The tensor form of the neutron-transport equation with application to finite differencing
Neutron transport equation in five dimensional tensor form, applying finite differencing by integration method via use of divergence theorem
Drawbaugh, D. W.↗
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Singular eigenfunction solution of the monoenergetic neutron transport equation for finite radially reflected critical cylinders.
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Westfall, R. M.↗
TRANSPORT THEORY OF NEUTRONS IN HEAVY GAS IN PLANE GEOMETRY. I. THE INFINITE MEDIUM AND CONSTANT TOTAL CROSS-SECTION
Solution of neutron transport equation in heavy gas in plane geometry in case of infinite medium and constant total cross section
REACTOR CONTROL↗
A Systematic Solution Approach for Neutron Transport Problems in Diffuse Regimes
A systematic solution approach for the neutron transport equation, based on a least-squares finite-element discretization, is presented. This approach includes the theory for the existence and uniqueness of the analytical as well as of the discrete solution, bounds for the discretization error, and guidance for the development of an efficient multigrid solver for the resulting discrete problem. To guarantee the accuracy of the discrete solution for diffusive regimes, a scaling transformation is applied to the transport operator prior to the discretization. The key result is the proof of the V-ellipticity and continuity of the scaled least-squares bilinear form with constants that are independent of the total cross section and the absorption cross section. For a variety of least-squares finite-element discretizations this leads to error bounds that remain valid in diffusive regimes. Moreover, for problems in slab geometry a full multigrid solver is presented with V(1, 1)-cycle convergence rates approximately equal to 0.1, independent of the size of the total cross section and the absorption cross section.
Manteuffel, T. A.↗
Consistent Pl Analysis of Aqueous Uranium-235 Critical Assemblies
The lethargy-dependent equations of the consistent Pl approximation to the Boltzmann transport equation for slowing down neutrons have been used as the basis of an IBM 704 computer program. Some of the effects included are (1) linearly anisotropic center of mass elastic scattering, (2) heavy element inelastic scattering based on the evaporation model of the nucleus, and (3) optional variation of the buckling with lethargy. The microscopic cross-section data developed for this program covered 473 lethargy points from lethargy u = 0 (10 Mev) to u = 19.8 (0.025 ev). The value of the fission neutron age in water calculated here is 26.5 square centimeters; this value is to be compared with the recent experimental value given as 27.86 square centimeters. The Fourier transform of the slowing-down kernel for water to indium resonance energy calculated here compared well with the Fourier transform of the kernel for water as measured by Hill, Roberts, and Fitch. This method of calculation has been applied to uranyl fluoride - water solution critical assemblies. Theoretical results established for both unreflected and fully reflected critical assemblies have been compared with available experimental data. The theoretical buckling curve derived as a function of the hydrogen to uranium-235 atom concentration for an energy-independent extrapolation distance was successful in predicting the critical heights of various unreflected cylindrical assemblies. The critical dimensions of fully water-reflected cylindrical assemblies were reasonably well predicted using the theoretical buckling curve and reflector savings for equivalent spherical assemblies.
Fieno, Daniel↗
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↗
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.↗
An Improved Neutron Transport Algorithm for Space Radiation
A low-energy neutron transport algorithm for use in space radiation protection is developed. The algorithm is based upon a multigroup analysis of the straight-ahead Boltzmann equation by using a mean value theorem for integrals. This analysis is accomplished by solving a realistic but simplified neutron transport test problem. The test problem is analyzed by using numerical and analytical procedures to obtain an accurate solution within specified error bounds. Results from the test problem are then used for determining mean values associated with rescattering terms that are associated with a multigroup solution of the straight-ahead Boltzmann equation. The algorithm is then coupled to the Langley HZETRN code through the evaporation source term. Evaluation of the neutron fluence generated by the solar particle event of February 23, 1956, for a water and an aluminum-water shield-target configuration is then compared with LAHET and MCNPX Monte Carlo code calculations for the same shield-target configuration. The algorithm developed showed a great improvement in results over the unmodified HZETRN solution. In addition, a two-directional solution of the evaporation source showed even further improvement of the fluence near the front of the water target where diffusion from the front surface is important.
Heinbockel, John H.↗
The cosmic ray electron spectrum and its modulation from 1968 through 1972
Over the past five years we have measured the energy spectrum of primary cosmic ray electrons with both a balloon-borne and a satellite absorption spectrometer. All of the balloon flights used identical equipment that was launched each summer from Fort Churchill, Manitoba, Canada. The satellite, OGO-5, has been in an eccentric orbit since March 1968. Together these instruments provide the electron spectrum over a range of energy from 20 MeV to 20 GeV. This wide range and the substantial span of time covered by the measurements permit a detailed study of the solar modulation of electrons. These results are compared with the modulation of the nuclear components as observed by a neutron monitor and interpreted using the cosmic ray transport equation.
Fulks, G.↗
Relativistic two-moment neutrino transport
We implement a general relativistic neutrino transport scheme appropriate to problems in stellar collapse and neutron star formation. In this scheme the spectral energy and momentum balance equations are solved as a coupled set. We perform transport calculations on two frozen stellar background models neglecting the feedback of the neutrinos on the matter. We compare with schemes that ignore either relativity or momentum balance or both. The work performed by the neutrino stress on the matter is taken into account and has an important effect. In particular, if there is a hydrodynamical shock in the system, this work contribution boosts the energy deposition behind the shock. This may enhance the chances for a successful neutrino driven supernova explosion.
Cernohorsky, J.↗
Energetic particle acceleration in spherically symmetric accretion flows and shocks
Steady state, spherically symmetric solutions of the cosmic-ray transport equation describing the acceleration of energetic particles in galactic accretion flows onto neutron stars, black holes, white dwarfs, and protostars are studied. The results indicate that astrophysical accretion flows can be partitioned into distinct classes depending upon whether the accretion rate lies above or below a critical value of a few times 10 to the -7th stellar masses/yr. When the particle transport is convection-dominated, both classes of accretion flows exhibit a spectral index appropriate for first-order Fermi acceleration at a plane shock in the absence of losses. As the particle transport becomes diffusion-dominated, both classes show a break and precipitous falloff in the particle spectrum due to the escape of these particles from the accretion flow. The precise nature of the spectrum depends on the relationship between the particle momentum and the spatial diffusion coefficient.
Webb, G. M.↗
Dibaryons in neutron stars
The effects are studied of H-dibaryons on the structure of neutron stars. It was found that H particles could be present in neutron stars for a wide range of dibaryon masses. The appearance of dibaryons softens the equations of state, lowers the maximum neutron star mass, and affects the transport properties of dense matter. The parameter space is constrained for dibaryons by requiring that a 1.44 solar mass neutron star be gravitationally stable.
Olinto, Angela V.↗
Exact expressions and improved approximations for interaction rates of neutrinos with free nucleons in a high-temperature, high-density gas
The exact expressions needed in the neutrino transport equations for scattering of all three flavors of neutrinos and antineutrinos off free protons and neutrons, and for electron neutrino absorption on neutrons and electron antineutrino absorption on protons, are derived under the assumption that nucleons are noninteracting particles. The standard approximations even with corrections for degeneracy, are found to be poor fits to the exact results. Improved approximations are constructed which are adequate for nondegenerate nucleons for neutrino energies from 1 to 160 MeV and temperatures from 1 to 50 MeV.
Schinder, Paul J.↗
The exact solution of the monoenergetic transport equation for critical cylinders
An analytic solution for the critical, monoenergetic, bare, infinite cylinder is presented. The solution is obtained by modifying a previous development based on a neutron density transform and Case's singular eigenfunction method. Numerical results for critical radii and the neutron density as a function of position are included and compared with the results of other methods.
Westfall, R. M.↗
A Burst Mode, Ultrahigh Temperature UF4 Vapor Core Reactor Rankine Cycle Space Power System Concept
Static and dynamic neutronic analyses have been performed on an innovative burst mode (100's of MW output for a few thousand seconds) Ulvahigh Temperature Vapor Core Reactor (UTVR) space nuclear power system. The NVTR employs multiple, neutronically-coupled fissioning cores and operates on a direct, closed Rankine cycle using a disk Magnetohydrodynamic (MHD) generater for energy conversion. The UTVR includes two types of fissioning core regions: (1) the central Ultrahigh Temperature Vapor Core (UTVC) which contains a vapor mixture of highly enriched UF4 fuel and a metal fluoride working fluid and (2) the UF4 boiler column cores located in the BeO moderator/reflector region. The gaseous nature of the fuel the fact that the fuel is circulating, the multiple coupled fissioning cores, and the use of a two phase fissioning fuel lead to unique static and dynamic neutronic characteristics. Static neutronic analysis was conducted using two-dimensional S sub n, transport theory calculations and three-dimensional Monte Carlo transport theory calculations. Circulating-fuel, coupled-core point reactor kinetics equations were used for analyzing the dynamic behavior of the UTVR. In addition to including reactivity feedback phenomena associated with the individual fissioning cores, the effects of core-to-core neutronic and mass flow coupling between the UTVC and the surrounding boiler cores were also included in the dynamic model The dynamic analysis of the UTVR reveals the existence of some very effectlve inherent reactivity feedback effects that are capable of quickly stabilizing this system, within a few seconds, even when large positive reactivity insertions are imposed. If the UTVC vapor fuel density feedback is suppressed, the UTVR is still inherently stable because of the boiler core liquid-fuel volume feedback; in contrast, suppression of the vapor fuel density feedback in 'conventional" gas core cavity reactors causes them to become inherently unstable. Due to the strength of the negative reactivity feedback in the UTVR, it is found that external reactivity insertions alone are inadequate for bringing about significant power level changes during normal reactor operations. Additional methods of reactivity control such as variations in the gaseous fuel mass flow rate, are needed to achieve the desired power level oontrol.
Dugan, E. T.↗
An Improved Elastic and Nonelastic Neutron Transport Algorithm for Space Radiation
A neutron transport algorithm including both elastic and nonelastic particle interaction processes for use in space radiation protection for arbitrary shield material is developed. The algorithm is based upon a multiple energy grouping and analysis of the straight-ahead Boltzmann equation by using a mean value theorem for integrals. The algorithm is then coupled to the Langley HZETRN code through a bidirectional neutron evaporation source term. Evaluation of the neutron fluence generated by the solar particle event of February 23, 1956, for an aluminum water shield-target configuration is then compared with MCNPX and LAHET Monte Carlo calculations for the same shield-target configuration. With the Monte Carlo calculation as a benchmark, the algorithm developed in this paper showed a great improvement in results over the unmodified HZETRN solution. In addition, a high-energy bidirectional neutron source based on a formula by Ranft showed even further improvement of the fluence results over previous results near the front of the water target where diffusion out the front surface is important. Effects of improved interaction cross sections are modest compared with the addition of the high-energy bidirectional source terms.
Clowdsley, Martha S.↗
Transport methods and interactions for space radiations
This report presents a brief history leading to the involvement of the Langley Research Center of the National Aeronautics and Space Administration (NASA) in space-radiation physics and protection. Indeed, a relatively complete summary of technical capability as of the summer of 1990 is given. The Boltzmann equations for coupled ionic and neutronic fields are presented and inversion techniques for the Boltzmann operator are discussed. Errors generated by the straight ahead approximation are derived and are shown to be negligible for most problems of space-radiation protection. A decoupling of projectile propagation from the target fields greatly simplifies the Boltzmann equations and allows an analytic solution of the target fragment transport. Analytic and numerical methods of solving the projectile transport equations are discussed. The nuclear physics underlying the coefficients in the Boltzmann equation is discussed. A coupled-channel optical model is found as a consequence of the loose binding of nuclear matter and closure of the nuclear states in high-energy reactions. Transport solutions with the developed data base are used with laboratory experiments to validate both the transport code and the data base. Numerical benchmarks and comparison with Monte Carlo calculations are also used for code validation.
Wilson, John W.↗
Extension of Case's method to the two-region critical cylinder and its application as an analytic standard
The normal mode expansion technique is applied to the transformed monoenergetic integral transport equation to develop a solution for the rotationally invariant and axially infinite, critical, two-region cylinder with a finite outer reflector boundary. Isotropic scattering and identical neutron mean free paths in the core and reflector regions are assumed. The solution in terms of singular integral equations is obtained by applying a completeness theorem found for the singular eigenfunctions. Numerical results for a variety of core and reflector multiplying properties and reflector thickness are presented and compared with the results of other method. An example of this type of application is given in a study of approximations inherent in the neutronic design analysis of a small, fast-neutron-spectrum reactor concept proposed as a space power source.
Westfall, R. M.↗