Computational methods for the HZETRN code

Asymptotic expansion has been used to simplify the transport of high charge and energy ions for broad beam applications in the laboratory and space. The solution of the lowest order asymptotic term is then related to a Green's function for energy loss and straggling coupled to nuclear attenuation providing the lowest order term in a rapidly converging Neumann series for which higher order collisions terms are related to the fragmentation events including energy dispersion and downshift. The first and second Neumann corrections were evaluated numerically as a standard for further analytic approximation. The first Neumann correction is accurately evaluated over the saddle point whose width is determined by the energy dispersion and located at the downshifted ion collision energy. Introduction of the first Neumann correction leads to significant simplification of the second correction term allowing application of the mean value theorem and a second saddle point approximation. The regular dependence of the second correction spectral dependence lends hope to simple approximation to higher corrections. At sufficiently high energy nuclear cross-section variations are small allowing non-perturbative methods to all orders and renormalization of the second corrections allow accurate evaluation of the full Neumann series.