The cascade theory with collision loss

Abstract

Electrons are assumed to suffer a constant energy loss β by collision, and the radiation loss and pair creation are taken to be described by the formulae of Bethe and Heitler valid for complete screening. With these assumptions a solution of the cascade equations is given in the form of a series, and it is shown that the series is so rapidly convergent that in general it is necessary only to calculate the first term. Collision loss enters into each of the terms in an essential way, and as a result the first term alone gives to a very considerable degree of accuracy the whole energy spectrum of electrons from the highest energy to energies far below the critical energy. For thicknesses greater than 1\cdot 5 in the characteristic unit of length the number of particles of energy E increases monotonically with decreasing E, but the spectrum gets flattened for energies below the critical energy. For thicknesses t below 1\cdot 5, the spectrum has a very different shape, decreasing first as E decreases from the primary energy and then increasing again to the smallest E, the flattening taking place now only for E < β ^t. It is shown that neglect of collision loss sometimes causes the number of electrons of even the critical energy to be as much as seven times too large. Tables of the spectra of cascade electrons due to primaries of different energies are given for five typical thicknesses. The solution is also valid when the energy of the primary electron starting the cascade is comparable with or lower than the critical energy, and gives in a compact form the complete solution of the problem of the absorption of a low-energy electron by collision loss and cascade production.