The adjoint neutron transport equation and the statistical approach for its solution

TL;DR

This paper reviews the adjoint neutron transport equation, derives an analytical solution for a specific case, and proposes a Monte Carlo sampling method to evaluate the importance function, validated against the analytical benchmark.

Contribution

It provides a new analytical solution for the adjoint neutron transport equation in a homogeneous medium and introduces a Monte Carlo sampling technique for its numerical evaluation.

Findings

Analytical solution serves as a benchmark for numerical methods.

The Monte Carlo sampling scheme accurately estimates the importance function.

The approach enhances understanding of neutron importance in reactor physics.

Abstract

The adjoint equation was introduced in the early days of neutron transport and its solution, the neutron importance, has ben used for several applications in neutronics. The work presents at first a critical review of the adjoint neutron transport equation. Afterwards, the adjont model is constructed for a reference physical situation, for which an analytical approach is viable, i.e. an infinite homogeneous scattering medium. This problem leads to an equation that is the adjoint of the slowing-down equation that is well-known in nuclear reactor physics. A general closed-form analytical solution to such adjoint equation is obtained by a procedure that can be used also to derive the classical Placzek functions. This solution constitutes a benchmark for any statistical or numerical approach to the adjoint equation. A sampling technique to evaluate the adjoint flux for the transport equation is then proposed and physically interpreted as a transport model for pseudo-particles. This can be done by introducing appropriate kernels describing the transfer of the pseudo-particles in phase space. This technique allows estimating the importance function by a standard Monte Carlo approach. The sampling scheme is validated by comparison with the analytical results previously obtained.