Past exam of the particle physics course of the University of Oxford / 2017 / 4 / d / Solution

In the binary branching model of an electromagnetic shower, after $n$ ideal equal-sharing generations, the model has $N_n=2^n$ particles of energy $E_n=E_0/2^n$, at depth $n{X}_0$. Stop the branching when $E_n\le E_c$. ThusThe continuous estimate is $13.29X_0$. At 13 generations the particles still have approximately $12.2\mathrm{MeV}$, whereas at 14 they have approximately $6.10\mathrm{MeV}$. The ideal branching model therefore suggests roughly $13$–$14X_0$ before multiplication stops.

This is not an accurate containment criterion. Real conversion and radiation distances fluctuate, bremsstrahlung energy sharing is broad rather than always one half, some energetic branches penetrate beyond the average, and particles below critical energy still have stopping and absorption ranges. The simplified stopping depth is closer to its own model's shower-maximum scale than a depth containing all deposited energy. A calorimeter needs additional longitudinal depth to control leakage and event-to-event tails. These effects explain why approximately $25X_0$ can be required even though the equal-sharing estimate is smaller. The toy model does not predict the realistic shower maximum or leakage fraction quantitatively.