An analytical discrete ordinates approach for the calculation of multiplicity moments in nuclear safeguards

Journal article, 2026

Analytical or semi-analytical calculation of the multiplicity moments of nuclear safeguards by accounting for the spatial transport of neutrons constitutes a challenging problem of one-speed transport theory. Application of standard one-speed transport methods is not trivial, given the fact that the moment equations are inherently of the backward (adjoint) type. In our previous work, a combination of a relationship between slab and spherical geometry with the Analytical Discrete Ordinates (ADO) method was applied to calculate the factorial moments of neutrons exiting from the item, due to one single source neutron. The formalism was applicable only with (Formula presented). In the present work, the method is extended and generalized in two ways; partly, to (Formula presented), and partly the factorial moments are calculated also for spatially distributed random source events (spontaneous fission and (Formula presented) neutrons). The extension for (Formula presented) with the ADO method requires the handling of complex eigenvalues and eigenfunctions and, consequently, the derivation of suitable particular solutions. The extension for spatially distributed random sources means that the formulae obtained for these factorial moments, referred to as “multiplicity moments”, are directly suitable for calculating the multiplicity rates, which are necessary to unfold the item parameters from the measured data. It is demonstrated that our method is vastly superior to the originally used collision number expansion method in terms of accuracy and speed of calculations. This is essential if the unfolding of the item parameters is performed with machine learning methods, which necessitate the calculation of a large number of training patterns.