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A diffusion approximation for a scalar neutron flux, with one prescribed energy group. Its terms balance current loss, absorption, nuclear fission production and a source. A solution also requires appropriate boundary conditions.

One energy group and a nearly isotropic local neutron distribution are the main approximations. The one-group neutron diffusion equation replaces the full energy-dependent transport problem by one scalar neutron flux and averaged coefficients. A single group cannot resolve spectral changes, resonance absorption or differences between rapidly moving and thermal neutrons; its effective parameters depend on the spectrum used for averaging.

The diffusion approximation also replaces angular transport by a current proportional to the negative neutron flux gradient. It works best where repeated scattering makes the angular distribution nearly isotropic over distances comparable to the field's variation length. It is less reliable close to a sharply localized source, a vacuum boundary, a strong absorber or an abrupt material interface, and in weakly scattering regions. Approximate boundary extrapolation does not restore the lost angular information.

Thus an apparently smooth solution of the one-group neutron diffusion equation need not accurately represent a transport field outside the assumptions that produced its coefficients and boundary conditions.

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.

For the prescribed homogeneous one-group model, number balance isUse steady state, $\partial n/\partial t=0$, and the diffusion approximation $\mathbf{J}=-D\nabla \varphi$. A spatially constant $D$ givesFor a spherically symmetric scalar field,Dividing by $D$ reproduces the requested radial equation. The assumptions are a steady source, spherical symmetry, homogeneous effective coefficients, a single energy-group description and a diffusion closure. In particular, replacing the general production term by the supplied $(\eta -1){\Sigma }_a\varphi$ is a model assumption, not a general identity for heterogeneous reactor materials.