Improved Calculations on Cascade Shower Theory

The diffusion equations of cascade shower theory are solved by means of a perturbation method. The first approximation is taken to be that solution obtained by Snyder using the so-called completely screened cross sections for the elementary processes of bremsstrahlung and pair production. The correction to this is then calculated using more refined approximations to the Bethe-Heitler cross sections.The results, which lend themselves to accurate numerical work only in the case of light elements and high incident energies, indicate with respect to Snyder's solution that (1) the shower maximum is decreased in height and slightly shifted to greater depths, (2) there is a decrease in the average number of electrons present at small absorber depths and a correspondingly larger number at large depths. The magnitude of the correction is larger than had been previously assumed. Numerical results are given so that the correction to the solution under the assumption of completely screened cross sections can be readily calculated for all light elements when the incident particle is either a single photon or a single electron.