Rational function approximation of the collision integrals

Simulation is often used to simplify and verify the interpretation of low energy ion scattering (LEIS) spectra. To obtain reliable results a large number (typically 10⁶) of ion tracks have to be calculated. The calculation of the collision integrals is a time consuming operation because these integrals cannot be expressed analytically and numerical integration has to be used. This gives accurate results but requires much computing time. In this work we present an approximation for the scattering angle that uses little computing time while maintaining a high accuracy. The present approximation is based on a rational function fit. It is shown that this approximation is one order of magnitude more accurate and about three times faster than other approximations in literature. A forthcoming generalization for the time integral will be discussed.