High Accuracy analytic expressions for the critical dimensions of reflected spheres and modified asymptotic diffusion theory

The success of the asymptotic diffusion approximation in calculating the critical thickness or radii in different geometries with high accuracy has been 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, such as spherical or cylindrical systems. In reflected systems, as the simplest case of heterogeneous media, the asymptotic diffusion fails due to the continuous conditions on the boundary between the core and the reflector. Discontinuous asymptotic diffusion approximation improves dramatically the accuracy of the calculated critical thickness or radii. In this work, we study the importance of the radius of curvature correction, which is 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 a general-depth reflector, in spherical geometry. The accuracy of the analytic expression is better than 1% accuracy compared to the calculated exact transport critical radii. The radius of curvature corrected discontinuous conditions give 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, yielding high accuracy with the analytic expression results.