What is the neutron current J and how is it related to the neutron flux φ in diffusion theory?

• A. J = φ ∇ D; represents net flow due to flux magnitude.
• B. J = -D ∇ φ; represents net flow of neutrons due to gradients in flux.
• C. J = D ∇ φ; represents outward leakage.
• D. J = -∇ φ; represents diffusion without material dependence.

Answer: (B)

In diffusion theory, how neutrons move is governed by Fick’s law: the current flows from regions of higher neutron flux to lower flux, and the strength of that flow is set by the diffusion coefficient. This gives the relation J = -D ∇ φ. The negative sign is crucial—it shows the current points in the direction opposite to the flux gradient, meaning it moves down the gradient of φ. The diffusion coefficient D encapsulates how easily neutrons spread through the material, depending on scattering and geometric properties.

The other forms don’t fit because they either ignore the role of the flux gradient or misplace the sign or the material dependence. For example, coupling J to φ ∇ D would imply current is driven by changes in material properties rather than the flux gradient, which isn’t how diffusion is described. Dropping the negative sign and writing J = D ∇ φ would spuriously suggest flow toward increasing φ. And omitting D entirely, as in J = -∇ φ, ignores the material-specific ease of diffusion.