M.M.R. Williams

The kinetic behaviour of simple neutronic systems with randomly fluctuating parameters

Abstract We have considered the problem of neutron density fluctuations in a nuclear reactor when the reactivity is a Gaussian, stationary random function. Point model kinetic equations have been employed and a range of situations studied by using: (i) no delayed neutrons, (ii) infinite delay time model and (iii) zero prompt generation time model. In each case we have found the mean value of the flux, arising from the random reactivity insertion, to be an increasing function of time. This leads to the conclusion that, despite its zero mean, a random fluctuation causes positivity reactivity feedback and can therefore lead to instability. Probability distributions for the resulting neutron densities have been computed and shown to be non-Gaussian. The precise shape depends on the power spectral density of the input reactivity fluctuations. By making use of the exact solutions obtained from the above models, we have assessed the accuracy of an approximation technique, based on closure methods, which proves useful for more complicated situations not amenable to exact solution.

Probability distributions in simple neutronic systems via the fokker-planck equation

Abstract The Fokker-Planck equation for the neutron density probability distribution in a multiplying system, possessing random source and random parametric excitation, is deduced. It is shown that white noise parametric excitation and white noise source excitation both have the same effect on the functional form of the correlation but that the parametric excitation increases its amplitude. A method for obtaining the correlation function directly from the conditional probability distribution is described and applied to a simple point model equation with one group of delayed neutrons.

A generalized energy exchange kernel for inelastic neutron scattering and thermonuclear reactions

Abstract We have obtained a general energy exchange kernel for use in the Boltzmann equation which will account for nuclear reactions of the general type A + B → X + Y . By appropriately choosing certain parameters, the kernel may be specialized to the case of inelastic neutron scattering by a nucleus with an arbitrary distribution of levels. Also, by averaging over the appropriate thermal distribution, we may calculate the differential cross-section for the reaction products in a thermonuclear plasma. For the special case of elastic neutron scattering this kernel reduces to that of the gas model of neutron thermalization. We also compute the neutron energy distribution emerging from an enclosed thermonuclear plasma and calculate the average neutron energy and the dispersion of the spectrum.

An application of slowing down kernels to thermal neutron density fluctuations in nuclear reactors

Abstract The backward equation for the space and energy-dependent probability generating function of the density fluctuations in a nuclear reactor, has been reduced to a more tractable form by the application of a slowing down kernel technique. Simple working expressions have been obtained for the variance, the correlation function and the power spectral density (P.S.D.) in terms of the infinite medium slowing down kernel. The method includes the effects of detectors, delayed neutrons and slowing down time. In addition, the frequency characteristics of the detection apparatus have been accounted for. It is shown that the dominant break frequency, of the fundamental mode component in the power spectral density, is markedly dependent on the slowing down lengths of the delayed neutrons, a result that can lead to appreciable corrections to the experimentally calculated neutron lifetime in certain systems. A general formalism accounting for slowing down, delayed neutrons and detector geometry is developed, and is applied to detectors of various shapes. Marked differences in the P.S.D. are noted for the different detectors in an infinite medium; however, for the finite medium, the effect of detector geometry is not so important and it is shown that, in certain circumstances, experiments can be analysed according to the simple, infinite medium-infinite detector formula.

A transport theory calculation of neutron flux, disadvantage factors and effective diffusion coefficients in square cells and slabs

Abstract An exact transport theory calculation has been carried out on a finite reactor lattice, and analytical results obtained for the flux distribution, disadvantage factor and anisotropy of diffusion. Spatially constant cross-sections have been assumed but a periodic source simulates accurately the cell flux that would arise in a more realistic situation. Exact results for the disadvantage factor show clearly the errors involved in cylindricalization; in particular, the minimum in the disadvantage factor vs. moderator/fuel ratio curve is shown to exist and not to be a consequence of the reflecting boundary condition. However, the minimum occurs at much lower moderator/fuel ratios than those considered by previous workers and certainly much lower than predicted by the reflecting boundary, cylindricalized cell approximation. Curves of the fluxes in the cell, together with the distortions arising from leakage, are given as well as detailed tables of numerical values of the disadvantage and streaming factors. These tables are offered as benchmark results against which approximate techniques may be compared.

The validity of the buckling concept and the importance of spatial transients in the pulsed neutron experiment

Abstract By solving numerically the exact integral transport equation for a pulsed slab of pure moderator, and the corresponding asymptotic equation, it has been possible to uniquely relate the buckling of the system to its physical size. In addition, the spatial transients, excited by the boundary, are found to have a marked effect on the flux curvature. Criteria are given for various moderators (graphite, beryllium, water) which will enable the experimentalist to judge the importance of dropping points near the boundary of the system when attempting to fit a cosine to experimental data. It is found that two distinct regions exist in a pulsed system. An interior region, corresponding to the dominance of the asymptotic solution, where the neutron spectrum exhibits the well-known cooling effect; but also a boundary or ‘buffer’ region, which has no simple relation with the diffusion cooling effect. The energy of neutrons in the buffer region is found to depend markedly on the variation of total cross section with energy, and can lead to either heating or cooling of the neutron spectrum, according to the moderator.

Space dependent noise spectra via the transport equation

Abstract Exact expressions for the cross power spectral density in a uniform, infinite medium have been calculated. One-speed transport theory is assumed, thereby enabling the deficiencies in diffusion theory for this type of problem to be assessed. It is noted that the exact transport solutions go over to the simple diffusion theory ones when the points under consideration are more than about two mean free paths apart and the absorption is weak. Moreover, the diffusion theory expressions fail at high frequency. We have also examined the nature of the cross correlation function when the points are of the order of a mean free path apart. It is shown that direct free flight correlations exist which could not possibly be predicted with diffusion theory. Specific expressions for the cross and auto power spectral densities in plane, spherical and cyclindrical detectors are given.

The application of a stochastic generator to determine the kinetic response of a neutronic system with randomly fluctuating parameters

Abstract A stochastic generator has been developed which produces a Gaussian random noise with any prescribed power spectral density. The computer programmes which form the basis of this generator have been extensively tested for a number of different power spectral densities and found to reproduce the statistical properties with some accuracy. The generator has been used to test the analytical solution for the autocovariance function arising from the noise generated by a plane source, fluctuating in intensity, situated in a moderating medium of infinite extent. The spatial dependence of the covariance and autocovariance functions for the analytical and directly simulated solutions are shown to be in close agreement. The stochastic generator is offered as a controlled source of noise which may be used in reactor situations of practical importance where no analytical results are available.

Extrapolation distances in thermal neutron systems-I. The pulsed source experiment

Abstract The effect of three-dimensional leakage on the extrapolated endpoint, Z 0 , is studied for non-multiplying, pulsed systems. The three-dimensional nature of the problem is treated accurately by modifying a method developed in an earlier publication on the diffusion length problem. Results are obtained using realistic scattering models for hydrogenous and polycrystalline moderators, and also in the 1-speed approximation, assuming isotropic scattering. The theory is compared with experimental values for particular systems. The variation of Z 0 with B 2 is demonstrated for cubical systems: Z 0 is found to decrease with increasing B 2 for hydrogenous and polycrystalline materials. This dependence is explained by considering the asymptotic spectra and the energy dependence of the neutron mean free path. Tables of Z 0 are included so that experimenters may deduce B 2 for non-cubic systems. The anisotropy of Z 0 is demonstrated by computing the value for each direction of a rectangular parallelepiped.

A NEW APPROACH TO ENERGY-DEPENDENT NEUTRON TRANSPORT THEORY.

Abstract A new approach to energy-dependent neutron transport theory is described which treats the asymptotic solution of the Boltzmann equation exactly and approximates the spatial transient by the separable kernel model. The method is illustrated by an application to the Milne and pulsed neutron problems, and exact expressions are obtained for the extrapolated endpoint and the emergent angular distribution. It is pointed out that the method can be extended to include criticality problems and three-dimensional geometries.