In this paper, we derive a nonstationary distribution function describing the energy distribution of the cascade of moving atoms taking into account their multiplication. The function was derived by solving the Boltzmann kinetic equation. The development of the cascade was considered for the materials consisting of atoms of the same type without taking into account the binding energy of atoms at the crystal lattice sites. The scattering of moving atoms is assumed to be elastic and spherically symmetrical in a center-of-inertia system, and the interaction cross-section is assumed to be constant. The use of these assumptions allows us to derive simple analytic formulas for the nonstationary energy distribution function for the cascade and analyze its main distinctive features. The results obtained allow evaluating the accuracy of various approximate solutions.
