The Calculation of Ion Ranges in Solids with Analytic Solutions

Abstract

There exist numerous approaches [1–7] to solve the transport equation for ions slowing down in amorphous matter. They use different expansions and iterations, but have in common the use of an approximation to the scattering cross section in combining the two variables e(energy) and sin θ /2 (θ scattering angle) into one single variable t^1/2=ε sin θ /2. As previously noted [8–10], this leads to deviations from exact values, particularly at low energies, where the effective potential becomes steeper and large deflection angles become more frequent. Some of the former approaches do not account for the straggling in nuclear energy loss, all neglect electronic straggling, and an exact treatment of compound targets often has difficulties. Recently, a new projected range theory was developed, which is based on well-known stopping powers and energy-loss stragglings, thus avoiding any uncertainties of prescribing differential scattering cross sections. It turned out that this can be accomplished by connecting directional angular spread to the nuclear energy loss directly, and that this approach yields good agreement with other existing theories and experimental results on projected ranges. The method has already been applied for tabulations of ion ranges in semiconductors, including multi-atomic composite materials [11,12].