Tamotsu Sekiya

Space and angle dependent collision probability in cell problems

Abstract The integral transport equation is formulated in terms of the generalized collision probability in a cylindrical lattice cell in one velocity. The anisotropy of the neutron angular distribution and the scattering is considered by representing them with spherical harmonics series respectively. The space dependence of the neutron distribution is taken into account by representing it with a Legendre series in each annular region. The isotropic return condition or white boundary condition is adopted for the generalized collision probability. One group disadvantage factors are calculated for cylindrical lattices in both cases of isotropic and linearly anisotropic scattering using the generalized first-flight collision probability.

Anisotropic Collision Probabilities in Cell Problems

A new first-flight collision probability that takes into account the effect of the anisotropic scattering in the moderator is derived in a cylindrical cell. This probability is obtained by expanding the scattering kernel, angular flux and angular source into spherical harmonics series and retaining the first two terms in the integral Boltzmann equation. Making use of a new reciprocity relation and the conservation law, we introduce the probability relevant to a lattice cell under the condition that all neutrons impinging on the cell boundary should reflect with isotropic distribution back into the original cell. This probability also satisfies both the usual reciprocity theorem and the conservation law. Though we have here treated only a 2-medium problem, the method can be easily extended to the problem of a cell containing many regions. As an example of application, we calculate the flux ratios in a two region cell by one-group theory and the neutron spectra in fuel and moderator using the Fermi-age kern...

Analysis of irreversible circulation in reactor noise

Abstract A new parameter α is proposed as a statistical variable of multi-dimensional reactor noise analysis. It indicates circulation in a space of state variables, while the well-known variance σ represents the spread of distribution of fluctuations. The pair of quantities σ and α are minimum necessary integral indices to characterize any power spectral density and are convenient for data retrieval. The α is less sensitive to the effect of low-cut filter than σ. The analysis of experimental data by using α is demonstrated and compared with the results of theoretical treatment.

An Improvement to the G. G. Approximation in Neutron Slowing Down through the Method of Generalized Function

The neutron slowing down problem in an infinite homogeneous medium is treated within the G.G. approximation through the theory of generalized function (g.f.). As test function for defining the g.f.s, the source importance for the slowing down is chosen. In place of the Taylor expansion of the collision term of slowing down equation in the G.G. method we expand the adjoint collision term of the importance equation. Solutions obtained with this method clearly reproduce the Placzek wiggles, which do not appear in corresponding solutions by the orthodox G.G. method using the same order of approximation, and our solutions are in very good agreement with the exact Placzek function.

EFFECT OF HIGHER ORDER ANISOTROPY ON DISADVANTAGE FACTOR IN A CELL.

In slab cells, the effect of higher order anisotropy of neutron scatterings on the one-speed disadvantage factor has been treated by Eccleston et al.(1) They used the singular eigenfunction expansion method. In plane geometry, their results show that for any given moderator-to-fuel volume ratio, the increase of the scattering anisotropy in the moderator under the assumption that the fuel scatters neutrons isotropically leads to a relative decrease in the disadvantage factor. For an infinite lattice of cylindrical cells, the problem is not so simple because of the coupling of anisotropic scattering and geometry effects(2). We evaluate the effect of higher order anisotropy by extending our method(3) which was based on the first-flight collision probability method. As shown in Ref.(1), it is sufficient to take into account effects up to quadratic anisotropy for the calculation of the disadvantage factor. Then we may expand the angular flux in a cylindrical cell in the form

A Stochastic Operator Method of Calculating the Time Interval Distributions in Neutron Detection

An operator method has been applied to formulate probability distribution functions of neutron counts in a reactor. Assuming all statistical events occurring in the reactor to be Markovian, an operator representation of the no count probability for a given time interval is given. The basic equation for the probability is derived from the Kolmogorov-Chapman equation, and the formal solution obtained thereof. Approximate expressions are also given, for two types of detectors—absorption and fission. The effects of moments of order higher than the second are evaluated numerically. Further development of the operator calculus has yielded relations connecting the waiting time distribution and the interval distribution with the no count probability.

Analysis of Large Liquid-Metal Fast Breeder Reactor Critical Experiments by Improved Methods

To solve the problems encountered in the analysis of the large homogeneous and heterogeneous fast critical assemblies, Zero-Power Plutonium Reactor (ZPPR) 9, 10, and 13, the authors have revisited the analysis using improved methods. Two-dimensional cell calculations, cell calculations using multidrawer cell models, and three-dimensional transport theory core calculations were introduced. Using these methods, the discrepancies in the calculation-to-experiment (C/E) values of kappa/sub eff/ for the fast critical assemblies was reduced. The use of the multidrawer model reduced the C/E spatial dependency of the control rod worths in the ZPPR-10 cores. To investigate the remaining problems of the spatial dependence of the C/E values of reaction rate distribution and control rod worth, the authors have adjusted a cross-section set obtained from the JENDL-2 library using the integral experiments. The cross-section changes, particularly for the diffusion coefficient, /sup 238/U scattering and capture, and /sup 239/Pu fission cross sections, have corrected the spatial dependence, as well as the overestimation of the /sup 239/U capture to /sup 239/Pu fission rate ratio and sodium void worth.

Calculation of the Anisotropic Diffusion Coefficient

The one-group anisotropic diffusion coefficient is calculated for slab and square lattice cells with use made of Benoists formula. In utilizing the integral transport theory, only several collisions suffered by a neutron have hitherto been considered. In this paper, we adopt the integral theory and take into consideration the effect of an infinite number of collisions suffered by a neutron, this being made possible by solving simultaneous equations. Further, we consider the anisotropic scattering through the generalized first-flight collision probability. Then we estimate the fundamental and the additional terms in the Benoist formula for slab and square lattice cells.

Approximate Treatments of Neutron Wave Propagation Problems in Three-Dimensional System

Neutron wave propagation problems, espetially the Milne and the albedo problems, are investigated by making use of the Wiener-Hopf technique for treating the Boltzmann equation with an isotropic one-term degenerate kernel in a semi-infinite prism. A pole of the solution of the integral-transformed Boltzmann equation corresponds to an eigenvalue—or physically, to a complex wave number varying on the two-dimensional complex plane with transverse buckling and wave oscillation frequency. In the Milne problem, the solution ceases to exist when the imaginary part of the complex wave number exceeds σmin. In the albedo problem, however, the solution always exists irrespectively of the oscillation frequency ω, and the discrete eigenvalue presents a continuous spectrum as soon as ω exceeds a critical frequency ωc. Detailed forms of solutions are derived for the case of constant velocity, and complex eigenvalues are evaluated numerically.

Radial diffusion coefficient in a cylindrical cell by P3 method

Abstract Radial diffusion coefficient for a cylindrical cell is calculated by the P 3 -approximation in one-group theory. Williams calculated the anisotropic diffusion coefficient in a cylindrical cell by the P 3 -approximation. His calculation was, however, limited to the axial diffusion coefficient. Therefore we calculate the radial diffusion coefficient by using the cylindrical approximation introduced by Williams and by following the concise procedure of the P N -method by Davison. As an example of numerical calculation we treat a cell consisting of two different regions, a fuel rod and a surrounding moderator. Results from the present method are compared with those from the diffusion theory and from the collision probability method.