The Angular and Lateral Distribution Functions for the Nucleon Component of the Cosmic Radiation

Abstract

A method is developed, whereby the angular and radial distribution functions for the nucleon component of the cosmic radiation in the atmosphere may be obtained in terms of the moments of the distributions. Essentially the method consists of an expansion of the unknown distribution functions in a series of derivatives of the Dirac delta-function, whose coefficients are identified with the angular and radial moments. The method gives results identical with those obtained by a more tedious procedure, and is of general applicability in reconstructing distribution functions when only their moments are known.The angular and radial distribution functions are found for various initial conditions and numerical results given for the case of an integral primary proton power law spectrum with exponent $\ensuremath{\gamma}=1.1$. Variation of atmospheric density with height is taken into account. Using a form of the differential cross section for nucleon-nucleon collisions, $R+{S}^{\ensuremath{'}} {cos}^{2}\ensuremath{\theta}$ in the center-of-mass frame of reference, predicted by most field theoretic treatments, it is shown that the calculated results are in disagreement by an order of magnitude with experimental data. The various assumptions on which our theory is based (power law, homogeneous form of total cross section, etc.) are then critically examined. It is concluded that the theory can be reconciled with experiment only when the number of scattered particles decreases exponentially from the direction of motion of the incident particle in the laboratory frame of reference. Using this result, the new angular and radial distribution functions are then calculated and found to be given at sea level essentially by $\mathrm{exp}{\ensuremath{-}166U(1\ensuremath{-}C)}$ and $\mathrm{exp}{\ensuremath{-}3.72{U}^{\frac{1}{2}}r}$, respectively, for particles of energy greater than $U$ Bev, where $C$ is the cosine of the angle with the shower axis and $r$ is the distance in kilometers from the shower axis. The half-value of the radial distribution occurs at a distance of 92 meters from the shower axis.