Accurate calculation of collision integrals using adaptive methods

Abstract

In this work, we have developed accurate and efficient methods for calculating the scattering angle, time integral and electronic energy loss for binary collisions in matter. In contrast to the conventional integration and the hard sphere approximations, our new time integral is obtained by calculating the difference between the distance to the target atom and the ion path length. To compute the scattering angle and time integral accurately and efficiently, an adaptive Simpson quadrature using a smoother integrand is adopted. The energy loss arising from the local electron-concentration-dependent electronic stopping power is obtained by tracing the ion trajectory. To reduce the computation time, a second-order differential equation for the ion motion using an adaptive Runge-Kutta-Fehlberg algorithm is developed.