The mathematical formulation of **neutron diffusion theory** is based on the **balance of neutrons** in a differential volume element. Since neutrons do not disappear (β decay is neglected), the following neutron balance must be valid in an arbitrary volume V.

**rate of change of neutron density = production rate – absorption rate – leakage rate**

where

Substituting for the different terms in the balanced equation and by dropping the integral over (because the volume V is arbitrary), we obtain:

where

**n**is the**density of neutrons**,**s**is the rate at which neutrons are emitted from sources per cm^{3 }(either from external sources (S) or from fission (**ν.Σ**)),_{f}.Ф**J**is the neutron current density vector**Ф**is the scalar neutron flux**Σ**is the macroscopic absorption cross-section_{a}

In steady state, when n is not a function of time:

## The Diffusion Equation

In previous chapters, we introduced **two bases for the derivation** of the diffusion equation:

**Fick’s law:**

which states that neutrons diffuse from high concentration (high flux) to low concentration.

**Continuity equation:**

which states that rate of change of neutron density = production rate – absorption rate – leakage rate.

We return now to the neutron balance equation and **substitute** the neutron current density vector by **J = -D∇Ф**. Assuming that ∇.∇ = ∇^{2} = Δ (therefore **div J = **-D div (∇Ф) = **-DΔФ**) we obtain the diffusion equation.