On the Theory of Cascade Showers, I

Abstract

185 The improved mathematical treatment for the calculation of cascade theory is presented. The solution corresponding to a single incident electron derived by this method is identical with that of Snyder and Scott. The method is mainly due to the principle of analytic continuation but somewhat different from that of Bhabha-Chakrabarty. It is also applicable to other inhomogenuous integro-differential equations and in particular $hould be useful for further works on cascade theory. For an example, it is shown the lateral and angular distribution functions are derivable analytically by this method. § 1. Introduction Since the original works on the theory of cascade showers by Bhabha and Heiderl) and by Carlson and Oppenheimer,2) many contributions have hitherto been published by various authors. 3) These cascade theories have become one of the indispensable and the most useful means for the interpretation of cosmic-ray phenomena. It has been required to get more accurate cascade functions for quantitative comparison between theory and the experimental results with increasing accuracy of the experiments. As the historical survey of these developments have been made by many authors/) we will not repeat them here. Snyder5) and Scott 6) have recendy shown that the exact series solution of-the diffusion equations of cascade showers can be derived by using the asymptotic forms for the cross section. Since the first terms in their expansions account for the total energy dissipated in the showers, they seem to be useful for the practical applications when the effects of spread of the showers caused by scattering 7) and the variation of cross sections R) with energy of the shower particles can be neglected. However, their mathematical treatments are somewhat complicated, and we present here the improved mathematical method which can derive the cascade functions identical with those of Snyder 5) and Scott 6) more easily. This treatment is mainly due to the principle of the analytic continuation and it is clear that the differences of the solutions between Snyder 5) and Bhabha-ChakrabartyH) come merely from the different analytic continuations used by each authors_ The method is also applicable to the problem of the lateral and angular spreads of the cascade showers which has so far scarcely been solved analytically. The detail features of these distribution functions, the effect of single scattering, comparison with experiments· .. will be given in the subsequent papers.