INELASTIC COLLISIONS AND THE MOLIERE THEORY OF MULTIPLE SCATTERING

The effect of inelastic collisions is often introduced in the Moli\`ere theory by replacing ${Z}^{2}$ with $Z(Z+1)$. It is pointed out that this procedure relies on the implied incorrect assumption that elastic and inelastic collisions have the same small-angle cut-off. Taking into account separately the cut-off of inelastic collisions, the Moli\`ere theory is shown to require the following modifications: (a) For incident electrons, replace ${Z}^{2}$ with $Z(Z+1)$ and increase the Moli\`ere $b$ by ${(Z+1)}^{\ensuremath{-}1}{\mathrm{ln}[0.160{Z}^{\ensuremath{-}\frac{2}{3}}(1+\frac{3.33Z{e}^{2}}{\mathrm{hv}})]\ensuremath{-}{u}_{\mathrm{in}}}$, where $\ensuremath{-}{u}_{\mathrm{in}}$ is defined as an integral over the incoherent scattering function whose value is about 5. (b) For incident heavy particles, leave ${Z}^{2}$ unaltered but increase $b$ by ${Z}^{\ensuremath{-}1}{\mathrm{ln}[1130{Z}^{\ensuremath{-}\frac{4}{3}}{(\frac{{c}^{2}}{{v}^{2}}\ensuremath{-}1)}^{\ensuremath{-}1}]\ensuremath{-}{u}_{\mathrm{in}}\ensuremath{-}\frac{1}{2}\frac{{v}^{2}}{{c}^{2}}}$.