In this research work, the neutron Boltzmann equation was separated into two coupled integro-differential equations describing forward and backward neutron fluence in selected materials. Linear B-splines were used to change the integro-differential equations into a coupled system of ordinary differential equations (O.D.E.'s). Difference approximations were then used to recast the O.D.E.'s into a coupled system of linear equations that were solved for forward and backward neutron fluences. Adding forward and backward fluences gave the total fluence at selected energies and depths in the material. Neutron fluences were computed in single material shields and in a shield followed by a target configuration. Comparison was made to Monte Carlo modeling of the Boltzmann equation for the same material configuration. Slabs of aluminum, copper, and Martian regolith served as single material shields. The Forward-backward model fluences are accurate for energies above 4 MeV at all depths in these media. There is less accuracy for energies below 4 MeV, but the forward-backward model results are more accurate than past calculations and provide approximately an order of magnitude improvement. An aluminum shield followed by a water target configuration was also investigated using the forward-backward model. The resulting fluence is accurate in the aluminum shield, but less accurate in the water target. It is suspected that this is largely due to inaccuracies in the material cross sections that were used in the modeling.
","abstract_html":"<p>In this research work, the neutron Boltzmann equation was separated into two coupled integro-differential equations describing forward and backward neutron fluence in selected materials. Linear B-splines were used to change the integro-differential equations into a coupled system of ordinary differential equations (O.D.E.'s). Difference approximations were then used to recast the O.D.E.'s into a coupled system of linear equations that were solved for forward and backward neutron fluences. Adding forward and backward fluences gave the total fluence at selected energies and depths in the material. Neutron fluences were computed in single material shields and in a shield followed by a target configuration. Comparison was made to Monte Carlo modeling of the Boltzmann equation for the same material configuration. Slabs of aluminum, copper, and Martian regolith served as single material shields. The Forward-backward model fluences are accurate for energies above 4 MeV at all depths in these media. There is less accuracy for energies below 4 MeV, but the forward-backward model results are more accurate than past calculations and provide approximately an order of magnitude improvement. An aluminum shield followed by a water target configuration was also investigated using the forward-backward model. The resulting fluence is accurate in the aluminum shield, but less accurate in the water target. It is suspected that this is largely due to inaccuracies in the material cross sections that were used in the modeling.</p>","abstract_has_math":false,"creators":["Feldman, Gary Alan"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["John Tweed","John H. Heinbockel","Martha S. Clowdslev","Fang Q. Hu"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2003,"date_issued":"2003-07-01T07:00:00Z","date_published":"2003-07-01T07:00:00Z","updated_at":"2026-07-24T03:35:15Z","subjects":["Boltzmann equation","Forward-backward fluence","Neutron Boltzmann equation","Neutron fluence","Shield material","Mathematics","Ordinary Differential Equations and Applied Dynamics"],"languages":[],"rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["9780496549399","https://digitalcommons.odu.edu/mathstat_etds/86"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this research work, the neutron Boltzmann equation was separated into two coupled integro-differential equations describing forward and backward neutron fluence in selected materials. Linear B-splines were used to change the integro-differential equations into a coupled system of ordinary differential equations (O.D.E.'s). Difference approximations were then used to recast the O.D.E.'s into a coupled system of linear equations that were solved for forward and backward neutron fluences. Adding forward and backward fluences gave the total fluence at selected energies and depths in the material. Neutron fluences were computed in single material shields and in a shield followed by a target configuration. Comparison was made to Monte Carlo modeling of the Boltzmann equation for the same material configuration. Slabs of aluminum, copper, and Martian regolith served as single material shields. The Forward-backward model fluences are accurate for energies above 4 MeV at all depths in these media. There is less accuracy for energies below 4 MeV, but the forward-backward model results are more accurate than past calculations and provide approximately an order of magnitude improvement. An aluminum shield followed by a water target configuration was also investigated using the forward-backward model. The resulting fluence is accurate in the aluminum shield, but less accurate in the water target. It is suspected that this is largely due to inaccuracies in the material cross sections that were used in the modeling.
"]},{"key":"dc:title","label":"Title","values":["A Forward-Backward Fluence Model for the Low-Energy Neutron Boltzmann Equation"]}]}],"canonical_facts":{"dc:contributor":["John Tweed","John H. Heinbockel","Martha S. Clowdslev","Fang Q. Hu"],"dc:creator":["Feldman, Gary Alan"],"dc:date.available":["2019-10-01T07:00:00Z"],"dc:description.abstract":["In this research work, the neutron Boltzmann equation was separated into two coupled integro-differential equations describing forward and backward neutron fluence in selected materials. Linear B-splines were used to change the integro-differential equations into a coupled system of ordinary differential equations (O.D.E.'s). Difference approximations were then used to recast the O.D.E.'s into a coupled system of linear equations that were solved for forward and backward neutron fluences. Adding forward and backward fluences gave the total fluence at selected energies and depths in the material. Neutron fluences were computed in single material shields and in a shield followed by a target configuration. Comparison was made to Monte Carlo modeling of the Boltzmann equation for the same material configuration. Slabs of aluminum, copper, and Martian regolith served as single material shields. The Forward-backward model fluences are accurate for energies above 4 MeV at all depths in these media. There is less accuracy for energies below 4 MeV, but the forward-backward model results are more accurate than past calculations and provide approximately an order of magnitude improvement. An aluminum shield followed by a water target configuration was also investigated using the forward-backward model. The resulting fluence is accurate in the aluminum shield, but less accurate in the water target. It is suspected that this is largely due to inaccuracies in the material cross sections that were used in the modeling.
"],"dc:identifier":["9780496549399","https://digitalcommons.odu.edu/mathstat_etds/86"],"dc:rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"],"dc:subject":["Boltzmann equation","Forward-backward fluence","Neutron Boltzmann equation","Neutron fluence","Shield material","Mathematics","Ordinary Differential Equations and Applied Dynamics"],"dc:title":["A Forward-Backward Fluence Model for the Low-Energy Neutron Boltzmann Equation"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:35:15Z"}