The Two-Direction Neutral-Particle Transport Model: A Useful Tool for Research and Education

Abstract

The two-direction transport model limits the transport of neutrons and photons to the +X and -X directions, resulting in a diffusion-type differential equation for the total flux. Anisotropic scattering can easily be incorporated. Analytical solutions of this equation can easily be obtained, as well as for the flux in either direction. This may help students to better understand transport and diffusion theory and the continuity or discontinuity of transport quantities and their derivatives at internal medium boundaries. As the two-direction transport model is a simplified but true transport model, it can easily be incorporated in a Monte Carlo code, and can then be used to test the operation and performance of the Monte Carlo code-especially to test methods for variance reduction, including zero-variance schemes.The adjoint equations for two different types of importance functions are derived, for which analytical solutions are possible. The less well-known importance function for particles entering a collision is essential in constructing Monte Carlo zero-variance sampling schemes.Several examples are worked out with analytical and numerical solutions, which demonstrate the use of the two-direction model; the application of the various boundary, interface, and source conditions; and the application of the adjoint functions. It also shows the unusual solution of an eigenfunction problem with anisotropic fission. The numerical results from analytical solutions are confirmed by Monte Carlo and discrete ordinates calculations.

Isotropic Scattering in a Flatland Half-Space

“…The study of linear transport theory [1,2] in lower-dimensional spaces serves a number of purposes. The simplicity of the one-dimensional rod model [3] makes it a useful tool for education [4] and occasionally the starting place for exploring new general transport processes [5]. Most rod model problems can be solved exactly and admit simple closed-form solutions, with diffusion encompassing the entire solution.…”

Rigorous Asymptotic and Moment-Preserving Diffusion Approximations for Generalized Linear Boltzmann Transport in Arbitrary Dimension

“…It is not purely a matter of mathematical interest to consider transport in various dimensions. Transport in a one-dimensional domain (the rod model [Wing 1962], also known as the Fermi model or the forward/backward model) has many application including transmission line theory [Redheffer 1962], for constructing the two-stream approximation in many three-dimensional problems [Peraiah 2002] for velocity jump processes in biology [Othmer and Hillen 2000] and for teaching and discovering new deterministic and Monte Carlo techniques [Hoogenboom 2008]. For classical transport in one dimension, diffusion happens to be an exact solution.…”

Analysis of the neutron noise induced by fuel assembly vibrations

“…In principle, the Green's function of Eq. (45) could be determined exactly, by slightly adapting the analytical methods proposed in (Hoogenboom, 2008), and the solution would then follow by quadrature formulas with respect to the noise source term Q ± (x, ω). In practice, the resulting expressions are rather cumbersome, so that a numerical solution by discretization and matrix inversion has been chosen for this work.…”