What is the mathematical expression for the net rate of neutron diffusion into a unit volume?

The net rate of neutron diffusion into a unit volume is represented by the mathematical expression R = -∇.j, where j is the neutron current density vector. This expression arises from Fick's Law of diffusion, which states that the diffusion flux is proportional to the negative gradient of concentration. In the context of neutron diffusion, the flow of neutrons into a given volume is driven by spatial variations in neutron density.

The term ∇.j indicates the divergence of the neutron current density, which quantifies how much neutron current is entering or leaving a particular volume. A negative divergence means that more neutrons are entering the volume than leaving, indicating a net accumulation of neutrons. The sign and magnitude of R therefore reflect the net rate at which neutrons are diffusing into the specified volume.

In practical terms, understanding this relationship is crucial for analyzing neutron behavior in nuclear reactors and ensuring effective control over the fission process by managing neutron populations within the reactor core.