To derive the neutron diffusion equation, we have to consider the following assumptions:

1. Use a one-speed or one-group approximation where the neutrons can be characterized by a single average kinetic energy.

2. Characterize the neutron distribution in the reactor by the particle density n(r, t), which is the number of neutrons per unit volume at a position r' at time t. Its relationship to the flux is

We consider an arbitrary volume V and write the balance equation as follows.

Time of change of the number of neutrons in V = Production rate in V - Absorptions in V - Net leakage from the surface of V.

The first term is expressed mathematically as

The production rate can be written as

The absorption term is
and the leakage term is

where we convert the surface integral to a volume integral by using Gauss theorem or the divergence theorem.

Substituting for the different terms in the balance equation, we get
Or,

Since the volume V is arbitrary, we may write Equation 2.29 as

We now use the relationship between J and ф (Fick's law) to write the diffusion equation

This is the basis of the development in reactor theory using diffusion theory.

Helmholtz Equation

The diffusion equation (2.30) is a partial differential equation of the parabolic type. It also describes the physical phenomena in heat conduction, gas diffusion and material diffusion.

This equation can be simplified in the case the medium is uniform or homogeneous such that D and T,_{a} do not depend on the position r' as (Hetrick 1971)

where we used the fact that the divergence of the gradient leads to the Laplacian operator:

The Laplacian operator V^{2} depends on the coordinates system used:

When the flux is not a function of time, we use the steady-state diffusion equation or the scalar Helmholtz equation

which is a partial differential equation of the elliptic type.