Validity of Fick’s Law
It must be emphasized that Fick’s law is an approximation and was derived under the following conditions:
- Infinite medium. This assumption is necessary to allow integration over all space but flux contributions are negligible beyond a few mean free paths (about three mean free paths) from boundaries of the diffusive medium.
- Sources or sinks. Derivation of Fick’s law assumes that the contribution to the flux is mostly from elastic scattering reactions. Source neutrons contribute to the flux if they are more than a few mean free paths from a source.
- Uniform medium. Derivation of Fick’s law assumes that a uniform medium was used. There are different scattering properties at the boundary (interface) between two media.
- Isotropic scattering. Isotropic scattering occurs at low energies, but is not true in general. Presence of anisotropic scattering can be corrected by modification of the diffusion coefficient (based on transport theory).
- Low absorbing medium. Derivation of Fick’s law assumes (an expansion in a Taylor’s series) that the neutron flux, φ, is slowly varying. Large variations in φ occur when Σa (neutron absorption) is large (compared to Σs). Σa << Σs
- Time – independent flux. Derivation of Fick’s law assumes that the neutron flux is independent of time.
To some extent, these limitations are valid in every practical reactor. Nevertheless Fick’s law gives a reasonable approximation. For more detailed calculations, higher order methods are available.