Further calculations on the cascade theory

Abstract

The paper summarizes the previous work, and briefly gives the steps by which the mathematical solution of the cascade equations given in our previous papers can be established rigorously. This solution has been used to calculate the number of shower particles for thicknesses between 0.25 characteristic unit and 20 characteristic units, and for primary energies from 2.7 times the critical energy (y₀=1) to 2.7×10¹⁰ times the critical energy (y₀=24). For this purpose, the second term of our series solution has also been calculated. The results are given in Table III and Fig. 3 in a form suitable for comparison with, and analysis of, experiment. These show that the second and higher terms of our series are negligible compared with the first for thicknesses less than three to four times that at which the shower reaches its maximum. A method has been developed which allows the integrals to be evaluated at very small thicknesses where transition effects are still of importance, so that it is now possible to trace a shower from its very beginning to large depths. It is shown that for very small showers started by particles of two to three times the critical energy the shower must penetrate to depths which are three times the maximum depth to which a single particle could penetrate as a result of collision loss alone. This is possible because part of the path of the shower is covered by photons alone which then materialize at a subsequent depth. A simple formula has been given (37) which enables one to calculate with considerable accuracy the spectrum of shower electrons of energy much below the critical energy. It is proved that this spectrum increases monotonically with decreasing energy at all thicknesses. Its form is approximately that of a modifi ed inverse square law at the maximum of the shower, the power of the law becoming higher with increasing depth.