Abstract
An analytical solution is obtained for the integro-differential equation of a cascade process for particles (knocked out atoms, etc.) in a three-dimensional model of an elementary act, into which a generalized cascade-probability function (GCPF) leads. It is shown that after the Laplace transformation, the equation passes into an integral one, which is then solved by the method of successive approximations. The basic physical properties of this function are established. With the constancy of the path to the interaction and the angle of departure of the particle after the collision, the GCPF goes into the simplest CPF. With small depths of registration, the probability that a particle will pass this depth tends to zero (both without collisions and when testing i interactions). At greater depths, these probabilities tend to zero. With a large i GCPF goes into a Stirling type formula. As the interaction path increases, the probability without interaction tends to unity, and the probability with i-interactions tends to zero.