Numerical Solution of Time-Dependent Two-Dimensional P 1 Equation of Neutron Transport

The time-dependent P ₁ equation for two-dimensional neutron transport is numerically solved by a finite difference approximation of the explicit form along the bicharacteristics of the P ₁ equation.

Applying von Neumann's stability condition to this numerical procedure in an infinite space, we can derive the condition necessary for the solution to be stable. This condition is that the mesh widths satisfy the inequality o<λ≦√3/2 with λ=time mesh δt/space mesh δ or δz, where the time t is measured in units of inverse neutron speed l/v. The sufficient stability condition on the ratio λ is to be determined by numerical experiments. It has been found that the upper bound of λ becomes larger for smaller values of space mesh width.

In respect of the stability of numerical solution, the P₁ approximation is more advantageous than the diffusion approximation.

Transient behavior of neutron flux distribution due to a stationary neutron source is numerically determined assuming zero initial values. After the transient state terminates, the steady state distribution is obtained.     

Applications of the multidimensional equations to complex fuel assembly problems

For the analysis of reactors with complex fuel assemblies or fine mesh applications as pin by pin neutron flux reconstruction, the usual approximation of the neutron transport equation by the multigroup diffusion equation does not provide good results. A classical approach to solve the neutron transport equation is to apply the spherical harmonics method obtaining a finite approximation known as the P_L equations. In this line, a nodal collocation method for the discretization of these equations on a rectangular mesh is used in this paper to analyse reactors with MOX fuel assemblies. Although the 3D P_L nodal collocation method becomes feasible due to the improvements in computer hardware, a complete treatment of the detailed structure of the fuel assemblies in actual three-dimensional geometry is still prohibitive, thus, an assembly homogenization method is necessary. A homogenization method compatible with our multidimensional P_L code is proposed and tested performing heterogeneous and homogenized calculations. In this work, we apply the method to 2D complex fuel assembly configurations.     

Analysis of Cavity Effect on Space- and Time-Dependent Fast and Thermal Neutron Energy Spectra

The effects of the presence of a central cavity on the space- and time-dependent neutron energy spectra in both thermal and fast neutron systems are analyzed theoretically with use made of the multi-group one-dimensional time-dependent S_n method. The thermal neutron field is also analyzed for the case of a fundamental time eigenvalue problem with the time-dependent P₁ approximation. The cavity radius is variable, and the system radius for graphite is 120 cm and for the other materials 7 cm.

From the analysis of the time-dependent S_n calculations in the non-multiplying systems of polythene, light water and graphite, cavity heating is the dominant effect for the slowing-down spectrum in the initial period following fast neutron burst, and when the slowing-down spectrum comes into the thermal energy region, cavity heating shifts to cavity cooling. In the multiplying system of ²³⁵U, cavity cooling also takes place as the spectrum approaches equilibrium after the fast neutron burst is injected.

The mechanism of cavity cooling is explained analytically for the case of thermal neutron field to illustrate its physical aspects, using the time-dependent P₁ approximation. An example is given for the case of light water.     

Calculation of extrapolation distances by low order Chebyshev polynomial approximation of transport equation in slab geometry

In this study, the problem of extrapolated end point has been studied in one-speed neutron transport equation with isotropic scattering by using the Chebyshev polynomial approximation which is called T_N method. Assuming neutrons of one speed, extrapolated end point are calculated for the uniform finite slab using Mark and Marshak type vacuum boundary conditions. It is shown that low order T_N method gives very good results of low order spherical harmonics approximation and diffusion theory for extrapolation of the flux of neutrons leaking from the medium. We present an alternative method which is similar to P₁ method to calculate the extrapolation distances z₀. Moreover, we prefer new solution of transport equation in one-dimensional slab geometry.     

An Analysis of Continuous Bi-Directional Reactor Refueling

An analysis of continuous bi-directional reactor refueling is developed by one group diffusion approximation. The fundamental integro—differential equation is first converted to second order nonlinear differential equation, which is further reconverted to first order nonlinear differential equation, to finally take the form (du/dx) ²=f(u). By applying the elliptic function theory, the analytical solution for neutron flux and eigen value is obtained, and their physical characteristics are examined. Numerical results for neutron flux distribution are presented for the case where the rate of fuel feed movement is not the same for the two directions.     

Beam Hole Configuration and Analysis of Medical Reactor for Boron Neutron Capture Therapy

This report presents an investigation of beam holes to be provided in a medical reactor for Boron Neutron Capture Therapy. The principal requirement for the beam holes is to deliver the therapeutic doses of thermal and epithermal neutrons in a modest time (30 to 60min) with minimal fast neutron and γ-contaminants. Characteristics of the beam holes have been evaluated by 2-dim. n-γ coupling S _N transport calculations. Reexaminations and revisions of the beam hole design have brought improvements of the characteristics, especially an increase of the thermal neutron flux at the horizontal thermal neutron beam port and a decrease of the fast neutron flux at the vertical epithermal neutron beam port. The design objectives for the beam holes set up in this study may be achievable even if the thermal power of the reactor is reduced from 2 to 1MW.     

Development of mathematical models and the dual-gauge principle for surface-type neutron moisture content gauges

Mathematical models are developed for the response of surface-type neutron moisture content gauges. Models are developed for both cadmium-covered and bare BF₃-filled proportional counter detectors so that the dual-gauge principle of measurement can be applied. This consists of the simultaneous solution of the two responses to minimize the effect of variations in sample composition.The response models are based on epithermal and thermal line and area flux models, where if flux as a function of radial distance r from the source is φ(r), then line flux is ∫φ(r) dr and area flux is ∫φ(r)r dr. The flux models are obtained from a solution of the three-group diffusion equation for a point source of fast neutrons at the surface of an infinite half-space and are expressed as inverse Hankel transforms. The models are checked by calculation and are verified with Monte Carlo and experimental results.To minimize the effects of variations in sample composition, represented by the weight fraction of thermal neutron absorber, two gauge response models are solved simultaneously for moisture and absorber weight fractions. This application of the dual-gauge principle is tested with experimentally obtained responses on samples of known density and composition. The results of this feasibility study are encouraging in terms of eventual commercial application of the response models and the dual-gauge principle.     

Some nonlinear problems of nuclear reactor theory

In reactor criticality calculations it is necessary to take into account the relationship between the tamperature and volume of the reactor and the neutron flux. It is frequently assumed for this, that k_eff is a function of the reactor power. The method proposed here is an approximate computation of this relationship by means of some functional of the flux (this functional can be expressed by the power averaged with respect to the reactor temperature, etc.), so that its calculation by means of k_eff is a special case of the proposed method. Results are given of calculations for a system of nonlinear equations which describe the neutron transport in one-group diffusion approximation in the plane of the reactor and the heat transfer by thermal conductivity. The results are analyzed with the object of comparing the precise and approximate solutions.Translated from Atomnaya Énergiya, Vol. 16, No. 4, pp. 304–309, April, 1964     

Design of an epi-thermal neutron flux intensity monitor with GaN wafer for boron neutron capture therapy

Boron neutron capture therapy (BNCT) is a promising cancer therapy. Epi-thermal neutron (0.5 eV < E_n < 10 keV) flux intensity is one of the basic characteristics for modern BNCT. In this work, based on the ⁷¹Ga(n,γ)⁷²Ga reaction, a new simple monitor with gallium nitride (GaN) wafer as activation material was designed by Monte Carlo simulations to precisely measure the absolute integral flux intensity of epi-thermal neutrons especially for practical BNCT. In the monitor, a GaN wafer was positioned in the center of a polyethylene sphere as neutron moderator covered with cadmium (Cd) layer as thermal neutron absorber outside. The simulation results and related analysis indicated that the epi-thermal neutron flux intensity could be precisely measured by the presently designed monitor.     

Spherical Harmonics Solutions of Multi-Dimensional Neutron Transport Equation by Finite Fourier Transformation

A method of solution of a monoenergetic neutron transport equation in P_L approximation is presented for x-y and x-y-z geometries using the finite Fourier transformation. A reactor system is assumed to consist of multiregions in each of which the nuclear cross sections are spatially constant. Since the unknown functions of this method are the spherical harmonics components of the neutron angular flux at the material boundaries alone, the three- and two-dimensional equations are reduced to two- and one-dimensional equations, respectively. The present approach therefore gives fewer unknowns than in the usual series expansion method or in the finite difference method. Some numerical examples are shown for the criticality problem.     

Neutron Economy of Transmutation of TRU in Thermal and Fast Neutron Fields

Neutron economy of the transmutation of TRU was examined in well thermalized, thermal and fast neutron fields. Burn-up chains of ²³⁷Np, ²⁴¹Am and ²⁴³Am, which are the main TRU nuclides in the high level waste, were calculated in the flux region from 10¹⁴ to 10¹⁷ n/cm².s. Numbers of neutrons absorbed and produced of each chain were calculated using JENDL-3. The net number of neutron produced n _net, which was obtained by the difference of the two numbers, largely varied with the neutron fields, the nuclides and the flux levels. The n _net value in the fast neutron field was positive (0.0–1.0) for ²³⁷Np, ²⁴¹Am, ²⁴³Am and TRU with the nuclide composition in the high-level waste generated by the conventional PWR. The transmutation of TRU by fission can be performed with producing neutrons in the fast neutron field. On the other hand, the n _net value was negative for the well thermalized and thermal neutron fields. For TRU in the high-level waste, the values in those fields were —1.0 at 10¹⁴ n/cm².s and 0.0 at 10¹⁶ n/cm².s. In the high flux region of 10¹⁶ n/cm².s, TRU in the high-level waste can be transmuted by fission without consuming additional neutrons. In the flux region about 10¹⁴ n/cm².s, the transmutation of TRU in the high-level waste by fission requires about one neutron.     

Development and verification of a nodal approach for solving the multigroup SP3 equations

The core model DYN3D which has been developed for three-dimensional analyses of steady states and transients in thermal reactors with quadratic or hexagonal fuel assemblies is based on nodal methods for the solution of the two-group neutron diffusion equation. Loading cores with higher content of MOX fuel, the increase of the fuel cycle length, and the consideration of new reactor types are challenging for these standard methods. A nodal expansion method for solving the equations of the simplified P₃ (SP₃) approximation of the multigroup transport equation was developed to improve the accuracy of the DYN3D code. The method described in the paper is verified with pinwise calculations of a steady state of the OECD/NEA and US NRC PWR MOX/UO₂ Core Transient Benchmark. The used 16-group cross section library was generated for DORT calculations with homogenized pin cells. Two different approximations of the diffusion coefficient which occurs in the within-group form of the SP₃ equations are investigated. Using the transport cross section for the calculation of the diffusion coefficient gives much better results than those obtained with the removal cross section. The improvement of the results in comparison to a pinwise diffusion calculation is shown. The results are compared with the DORT and the heterogeneous reference solution of the code DeCART. Concerning the SP₃ calculation using the diffusion coefficient based on the transport cross section (DYN3D-SP3-TR) the deviations of the eigenvalue k_eff and the assembly powers from the transport solutions of DORT and DeCART are in the same order as those between the two transport solutions themselves. The improvement of the DYN3D-SP3-TR results in comparison to the diffusion calculation is presented. As the DYN3D-SP3-TR and DORT calculations are performed with homogenized pin cells, the pin powers of the two calculations are closer to each other than to the pin powers of the DeCART solution. To estimate the contribution of higher flux-moments, the ratio of the second flux moment to the zeroth flux moment is investigated along a horizontal row of pins. It is shown that it is low in the fuel assemblies with small peaks at the water channels. Considering the baffle and the water reflector regions higher values are obtained in the fast neutron groups.     

Diffusion cooling of thermal neutrons in the outer Plexiglas layer of a two-zone system with non-hydrogenous material inside

Diffusion cooling of thermal neutron flux in a finite volume of a medium results from the leakage of neutrons. The cooling effect is particularly complex when the system consists of zones with different neutron scattering properties. A study is presented on the diffusion cooling in Plexiglas (hydrogenous material), which constitutes an outer cylindrical shell surrounding an inner cylinder made of a non-hydrogenous substance. The pulsed neutron method has been used. Numerous series of the desired pulsed neutron experiments in two-zone cylindrical systems have been substituted by Monte Carlo simulations. The decay constant of the fundamental mode in the time distribution of the pulsed thermal neutron flux has been determined in each experiment. The so-called theoretical decay constants (defined under a particular assumption) have been calculated as a function of the system geometry and neutron dynamic parameters including the diffusion cooling coefficient of the outer Plexiglas shell. The variability of this parameter has been determined from a combination of experimental and theoretical results. For a system with a theoretical decay constant varying between 12,000 and 34,000 s⁻¹, the diffusion cooling coefficient of Plexiglas in the outer layer varies between 9000 and 4800 cm⁴ s⁻¹ with the latter being close to the value for homogeneous material. A function has been obtained and successfully applied in interpreting real experiments using Czubek’s method of measuring the absorption cross-section of small samples.     

Calculation of thermal neutron self-shielding correction factors for aqueous bulk sample prompt gamma neutron activation analysis using the MCNP code

In this work thermal neutron self-shielding in aqueous bulk samples containing neutron absorbing materials is studied using bulk sample prompt gamma neutron activation analysis (BSPGNAA) with the MCNP code. The code was used to perform three dimensional simulations of a neutron source, neutron detector and sample of various material compositions. The MCNP model was validated against experimental measurements of the neutron flux performed using a BF₃ detector. Simulations were performed to predict thermal neutron self-shielding in aqueous bulk samples containing neutron absorbing solutes. In practice, the MCNP calculations are combined with experimental measurements of the relative thermal neutron flux over the sample’s surface, with respect to a reference water sample, to derive the thermal neutron self-shielding within the sample. The proposed methodology can be used for the determination of the elemental concentration of unknown aqueous samples by BSPGNAA where knowledge of the average thermal neutron flux within the sample volume is required.     

Critical Experiments on Light-Water Moderated PuO2-UO2 Lattices

A series of critical experiments using mixed-oxide (PuO₂-UO₂) plutonium fuels was carried out at the Tank type Critical Assembly (TCA) in the Japan Atomic Energy Research Institute in cooperation with the Power Reactor and Nuclear Fuel Development Corporation. Isotopie compositions of the Pu were 68, 22, 7 and 2w/0 for ²³⁹Pu, ²⁴⁰Pu, ²⁴¹Pu and ²⁴²Pu, respectively, and the U was natural uranium.

Critical mass, power distribution and neutron activation distribution of Au wire were measured for five kinds of lattices in which the ratios of atomic density of H to Pu were 295, 402, 494, 704 and 922, respectively. The codes LASER, UGMG42-THERMOS and GTB-2 were used for obtaining three neutron group constants and the PDQ-5 code was used in two-dimensional diffusion calculations for the lattices.

The maximum differences of the calculated multiplication factors from the measured values distributed from –1.32 to 1.72%δk/k. Calculated thermal neutron density, epithermal neutron flux and power distributions showed good agreement with measured data.     

Neutron Capture Cross Section Measurement of Praseodymium in the Energy Region from 0.003 eV to 140 keV

The neutron capture cross section of praseodymium (¹⁴¹Pr) has been measured relative to the ¹⁰B(n,αγ) standard cross section in the energy region from 0.003 eV to 140 keV by the neutron time-of-flight (TOF) method with a 46-MeV electron linear accelerator (linac) of the Research Reactor Institute, Kyoto University (KURRI). An assembly of Bi₄Ge₃O₁₂ (BGO) scintillators was used for the capture cross section measurement. In addition, the thermal neutron cross section (2,200 m/s value) of the ¹⁴¹Pr(n, γ)¹⁴²Pr reaction has been also measured by an activation method at the heavy water thermal neutron facility of the Kyoto University Reactor (KUR). The thermal neutron flux was monitored with the ¹⁹⁷Au(n, γ)¹⁹⁸Au standard cross section. The above TOF measurement has been normalized to the current activation data (11.6±1.3 b) at 0.0253 eV.

The evaluated data in JENDL-3.3, ENDF/B-VI, and JEF-2.2 have been in general agreement with the current result, except that the JENDL-3.3 and the JEF-2.2 values are clearly lower than the measurement in the cross section minimum region from about 10 to 500 eV.     

A Two-Dimensional Benchmark Experiment for Neutron Transport in Water

For the purpose of providing standard data for checking two-dimensional neutron penetration calculations, fast neutron spectra as well as thermal and epithermal neutron fluxes were measured over a two-dimensional (R, Z) space in water shield using an activation method. Threshold reaction rates were converted to fast neutron scalar flux spectra with the aid of the SAND-II code. These results agree within a factor of 2 with the calculations by a two-dimensional discrete ordinates code PALLAS-2D. Thermal and epithermal neutron fluxes obtained with the Westcott's method agree quite well with the calculated values by the PALLAS-2D code in which the diffusion equation was adopted for dealing with low energy neutrons to reduce the computing time. All experimental results are given in the absolute values.     

Thermal Neutron Spectra in Light Water Poisoned with Cadmium and/or Indium

The thermal neutron spectra in light water of slab geometry poisoned with Cd and/or In were measured by the time of flight method, using a 20-MeV electron linear accelerator. The thermal spectra were simulated to those in the Pu built-up core of a commercial light water reactor corresponding to a fuel burnup of about 15,000 MWD/T. The results of measurements were compared with calculations based on the S ₄ method using the Haywood scattering law. Fairly good agreement was obtained between the calculated and measured results except in a limited range of energy above the 0.176 eV resonance of Cd. It is concluded that the P ₁ components of the source neutrons as well as the neutron scattering kernel play a significant role in the calculation of the thermal neutron spectra with large flux gradients, and that the scattering kernel of light water based on the Haywood model will be accurate enough to evaluate the infinite multiplication constant k_∞ of light water reactor cores with high fuel burnup within an error of about — 0.17%, as estimated from the uncertainty in the spectrum calculation in the region above the Cd resonance. It is also emphasised from the two- dimensional S ₄ calculations that the effect of reentrant hole perturbation should be evaluated quantitatively in the interpretation of the measured angular neutron spectra produced within finite media.     

Analysis of the Rossi-α Experiment through Pál-Bell's Equation

Pál -Bell's equation for the probability generating function of neutron counts has been analytically solved in the case of three time states, using two-forked approximation. From this solution it is found that all experimental data on neutron fluctuation consist individually of only three basic parameters. The average counting rate C, the decay constant α and the chain register rate C_r are in this instance chosen for the three fundamental measures of correlation. The original observation is presented that C_r can be obtained precisely by determining the waiting time distribution for the triggering of the time analyzer.

The correlated and uncorrelated parts of the Rossi-α data in a thermal system have been analyzed by this three-parameter scheme, and a consistent explanation is given of the results obtained.     

Estimation of time-dependent neutron transport from point source based on Monte Carlo power iteration

This study proposes a method for calculating time-dependent neutron transport from a point source with a continuous-energy Monte Carlo code. To deal with neutron multiplication and attenuation in orders of magnitude, the power iteration method conventionally used to estimate the effective multiplication factor k_eff was utilized. The time of a neutron flying in a cycle from emission of its ancestor at the point source was estimated. In the estimation, the decay time of the delayed neutron precursor was considered. The neutron flux was tallied in time bins in each cycle. The source strength in the cycle was considered as the product of k_eff estimators from the first to the previous cycle. By summing up the tallied flux multiplied by the strength, the neutron flux variation with time was obtained. This method was verified against a UO₂ fuel lattice moderated and reflected by light water.