Asymptotic Flux Research Articles

Asymptotic Neutron Distribution in a Finite Cylinder

Calculations have been performed on the asymptotic angular neutron flux, critical thickness and extrapolation distance in the axial direction for a cylindrical system of finite length. Neutrons are assumed to be monoenergetic. Applying the finite Laplace transformation in the axial direction and buckling approximation in the radial, we find a solution of the transport equation that satisfies the boundary conditions of no incoming neutron and symmetry. This method is an extension of Kobayashi's Laplace transform method for slab problem, and may easily be applied to other geometries. For a rectangular prism system, numerical results based on our method are compared with results from P3-approximation.

Axial Propagation of Neutron Waves in Heterogeneous Media

A theoretical study of the axial propagation of plane-thermal-neutron waves in a heterogeneous system is performed in the framework of the P-1 approximation to the Boltzmann equation. The method is based on a modified form of heterogeneous reactor theory due to Feinberg and Galanin. The analysis predicts that the phase interference between the modes of propagation in the axial direction may give rise to resonances in the frequency response of the asymptotic moderator flux. A standing wave pattern is also predicted in the amplitude distribution of the oscillating part of the moderator flux in the axial direction. The relationships between the resonances and the system parameters are investigated. An experimental method that can be useful for the determination of the effective values of the diffusion parameters and the slowing down time is suggested.Numerical calculations for a heavy-water-moderated natural uranium system containing four identical fuel rods are presented in the frequency range from 0 to 1500 Hz. Two resonances are predicted in the transfer function of such a system in this frequency range. A comparison is made with the experimental results published in the literature for a similar system. The complex relaxation length for this system is also calculated numerically in order to study the effect of the resonances in the transfer function on the complex relaxation length. The results show existence of “loops” in the plot of the complex relaxation length.

Asymptotic Solutions to the Transport Equation for a Plane Lattice

A transport calculation of the lattice diffusion length, yielding the “gross” decay of the asymptotic flux in a lattice, is made using the method of K. M. Case. Refinements over the diffusion calcu...

Equilibrium Spectra and Diffusion Lengths of Neutrons in Semi-Infinite Moderator Block: Part II: Effect of ‘Trapped’ and Cold Neutrons

In a semi-infinite block of beryllium at room temperature, the effect of ‘trapped’ and cold neutrons on the asymptotic equilibrium neutron flux distribution and on diffusion length is investigated by considering various forms for the energy dependence of the transport cross-section. It is found that in static experiments, cold neutrons play a dominant role. The effect of non-l/v absorbers on the above properties has also been studied.

The Slowing Down of Neutrons in Hydrogeneous Mixtures

The spatially-independent spectrum of neutrons slowing down in an infinite medium with constant cross sections is calculated from both the Laplace transform exactitude (LTE) and a generalized synethtic kernel approximation (SKA). The fluxes are expressed as sums of exponentials in lethargy and compared asymptotically. For hydrogenous mixtures, two of an infinite number of terms from the LTE are non-oscillatory, both dominate all others at large lethargies, and one vanishes whenever hydrogen is a sole or missing constituent. The SKA yields a solution consisting of as many exponentials as isotopes present. The longest-lived terms are generally most accurate, but even the dominant one can be exact only if there is no absorption or if hydrogen is the sole moderator. For binary mixtures, both terms in the SKA fluxes are non-oscillatory, and the secondary one vanishes for the same concentrations that make the corresponding term in the LTE vanish. Analytic expressions for errors in the asymptotic flux from the SKAs are given as a function of lethargy, all the cross sections, and masses. For every instance observed, the exact asymptotic flux is bounded on different sides by values from the Greuling-Goertzel and Selengut-Goertzel SKAs.

Transport Effects in Diffusion Theory

A diffusion theory for the asymptotic transport scalar flux is derived from the monoenergetic transport equation in slab geometry. By allowing the scalar flux to be discontinuous at a material prop...

Preliminary Calculation for Experiments Using Osaka University HITS, (I)

In order to discuss the ambiguity inherent in the diffusion approximation applied to a small-sized medium including a pulsed neutron source, the time variation and the spatial distribution of thermal neutron angular flux are calculated in P 3-approximation of the time-dependent Boltzmann transport equation. With the discussion on the existence of a set of real and complex eigenvalues, the real eigenvalues for the lowest and some higher modes are computed numerically and the corresponding eigenfunctions are derived under Marshak's boundary condition. The results are compared with those obtained in the diffusion approximation and the physical interpretations of resultant asymptotic and non-asymptotic thermal neutron flux are given.

Some Transient Phenomena in Thermalization I. Theory

We discuss the behaviour of the asymptotic thermal flux in the “pulsed” and “exponential” experiments. In particular, we give a pair of theorems which bear upon the possibility of measuring λ0 and ...

On the use of case’s general solution of the transport equation in neutron transport problems

Case’s general solution of the one-velocity transport equation is applied to the following problems: limiting behaviour of the asymptotic flux in a noncapturing medium; Milne’s problem and extrapolation distance; connection at an interface of asymptotic flux and current. Case’s general solution is then compared with that obtained by thePL approximation. Finally, an extension is given to the case of anisotropic scattering.

Generalization of the singular integral method for an anisotropic scattering medium

The singular integral method suggested by C case (1960) for solving neutron diffusion problems in an isotropic scattering medium is generalized for a linearly anisotropic medium. The formulation of the anisotropic problem is the same as that of the isotropic scattering problem except for small correction terms. These correction terms are proportional to the absorption probability. In the case of a non-absorbing medium, it is shown that the anisotropic scatterings affect only the asymptotic flux distribution and not the transient fluk distribution.

Streaming due to holes in a reactor

The effect of holes on a reactor is considered by using appropriate boundary conditions on the asymptotic neutron flux at the surface of the holes. For a flux variation along the hole, this method gives the same streaming correction as BEHRENS, but for a flux variation across the hole, the streaming correction is smaller than that given by BEHRENS' formula. Values of the radial streaming correction factor are compared with experimental and Monte Carlo results and also with values obtained from the iterative method of Benoist.

Thermal Neutron Flux Distributions in Space and Energy

The steady-state space-energy distribution of thermal neutrons in homogeneous media is considered in the diftusion approximation. From the general form of the solution it is shown that the asymptotic (in space) distribution depends upon the source distribution and under different circumstances can be qualitatively different. The relation of the asymptotic flux to the usually found infinite medium spectrum is exhibited. For a heavy gas moderator the asymptotic flux from a source of limited extent is shown to be much softer than an infinite medium spectrum in the same material. (auth)

Some Monte Carlo and Analytical Results for Resonance Capture in Lattices

Detailed Monte Carlo studies of the resonance capture in the lowest U/ sup 238/ resonances were made for uraniumwater lattices and a uranium-graphite lattice. Direct comparison with calculated values was considered a gocd test of the validity of the assumptions made in obtaining the theoretical estimates. Good agreement was obtained for the over-all capture in a single resonance and for the energy distribution of absorptions. Monte Carlo tests also showed that the asymptotic flux was fairly well reestablished between the two strongest U/sup 238/ resonances, and that the number of neutrons captured in this energy region was only slightly affected by Doppler broadening and interference between resonance and potential scattering. Finally the calculated resonance escape probability in a uranium-water lattice for 26 resolved resonances was compared with the Monte Carlo value. There was quite close agreement. (auth)

A Time-Dependent Analysis of Spatial Flux Distributions

A technique is exhibited which permits an investigation of the changes in flux shape which occur when reactivity is inserted locally in an inhomogeneous reactor system and the power level rises. Thus, transient flux shapes at any time may be found as well as the asymptotic flux shape which is eventually attained. The reactor kinetics study in this article is motivated by a method of harmonics which does not employ the conventional assumption of separability of the flux into a product of a function of position alone and a function of time alone. From the point of view of practical applications the method is restricted to systems of rather simple geometry in which the slowing down is everywhere uniform and in which there are no nonlinear feedback effects. (auth)

The Dynamic Reactivity Interpretation of Pulsed Neutron Measurements

A dynamic reactivity, not the reactivity in general use, is defined relative to prompt critical as ΔK = −lα, where α is the asymptotic (prompt) flux decay rate observed in a pulsed neutron experime...