The Fick's law

The Fick's law relates the current $\overrightarrow{J} (\overrightarrow {r} ,v,t)$ to the flux $\varphi(\overrightarrow{r} ,v,t).$ It reads:

$$\overrightarrow{J} (\overrightarrow{r} ,v,t)=-D\overrightarrow{grad} (\varphi(\overrightarrow{r} ,v,t))$$ (4.21)

We give a simple derivation of this relation, following Lamarsh[34]. We assume that $\varphi(\overrightarrow{r} ,v,t)$ over a neutron mean free path can be considered as linear^4.5. For simplicity we assume that we compute the current at the origin. Thus:

$$\varphi(\overrightarrow{r} ,v,t)=\varphi_{0}+x\left( \frac{\p... ...t) _{0}+z\left( \frac{\partial\varphi}{\partial z}\right) _{0}$$

We compute the neutron current through a surface dA_z, situated at the origin, and perpendicular to the z axis. We use the spherical coordinates $x=r\sin(\theta)\cos(\phi)$, $y=r\sin(\theta)\sin(\phi)$, $z=r\cos(\theta).$ It is, then, clear that any integral over x and y of a linear functional of $\varphi$ will only involve $\varphi_{0}+r\cos(\theta)\left( \frac{\partial\varphi}{\partial z}\right) _{0}$. Furthermore, the neutron currents originating from z>0 and z<0, have opposite direction. This means that only odd terms will be involved in the total neutron current^4.6, i.e. the term $r\cos(\theta)\left( \frac {\partial\varphi}{\partial z}\right) _{0}$. Terms from z<0 and z>0 are equal, so that it is enough to evaluate the contribution of the z<0 region and double it. In the absence of external sources, neutrons crossing the surface dA_z come, either from a scattering or a fission event. In a first approach we neglect the contribution of fission neutrons, assuming that $\Sigma_{s}>>\Sigma_{f}$. We also assume that the time delay between neutron scattering and the arrival of the neutron at the origin is negligible.

The number of neutrons scattered in an elementary volume dV is $\Sigma _{s}\varphi(\overrightarrow{r} ,v,t)$, and those heading towards the surface dA_z, assuming isotropic laboratory scattering, is $\frac{\Sigma _{s}\varphi(\overrightarrow{r} ,v,t)\cos(\theta)dA_{z}dV}{4\pi r^{2}}$. Along their path towards the surface dA_z, these neutrons may undergo a reaction which takes them out, thus the number of neutrons reaching the surface is $\frac{e^{-\Sigma_{T}r}\Sigma_{s}\varphi(\overrightarrow{r} ,v,t)\cos (\theta)dA_{z}dV}{4\pi r^{2}}$. Taking into account the remarks of the preceeding paragraph we obtain the neutron current:

$$\overrightarrow{J_{z}} =\frac{\Sigma_{s}}{2\pi}\left( \frac{\... ...heta)\sin(\theta)d\theta\int_{r=0}^{\infty}re^{-\Sigma_{T}r}dr$$

and:

$$\overrightarrow{J_{z}} =-\frac{\Sigma_{s}}{3\Sigma_{T}^{2}}\left( \frac{\partial\varphi}{\partial z}\right) _{0}$$ (4.22)

Similar relations hold for the other components of $\overrightarrow{J} ,$ so that we get the Fick's law 3.24 with

$$D=\frac{\Sigma_{s}}{3\Sigma_{T}^{2}}$$ (4.23)

Note that, for $\Sigma_{a}<<\Sigma_{s}$, $D=\frac{1}{3\Sigma_{s}}$.

Although our derivation of the Fick's law assumed isotropic scattering in the laboratory frame, it is, in fact, possible to extend its validity to the case of moderetely anisotropic scattering. In particular if $\Sigma_{a}<<\Sigma_{s}$, $D=\frac{1}{3\Sigma_{s}(1-\overline{\mu})}$ where $\overline{\mu}$ is the average value of the cosine of the scattering angle.

The derivation also assumed an infinite, homogeneous medium. It is, in fact valid, when applied in regions several mean free paths away from the medium's boundary. It is, even, valid at the frontier between two media, provided the absorption cross-section is small.

Similarly Fick's law is valid in the presence of external sources in regions sufficiently far from the sources (several mean free paths).

One of the most serious limitation of Fick's law is that it assumes no velocity modification after scattering. This is true in thermal reactors where neutron spectra can be considered to have reached an equilibrium in which up-scattering is as probable as down-scattering. It may also be true for low lethargy^4.7 fast flux reactors. In both cases the Fick's law can be used to simplify the Boltzmann equation into a diffusion equation.