Neutrons as a Diffusing Gas
In a nuclear reactor, neutrons scatter off nuclei millions of times before being absorbed or leaking out. This random-walk behavior is well described by diffusion theory — the same mathematics that governs heat conduction and chemical diffusion. The neutron diffusion equation relates the spatial distribution of neutron flux to the material properties of the reactor, providing the foundation for reactor core design.
The One-Group Approximation
Real neutrons span energies from 10 MeV (fast) to 0.025 eV (thermal), and cross-sections vary wildly with energy. The one-group approximation collapses this complexity into a single effective energy group with averaged parameters. While crude, it captures the essential physics of criticality and spatial flux distribution, and remains the standard pedagogical tool for understanding reactor behavior.
Boundary Conditions and Critical Size
Neutrons that reach the reactor surface can leak out and never return. The diffusion equation requires the flux to vanish at the extrapolated boundary — a distance of about 0.71 transport mean free paths beyond the physical surface. For a given set of material properties, there exists exactly one slab thickness where the production of neutrons by fission exactly balances absorption and leakage — the critical size.
Exploring the Flux Profile
This simulation solves the one-group diffusion equation for a bare slab reactor and displays the cosine-shaped flux profile. Adjust the slab width, diffusion coefficient, and cross-sections to observe how the reactor transitions between subcritical and supercritical states. Notice that increasing the slab width reduces geometric buckling and leakage, pushing k-effective upward, while increasing absorption cross-section pulls it down.