A simple model for straggling evaluation

Validation of the HZETRN code for laboratory exposures with 1A GeV iron ions in several targets

A new version of the HZETRN code capable of validation with HZE ions in either the laboratory or the space environment is under development. The computational model consists of the lowest order asymptotic approximation followed by a Neumann series expansion with non-perturbative corrections. The physical description includes energy loss with straggling, nuclear attenuation, nuclear fragmentation with energy dispersion and downshift. Measurements to test the model were performed at the Alternating Gradient Synchrotron and the NASA Space Radiation Laboratory at Brookhaven National Laboratory with iron ions. Surviving beam particles and produced fragments were measured with solid-state detectors. Beam analysis software has been written to relate the computational results to the measured energy loss spectra of the incident ions for rapid validation of modeled target transmission functions.

A procedure for benchmarking laboratory exposures with 1 A GeV iron ions

A new version of the HZETRN code capable of simulating HZE ions with either laboratory or space boundary conditions is under development. The computational model consists of combinations of physical perturbation expansions based on the scales of atomic interaction, multiple scattering, and nuclear reactive processes with use of asymptotic/Neumann expansions with non-perturbative corrections. The code contains energy loss with straggling, nuclear attenuation, nuclear fragmentation with energy dispersion and downshifts, and off-axis dispersion with multiple scattering under preparation. The present benchmark is for a broad directed beam for 1 A GeV iron ion beams with 2 A MeV width and four targets of polyethylene, polymethyl metachrylate, aluminum, and lead of varying thickness from 5 to 30 g/cm2. The benchmark quantities will be dose, track averaged LET, dose averaged LET, fraction of iron ion remaining, and fragment energy spectra after 23 g/cm2 of polymethyl metachrylate.

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.

A space radiation transport method development

Improved spacecraft shield design requires early entry of radiation constraints into the design process to maximize performance and minimize costs. As a result, we have been investigating high-speed computational procedures to allow shield analysis from the preliminary design concepts to the final design. In particular, we will discuss the progress towards a full three-dimensional and computationally efficient deterministic code for which the current HZETRN evaluates the lowest-order asymptotic term. HZETRN is the first deterministic solution to the Boltzmann equation allowing field mapping within the International Space Station (ISS) in tens of minutes using standard finite element method (FEM) geometry common to engineering design practice enabling development of integrated multidisciplinary design optimization methods. A single ray trace in ISS FEM geometry requires 14 ms and severely limits application of Monte Carlo methods to such engineering models. A potential means of improving the Monte Carlo efficiency in coupling to spacecraft geometry is given in terms of re-configurable computing and could be utilized in the final design as verification of the deterministic method optimized design.

An improved Green's function for ion beam transport

Ion beam transport theory allows testing of material transmission properties in the laboratory environment generated by particle accelerators. This is a necessary step in materials development and evaluation for space use. The approximations used in solving the Boltzmann transport equation for the space setting are often not sufficient for laboratory work and those issues are the main emphasis of the present work. In consequence, an analytic solution of the linear Boltzmann equation is pursued in the form of a Green's function allowing flexibility in application to a broad range of boundary value problems. It has been established that simple solutions can be found for high charge and energy (HZE) ions by ignoring nuclear energy downshifts and dispersion. Such solutions were found to be supported by experimental evidence with HZE ion beams when multiple scattering was added. Lacking from the prior solutions were range and energy straggling and energy downshift with dispersion associated with nuclear events. Recently, we have found global solutions including these effects providing a broader class of HZE ion solutions.