One-group analysis of the spatial variation of the flux in a parallelepiped, diffusing, non-source medium can be easily solved. If a source of neutrons is located in the x = y = 0 plane, the solution away from the boundaries takes the form:
(x,y,z) = C sinh([c-z]) cos(x/a) cos(y/b)
where: a and b are the extrapolated transverse dimensions of the medium and c is the extrapolated column height.
Away from the boundaries, the expression becomes:
(x,y,z) = C exp(-z) cos(x/a) cos(y/b)
In both cases, the coefficient C reflects the source conditions.
It is common to provide this geometry with a diffusing medium such as graphite adjacent to an operating test or research reactor core. Both fast and thermal neutrons enter the pile at the source face, and thermalization of the fast component occurs rapidly with penetration in the Z direction. Such a facility provides an excellent source of thermal neutrons for experimentation.
At some point, the thermal flux begins to behave according to the above model, and from that location onward, we have what is known as an EXPONENTIAL PILE. Neutrons in this region have the characteristic Maxwellian distribution associated with thermal equilibrium of neutrons within the host medium.
It is common to append experiments to the far face (z = c) to take advantage of a well-behaved source of thermal neutrons crossing into the experiment. Indeed, this is the manner in which the diffusion length of many materials may be measured.
When studying the diffusion length of graphite, which is the pile medium, measurements may be made directly in the pile. The thermal flux distribution may be measured throughout the pile, and the axial (Z-direction) thermal flux plotted as a function of distance on semi-log paper. That region which yields a straight-line plot conforms to the diffusion model and permits evaluation of the constant . A set of foils in the transverse directions X and Y (in that same region) establishes the a and b extrapolated dimensions of the pile. The diffusion length is then found from the derived relationship:
1/L2=2 - 2 - 2 where: = /a and = /b
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Foils of gold or indium are widely used in reactor measurements. Both are available at the reactor facility in metallic foil forms of uniform thickness and standard diameters. Both metals are easily handled, are 1/v absorbers, and have relatively large neutron absorption cross-sections. Unfortunately, both gold and indium have strong low-energy capture resonances above the thermal region: gold at 4.9 eV, and indium at 1.41 eV.
The activation of the foil reflects both thermal capture and resonance capture. The thermal flux must be inferred by the following procedure, which depends upon the use of cadmium which has a high neutron absorption cross section for neutron energies below 0.4 eV, and a very small cross section above that energy.
Two measurements are required: one with a cadmium covered foil, another with a bare foil..... for every location of interest. The irradiation of each foil must be carried out separately (at each location). The activity due to the thermal neutron flux is the difference between both foil measurements, corrected for power, cooling time, and weight.
If we accept the premise that the neutrons will be thermalized after a short distance into the graphite, and assume that the activation due to resonance capture is not significant, one run using only bare foils could be made. The data will, of course, drift from the proper values, especially as the core is approached.
Use gloves to avoid contamination of the graphite and your hands when handling the logs. Also, handle the logs carefully to prevent mechanical damage.
