The Excess-Path-Length Distribution of Electrons in Electromagnetic Cascade Showers

The excess-path-length distribution of cascade-shower particles due to the multiple Coulomb scattering of shower electrons is investigated analytically. A diffusion equation for the distribution under the Landau approximation is proposed and solved under the conditions of neglecting and including the ionization loss. Contributions of the Rutherford single scattering on the distribution are also investigated and found to dominate in very large excess regions of path length. The excess-path-length distribution of shower electrons is observed as an arrival-time distribution at a given thickness of traverse. The present results will be valuable for qualitative and quantitative analyses of cascade showers observed by the arrival-time distribution.