a requirement of compatibility among the individual inelastic scattering cross sections; for example, together with the experimental ionization cross section, it yields the total excitation cross section, which is very poorly known at the present time. An accurate evaluation is made for the helium atom. The resulting total inelastic scattering cross section, ${\sigma }_{\mathrm{tot}}=\frac{z^2}{{\beta }^2}\left\{1.410\left[\ln \left(\frac{{\beta }^2}{1-{\beta }^2}\right)-{\beta }^2\right]+15.118\mathrm{}\right\}\times {10}^{-20\mathrm{}} {\mathrm{cm}}^2,$ is in excellent agreement with experimental data.