Existence of a Diffusion Length in a Finite Prism of Pure Moderator

An exact integral transport equation is derived for the neutron flux in a finite prism of pure moderator. Assuming that the prism is infinite in the z-direction, a trial solution expi(-vz) is inserted, and an eigenvalue problem for v, the spatial relaxation constant, results. The eigenvalue v is studied as a function of the absorption and dimensions of the prism. It is shown that the discrete eigenvalues are bounded by Σmin, and that a minimum dimension exists below which a diffusion length can no longer be found. This result is much more restrictive than the maximum absorption theorem of Comgold, and the minimum dimension of the prism is shown to correspond to transverse bucklings of 10-3 cm2 for beryllium, and 1.5 x 10-3cm2 for graphite, at 300°K. For poisoned and low-temperature moderators these values will be even smaller. The angular distribution of the neutrons in the moderator has been calculated perpendicular and parallel to the axis, and it is found that a marked anisotropy of the spectrum exists, particularly in the neighbourhood of the Bragg cut-off. On the basis of degenerate kernel approximation to the scattering process, it is shown that when the system is such that no diffusion length exists, then the asymptotic transverse flux shape is no longer of the cosine type, but becomes progressively flatter as the distance from the source increases. This is due to an increase in the proportion of cold neutrons which have longer mean free paths than the average. The well-known cold neutron filtering effect can be explained in terms of this phenomenon. The results obtained from the present analysis are in quantitative agreement with recent experiments performed by De Juren and Swanson. (author)