Fick’s law was introduced in the frame of the kinetic theory of gases. It relates the particle flux or current to the gradient of the particle density,

j =-D grad (p). (3.20)

In neutron physics the density is replaced by the neutron flux and the particle flux by the neutron current. For those used to the theory of gases this change of definitions may be the source of some confusion.

Fick’s law relates the current J(r, v, t) to the flux ‘(r, v, t). It reads:

J(r, v, t) = — Dgrad (‘(r, v, t)). (3.21)

We give a simple derivation of this relation, following Lamarsh [55]. We assume that ‘(r, v, t) over a neutron mean free path can be considered as linear.[13] For simplicity we assume that we compute the current at the

origin. Thus,

We compute the neutron current through a surface dAz, placed at the origin, and perpendicular to the z axis. We use the spherical coordinates x = r sin(e) cos(^), y = r sin(e) sin(^), z = rcos(d). It is then clear that any integral over x and y of a linear functional of ‘ will only involve ‘0 + rcos(d)(d’/dz)0. Furthermore, the neutron currents originating from z > 0 and z < 0 have opposite directions. This means that only odd terms will be involved in the total neutron current,* i. e. the term rcos(0)(d’/dz)0. Terms from z < 0 and z > 0 are equal, so that it suffices to evaluate the con­tribution of the z < 0 region and double it. In the absence of external sources, neutrons crossing surface dAz come either from a scattering or a fission event. In a first approach we neglect the contribution of fission neutrons, assuming that Xs ^ Xf. 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 Xs'(r, v, t), and those heading towards the surface dAz, assuming isotropic laboratory scattering, is

Xs'(r, v, t) cos(6) dAz d V
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On their way towards surface dAz, these neutrons may undergo a reaction
which takes them out, thus the number of neutrons reaching the surface is

e_SrrXs'(r, v, t) cos(0) dAz d V
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Taking into account the remarks of the preceding paragraph we obtain the neutron current,

and

r = M

z 3XT dz J0

Similar relations hold for the other components of J, so that we get Fick’s law, equation (3.21), with

D=X • (323)

Note that, for Xa ^ Xs, D = 1/3Xs.

* This circumstance is responsible for the validity of Fick’s law up to second order.

Although our derivation of Fick’s law assumed isotropic scattering in the laboratory frame, it is in fact possible to extend its validity to the case of mod­erately anisotropic scattering. In particular if Xa ^ Xs, D = 1/3Es(1 — p) where p 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 boundary 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 limitations of Fick’s law, in its simple, mono­group, form, 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[14] fast flux reactors. In both cases Fick’s law can be used to simplify the Boltzmann equation to a diffusion equation.