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

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.

Comparison of Monte Carlo methods for adjoint neutron transport

“…In a recent paper [22], a model (M 3 ) in which the functionsp j are equal to the forward macroscopic cross section of the j-th reaction has been proposed, which yields…”

“…An analytical solution to Eq. (22) has been explicitly derived for the case of elastic scattering alone with general cross-section Σ e (E) [22], based on adjoint Placzek functions [27]. The forward transfer function for elastic scattering reads…”

“…The solution of Eq. (29) can be recursively obtained by closely following the arguments in [22], and yields:…”

“…For elastic scattering, model M 3 is defined such thatp M3 e = Σ e andΣ e = Σ e [22]. The scattering law and the weight multiplication are also equivalent to those of model M 1 .…”

“…Despite these difficulties, adjoint Monte Carlo methods are still a subject of active research per se [20][21][22] and in view of providing efficient variance reduction strategies for forward methods [23][24][25]. Bearing in mind the possibility of introducing adjoint methods into Tripoli-4 R , the production Monte Carlo transport code developed at CEA [26], in this work we illustrate a comparison of different adjoint simulation models.…”