The nucleon-air nucleus interaction probability law with rising cross section

The diffusion equation of cosmic-ray nucleons is exactly integrated using t he successive approximation method for a general distribution of the primar y component, and taking into account the rising nucleon-air cross sections with energy. The interaction probability law for the nucleon in the atmosph ere is obtained as a consequence of the respective diffusion equation. If t he nucleon-air cross sections rise logarithmically, this probability law as sumes a binomial form, and for the constant cross section, it is purely Poi ssonian. The well known approximate solution is compared with our exact sol ution. It is found that the former always gives a nucleon number greater th an ours by, for example, 15-25% in the energy region 30-10000 GeV at sea le vel in the case of the mean inelasticity [K] = 0.60. It is also shown that a fairly accurate description of nucleon flux at sea level (1030 g cm(-2)) and hadron intensities at 840 g cm-2 and at 1030 g cm(-2) are obtained with [K] varying between 0.55 and 0.60.