## Fick’s Law

**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 a law (

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

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

*If the concentration of a solute in one region is greater than in another of a solution, 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**.

**The Fick’s law in reactor theory stated 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 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 unit of cm and it is given by:*φ*is the neutron flux, which is 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

[xyz-ihs snippet=”diffusion”]The physical interpretation is similar to 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 as the result of 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 over all space but flux contributions are negligible beyond a few mean free paths (about three mean free paths) from 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 two media.**Isotropic scattering**. Isotropic scattering occurs at low energies, but is not true in general. Presence of anisotropic scattering can be corrected by modification of the diffusion coefficient (based on transport theory).**Low absorbing medium**. Derivation of Fick’s law assumes (an expansion in a 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.

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