A set of coupled linear integral equations for the energy distributions of rapidly moving atoms in an irradiated medium composed of many species is investigated. It is assumed that only binary collisions occur in which an energetic atom interacts with a thermal atom. Solutions are obtained for the case of two species where scattering is isotropic in the center of mass system and all collision cross sections have the same energy dependence. When the massm1of species 1 is much less thanm2the collision densities have the formf(E)=aE−2+bE−2+4(c1+c2)m1/m2, wherec1andc2are related to the collision cross sections. Whenm1=m2, thenf(E)=aE−2+bE−1/(1+c1)−1/(1+c2). Similar solutions are found for intermediate mass ratios.