Neutron flux as a function of position near a free surface according to diffusion theory and transport theory.
The diffusion equation is mostly solved in media with high densities such as neutron moderators (H2O, D2O or graphite). The problem is usually bounded by air. The mean free path of neutron in air is much larger than in the moderator, so that it is possible to treat it as a vacuum in neutron flux distribution calculations. The vacuum boundary condition supposes that no neutrons are entering a surface.
If we consider that no neutrons are reflected from the vacuum back to the volume, the following condition can be derived from the Fick’s law:
Where d ≈ ⅔ λtr is known as the extrapolated length. For homogeneous, weakly absorbing media, an exact solution of the mono-energetic transport equation in this case yields d ≈ 0.7104 λtr. The geometric interpretation of the previous equation is that the relative neutron flux near the boundary has a slope of -1/d, i.e., the flux would extrapolate linearly to 0 at a distance d beyond the boundary. This zero flux boundary condition is more straightforward and is can be written mathematically as:
If d is not negligible, physical dimensions of the reactor are increased by d and extrapolated boundary is formulated with dimension Re = R + d and this condition can be written as Φ(R + d) = Φ(Re) = 0.
It may seem the flux goes to 0 at an extrapolated length beyond the boundary. This interpretation is not correct. The flux cannot go to zero in a vacuum, because there are no absorbers to absorb the neutrons. The flux only appears to be heading to the zero value at the extrapolation point.
Note that, the equation d ≈ 0.7104 λtr is applicable to plane boundaries only. The formulas for curved boundaries can differ slightly, however, the difference is small unless the radius of curvature of the boundary is of the same order of magnitude as the extrapolated length.
Typical values of the extrapolated length:
The most common moderators have following diffusion coefficients (for thermal neutrons):
D(H2O) = 0.142 cm
D(D2O) = 0.84 cm
D(Be) = 0.416 cm
D(C) = 0.916 cm
The thermal neutron extrapolated lengths are given by:
d ≈ 0.7104 λtr = 0.7104 x 3 x D
H2O: d ≈ 0.30 cm
D2O: d ≈ 1.79 cm
Be: d ≈ 0.88 cm
C: d ≈ 1.95 cm
As can be seen, this approximation is valid when the dimension L of the diffusing medium is much larger than the extrapolated length, L >> d.