The Derivation and Moments Solution of Approximate Transport-Equations for the Implantation of Ions into Amorphous Targets

Commencing with the LSS integro-differential equation, an approximate transport equation is derived from which the moments of the range distribution may be obtained. The resulting equation set is known as the Rent Range Algorithm (KRAL). The method for numerical solution of these equations, when written as a set of coupled second order ordinary differential equations (ODEs) of the initial value type, is then outlined. Solution is achieved by recasting the equation set in the form of first order ODEs designed for iterative solution. The technique used is an iterative refinement (or residual correction) procedure and the set of first order ODEs is called the Rent Optimised Range Algorithm (KORAL). Finally, the first three moments from KORAL, first and second order PRAL codes and the full transport equation code KUBBIC-91 are compared with Monte Carlo data obtained from a TRIM code modified to treat targets of infinite extent. Comparisons are performed using consistent nuclear and electronic energy loss models.