Lateral Spread of the Cascade Developed in the Earth's Atmosphere

Abstract

An analytical treatment is developed to calculate the mean squares of the lateral spreads of electrons in the electromagneti~ cascades developed in the- earth's atmosphere. The numerical results derived by using this analytical method are compared with those worked out by other workers. Effects of density variation of the· atmosphere and the variation of ' starting point of the cascade are shown by comparing the present. results with those for homogeneous matter. §I. Introduction The cascade theory for homogeneous matter has been developed by many ingeneous workers, and it is now of great use m the studies of the cosmic rays and the high energy physics. The electromagnetic cascades m extensive air showers, however, develop through the earth's atmosphere in which density of air varies with depth. Kamata and Nishimura 1l proved that the results of the cascade theory for homogeneous ·matter are approximately valid to the atmosphere if density of homogeneous matter is taken to be that of the atmosphere at the depth a few cascade units above the observation point. The direct and rigorous approach to the cascade theory for inhomogeneous matter is possible as attempted by ,several workers. 2 l~ 4 l In view of recent develop­ ment of the theoretical and experimental studies of air showers, it would be of great importance to develop the cascade theory for the earth's atmosphere: In this paper, the mean squares of the lateral spread of shower electrons are' discussed under approximation A. The mean squares of the lateral spread, in theoretical aspect, are very suitable for seeing the effect of density variation and the starting point of showers. Also, from experimental interest, applications will be found to the studies· of high energy electromagnetic component in air showers. Throughout the paper, definitions of variables, constants, equations and func­ tions follow Nishimura, 5l whenever they are common to those of the ordinary cascade theory for homogeneous matter.