Monte Carlo and Deterministic Implementations of the Zeta-Eigenvalue Equation for Efficient Criticality Searches

The zeta (ζ)-eigenvalue equation is a formulation of the neutron transport equation that ensures criticality by scaling the density of the nuclides or materials of choice with the parameter ζ. As a consequence, the value of ζ provides quantitative information about what material density or nuclide concentration can make a system critical. This paper proposes a Monte Carlo algorithm to solve the ζ-eigenvalue equation. Additionally, it tackles practical challenges in the deterministic implementation of the method. The Monte Carlo algorithm, based on a fixed-point iteration scheme, is implemented in the code SCONE and tested on some cases of practical utility. Examples include the search of critical gadolinium concentration in a pressurized water reactor assembly and the search of critical boron in BEAVRS. The Monte Carlo implementation is successful at finding the critical densities requested and is faster than state-of-the art iterative searches based on k-eigenvalue calculations. The second part of the paper illustrates the impact of the density scaling on both the generation of multigroup constants and spatial self-shielding effects, relevant to the deterministic implementation. A computational scheme exploiting the ζ formulation to get reliable results in a deterministic framework is also suggested.