Calculations on the Cascade Theory of Showers

Abstract

The solution of the cascade equations, obtained previously by Bhabha and Chakrabarty, is rearranged in a form suitable for numerical calculations. Although the solution is still in the form of an infinite series, the first term alone gives practically the entire contribution to the number of particles in a shower for all values of the energy of the shower particles. The results are compared with the values given previously by Bhabha and Chakrabarty and also by Snyder. The defects in the analysis of Snyder are discussed.Values of $N(E, t)$, the total number of particles in a shower having energies greater than $E$, are obtained for different values of $E$, $t$, and ${E}_{0}$. By the evaluation of a single integral it is now possible to obtain the values of $N(E, t)$ for any value of $E$ in the entire range (0, ${E}_{0}$), and also the nature of the energy spectrum of the shower electrons at different depths. Asymptotic values to which $P(E, t)$ and $N(E, t)$ merge, when $E$ tends to zero and infinity, are derived from the general expression. It is shown that the values of ${N}_{0}(t)+{N}_{2}(t)$, derived previously by Bhabha and Chakrabarty, is a fair approximation to the value of $N(E, t)$ if we take $E=2m{c}^{2}$.