One-speed Transport Research Articles

Transport Calculation of the Multiplicity Moments for Cylinders

In a previous paper by Pázsit and Pál [“Multiplicity Theory Beyond the Point Model,” Ann. Nucl. Energy, Vol. 154 (2021)], a general transport theory calculation of the factorial moments of the number of neutrons emitted spontaneously from a sample was elaborated. In contrast to the original derivations by Hage and Cifarelli [“On the Factorial Moments of the Neutron Multiplicity Distribution of Fission Cascades,” Nucl. Instrum. Meth. Phys. Res. A, Vol. 236 (1985)] and Böhnel [“The Effect of Multiplication on the Quantitative Determination of Spontaneously Fissioning Isotopes by Neutron Correlation Analysis,” Nucl. Sci. Eng., Vol. 90 (1985)], also referred to as the point model, in the transport model the spatial and angular dependence of the internal fission chain is taken into account with a one-speed transport theory treatment. Quantitative results were given for a spherical item, and the bias of the point model regarding the estimation of the fission rate as compared to the more exact space-dependent model was estimated as a function of the size of the sphere and the factor. In the present paper the formalism and the quantitative work are extended to the treatment of items with cylindrical shapes, which are more relevant in many practical applications. Results are presented for both square cylinders () and for tall () and flat () cylinders. This way the differences between the cylinder and the sphere on one hand and those between the various cylinder shapes on the other hand can be estimated. The results show that the bias depends on the geometry of the cylinder quite moderately, but similarly to the case of the sphere, the bias of the point model is quite significant for larger item sizes and values, and it is nonconservative (underestimates the fissile mass) as well.

On the spectrum of discrete-ordinates neutron transport problems

Over the last six decades, the discrete spectrum of the neutron transport operator has been widely studied. Important theoretical results can be found in the literature regarding the one-speed linear transport equation with anisotropic scattering. In this work, the discrete-ordinates (SN) transport problem with anisotropic scattering has been considered and the discrete spectrum results in multiplying media have been corroborated. The numerical results obtained for the dominant SN eigenvalues agreed with the ones for the analytic problem reported in the literature up to a triplet scattering order. A compact methodology to perform the spectral analysis to multigroup SN problems with high anisotropy order in the scattering and fission reactions is also presented in this paper.

Discrete Eigenvalues of Case Spectrum with Anisotropic Scattering

Discrete eigenvalues of a one-speed linear transport equation with anisotropic scattering are studied. It is shown that there is only one pair of real discrete eigenvalues for linear, quadratic, or triplet scattering for a nonmultiplicative medium. For a multiplicative medium there is one imaginary pair of eigenvalues or at most four eigenvalues. These can form one real and one imaginary pair, two imaginary pairs, or a quartet. The range of parameters for these different situations is derived analytically. These are then supported by numerical results that are tabulated in tables for each type of scattering.

Generalized linear Boltzmann equation, describing non-classical particle transport, and related asymptotic solutions for small mean free paths

In classical kinetic models a particle free path distribution is exponential, but this is more likely to be an exception than a rule. In this paper we derive a generalized linear Boltzmann equation (GLBE) for a general free path distribution in the framework of Alt’s model. In the case that the free path distribution has at least first and second finite moments we construct an asymptotic solution to the initial value problem for the GLBE for small mean free paths. In the special case of the one-speed transport problem the asymptotic solution results in a diffusion approximation to the GLBE.

On spectral geometry for the one-speed particle transport operator

It is proved that, among all domains of identical measure, the ball minimizes the first eigenvalue of the one-speed particle transport operator in a multidimensional Euclidean space.

On the dispersion function for complex values of the parameter c

The dispersion function relevant to one-speed transport theory with isotropic scattering is analyzed for the case of complex values of c, and an explicit expression is given for the discrete eigenvalue ν0.

Alpha eigenvalue calculations with Tripoli-4®

Characterizing the time behavior of neutron transport is crucial in several technological issues, such as for instance reactor start-up analysis, reactivity measurements, or kinetics studies of accelerator-driven systems. Basically, this amounts to computing the spectrum of the Boltzmann operator: analytical techniques, though quite ingenuous, are often restricted to simple geometries and one-speed transport, so that numerical solutions become mandatory to this aim. In this paper we develop a Monte Carlo method for determining the dominant eigenvalue of the Boltzmann operator and the associated fundamental mode for arbitrary geometries, materials, and boundary conditions; this is done in view of adding such capabilities to the Monte Carlo code Tripoli-4®. Extensive verification and validation tests are performed on multiplying and moderating media, and a critique of the algorithm is proposed.

Consistent Algorithms Marching Along Characteristics for the Numerical Solution of the Boltzmann Equation

We introduce algorithms marching over a polygonal mesh with elements consistent with the propagation directions of the particle (radiation) flux. The decision for adopting this kind of mesh to solve the one‐speed Boltzmann transport equation is due to characteristics of the domain of the transport operator which controls derivatives only in the direction of propagation of the particles (radiation) flux in the absorbing and scattering media. This a priori adaptivity has the advantages that it formulates a consistent scheme which makes appropriate the application of the Lax equivalence theorem framework to the problem. In this work, we present the main functional spaces involved in the formalism and a description of the algorithms for the mesh generation and the transport equation solution. Some numerical examples related to the solution of a transmission problem in a high‐contrast model with absorption and scattering are presented. Also, a comparison with benchmarks problems for source and reactor criticality simulations shows the compatibility between calculations with the algorithms proposed here and theoretical results.

Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique

Much of the literatures are directed toward the development of a mathematical formalism for a rigorous estimation of the ensemble average of the solution process of a stochastic differential equation (SDE). The Random Variable Transformation technique (RVT) is a powerful technique to get the complete solution for the SDE represented by the probability-density function of the solution process. In this paper, the RVT technique together with a simple integral transformation to the input stochastic process are implemented to get the complete solution of the one-speed transport equation for neutral particles in a semi-infinite stochastic medium with linear anisotropic scattering. The extinction function of the medium (input stochastic process) is assumed to be a continuous random function of position. The probability-density function and hence, the higher order statistical moments of the solution process are presented. Numerical results are given for different distributions of the input stochastic process.

Green's Functions for the Non-Stationary Kinetic Equation with Constant Speed

Green's functions for the one-speed non-stationary and stationary transport equation in a homogeneous medium are considered. For an anisotropically scattering medium, in which we represent the scattering phase function by a truncated series of Legendre polynomials, Green functions are determined for isotropic and monodirectional plane source emissions. These solutions are represented by contour integrals in the complex plane with a cut on [−1, 1]. Numerical results are given.

Transport Of Neutral Particles In A Finite Continuous Stochastic Medium With Gaussian Statistics

The stationary solution of the one-speed transport equation in a finite stochastic medium with linear anisotropic scattering is considered. The solution is presented for an arbitrary absorption and scattering cross sections. The total cross section of medium is assumed to be a continuous random function of position, with fluctuations about the mean taken as Gaussian distributed. The Pomraning-Eddington technique is used at first to solve the problem in the deterministic case. Two correlated random variables appear in the solution; namely, the optical space variable and the optical thickness of the medium. The dual Gaussian-probability density function of these two random variables is derived from which the ensemble-averaged solution is calculated for an arbitrary correlation function. The first and the second statistical moments of some quantities of interest, such as radiant energy, net flux, reflectivity, and transmissivity, are calculated. The problem is treated with specular-refecting boundaries and an incident flux of particles on the medium from the left and with no flux from the right.

Criticality of bare and reflected spheres by [formula omitted] method

The criticality problem of the one-speed transport equation with isotropic scattering for bare and reflected spheres is studied by PN method. The divergence of the PN method for sufficiently small radius in high order of approximation are discussed. A robust numerical method to overcome the difficulty in accurately finding the roots of the determinantal equation has been proposed. A range of critical radii have been obtained for bare and different reflected coefficient by the PN method. Results are compared with published data where possible.

The transport and diffusion theory of a line source in an infinite half-space with internal reflection

A closed form analytical solution is obtained for a three-dimensional transport theory problem, namely that of a line source in a half-space in one-speed transport theory with an internally reflecting surface. The theory is developed by using Fourier transforms in the transverse directions and a Laplace transform in the axial direction, together with the Wiener–Hopf technique. Both specular and diffuse internal reflection are considered and it is shown that a closed form solution is not available for the specular case but is available for the diffuse case. The surface particle scalar intensity and current are obtained and their sensitivity to absorption and the reflection coefficient are assessed. We also obtain a number of analytic asymptotic estimates for the intensity valid at large distances from the source. Diffusion theory is also employed and compared numerically with the transport solution. In general, diffusion theory gives very satisfactory results within its regions of limitation. We also offer this paper as an exercise in analytical transport theory; an aspect of reactor physics which is not often seen in recent times.

An Analytical Study of Particle Transport in Spatially Random Media in One Dimension: Mean and Variance Calculations

Statistical aspects of particle transport in spatially random media in one dimension are studied analytically using the dichotomic and the binomial binary random processes as two models for the spatial randomness in the particle density. The mean and the variance of the flux are expressed in terms of the moments <U(x)n> of the stochastic propagator U(x) = exp[–z(x)], where z(x) is the optical path length. The moments <U(x)n> are rigorously calculable with the above random processes. In the case of semi-infinite media, the calculations are carried out using the one-dimensional, one-speed transport equation. In finite media, one-speed diffusion theory is used to calculate the mean and variance of the flux within the slab. In particular, the statistics of the albedo and the exit current are investigated. The mean of the local reaction rates within the slab is also calculated.

Criticality of reflected spheres by PN method

The criticality problem of the one-speed transport equation with isotropic scattering for reflected spheres is studied using the P N method. It is observed that the method diverges as one increases the order of the P N approximation. Before this divergence, the method yields results that agree with those of the integral equation.

The flat flux problem in one-speed neutron transport theory

The classic flat flux problem in nuclear reactor physics, first solved by Goertzel using diffusion theory, is extended to one-speed transport theory. A numerical solution is obtained for a general two region problem in which the inner region has a flat flux and the outer one a spatially variable flux. For the exact solution, plane geometry is employed with the optical path representation. This solution allows the accuracy of a more flexible but approximate method to be assessed. A feature of the problem which has not been noted before is that in the region of flat flux the current is not zero. The reason for this is explained and is used to assess the accuracy of the approximation method. Diffusion theory is shown to be a poor but not unreasonable approximation. However, to make it viable it is necessary to introduce generalised function solutions to the diffusion equation. The theory is illustrated in detail by numerical calculations. It is shown that a flat flux in one speed theory corresponds to a maximum fuel loading. This is in contrast to the multigroup case for which it is a minimum loading.

DIFFUSION OF PARTICLES IN THE PRESENCE OF STRONG FORWARD–BACKWARD ANISOTROPIC SCATTERING

ABSTRACT We consider one-speed transport problems with forward–backward-isotropic scattering in an infinite medium and study its relationship with the diffusion equation. Both time dependent and time independent transport equations are considered. We also obtain the Greens function for this scattering model.

Spectral analysis of transport operator: Functional model approach

This paper presents the spectral analysis of the dissipative one-speed transport operator for the cases of a finite slab and of a finite body, by means of the Szokefalvi-Nagy - Foias functional model. It is shown that the essential spectrum of the operator is absolutely continuous with one possible spectral singularity at point 0. The absolutely continuous component of the operator is studied in detail in the nontrivial case, when the spectral singularity does occur.

A NOTE ON THE PROPAGATION OF BOUNDARY INDUCED DISCONTINUITIES IN KINETIC THEORY

In this paper, we study, on a very simple kinetic model, the flow structure induced by a discontinuity of the boundary data. The model considered is a stationary one-speed transport equation posed in a half-plane; for simplicity, the boundary data consist of the number density of incoming particles. The propagation of singularities is studied with the velocity averaging method.11

Calculating reactor transfer functions by Padé approximation via Lanczos algorithm

An effective method of solving the one-speed transport equation in frequency domain is demonstrated in this paper, the so-called Padé approximation via Lanczos algorithm (PVL). The advantage of the PVL method is that implementing the calculation process over a considerably reduced model yields a pseudo-analytical expression of the transfer function over a fairly large range of frequency. As a particular application, the dynamic transfer function of a reactor, i.e. the neutron noise induced by a localised perturbation is calculated in one-speed transport theory. The problem is essentially the same as that of the “detector-field-of-view”; studied by other authors. The PVL algorithm is demonstrated through the solution of the problem and its advantages are described. The quantitative results show that although one-speed theory was used, a local component was found, and thus the local-global decomposition could be reconstructed. This shows that unlike in diffusion theory where at least two-group theory is necessary, the local behaviour can be described already by a one-speed equation in transport theory.