High-accuracy critical radii analytic expressions for spherical reflected systems based on discontinuous asymptotic diffusion theory

The success of the asymptotic diffusion approximation in calculating the critical depth in different geometries in high accuracy is well-known for decades. The high accuracy is achieved by taking into account the radius of curvature in the boundary condition in curvilinear coordinate systems. In reflected systems, as the simplest case of heterogeneous media, the asymptotic diffusion fails due to the continuous conditions of the scalar flux and current on the boundary between the core and the reflector. Discontinuous asymptotic diffusion approximation improves dramatically the accuracy of the calculated critical radii. In this work we study the importance of the radius of curvature correction, applied to the discontinuous jump conditions between the core and the reflector in simple mono-energetic (one-velocity) reflected spheres. We find a new one-velocity high-accuracy analytic expression for the critical radii, coated by general-depth reflector in spherical geometry. The accuracy of the analytic expression is better than 1% accuracy comparing to the calculated exact transport critical radii. The radius of curvature corrected discontinuous conditions gives rise to a new modified diffusion-like equation that reproduces the high-accuracy of the critical radii of reflected systems. The new modified equation is tested via numerical simulations and yields high accuracy with the analytic expression results.