Phase-amplitude algorithms for atomic continuum orbitals and radial integrals

We present a new, fast and accurate phase-amplitude algorithm for the calculation of atomic continuum orbitals needed for cross sections computations of various atomic processes in plasmas. A coarse, energy independent, mesh is sufficient to achieve high accuracy. A straightforward application of a predictor-corrector procedure to the non-linear differential equations would fail, in particular for high energy free electrons in any atomic potential. The present algorithm overcomes this problem. In addition, we describe a novel method for calculating the radial integrals by integration over the phases instead of r. With the use of Gaussian trigonometric formulas over half periods, the integrals are expressed as alternating series. Levin's transform for convergence acceleration then provides the sum of the series with a few terms only. These methods are applicable in a relativistic framework as well as non-relativistic.