**Fick’s law**, which states that solute diffuses (

**neutron current**) from high concentration (high flux) to low concentration. As can be seen, we have to investigate the relationship between the

**flux (φ)**and the

**current (J)**. This relationship between the flux (φ) and the current (J) is identical in form to the law (

**Fick’s law**) used in the study of physical diffusion in liquids and gases.

**In chemistry, Fick’s law states that**:

*Suppose the concentration of a solute in one region is greater than in another of a solution. In that case, the solute diffuses from the region of higher concentration to the region of lower concentration, with a magnitude that is proportional to the concentration gradient.*

In one (spatial) dimension, the law is:

where:

*J*is the diffusion flux,*D*is the**diffusion coefficient,***φ*(for ideal mixtures) is the concentration.

The use of this law in **nuclear reactor theory** leads to the **diffusion approximation**.

**Fick’s law in reactor theory states that**:

*The current density vector J is proportional to the negative of the gradient of the neutron flux. The proportionality constant is called the diffusion coefficient and is denoted by the symbol D.*

In one (spatial) dimension, the law is:

where:

(neutrons.cm*J*is the neutron current density^{-2}.s^{-1}) along the x-direction, the net flow of neutrons that pass per unit of time through a unit area perpendicular to the x-direction.*D*is the**diffusion coefficient,**it has the unit of cm, and it is given by:*φ*is the neutron flux, the number of neutrons crossing through some arbitrary cross-sectional unit area in**all directions**per unit time.

The generalized Fick’s law (in three dimension) is:

where **J** denotes the **diffusion flux vector**. Note that the gradient operator turns the neutron flux, which is a **scalar quantity** into the neutron current, which is a **vector quantity**.

## Physical Interpretation

The physical interpretation is similar to the fluxes of gases. The neutrons exhibit a net flow in the direction of least density. This is a natural consequence of **greater collision densities** at positions of **greater neutron densities**.

Consider neutrons passing through the plane at x=0 from left to right due to collisions to the left of the plane. Since the concentration of neutrons and the flux is larger for negative values of x, there are **more collisions per cubic centimeter on the left**. Therefore more neutrons are scattered from left to right, then the other way around. Thus the neutrons naturally diffuse toward the right.

## Validity of Fick’s Law

**It must be emphasized that Fick’s law is an approximation and was derived under the following conditions:**

**Infinite medium.**This assumption is necessary to allow integration of overall space. Still, flux contributions are negligible beyond a few mean free paths (about three mean free paths) from the boundaries of the diffusive medium.**Sources or sinks.**Derivation of Fick’s law assumes that the contribution to the flux is mostly from elastic scattering reactions. Source neutrons contribute to the flux if they are more than a few mean free paths from a source.**Uniform medium.**Derivation of Fick’s law assumes that a uniform medium was used. There are different scattering properties at the boundary (interface) between the two media.**Isotropic scattering**. Isotropic scattering occurs at low energies but is not true in general. Anisotropic scattering can be corrected by modification of the diffusion coefficient (based on transport theory).**Low absorbing medium**. Fick’s law derivation assumes (an expansion in Taylor’s series) that the neutron flux,*φ,*is slowly varying. Large variations in φ occur when Σ_{a}(neutron absorption) is large (compared to Σ_{s}).**Σ**_{a}**<< Σ**_{s}**Time-independent flux.**Derivation of Fick’s law assumes that the neutron flux is independent of time.

To some extent, these limitations are valid in every practical reactor. Nevertheless Fick’s law gives a reasonable approximation. For more detailed calculations, higher order methods are available.