Note on the complete stochastic treatment of electron cascades

Abstract

The present note extends to the actual cascade process a result already established by Scott and Uhlenbeck for a model in which there is only one type of physical entity involved. It is shown that with the sole approximation of neglecting the angular divergence of showers complete information about the stochastic process of cascade generation can be calculated. Theqth order correlation functions are expressed as integrals involving the (q-1)th order correlation functions and the two independent elementary solutions of the cascade equations (8). The latter are the usual functions giving the mean number of electrons and photons in cascades excited respectively by a single electron or photon of some definite energy. If the latter are known exactly, then all the higher order correlation functions can be calculated by repeated integration. One thus obtains explicit expressions for the mean of theqth power of the number of electrons or photons in some finite energy interval, and from this again, through a use of equation (5), of the probability of finding N particles in this energy interval at a deptht in a cascade started by an electron or photon of some given energy.