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All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

The linear Boltzmann equation in slab geometry - Development and verification of a reliable and efficient solution

The linear Boltzmann equation can be cast in a form mathematically identical to the radiation-transport equation. A multigroup procedure is used to reduce the energy (or velocity) dependence of the transport equation to a series of one-speed problems. Each of these one-speed problems is equivalent to the monochromatic radiative-transfer problem, and existing software is used to solve this problem in slab geometry. The numerical code conserves particles in elastic collisions. Generic examples are provided to illustrate the applicability of this approach. Although this formalism can, in principle, be applied to a variety of test particle or linearized gas dynamics problems, it is particularly well-suited to study the thermalization of suprathermal particles interacting with a background medium when the thermal motion of the background cannot be ignored. Extensions of the formalism to include external forces and spherical geometry are also feasible.

Stamnes, K.↗
NASA NTRS2004

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. c2004 COSPAR. Published by Elsevier Ltd. All rights reserved.

NASA Center LaRC↗
NASA NTRS2003

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 the high charge and energy (HZE) 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.

Tweed, J.↗
NASA NTRS1974

Transport coefficients of gaseous ions in an electric field

A general theory of ion mobility formulated by Kihara (1953) is extended to ion diffusion and to mixtures of neutral gases. The theory assumes that only binary collisions between ions and neutral particles need to be taken into account and that the velocity distribution function of the neutral particles is Maxwellian. These assumptions make it possible to use a linearized Boltzmann equation. Questions of mobility are considered along with aspects of diffusion and deviations from Fick's law of diffusion.

Whealton, J. H.↗
NASA NTRS1991

Computational attributes of the integral form of the equation of transfer

Difficulties can arise in radiative and neutron transport calculations when a highly anisotropic scattering phase function is present. In the presence of anisotropy, currently used numerical solutions are based on the integro-differential form of the linearized Boltzmann transport equation. This paper, departs from classical thought and presents an alternative numerical approach based on application of the integral form of the transport equation. Use of the integral formalism facilitates the following steps: a reduction in dimensionality of the system prior to discretization, the use of symbolic manipulation to augment the computational procedure, and the direct determination of key physical quantities which are derivable through the various Legendre moments of the intensity. The approach is developed in the context of radiative heat transfer in a plane-parallel geometry, and results are presented and compared with existing benchmark solutions. Encouraging results are presented to illustrate the potential of the integral formalism for computation. The integral formalism appears to possess several computational attributes which are well-suited to radiative and neutron transport calculations.

Frankel, J. I.↗
NASA NTRS1993

Development of deterministic transport methods for low energy neutrons for shielding in space

Transport of low energy neutrons associated with the galactic cosmic ray cascade is analyzed in this dissertation. A benchmark quality analytical algorithm is demonstrated for use with BRYNTRN, a computer program written by the High Energy Physics Division of NASA Langley Research Center, which is used to design and analyze shielding against the radiation created by the cascade. BRYNTRN uses numerical methods to solve the integral transport equations for baryons with the straight-ahead approximation, and numerical and empirical methods to generate the interaction probabilities. The straight-ahead approximation is adequate for charged particles, but not for neutrons. As NASA Langley improves BRYNTRN to include low energy neutrons, a benchmark quality solution is needed for comparison. The neutron transport algorithm demonstrated in this dissertation uses the closed-form Green's function solution to the galactic cosmic ray cascade transport equations to generate a source of neutrons. A basis function expansion for finite heterogeneous and semi-infinite homogeneous slabs with multiple energy groups and isotropic scattering is used to generate neutron fluxes resulting from the cascade. This method, called the FN method, is used to solve the neutral particle linear Boltzmann transport equation. As a demonstration of the algorithm coded in the programs MGSLAB and MGSEMI, neutron and ion fluxes are shown for a beam of fluorine ions at 1000 MeV per nucleon incident on semi-infinite and finite aluminum slabs. Also, to demonstrate that the shielding effectiveness against the radiation from the galactic cosmic ray cascade is not directly proportional to shield thickness, a graph of transmitted total neutron scalar flux versus slab thickness is shown. A simple model based on the nuclear liquid drop assumption is used to generate cross sections for the galactic cosmic ray cascade. The ENDF/B V database is used to generate the total and scattering cross sections for neutrons in aluminum. As an external verification, the results from MGSLAB and MGSEMI were compared to ANISN/PC, a routinely used neutron transport code, showing excellent agreement. In an application to an aluminum shield, the FN method seems to generate reasonable results.

Ganapol, Barry↗
NASA NTRS1984

Broadening of the Interplanetary Helium Cone Structure Due to Elastic Collisions of LISM Helium Atoms with Solar Wind Ions

Neutral interstellar particles penetrating into the heliosphere, besides being subject there to specific loss processes, suffer elastic collisions with KeV-solar wind ions. The momentum transfer to the neutrals connected with these collisions leads to a loss of angular momentum with respect to the Sun and to a fractional compensation of the effective solar gravity. The dynamical particle trajectories hence are changed into non-Keplerians leading to density and temperature distributions differing from those calculated in the past. This is found from a solution of the Boltzmann equation that linearizes the effect of this additional force. It is shown that the HeI-584A resonance glow of the heliospheric helium cone lead to substantially lower interstellar helium temperatures if re-interpreted on the basis of this revised theory. These temperatures now seem to be in accordance with the derived temperatures for interstellar hydrogen.

Fahr, H. J.↗
NASA NTRS1991

Photon interaction cross sections for aggregations of finite-dimensional leaves

The Ross (1981) plate turbid-medium theory, which abstracts the leaf canopy as a gaseous mixture of oriented planar nondimensional plates that are randomly distributed in the configuration space, does not account for the 'hot spot' effect typical of all layered media with finite-dimensional scattering centers. The concept of 'particle distribution functions' is presently used to derive the interaction cross sections. This formalism is statistical-mechanical in nature, since the radiative transfer equation is essentially a linearized form of the Boltzmann equation for N-body systems.

Myneni, Ranga B.↗
NASA NTRS1968

Slip coefficient of a gas.

Slip coefficient of gas calculated from linearized Boltzmann-Bhatnagar-Gross-Krook equation for slip velocity problem

Huang, A. B.↗
NASA NTRS2000

Higher Pole Linear Traps for Atomic Clock Applications

We investigate experimentally and theoretically higher pole linear ion traps for frequency standard use. We have built a 12-pole trap and have successfully loaded ions into it from a linear quadrupole trap. By solving the Boltzmann equation describing large ion clouds where space charge interactions are important, we show that clock frequency changes due to ion number fluctuations are much smaller in ion clocks based multipole traps than comparable clocks based on quadrupole linear traps.

Prestage, John D.↗
NASA NTRS1998

Locations of Halide Ions in Tetragonal Lysozyme Crystals

Anions play an important role in the crystallization of lysozyme, and are known to bind to the crystalline protein. Previous studies employing X-ray crystallography had found one chloride ion binding site in the tetragonal crystal form of the protein and four nitrate ion binding sites in the monoclinic form. Studies using other approaches have reported more chloride ion binding sites, but their locations were not known. Knowing the precise location of these anions is also useful in determining the correct electrostatic fields surrounding the protein. In the first part of this study the anion positions in the tetragonal form were determined from the difference Fourier map obtained from the lysozyme crystals grown in bromide and chloride solutions under identical conditions. The anion locations were then obtained from standard crystallographic methods and five possible anion binding sites were found in this manner. The sole chloride ion binding site found in previous studies was confirmed. The remaining four sites were new ones for tetragonal lysozyme crystals. However, three of these new sites and the previously found one corresponded to the four unique binding sites found for nitrate ions in monoclinic crystals. This suggests that most of the anion binding sites in lysozyme remain unchanged, even when different anions and different crystal forms of lysozyme are employed. It is unlikely that there are many more anions in the tetragonal lysozyme crystal structure. Assuming osmotic equilibrium it can be shown that there are at most three more anions in the crystal channels. Some of the new anion binding sites found in this study were, as expected, in pockets containing basic residues. However, some of them were near neutral, but polar, residues. Thus, the study also showed the importance of uncharged, but polar groups, on the protein surface in determining its electrostatic field. This was important for the second part of this study where the electrostatic field surrounding the protein was accurately determined. This was achieved by solving the linearized version of the Poisson-Boltzmann equation for the protein in solution. The solution was computed employing the commercial code Delphi which uses a finite difference technique. This has recently become available as a module in the general protein visualization code Insight II. Partial charges were assigned to the polar groups of lysozyme for the calculations done here. The calculations showed the complexity of the electrostatic field surrounding the protein. Although most of the region near the protein surface had a positive field strength, the active site cleft was negatively charged and this was projected a considerable distance. This might explain the occurrence of "head-to-side" interactions in the formation of lysozyme aggregates in solution. Pockets of high positive field strength were also found in the vicinity of the anion locations obtained from the crystallographic part of this study, confirming the validity of these calculations. This study clearly shows not only the importance of determining the counterion locations in protein crystals and the electrostatic fields surrounding the protein, but also the advantage of performing them together.

Lim, Kap↗