Cascade Theories with Ionization Loss

Analytical solutions have previously been given for the number distribution functions and for the general moments of the electron-photon and nucleon cascades neglecting ionization losses (approximation $A$). Solutions are now given for the moments of the electron-photon and proton-neutron cascades taking into account energy loss, via ionization, by electrons and protons (approximation $B$). The diffusion equations for the differential moment functions, which yield the required factorial moments by a simple integration over the energy variables, are transformed by Laplace-Mellin transforms to matrix recurrence relations, the general solution of which is obtained in the form of power series. From these series, solutions for the moments in a form suitable for numerical calculations are obtained by a generalization of the method used by Bhabha and Chakrabarty for the first moments of the electron-photon cascade and by Messel in the proton-neutron cascade. To a first approximation, the solutions for the moments in approximation $B$ are expressed as a correction factor multiplying the solutions obtained in approximation $A$.