 Solution

The one-group particle balance with a prescribed external volume source $S$ can be writtenHere $\varphi$ is scalar neutron flux, $D$ is the neutron diffusion coefficient, and ${\Sigma }_a,{\Sigma }_f$ are macroscopic cross sections. The terms describe accumulation, current divergence, absorption, nuclear fission production and external input.

For a steady state, set ${\partial }_t\varphi =0$; source-free means $S=0$. In a homogeneous medium $D$ is spatially constant, so substitution givesThe supplied notation defines $\eta$ as the effective production per absorption, $\eta =\nu {\Sigma }_f/{\Sigma }_a$. ThereforeThis identification uses the absorption accounting for the homogeneous model; a fuel-only reproduction factor with other absorbers excluded would not automatically equal this $\eta$. The derivation assumes one prescribed energy group, constant effective coefficients and the diffusion approximation; all neutron production included here is represented by the steady nuclear fission term.