A Monte Carlo Implementation of the ζ-Eigenvalue Equation

The 𝜁-eigenvalue equation is a formulation of the neutron transport equation which 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. The method, 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 PWR assembly, and the search of critical boron in BEAVRS. The Monte Carlo implementation is successful at finding the critical densities requested, and in the cases tested the runtime of a 𝜁 calculation is generally comparable to that of a 𝑘-eigenvalue calculation.