Inelastic Collisions and the Molière Theory of Multiple Scattering

The effect of inelastic collisions is often introduced in the Moli\`ere theory by replacing Z² 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 Z² with $Z(Z+1)$ and increase the Moli\`ere b by (Z+1)^-1ln[0.160Z^-2/3(1+3.33Ze²hv)]-uᵢₙ, where -uᵢₙ is defined as an integral over the incoherent scattering function whose value is about 5. (b) For incident heavy particles, leave Z² unaltered but increase b by ${Z}^{{-}1}{ln[1130{Z}^{{-}4/3}{({{c}²}{{v}²}{-}1)}^{{-}1}]{-}{u}ᵢₙ{-}1/2{{v}²}{{c}²}}$.