Calculation of projected ranges — analytical solutions and a simple general algorithm

Abstract The concept of multiple scattering is reconsidered for obtaining the directional spreading of ion motion as a function of energy loss. From this the mean projection of each pathlength element of the ion trajectory is derived which — upon summation or integration — leads to the desired mean projected range. In special cases, the calculation can be carried out analytically, otherwise a simple general algorithm is derived which is suitable even for the smallest programmable calculators. Necessary input for the present treatment consists only of generally accessable stopping power and straggling formulas. The procedure does not rely on scattering cross sections, e.g. power potential or f(t 1 2 ) approximations. The present approach lends itself easily to include electronic straggling or to treat composite target materials, or even to account for the so-called “time integral”.